Disproof of the Yau–Tian–Donaldson conjecture
Abstract.
We construct a polarized smooth projective fivefold and prove that it is K-polystable but does not admit a constant scalar curvature Kähler metric. This disproves the Yau–Tian–Donaldson conjecture for constant scalar curvature metrics.
The main result of this paper was obtained using generative AI, particularly GPT-5.6-sol, Fable 5, and the Danus system. A detailed report on the use of generative AI in this paper is enclosed in the appendix, joint with Bin Dong and Guoxiong Gao.
Key words and phrases:
Yau–Tian–Donaldson conjecture, K-polystability, uniform K-stability, constant scalar curvature Kähler metric, test configuration, projective bundle2020 Mathematics Subject Classification
53C55, 14L24, 32Q201. Introduction
1.1. Background
Calabi’s program asks for an extremal Kähler metric in a prescribed Kähler class [Cal82, Cal85]. The K-energy of Mabuchi and the moment-map interpretation of scalar curvature due to Fujiki and Donaldson gave analytic form to the expectation that the existence problem should be governed by an algebro-geometric stability condition [Mab86, Fuj92, Don97, Don99]. The obstructions of Matsushima and Futaki are recorded in [Mat57, Fut83]. In the setting where the first Chern class is positive, the expectation that Kähler–Einstein existence should be governed by algebraic stability goes back to Yau [Yau93]. Building on the generalized Futaki invariant of Ding and Tian [DT92], Tian introduced K-stability for special degenerations of Fano manifolds, proved the necessity direction, and formulated the Kähler–Einstein conjecture [Tia97, Definition 1.1, Theorem 1.2, and Conjecture 1.4].
Donaldson introduced test configurations of arbitrary positive exponent for general polarized varieties [Don02, Definition 2.1.1]. He conjectured that a smooth polarized variety admits a constant scalar curvature Kähler (cscK) metric precisely when it satisfies the resulting stability condition [Don02, p. 290]. His balanced-embedding theorem had already proved a finite-dimensional consequence in geometric invariant theory (GIT) of cscK existence when the automorphism group is discrete [Don01, Corollary 4]. In the terminology now in use, Donaldson’s product-equality clause [Don02, Definition 2.1.2] is a polystability condition. We state it in the form relevant to this paper.
Conjecture 1.1 ([Don02, Conjecture], Yau–Tian–Donaldson conjecture).
Let be a smooth polarized projective complex variety. Then is K-polystable with respect to all normal ample algebraic test configurations of every positive exponent if and only if contains a constant scalar curvature Kähler metric.
Remark 1.2.
The normality convention in Conjecture 1.1 is the standard repair of Donaldson’s original scheme-theoretic formulation; otherwise, the conjecture would be trivially false. See Corollay 1.6 below. We recall the history of this repair here.
Ross and Thomas developed the slope and Hilbert–Mumford viewpoints and clarified the distinction between product and trivial configurations [RT06, RT07]. Li and Xu exhibited a nonproduct zero-invariant configuration with an embedded point [LX14, Section 8.2, especially Example 4 and Remark 4]. Stoppa called a test configuration trivial in codimension two when it is equivariantly a product away from a closed subscheme of codimension at least two, and modified the equality condition accordingly [Sto11, Definitions 1–2]. Odaka used the term almost trivial for the corresponding codimension-one condition [Oda15, Definitions 3.3–3.4]. Boucksom–Hisamoto–Jonsson later used almost trivial to mean that the normalization is trivial, and characterized this condition by vanishing of the -norm [BHJ17, Corollary B]. These conditions are not equivalent in general: triviality in codimension one implies almost triviality in the normalization sense, but the converse can fail [BJ22, Definition 2.36, Example 2.37, and Theorem 2.38(iii)]. Stoppa observed that his repaired K-stability condition can equivalently be tested on normal test configurations, excluding only the trivial configuration [Sto11, p. 2]. Li and Xu likewise formulated K-stability using normal test configurations [LX14, Definition 6 and Remark 2(2)].
We use the K-polystability convention of Codogni–Stoppa, restricting to normal total spaces, [CS19, Definition 18]. We remark that Codogni and Stoppa conjectured that, for every reductive subgroup , K-polystability is equivalent to nonnegativity of the Donaldson–Futaki invariant on every -equivariant test configuration, with equality if and only if its normalization is a product [CS19, Conjecture 1].
1.2. History of the Yau–Tian–Donaldson conjecture
Several parts and variations of the Yau–Tian–Donaldson conjecture (Conjecture 1.1) are theorems. In this subsection we briefly recall the historical progress towards this conjecture.
The necessity part. On the necessity side, Donaldson proved K-semistability [Don05b, Theorem 1], and Stoppa proved K-stability when the automorphism group is discrete [Sto09, Theorem 1.2], as corrected in [Sto11, p. 1]. For normal test configurations, Mabuchi gave the first general proof of K-polystability [Mab08, Main Theorem]; a missing step in that preprint was supplied in [Mab09, p. 2]. Berman–Darvas–Lu later gave a different proof [BDL20]. Székelyhidi introduced relative K-stability [Sze07, Definition 2.2] and formulated the extremal correspondence [Sze07, Conjecture 1.1], while Stoppa–Székelyhidi proved its necessity direction [SS11, Theorem 4]. Kähler and transcendental extensions were developed in [DR17b, SD18, SD20].
The Fano case. For smooth Fano manifolds with the anticanonical polarization, the sufficiency direction was proved by Tian [Tia15a, Tia15b] and Chen–Donaldson–Sun [CDS15a, CDS15b, CDS15c]. Li and Xu’s reduction to special test configurations [LX14, Corollary 1] and Berman’s necessity theorem [Ber16, Theorem 1.1] place this result in the modern K-polystable framework. Further analytic proofs are due to Datar–Székelyhidi [DS16, Theorem 1]. Chen–Sun–Wang obtained another proof [CSW18, Theorem 1.2]. For singular log Fano pairs, variational and equivariant methods relate K-polystability, reduced uniform K-stability, and weak Kähler–Einstein existence [BBJ21, LTW21, LTW22, Li22a]. The finite-generation input is due to Liu–Xu–Zhuang [LXZ22, Theorems 1.1 and 1.6].
The algebraic theory also includes log discrepancies, local volumes, valuative criteria, and stability-threshold methods [Oda12, Oda13, Fuj19, Li17, FO18, BJ20]. Blum–Xu proved uniqueness for K-polystable Fano degenerations [BX19, Theorem 1.1(2)]. Zhuang proved that equivariant K-polystability for a reductive group implies geometric K-polystability [Zhu21, Theorem 1.1(2)]. Related quantization arguments are due to K. Zhang [Zha24]; see also Xu’s survey and monograph [Xu21, Xu25].
The toric theory is also extensive. Wang–Zhu treated toric Fano solitons [WZ04]. Donaldson developed an analytic program [Don05a, Don08] and proved cscK existence for polarized toric surfaces satisfying his toric K-stability condition [Don09, Corollary 1]. Chen–Li–Sheng proved a uniform condition necessary for toric extremal metrics [CLS14].
We also remark that Li proved the uniform correspondence for polarized toric manifolds in every dimension [Li22b, Theorem 1.12].
Progress on general polarizations. For general polarizations, later work has increasingly separated K-polystability from quantitative and completed notions. Székelyhidi introduced uniform K-stability in his thesis [Sze06, Section 3.1.1]. Relations among test configurations, geodesic rays, Okounkov bodies, and concave transforms were developed in [PS07, PS09, WN12, BC11]. Dervan introduced the minimum norm [Der16]. The non-Archimedean slope formalism is developed in [BHJ19, BHJ22], and Hisamoto introduced a reduced form adapted to automorphisms [His16]. Dervan–Reboulet characterized coercivity on fixed Fubini–Study spaces by uniform arc K-polystability [DR24, Theorem 1.1]. Darvas–Rubinstein formulated a general existence–properness principle [DR17a, Theorem 3.4] and applied it conditionally to the cscK problem [DR17a, Theorem 10.1]. Darvas–Lu developed the metric and geodesic structure of spaces of rays [DL20, Theorems 1.3 and 1.4]. Chen–Cheng obtained the estimates used in their program [CC21a, Theorem 1.1], proved the direction from properness to existence in the discrete-automorphism case [CC21b], and extended it to general in [CC18, Theorem 1.6].
Finally, stronger correspondences use completed or quantitative stability conditions. Boucksom–Jonsson use -polystability on the finite-energy non-Archimedean completion [BJ25b, Theorem A]. Darvas–Zhang characterize the existence of a unique cscK metric by uniform -stability for some and obtain an automorphism-equivariant criterion through models defined by log discrepancies [DZ25, Theorems 1.1 and 1.6]. The entropy-regularization program is formulated in [BJ18, Conjecture 2.5], and the model criterion in [Li22b, Theorem 1.10 and Conjecture 1.8]. Trusiani’s special Fujita approximation theorem solves the regularization problem and yields an equivalence with -uniform K-stability [Tru26, Theorem A, Theorem B, and Corollary A]. Kähler and transcendental variants are due to Mesquita-Piccione and Mesquita-Piccione–Witt Nyström [MP25, MPWN25]. Further existence, properness, and threshold results appear in [SW08, He19, JSS19, DZ24, Tru24]. Further parts of the non-Archimedean program appear in [BJ22, BJ23, BJ25a]. For flat families of smooth polarized varieties with finite automorphism groups, Dervan proved that the cscK locus is very general [Der25, Theorem 1.1].
1.3. The counterexample
Despite all the aforementioned progress, it is worth mentioning that the full version of the original Yau–Tian–Donaldson conjecture [Don02, Conjecture] (Conjecture 1.1) has remained open up to now. Indeed, some experts are skeptical of [Don02, Conjecture] and expect that (a version of) uniform K-stability is necessary (cf. [Sze06]). It is particularly worth mentioning that [ACG+08] constructed polarized smooth projective fourfolds whose Ross–Thomas slope degenerations of the zero and infinity sections all have positive modified Futaki invariant, but which do not admit any extremal Kähler metric, hence do not admit any cscK metric. However, proving that an example produced by this mechanism is K-polystable (or even K-semistable) is a very difficult task, as one needs to consider all normal ample algebraic test configurations instead of only the ones obtained from the Ross–Thomas degenerations. As mentioned in [ACG+08], verbatim:
While we cannot prove that there is no other (algebraic) test configuration which would detect this instability, it is difficult to imagine how such a test configuration could be constructed.
In other words, [ACG+08] provided a mechanism for the construction of potential counterexamples to the Yau–Tian–Donaldson conjecture, i.e. well-described polarized smooth projective varieties. However, proving the K-polystability of polarized smooth projective varieties constructed in such a way essentially requires new mathematical input, and is not a streamlined verification.
Main Theorem. The main theorem of this paper is the construction of a polarized smooth projective fivefold via (a variation of) the mechanism of [ACG+08], and the proof that this polarized smooth projective fivefold is K-polystable but does not admit any extremal Kähler metric.
As a consequence, the Yau–Tian–Donaldson conjecture (Conjecture 1.1) is false. We emphasize that the Yau–Tian–Donaldson conjecture for (log) Fano varieties (cf. [Tia15a, Tia15b, CDS15a, CDS15b, CDS15c, LXZ22]) and the uniform and completed K-stability variations of the cscK Yau–Tian–Donaldson conjecture (cf. [BJ25b, DZ25, Tru26]) are not affected by this counterexample.
Theorem A (The counterexample).
There exist smooth projective connected complex curves of genera
| (1.1) |
such that
| (1.2) |
Choose line bundles on with
| (1.3) | ||||
Put
| (1.4) |
and, in the quotient convention, put
| (1.5) |
Then the following statements hold.
- (1)
is a smooth polarized complex projective fivefold, and acts by fiber scaling.
- (2)
For every positive integer , every normal ample algebraic test configuration with generic polarized fiber has nonnegative Donaldson–Futaki invariant. Equality holds only for a polarized product test configuration induced by an integral one-parameter subgroup of fiber scaling together with a scalar character on the polarization.
- (3)
The class contains no extremal Kähler metric and hence no constant scalar curvature Kähler metric.
Consequently, is an explicit counterexample to Conjecture 1.1.
To the author’s knowledge, Theorem A gives the first smooth polarized projective variety which is K-polystable in the Donaldson–Futaki sense with respect to every normal ample algebraic test configuration of every positive exponent, but whose polarization class contains no cscK metric. The K-polystability assertion includes both global nonnegativity and product rigidity in the zero-invariant case, and it imposes no equivariance assumption.
The construction continues a line of examples originating in the ruled-surface work of Tønnesen-Friedman [TF98], the admissible projective-bundle theory [ACG+11], and the splitting theorem of Apostolov–Huang [AH15]. The authors of [ACG+08] constructed polarized fourfolds whose admissible extremal polynomial is positive at every rational point but has an irrational repeated interior zero. The detecting degeneration in that example was not algebraic [ACG+08, Example 1, pp. 580–581], and the authors left open whether an algebraic test configuration detects the instability [ACG+08, p. 551]. For projectivizations over a curve, Jubert–Yin recently proved a relative uniform correspondence with respect to compatible test configurations [JY26, Theorem A and Corollary A]. Moreover, the Futaki character of the fourfold constructed in [ACG+08, Example 1] is nonzero [ACG+08, Proposition 6 and the discussion following Proposition 8]. Székelyhidi later interpreted the irrational degeneration as a non-finitely-generated filtration in work with an appendix by Boucksom [Sze15, Section 4], while Dervan proved instability in the transcendental Kähler framework [Der18, Example 2.15]. Apostolov–Pym–Streets continued to describe the general polarized problem as open [APS26, Section 1.1 and Remark 4.6(5)].
It is worth mentioning that our construction, although it generally follows the mechanism of [ACG+08], has a slight difference: the construction in [ACG+08] provides a fourfold, while our example is a fivefold. The five-dimensional construction in Theorem A has a four-dimensional base and one projective-line direction. The large genera and degrees in (1.1)–(1.3) give an integral realization of the required boundary polynomial. The fourth curve factor and the vanishing in (1.2) also give the rigidity and multiplication properties needed to classify arbitrary algebraic test configurations. Therefore, although the extra dimension does not alter the irrational destabilizing mechanism, it makes the full K-polystability assertion provable.
Remark 1.3.
It is also worth mentioning that the identity component of the automorphism group of in Theorem A is ; in particular, the automorphism group of is infinite. It remains interesting to ask whether the Yau–Tian–Donaldson conjecture holds under the extra condition that is trivial (cf. [Der25, Theorem 1.1]) or finite.
1.4. The equality case
The proof of Theorem A consists of three parts: (i) the non-existence of extremal Kähler metrics, (ii) the K-semistability, and (iii) the passage from K-semistability to K-polystability. The major difficulty of the proof, which requires essential new mathematical input, is in part (iii), as one needs to consider all normal ample test configurations with zero Donaldson–Futaki invariant. Our way of doing this is to make a complete classification of all of them for the precise example in Theorem A.
Theorem B (Zero-invariant test configurations are products).
Let and the data be as in Theorem A. Let be a positive integer, and let be a normal ample algebraic test configuration whose generic polarized fiber is . Then . If , then there exist integers with the following property. For and , put
| (1.6) |
The increasing filtration of , in the convention of Definition 2.1, has the single jump
| (1.7) |
on every block . The configuration is the polarized product in which fiber scaling contributes and the scalar character contributes to this jump.
Theorem B essentially says that the Codogni–Stoppa conjecture [CS19, Conjecture 1] holds in the normal category for the example in Theorem A. Let be the fiber-scaling torus. For , Theorem B proves that every normal zero-invariant test configuration, without an equivariance assumption, is the product induced by a one-parameter subgroup of , up to a scalar character. This confirms [CS19, Conjecture 1] for this specific example.
Note that no counterpart of the Fano reduction to special test configurations is available here. The proof of Theorem B, therefore, is essentially new: the idea is to compare every graded space of sections together with the marking of its generic fiber, rather than restricting the class of degenerations.
We remark that Corollary 1.5 extends this result to arbitrary ample test configurations: equality holds precisely when the normalization is such a product and the normalization morphism is an isomorphism away from codimension two.
1.5. The reduced Donaldson–Futaki quotient
We next explain why K-polystability has no positive uniform margin in this example, i.e. the polarized manifold is not uniformly relatively K-polystable. For the family in Theorem C, the reduced Donaldson–Futaki quotient has infimum zero, and the family violates every positive uniform relative inequality.
Theorem C (Vanishing of the reduced Donaldson–Futaki quotient).
Let be the polarized fivefold in Theorem A, and put . Let be the Fibonacci sequence defined by
| (1.8) |
For , put
| (1.9) |
For every and every sufficiently divisible positive integer , there exists a normal nonproduct -equivariant algebraic test configuration of exponent for whose single increasing filtration entry, in the convention of Definition 2.1, on
| (1.10) |
is
| (1.11) |
Moreover,
| (1.12) |
Writing and for the reduced non-Archimedean -functional of [NS21, Definitions 3.6.1 and 3.6.3], we have
| (1.13) |
where
| (1.14) |
Consequently,
| (1.15) |
Thus the reduced Donaldson–Futaki quotient of this explicit family has infimum zero. The pair is not uniformly relatively K-polystable in the sense of [NS21, Definition 3.7.1(4)].
Theorems A and C separate K-polystability and uniform K-stability explicitly. K-polystability excludes nonproduct zero-invariant configurations, while the positive gap collapses along the rational Fibonacci approximants to the irrational crease. Uniform relative K-stability therefore rejects the example which K-polystability accepts.
Remark 1.4.
Hattori exhibited K-stable but not uniformly K-stable examples among polarized normal pairs [Hat26, Corollary 7.5] that are not klt (but log canonical), and among connected deminormal surfaces [Hat26, Corollary 7.7]. These are algebraic separation results and do not assert the nonexistence of cscK metrics. There are also smooth examples concerning J-stability [Hat26, Theorem 7.3]. Hattori also conjectured that K-stable but not uniformly K-stable examples should exist for normal polarized varieties [Hat26, Conjecture 7.6]. However, the construction of Hattori essentially used the strictly log canonical property and seems difficult to generalize to the smooth/klt category. The construction of our paper is inspired by [ACG+08] instead.
1.6. Scheme-theoretic consequences
Finally, as we discussed in Remark 1.2, the definition of K-polystability via test configurations has changed throughout history. In Theorem A, we only consider normal ample test configurations, which seems to be the standard category of test configurations to consider in modern terminology. Another modern convention is to consider arbitrary ample test configurations and to allow zero Donaldson–Futaki invariant exactly for those that are products in codimension two, in the spirit of [Sto11]. This condition should not be identified with the weaker zero-norm or almost-trivial condition in the current terminology. We show that, even adopting this convention, the Yau–Tian–Donaldson conjecture still fails.
Corollary 1.5 (Scheme-theoretic extension).
Let be the polarized fivefold in Theorem A. Let be a positive integer, and let be an ample algebraic test configuration whose generic polarized fiber is . No normality assumption is imposed on , and the central fiber may be nonreduced. Let denote the normalization. Then
| (1.16) |
Equality holds if and only if is the polarized product test configuration induced by an integral one-parameter subgroup of fiber scaling together with a scalar character, and is an isomorphism away from a closed subset of of codimension at least two.
For completeness, we provide the following well-known corollary, which indicates that allowing codimension-two defects in the equality clause is necessary for the Yau–Tian–Donaldson conjecture.
Corollary 1.6.
Under the unrepaired scheme-theoretic convention in which zero Donaldson–Futaki invariant is permitted only for an actual product test configuration, K-polystability is not necessary for the existence of a cscK metric. The polarized curve carries a cscK metric but does not satisfy this literal equality condition.
1.7. Sketch of the proofs
We now explain the three parts of the proof. We first construct the polarized fivefold and compute an admissible boundary polynomial with a unique irrational interior zero. Smooth admissible metrics whose error in scalar curvature tends to zero, together with Donaldson’s lower bound, give K-semistability at every exponent. Circle invariance, equivariant openness, uniqueness, and naturality identify nearby extremal metrics with explicit admissible profiles. Their normalized momenta and profiles converge uniformly. At the interior zero of the limiting profile, this would make a nonzero fiber-scaling vector field have zero length for the limiting extremal metric, which is impossible. In particular, this argument does not require a classification of the limiting metric.
For Theorem B, we begin with an arbitrary normal ample test configuration for with and specialize its filtration in the two directions of the fiber-scaling action. The two convex transforms are affine in the fiber coordinate, and their coefficients coincide and are rational. After a ramified base change clears the denominators, the normalization of is compared with an integral product test configuration by Proposition 5.9. Lemma 6.8 identifies the relative Smith integers with the residual jumps in the correct direction, and Proposition 6.4 upgrades their asymptotic vanishing to equality of the marked section algebras. Descending through the base change leaves a possible rounding in the affine weights; Lemma 4.13 excludes that rounding and proves that the original test configuration is a product.
Finally, Theorem C follows from a separate explicit calculation. Rational Fibonacci creases converge to the irrational double zero. The Fibonacci recurrence gives cubic decay of the Donaldson–Futaki invariant in the denominator, while the reduced non-Archimedean -functional remains of linear size.
1.8. Structure of the paper
Section 2 fixes the conventions used throughout the paper. Section 3 constructs the polarized fivefold and proves the analytic assertions. Section 4 proves affine rigidity for the two opposite initial filtrations. Section 5 identifies their common rational profile and constructs the integral product comparator after base change. Section 6 proves the equality classification. Section 7 proves Theorem A and derives Corollaries 1.5 and 1.6. Section 8 constructs the rational single-crease test configurations and proves Theorem C.
Remark 1.7.
The main result of this paper was obtained using generative AI, particularly GPT-5.6-sol, Fable 5, and the Danus system. Danus is a specialized agent built on the Rethlas system and is substantially more capable of conducting fundamental mathematical research. See [Liu+26] and [Ju+26] for detailed introductions to the Danus system and the Rethlas system, respectively. The author is fully responsible for the correctness of the paper.
The use of generative AI in this paper is discussed in detail in Appendix A, which is joint work with Bin Dong and Guoxiong Gao.
Acknowledgements
The work was partially supported by the National Key R&D Program of China #2024YFA1014400.
The author would like to thank Bin Dong, Guoxiong Gao, Chi Li, Gang Tian, and Kewei Zhang for their enormous efforts in assisting with the verification of the paper and for many useful comments. The author would like to thank Bin Wu for carrying out the comparison tests with the other agent systems. The author would like to thank other members of the Rethlas and Danus team (namely Leheng Chen, Guoxiong Gao, Jiedong Jiang, Haocheng Ju, Shurui Liu, Zeming Sun, Yuefeng Wang, Bin Wu, Liang Xiao, and Bin Dong) for their contributions to the development of these systems. The author would like to thank Ruochuan Liu and Gang Tian for constant support and encouragement.
2. Preliminaries
We recall the conventions for test configurations and scalar curvature that will be used throughout the paper. We work over the field of complex numbers .
2.1. Test configurations and K-polystability
In this subsection, we fix the stability convention used in the statements of the main results.
Definition 2.1 (Test configurations).
Let be a polarized projective variety. A test configuration of exponent for means an algebraic test configuration in the sense of [Don02, Definitions 2.1.1–2.1.2] whose generic polarized fiber is . It is normal ample if its total space is normal and its relative polarization is ample. A polarized product test configuration is one induced by an algebraic one-parameter subgroup of , together with a scalar character on the polarization. After fixing an equivariant product identification over , let be the largest integer for which extends over the test configuration. This is the jump of the decreasing extension-order filtration. We use the increasing entry ; equivalently, precisely when extends. These entries are the negatives of the algebraic weights on the central fiber. We therefore label product data by their contributions to the increasing entries: fiber scaling contributes on the block indexed by , and the scalar character contributes in section degree . The algebraic weights on sections of the central fiber are and . This convention fixes all character signs used throughout the paper. The probability measure obtained from the normalized increasing entries is called the entry law. It is the reflection under of the standard central-weight Duistermaat–Heckman law.
Definition 2.2 (K-polystability).
A polarized projective variety is K-semistable with respect to normal ample algebraic test configurations if every such test configuration of every positive exponent has nonnegative Donaldson–Futaki invariant. It is K-polystable if, in addition, equality occurs only for polarized product test configurations.
2.2. Scalar curvature
In this subsection, we record the normalization used in the analytic and algebraic formulas.
Notation 2.3 (Scalar-curvature normalization).
Throughout the paper, denotes the Riemannian scalar curvature divided by . This is the normalization used in the Donaldson–Futaki formulas in this paper; it does not affect whether a Kähler metric has constant scalar curvature or is extremal.
3. The fivefold and its Kähler geometry
In this section, we construct the polarized fivefold of Theorem A, compute its admissible boundary polynomial, and prove Propositions 3.12 and 3.13.
3.1. The polarized fivefold
In this subsection, we choose four curves with pairwise Hom-orthogonal Jacobians and construct a smooth polarized fivefold whose connected automorphism group consists of fiber scalings.
Theorem 3.1 (Hyperelliptic Jacobians, [Zar00, Theorem 2.1, p. 124]).
Let be a field of characteristic zero, and let be an irreducible polynomial of degree at least five. Assume that the Galois group of is either the full symmetric group or the alternating group. If is the Jacobian of the smooth projective model of , then
| (3.1) |
In particular, is absolutely simple.
Theorem 3.2 (Chow base change, [Yu19, Theorem 1.2]).
Let be a primary field extension, and let and be semi-abelian varieties over . Then base change induces an isomorphism
| (3.2) |
Lemma 3.3.
There exist smooth projective connected curves of genera
| (3.3) |
such that
| (3.4) |
Proof.
We construct the curves over finitely generated fields and then embed those fields into . Theorem 3.1 will give simple Jacobians, and the distinct genera will exclude isogenies. For each , put . Choose algebraically independent variables over , let be the monic polynomial whose roots are those variables, and let be the field generated by the coefficients of . Let be the rational function field generated by the roots. Every permutation of the algebraically independent roots defines a distinct -automorphism of . Conversely, is the splitting field of , and every -automorphism of permutes those roots. Thus is separable and has Galois group . Since this group acts transitively on the roots, is irreducible. The smooth projective model of
| (3.5) |
has genus . By Theorem 3.1,
| (3.6) |
Thus each Jacobian is absolutely simple.
We next pass from the fields to the complex numbers. Choose embeddings , extend them to , and base change the curves. Since is a primary extension, Theorem 3.2, applied to the Jacobian over , gives
Hence the four resulting complex Jacobians remain simple.
It remains to prove the vanishing of the homomorphism groups in (3.4). Suppose that and that
| (3.7) |
is nonzero. Simplicity of the target implies that is surjective, while simplicity of the source implies that the identity component of is trivial. Therefore is an isogeny. This is impossible because . We obtain (3.4). ∎
Lemma 3.4.
Let be a smooth projective connected curve of genus for , put , and let and be line bundles on . Fix a positive integer . Assume that
| (3.8) |
In the quotient convention, put
| (3.9) |
Then is ample.
Proof.
Put
| (3.10) |
For and , linearity of the degree in gives
| (3.11) |
Choose so large that the right-hand side is at least for every . If is a line bundle of degree at least on and has length two, then the dual of is the space of sections of a line bundle of degree
| (3.12) |
Thus is surjective, so is very ample and hence globally generated. It follows that every line bundle in the decomposition
| (3.13) |
is then globally generated, while the two endpoint line bundles and are very ample on .
The evaluation maps of the summands in (3.13) are surjective at every point of . Hence the complete linear system of restricts to the complete linear system of on every fiber of and separates points in each fiber. If two points lie over distinct base points, choose at the first point a nonzero endpoint fiber coordinate. The corresponding endpoint summand, or , contains a base section which is nonzero at the first base point and vanishes at the second, so it separates the two points even when they lie in different fiber charts.
We next separate tangent vectors. Let be a tangent vector at . If , choose an endpoint monomial which is nonzero at and a base section which vanishes at and satisfies . Then
because , so the derivative of contributes nothing. If , then is vertical and is separated by the complete fiber system. Thus separates points and tangent vectors on , so it is very ample. Therefore is ample. ∎
Theorem 3.5 (Finite descent of ampleness, [Sta26, Tag 0B5V]).
Let be a finite surjective morphism of Noetherian schemes proper over , and let be a line bundle on . Then is ample if and only if is ample.
Proof.
This is [Sta26, Tag 0B5V]. ∎
Lemma 3.6.
Let be a Noetherian scheme proper over , with irreducible components in its reduction, and let be a line bundle on . If is ample for every , then is ample on .
Proof.
Construction 3.7.
Choose curves as in Lemma 3.3. For , choose line bundles and on with degree vectors
| (3.15) | ||||
Put
| (3.16) |
Using the Grothendieck quotient convention, define
| (3.17) |
Proposition 3.8.
In Construction 3.7, is a smooth projective fivefold, is ample, and
| (3.18) |
The group acts by scaling the -summand relative to the -summand.
Proof.
We first prove smoothness and ampleness. We then show that every connected automorphism acts trivially on and compute the relative automorphism group from the Euler sequence.
Step 1. In this step, we prove that is a smooth projective fivefold and that is ample. The degree vectors at the two ends of the fiber interval are
| (3.19) |
and
| (3.20) |
Every entry in (3.19) and (3.20) is positive. Lemma 3.4, applied with and , therefore shows that is ample. Since is a smooth projective fourfold and is the projectivization of a rank-two vector bundle on , the variety is smooth, projective, and .
Step 2. In this step, we show that the connected automorphism group acts trivially on . The signs in (3.15) imply that
| (3.21) |
The tensor-product decomposition of sections gives both vanishings because and have negative degree.
Every map from to a curve is constant because . Hence every rational curve in is contained in a fiber of . Conversely, any two points of a fiber are joined by that fiber. The fibers of are therefore exactly the equivalence classes generated by chains of rational curves. Every automorphism of preserves this equivalence relation and descends, using , to an automorphism of . Moreover,
| (3.22) |
because and each has negative degree. The identity component of the automorphism scheme of the smooth projective variety has this space as its Lie algebra. In characteristic zero it is smooth, and hence its zero Lie algebra implies . It follows that acts trivially on and has Lie algebra .
Step 3. We compute and conclude the proof in this step. In the quotient convention, the relative tangent bundle fits into
| (3.23) |
The projective-bundle formula gives and . Pushing forward (3.23) and using therefore give
| (3.24) |
By (3.21), . The diagonal automorphisms induce an effective copy of in . A connected one-dimensional algebraic group containing this copy equals it, and (3.18) follows. ∎
3.2. The admissible boundary polynomial
In this subsection, we compute the one-variable polynomial that controls both the algebraic and analytic threshold of the polarization .
Set-up 3.9.
For , put
| (3.25) |
For , define
| (3.26) |
The four affine factors are
| (3.27) | ||||||
They are positive on . Finally, set
| (3.28) |
Lemma 3.10.
Let be the polynomial determined by
| (3.29) |
Then
| (3.30) |
Moreover,
| (3.31) |
and the unique zero of in is
| (3.32) |
with multiplicity two.
Proof.
We verify an explicit candidate by interpolation at the four roots of the primitive affine factors. We then identify its differential-equation coefficient with and determine its interior zero.
Step 1. In this step, we define the candidate and compute its boundary data. Put
| (3.33) |
Write the primitive affine factors in (3.27) as , , , and . Their products at and both equal , and hence
| (3.34) |
Define
| (3.35) |
Then
| (3.36) |
The factorization in (3.35) gives
| (3.37) |
Step 2. In this step, we prove the required differential equation by polynomial interpolation. The roots of are
| (3.38) |
Substitution in (3.36) gives
| (3.39) | ||||||
Since
| (3.40) |
the identities in (3.39) are equivalent to
| (3.41) |
Set and
| (3.42) |
By (3.41), for every . The leading coefficients of and are and , while . Thus the coefficient of degree four in equals
| (3.43) |
We have , and the four distinct zeros force . Therefore
| (3.44) |
Step 3. In this step, we identify with the coefficient in Set-up 3.9. Integrating (3.44) over and using (3.37), we obtain
| (3.45) |
Since , (3.28) implies that . Hence satisfies (3.29). Uniqueness follows because the difference of two solutions has zero second derivative, value, and first derivative at .
Step 4. We determine the zero set of and conclude the proof in this step. The factor is nonnegative on , while the remaining factor in (3.30) is a square. The roots of are and . Only the first root belongs to , and its multiplicity in is two. ∎
3.3. Approximation by admissible metrics
In this subsection, we use the boundary polynomial to construct smooth Kähler metrics whose scalar curvatures converge in to the average scalar curvature.
Theorem 3.11 (Admissible metric formulas, [ACG+08, Theorem 1, Section 1.3, and Proposition 6, especially eq. (10)]).
Let be smooth on , positive on , and assume that
| (3.46) |
Then the admissible construction with profile defines a Kähler form in . Put
| (3.47) |
Let be the constant-curvature form on , and let be the connection one-form in the admissible construction. There exists a constant , independent of , such that
| (3.48) | ||||
Here has the normalization fixed in Notation 2.3.
Proposition 3.12.
There exist smooth admissible Kähler metrics , indexed by , such that
| (3.49) |
In particular,
| (3.50) |
Proof.
We perturb by a positive polynomial whose value and first derivative vanish at both boundary points. The scalar-curvature formula then makes the dependence of the error on the perturbation parameter explicit. We first construct smooth admissible profiles in the fixed Kähler class. Put
| (3.51) |
The function is positive on and satisfies
| (3.52) |
Lemma 3.10 shows that and that its only interior zero is . Hence on , and satisfies (3.46). By Theorem 3.11, it defines a smooth admissible metric .
3.4. Nonexistence of extremal metrics
We prove Proposition 3.13 by deforming a hypothetical extremal metric to nearby admissible classes and then passing the explicit momentum-profile identity to the limit.
Proposition 3.13.
The class contains no extremal Kähler metric. In particular, it contains no cscK metric.
We first state the external analytic inputs and the local computations used to prove Proposition 3.13.
Theorem 3.14 (Constructive admissible-extremal theorem, [ACG+08, Proposition 1, pp. 553–554; Lemma 5, p. 567; Proposition 8 and the paragraph following it, pp. 569–570]).
Let be an admissible Kähler class on a split projective-line bundle over a local product of cscK factors, with momentum coordinate . Let
| (3.56) |
where and are the admissible parameters and multiplicities. There exists a unique polynomial satisfying the extremal differential equation and the conditions
| (3.57) | ||||||
If on , then defines a smooth admissible extremal metric in . Conversely, every admissible extremal metric in has this profile and requires on . Let be the real generator of the fiber circle in the normalization of [ACG+08, eq. (1)], and denote the two fixed sections by and according to the value of the momentum coordinate. If and are the resulting Riemannian and Kähler metrics, then
| (3.58) |
Theorem 3.15 (Maximal compactness, [Cal85, Theorem 3]; see also [CPZ15, Theorem 4.1]).
Let be a compact complex manifold with an extremal Kähler metric. Then the identity component of the holomorphic isometry group is a maximal compact connected subgroup of .
Theorem 3.16 (Equivariant openness of extremal metrics, [LS93, Section 5, pp. 263–269, especially Propositions 7–8]).
Let be a compact extremal Kähler manifold, and let be the identity component of the isometry group of . There exists a neighborhood of in the space of -invariant Kähler classes such that every contains a -invariant extremal metric . After fixing the gauge, if , then these metrics may be chosen so that in .
Proof.
Put and fix an integer . Let be the finite-dimensional space of real -harmonic -forms. The connected group acts trivially on cohomology and by isometries, so every element of is -invariant. Let be the real -invariant Sobolev space of order , and let be the -orthogonal complement of the kernel of the Lichnerowicz operator of . This is the gauge which removes the holomorphy-potential directions.
Let and denote the -projectors onto the holomorphy potentials for and , respectively, and let be the scalar curvature of the latter metric. On a neighborhood of in , [LS93, Section 5] constructs the map
| (3.59) |
The kernel identity in equation (5.3) of that paper shows that the second component in (3.59) vanishes exactly when the corresponding metric is extremal. Proposition 7 there proves that this map is and identifies its linearization; Proposition 8 proves that the linearization is an isomorphism and applies the Banach inverse function theorem. Restricting to therefore gives a map near the origin, with , such that
| (3.60) |
is a -invariant extremal metric. This proves the equivariant existence assertion without changing the group or the gauge.
If in , continuity of the inverse map gives in . Since the real dimension is and , Sobolev embedding gives . Taking two derivatives in (3.60) yields in , as asserted. ∎
Theorem 3.17 (Uniqueness of extremal metrics, [BB17, Theorem 4.15]).
Let be a compact complex manifold, and let and be extremal Kähler metrics in one Kähler class. Then there exists such that
| (3.61) |
Set-up 3.18.
Put
| (3.62) |
For , define
| (3.63) |
Then . Define
| (3.64) |
For , put
| (3.65) |
and, for , put
| (3.66) |
Let be determined by
| (3.67) |
and define
| (3.68) |
This is the extremal polynomial of in the momentum coordinate .
Lemma 3.19.
Proof.
Expanding (3.64) gives
| (3.72) | ||||
Termwise integration gives
| (3.73) | ||||
Substitution in (3.67) gives (3.69). The determinant is nonzero because is the strictly positive variance of for the positive weight . Substitution in (3.68) and termwise integration give (3.70). The quadratic factor has roots and , and only the first belongs to . ∎
Set-up 3.20.
Let
| (3.74) |
For , let be the polynomial obtained from Set-up 3.18 by replacing with and keeping the numbers fixed, and put
| (3.75) |
For , put
| (3.76) |
Lemma 3.21 (The nearby admissible classes).
Let
| (3.77) |
where is the pullback to of the positive generator of . For , put
| (3.78) |
Then is an admissible Kähler class with a representative invariant under the fiber circle. Its restrictions to the two fixed sections and its fiber integral are
| (3.79) |
Here and are the sections defined by the quotients and , respectively. The extremal polynomial of in the coordinate is . Moreover,
| (3.80) |
Proof.
Choose on each the signed constant-curvature form with cohomology class , and suppress pullbacks to and . Let be the connection in the admissible construction, normalized by
| (3.81) |
On the complement of the two fixed sections, the admissible Kähler form has the expression
| (3.82) |
For , the numbers and have equal signs and . Thus every is positive and for . The function is positive on , vanishes at the endpoints, and has derivatives and there. The canonical admissible metric and its compactification in [ACG+08, pp. 553–555, Sections 1.2–1.3] therefore extend (3.82) to a smooth Kähler form on invariant under the fiber circle.
The quotient convention is the line convention of [ACG+08, Section 1.3, pp. 554–555] applied to . Thus on and on . At these endpoints, the restrictions of (3.82) are the two classes in (3.79). The circle has period , so the integral of along a projective-line fiber is . In the quotient convention, is trivial on , restricts to on , and has degree one on every fiber. The projective-bundle formula now identifies with (3.78).
For the specialization, and . The definition of gives
| (3.83) |
Substitution in (3.78) yields . The convergence in (3.80) follows directly from (3.78) and . It remains to identify the extremal polynomial. The moment system defining in Set-up 3.20 is the extremal moment system for . Uniqueness in Theorem 3.14 identifies its extremal polynomial with . ∎
Lemma 3.22.
There exists such that
| (3.84) |
At the double zero , the transverse derivative is
| (3.85) |
Define
| (3.86) |
Then
| (3.87) |
Proof.
We first prove real-analytic dependence on and verify the positive transverse derivative. We then treat a neighborhood of the double zero, the two boundary neighborhoods, and the remaining compact set separately.
Step 1. In this step, we prove real-analytic dependence and the positivity in (3.85). For , the polynomial in (3.75) is positive on . The determinant of the corresponding moment system is the negative of the strictly positive weighted variance
| (3.88) |
Hence every coefficient of depends real-analytically on . Differentiating the defining integrals gives (3.85). Its sign follows from
| (3.89) |
Step 2. In this step, we prove positivity near the interior double zero. Let
| (3.90) |
By Lemma 3.19, is nonnegative on and vanishes only at , , and . Joint real-analyticity and (3.85) imply that remains positive on a neighborhood of for all sufficiently small . Integrating from to proves that on this neighborhood when .
Step 3. We prove positivity on the complement of the double-zero neighborhood and conclude the proof in this step. The moment system preserves the four boundary conditions. Since remains uniformly positive, there exist such that, after decreasing the upper bound on ,
| (3.91) |
For , integration of the derivative bounds gives
| (3.92) |
On the remaining compact subset, has a positive minimum, which remains positive for small . Taking the minimum of the resulting upper bounds on proves (3.84). Real-analytic coefficient dependence gives uniformly on , while uniformly. The polynomial is strictly positive on , so the denominators in (3.86) are uniformly bounded away from zero for small . This proves (3.87). ∎
Lemma 3.23.
Let be an admissible Kähler metric on , with Kähler form , profile , and normalized momentum coordinate as in (3.58). If , then is admissible for the split bundle in (3.17) and the original fiber circle. Its profile is , its normalized momentum coordinate is , and
| (3.93) |
The values of on and are and , respectively. If is extremal, then is extremal.
Proof.
By Proposition 3.8, is a fiber scaling. Hence covers the identity on , fixes and , and commutes with the fiber circle. In particular, . Pulling back (3.58) gives
and
The fixed-section values and the compactification data are unchanged, so is admissible with the asserted profile. Scalar curvature and holomorphicity of its gradient are natural under biholomorphic pullback, which proves the last assertion. ∎
Proof of Proposition 3.13.
Positive rescaling preserves extremality, so it is enough to consider . Suppose that contains an extremal metric . We construct admissible extremal metrics in nearby classes and pass their momentum-profile identity to the limit.
Step 1. In this step, we construct admissible extremal metrics converging to . By Proposition 3.8, the fiber circle is the unique maximal compact connected subgroup of . Theorem 3.15 implies that this circle acts isometrically on . Let be its period- real holomorphic generator, oriented as in (3.58).
Choose a sequence as in Lemma 3.22, and put . Lemma 3.21 shows that these are circle-invariant Kähler classes with fiber integral and that . By Theorem 3.14, there exists an admissible extremal metric with profile . By Theorem 3.16, there also exist circle-invariant extremal metrics such that
| (3.94) |
Theorem 3.17 gives such that
| (3.95) |
Let and put . Lemma 3.23 gives
| (3.96) |
Step 2. In this step, we prove uniform convergence of the normalized momentum coordinates. Fix a background Riemannian metric on , let be its diameter, and choose . The convergence in (3.94) implies that uniformly. Hence is uniformly Cauchy. Since , for all and all , integration along a minimizing background geodesic from to gives
| (3.97) |
Thus converges uniformly to a continuous function . Moreover,
| (3.98) |
Step 3. We use the interior zero of the limiting profile to obtain a contradiction. Fix . The fiber meets and , where has values and . The intermediate value theorem therefore gives a point such that
| (3.99) |
This value lies in , so is not on either fixed section. On each fiber the circle acts by and its generator vanishes only at and . Hence
| (3.100) |
| (3.101) |
The absolute value of the difference in (3.101) is at most plus . Passing to the limit in (3.96) and using Lemma 3.19, we obtain
| (3.102) |
This contradicts the positive definiteness of and (3.100). Thus contains no extremal metric. Rescaling by proves that contains no extremal metric, and a cscK metric is extremal. ∎
4. Affine transforms of the two fiber-scaling initial filtrations
This section defines the two initial filtrations obtained from the opposite directions of fiber scaling and proves that either convex transform is affine in the fiber coordinate.
4.1. Fixed Veronese gradings and rational rays
We first fix the degree convention and identify the limiting measures on the five-dimensional Newton–Okounkov body and on its rational slices.
Set-up 4.1.
Notation and conditions are as in Construction 3.7. Thus
| (4.1) |
The genera and the degrees are
| (4.2) | ||||
| (4.3) | ||||
| (4.4) |
The Jacobians satisfy
| (4.5) |
We use the quotient convention in
| (4.6) |
For an actual power of , fiber scaling gives
| (4.7) |
Fix a positive integer , and let be a normal ample algebraic test configuration whose generic polarized fiber is and such that . Exponent-one degree for is called the section degree; it corresponds to the actual -degree
| (4.8) |
The supported section ring is
| (4.9) |
Let be the subspace whose increasing entries are at most . Define two one-parameter subgroups on (4.7) by
| (4.10) |
For and every , put
| (4.11) |
where the limit is taken in the Grassmannian of subspaces of the fixed dimension. These filtered pieces define the two fiber-scaling initial filtrations . Neither initial filtration is recentered or shifted by a character. Denote the convex transform of in actual degree by . If is a decreasing-filtration jump, we use the increasing-entry convention
| (4.12) |
For , put
| (4.13) | ||||
| (4.14) |
All four functions are positive on . Let be normalized five-dimensional Lebesgue measure on , and define
| (4.15) |
The -marginal of is .
When one sign is fixed, we suppress it from the notation. Write for the entries of on , counted with multiplicity, and put
| (4.16) |
We record five external results in the precise forms used in this section.
Theorem 4.2 (Filtered Okounkov equidistribution, [BC11, Theorem 1.11 and Remark 1.12(i)]).
Let a graded linear series containing an ample series carry a multiplicative, linearly bounded filtration, and choose a valuation with one-dimensional leaves. The normalized filtration measures converge to the pushforward of normalized Lebesgue measure on the Newton–Okounkov body by the associated concave transform. After reversing the sign of the filtration jumps, this convergence statement holds for the convex transform in the increasing-entry convention.
Theorem 4.3 (Normal generation on a smooth curve, [FT14, Theorem 4.2]).
Let be a smooth projective curve of genus , and let be a line bundle on with . Then the complete section ring of is generated in degree one.
Mumford’s curve theorem builds on Gieseker’s method; see [Mum77, Theorem 4.15] and Gieseker’s later account [Gie82].
Theorem 4.4 (Chow stability of embedded curves, [Mum77, Theorem 4.15]).
Let be a smooth projective curve of genus , and let be a line bundle of degree at least . The embedding defined by the complete linear system of is Chow stable.
The balanced-metric criterion is due independently to Luo and Zhang [Luo98, Zha96]. We use the modern formulation in [AH15, Theorem 2].
Theorem 4.5 (Luo–Zhang balanced-metric criterion, [AH15, Theorem 2]).
Let be a compact polarized projective manifold. Then is Chow polystable if and only if admits a balanced Hermitian metric.
Theorem 4.6 (Asymptotic Chow stability, [Don01, Corollary 4]).
Let be a polarized manifold admitting a cscK metric in , and assume that its polarized automorphism group is discrete. Then is Chow stable for every sufficiently large .
Lemma 4.7 (Uniform Chow polystability of the base blocks).
Notation and conditions are as in Set-up 4.1. There exists an integer such that, whenever and , the complete linear-system embedding of defined by is Chow polystable.
Proof.
Put
| (4.17) |
The degree of the restriction of to is
| (4.18) |
Choose so that for every . Theorem 4.4 shows that all four factor embeddings are Chow stable. By Theorem 4.5, each factor polarization admits a balanced Hermitian metric .
Let on . The tensor-product isomorphism identifies an external tensor product of orthonormal factor bases with an orthonormal basis of the product section space. The Bergman function of this basis is the product of the four constant factor Bergman functions, and is therefore constant. Thus is balanced. A second application of Theorem 4.5 proves the Chow polystability of the product embedding. ∎
The following normalization ties the Chow stability statements above to the filtration invariants used in the rest of this section. Let be a polarized projective variety of dimension , and let be a multiplicative linearly bounded filtration of the complete section ring with rational entries. Fix such that the Veronese subring is generated in degree one, and generate a filtration of this subring from the weighted space . Regrading into degree , the Hilbert and increasing-entry total functions of the generated filtration admit two-term expansions
| (4.19) |
because the total weight function of an ample test configuration is a polynomial of degree at most in all sufficiently large degrees [Don02], [BHJ17, Theorem 3.1]. Writing for the mean entry of on , the degree- Chow weight and the intrinsic asymptotic Chow invariant of are
| (4.20) |
Adding a constant to all entries in degree shifts and by the same amount, and a positive rescaling of the entries rescales both terms linearly, so is computed by the centered integral flag obtained from the degree- entries by subtracting the mean and clearing denominators. The one-parameter subgroup associated with this centered flag acts on with weights the negatives of the centered entries, in the convention of Definition 2.1; its induced degeneration of the section ring is the generated filtration, with total central weight in degree , and by Mumford’s weight formula the Hilbert–Mumford weight of the Chow point of the embedded cycle , computed on this degeneration, is a positive multiple of its leading normalized coefficient, that is, of [Mum77, Theorem 2.9]. Chow semistability of the embedding defined by , and in particular the Chow stability and polystability statements above, therefore implies
| (4.21) |
for every such flag supported in degree . This is the normalization in which the strictness theorem of [Sze15] is quoted in Section 4.4.
Lemma 4.8 (Fixed-Veronese normalization).
Notation and conditions are as in Set-up 4.1. Fix and write . Then is block preserving with respect to (4.7), multiplicative, integer valued, and two-sided linearly bounded on . In every supported actual degree , its entry multiset agrees with that of .
Normalize valuation coordinates and filtration entries by . The Newton–Okounkov body is , and has a finite convex transform . The entry probability laws in actual degree of and are both
| (4.22) |
If is the transform in the exponent-one grading of , then its body is and
| (4.23) |
Consequently, the exponent-one entry law is
| (4.24) |
Let , where may be replaced by a common positive multiple so that
| (4.25) |
Then the exact-ray ring
| (4.26) |
is graded by the integer in this display; we call its ray degree. The ring is generated in ray degree one. The restricted filtration has transform on , nonnegative asymptotic Chow invariant, and squared norm
| (4.27) |
where is normalized four-dimensional Lebesgue measure on .
Proof.
We prove the statement in three steps. The first concerns the degreewise initial filtration, the second compares the two gradings, and the third restricts to a rational ray.
Step 1. In this step, we identify the body and the limiting law in actual degree. In a fixed section degree, torus specialization is a Grassmannian limit of every filtered subspace. Such a limit preserves dimension. Taking successive differences of filtered-piece dimensions shows that the complete entry multiset is unchanged. The limiting subspaces are invariant under fiber scaling and hence are direct sums of the blocks . The multiplication inclusions defining a multiplicative filtration are closed conditions on the relevant Grassmannians, so they pass to the limit. Integrality and the common linear bounds are also unchanged.
The valuation defined by the fiber index and product points has one-dimensional leaves. In actual degree , its normalized value semigroup has closed convex body . Restricting the degree semigroup to does not change the normalized cone or its degree-one slice. Theorem 4.2 identifies the limiting entry law with . Equality of the degreewise entry multisets identifies this law with that of .
Step 2. In this step, we compare actual degree with exponent-one degree. Exponent-one degree for is actual degree . Dividing a valuation vector and an entry by , rather than by , multiplies both by . This proves (4.23) and (4.24).
Step 3. In this step, we identify the rational-ray transform and its norm. Choose as in (4.25), and put . Every factor of has degree at least . Theorem 4.3, the tensor-product decomposition, and the factorwise multiplication maps show that the complete section ring of is generated in degree one.
Write in lowest terms and . For the body and transform, compare the exact-ray filtered cone with the -slice of the supported global filtered cone, working temporarily with decreasing jumps. Represent an arbitrary point of the slice by supported sections , where , recorded at levels , and with . Put . Choose so that both endpoint blocks and are nonzero for every ; this is possible because and are ample.
If , choose , using when , and form
| (4.28) |
If , put , choose , and form
| (4.29) |
These products are nonzero because the section ring is a domain. Record them at the levels supplied by multiplicativity, choosing for an admissible level whose absolute value is bounded linearly by its degree. Since , the endpoint correction has sublinear degree, valuation, and filtration cost relative to the powered main term. Consequently, the normalized exact-ray points converge to the original slice point. The opposite inclusion is tautological, so the closed filtered cones agree. Reflecting back to increasing entries and normalizing by ray degree , rather than actual degree , gives the body and the transform . The product of constant-curvature metrics on the four curves is cscK in , and the polarized automorphism group is finite. Theorem 4.6 gives Chow stability of the embedding defined by for every sufficiently large , so the Hilbert–Mumford bridge (4.21) gives for every sufficiently large , and the intrinsic asymptotic Chow invariant (4.20) is nonnegative. Theorem 4.2, applied to the exact ray, gives (4.27). All degrees used here are divisible by . ∎
4.2. Multiplication along fixed rational pairs
Lemmas 4.9 and 4.10, together with Lemma 4.11, compare the mean entries on two fixed rational blocks with the mean entry on their midpoint block, without requiring the two filtration flags to split across the four curve factors.
Lemma 4.9 (The one-curve multiplication projector).
Let be a smooth projective curve of genus , and let be positive line bundles on . Choose constant-curvature Hermitian metrics on , and , and use the resulting -products. Put
| (4.30) |
and let be multiplication. For all sufficiently large , the map is surjective. Put
| (4.31) |
Let be the orthogonal projector onto . Then there exist such that
| (4.32) |
Proof.
We prove Lemma 4.9 in five steps. The first fixes the metrics, the next two construct the compact Bergman kernels from the disk kernels, the fourth compares their products, and the fifth derives the partial-trace estimates.
Step 1. In this step, we choose constant-curvature metrics and compute the dimensions. Normalize the hyperbolic area form by
| (4.33) |
For a positive line bundle of degree , start with a Hermitian metric and put . Since , the scalar Poisson equation gives a smooth real function such that
| (4.34) |
Replacing by , with the corresponding Chern curvature convention, gives
| (4.35) |
We carry out this construction for and , and use the product metric on . If and , then, for all sufficiently large , the three relevant degrees exceed and the dimensions are
| (4.36) |
Step 2. In this step, we compute the exact disk kernel. By uniformization [Jos06, Theorem 4.4.1], after identifying the upper half-plane with the disk, write , where is torsion-free and cocompact, and normalize
| (4.37) |
For a degree- constant-curvature line bundle, the pullback to is holomorphically trivial: after choosing a smooth frame, we solve its scalar -equation on the disk. Fix a holomorphic frame . With the convention , the identity , together with (4.35) and (4.37), shows that is harmonic. Since is simply connected, write with holomorphic. Replacing by gives a holomorphic frame in which
| (4.38) |
Because the metric and bundle are pulled back from , every deck transformation acts holomorphically and unitarily. In the frame , these actions are given by holomorphic factors of automorphy satisfying the cocycle law; Step 3 uses these deck identifications invariantly through the maps . The lifted norm of is
| (4.39) |
The monomials are orthogonal and satisfy
| (4.40) |
Summing the binomial series gives the exact weighted disk kernel
| (4.41) |
Its invariant norm is
| (4.42) |
In compatible product frames,
| (4.43) |
Step 3. In this step, we justify reproduction and periodize the disk kernel. Put
| (4.44) |
If is holomorphic and belongs to , the kernel reproduces . For polynomials this follows by expanding the kernel and using angular integration and (4.40). For a general , let . Rotation invariance of the weight, strong continuity of rotations on weighted , and the integral representation
show that in the weighted -norm as . Each is holomorphic on a disk larger than the closed unit disk, so its Taylor polynomials converge uniformly there. Since the scalar kernel is bounded in for fixed , the polynomial identity passes first to and then to . Thus
| (4.45) |
If is the lift of a section on , its invariant pointwise norm is bounded, and hence
| (4.46) |
For , the right-hand side belongs to . Therefore (4.45) applies to all lifted compact sections in the sufficiently positive degrees used in Steps 4 and 5 of this proof.
Let
be the unitary deck identification for a lifted constant-curvature bundle , and define
| (4.47) |
The compact Bergman kernel is
| (4.48) |
Packing disjoint hyperbolic balls gives the orbit estimate
| (4.49) |
Together with (4.42), this gives normal convergence in sufficiently positive degrees. For a fundamental domain and a lifted compact section , absolute convergence, automorphy, and (4.45) give
| (4.50) | ||||
The cocycle law gives descent, and replacing by gives Hermitian symmetry. This proves (4.48).
Step 4. In this step, we compare the product of the two compact kernels with the kernel of the tensor product. Use the periodization (4.48) for , , and . The terms with equal deck indices satisfy (4.43), and therefore
| (4.51) |
Let . If , then , so at least one of and is at least . In that factor of (4.42), split the exponent in half. To display the double summation, after decreasing two positive constants if necessary, put
The kernel prefactors are , while (4.49) gives, uniformly in and large ,
Partition the unequal pairs according to which of the two distances is at least . One half of the corresponding far factor is at most , and therefore
| (4.52) | ||||
The polynomial prefactor has been absorbed by decreasing the exponential rate. This estimate is uniform on . Hence
| (4.53) |
Step 5. In this step, we pass from kernels to partial traces. The Schwartz kernel of is the product of the first two Bergman kernels. Since has total mass one, the uniform bound in (4.53) also bounds the Hilbert–Schmidt norm of the error operator and gives
| (4.54) |
Since , the operator is invertible for large , so is surjective. The required projector is
| (4.55) |
On the diagonal of (4.48), the term indexed by the identity deck transformation is the dimension, and the other terms are exponentially small. Thus
| (4.56) |
For an orthonormal basis of , direct expansion gives
| (4.57) |
Consequently,
| (4.58) |
Write . By (4.54), . Expanding as its convergent geometric series and using give
| (4.59) |
A partial trace costs at most the dimension of the traced factor. Combining (4.58) and (4.59) with and proves (4.32). ∎
Lemma 4.10 (A Plücker estimate for quotient filtrations).
Let be a surjective map of finite-dimensional Hermitian spaces of dimensions . Give arbitrary increasing weighted flags with mean jumps . If and are their filtered pieces, give the quotient filtration
| (4.60) |
Let be its mean jump, and let be the centered self-adjoint flag generators. If is the orthogonal projector onto , put
| (4.61) |
Then
| (4.62) |
Proof.
Orthogonally split the two flags, choose orthonormal eigenbases and , and set
The columns have centered costs
Scanning these costs in increasing order shows that the quotient dimension at a threshold is the rank of the columns available at that threshold. Hence
| (4.63) |
where ranges over the column sets giving a basis of .
On , set
For a nonzero Plücker vector , its coordinate indexed by is nonzero exactly when the corresponding maximal minor of is nonzero, and its -weight is . Therefore the largest occurring weight is
| (4.64) |
After normalizing , the expectation of the additive exterior-power generator is a convex combination of its occurring weights. Hence
| (4.65) | ||||
In particular,
| (4.66) |
The scalar parts vanish because are trace free. Estimating the two remaining traces by operator norms and dividing by gives (4.62). ∎
Lemma 4.11 (The fixed-pair mean inequality).
Notation and conditions are as in Set-up 4.1. Fix with , and choose a positive integer divisible by and by the reduced denominators of . There exist constants such that, for all sufficiently large , with ,
| (4.67) |
The constants may depend on the fixed test configuration, initial filtration, denominator, rational pair, and Hermitian data, but not on . No uniformity for a moving pair is asserted.
Proof.
We prove the inequality in three steps. The first identifies the three section spaces, the second tensorizes the one-curve projector estimates, and the third applies Lemma 4.10 to the actual filtration flags.
Step 1. In this step, we fix the rational-slope bundles. Put
| (4.68) |
Every has positive degree. With , we have
| (4.69) | ||||
The relevant multiplication is the ordinary multiplication map among these three complete section spaces.
Step 2. In this step, we tensorize the multiplication projectors. For each , Lemma 4.9, applied to , gives the factorwise estimate. The tensor-product decomposition and canonical regrouping identify the global spaces and maps as
| (4.70) |
| (4.71) |
Partial trace commutes with tensor products. The scalar ratios in (4.32) are bounded, and only four factors occur. Expanding the tensor products gives
| (4.72) |
The finite-dimensional flag estimate is expressed in the dual space. Let be the antiunitary Riesz maps. Then
| (4.73) |
Thus the orthogonal projector onto is the antiunitary conjugate of , and its partial traces have the errors in (4.72). This conclusion concerns the full groups , so it applies to arbitrary flags on the complete section spaces.
Step 3. In this step, we use the Plücker estimate for the initial-filtration flags. Give the two source spaces their actual increasing weighted flags and the target the quotient filtration defined in (4.60). Let be its mean entry. Linear boundedness gives
where are the centered splitting generators. The exact linear dimension formula on each curve factor and the tensor-product decomposition give
| (4.74) |
The combination of (4.72) and (4.73), followed by Lemma 4.10, yields
| (4.75) |
Finally, multiplicativity in the convention (4.12) gives
| (4.76) |
Thus every jump of the target filtration is at most the corresponding jump of the quotient filtration. Hence , and (4.67) follows. ∎
4.3. The scalar profile and a summability estimate
We next use the zero Donaldson–Futaki equality to locate the curvature of the slice-average profile and to bound the accumulated blockwise error.
Lemma 4.12 (Endpoint trapezoidal summation).
Let be a finite continuous convex function on , and let . Then
| (4.77) |
Proof.
For , the exact error kernel after multiplying the trapezoidal error by is
| (4.78) |
It satisfies and . Distributionally,
| (4.79) |
Finite convexity gives and . Dominated convergence applies to the term, while the uniform bound applies to the remaining finite measure. This proves (4.77). ∎
Lemma 4.13 (One-crease profile and a summability estimate).
Notation and conditions are as in Set-up 4.1. Fix one of the two filtrations , and let be its convex transform after division by the actual degree. For , restrict to the exact endpoint ring
| (4.80) |
and let be the transform of this restricted filtration on , with valuation coordinates and entries divided by the actual degree . For , define
| (4.81) |
At the endpoints, define
| (4.82) |
Then is finite and convex on , satisfies
| (4.83) |
and has the form
| (4.84) |
For all sufficiently large supported degrees , the complete section ring of is generated by its degree-one piece for every . Generate a filtration of this ring from the weighted space , and write
| (4.85) | ||||
| (4.86) |
The two-term expansions hold as in (4.19). Define
| (4.87) | ||||
| (4.88) | ||||
| (4.89) |
There exist and such that, for every supported actual degree ,
| (4.90) |
Proof.
We prove Lemma 4.13 in four steps. The first proves the pointwise signs, the second compares the exact endpoint rays with the global filtered cone, the third identifies the equality profile, and the fourth proves the uniform summability in (4.90).
Step 1. In this step, we prove the nonnegativity of and . Center the entries on by . After clearing their common scalar denominator, we obtain an integral one-parameter subgroup. In the algebraic weight convention, the weights are the negatives of the increasing entries. After increasing , (4.18) and Theorem 4.3, together with the tensor-product decomposition, show that every block ring in the statement is generated by . Lemma 4.7 and the Hilbert–Mumford bridge (4.21) therefore give
| (4.91) |
The filtration generated by uses only products of sections from that block. The exact-ray filtration may also use sections from other supported degrees. Subadditivity in (4.12) therefore implies that the leading mean of the generated filtration is at least the leading mean of the exact-ray filtration. For , Lemma 4.8 identifies the leading exact-ray mean with . At , the ring generated from or is the -th Veronese of the corresponding exact endpoint ray, so both valuation coordinates and filtration entries are multiplied by . The leading exact-ray mean is therefore at the endpoints as well. Thus
| (4.92) |
This proves and .
Step 2. In this step, we compare the endpoint values. For , the change of variables in (4.81) gives
| (4.93) |
For fixed , the argument of is affine in , because every is affine. Convexity of , followed by integration over the cube, proves that is convex on . Work temporarily with decreasing jumps. Let be the closed downward-saturated cone generated by the filtered tuples
| (4.94) |
The cone generated by the exact tuples with lies in . Its upper boundary is therefore at most the upper boundary of the global face. For , consider the affine radial path
| (4.95) |
The upper boundary restricted to this path is finite and concave. Closedness shows that its face value is at least its interior limit, while convexity of the hypograph with a fixed interior point gives the reverse inequality. Thus the face value is the interior radial limit. Reversing the sign from decreasing jumps to increasing entries shows that the exact endpoint transform at is at least the interior radial limit. For the radial path oriented from toward the interior, closedness makes the face value at least its interior limit, while convexity of the hypograph makes it at most that limit. After reversing the filtration sign, the exact endpoint transform at is at least its interior radial limit. Averaging over the fixed unit cube and using the common linear bound gives
| (4.96) |
Raising the endpoint values of a convex function preserves convexity. Hence the endpoint extension of is finite and convex.
Step 3. In this step, we determine the support of . Set
| (4.97) | ||||
| (4.98) |
Write
| (4.99) |
The boundary polynomial from Lemma 3.10 is
| (4.100) |
and satisfies
| (4.101) |
It is nonnegative on , and its unique interior zero is the double zero .
Let be the interior endpoint limits of , and put
Let denote the continuous convex extension of to , so that . Distributional integration by parts gives
| (4.102) |
Both terms on the right are nonnegative.
For a sufficiently large supported actual degree , define the increasing-entry total of the algebraic source by
Put
| (4.103) |
The first equality is the definition of , and the second follows from degreewise preservation of the entry multiset. In particular, . Uniformly in , every curve-factor line bundle has degree greater than for sufficiently large , and hence has dimension equal to its degree plus . The tensor-product decomposition gives
| (4.104) |
Define
Lemma 4.12, first applied to and then to , together with (4.104), gives
| (4.105) |
The last two summands in are and .
Write the increasing-entry expansion of the algebraic source as
| (4.106) |
Equality of the limiting entry laws, equivalently equality of their Duistermaat–Heckman pushforwards, gives
| (4.107) |
Substitution in (4.103), followed by along all sufficiently large supported degrees, gives
| (4.108) |
In the increasing-entry convention, put
| (4.109) |
In the exponent-one grading of , the Hilbert coefficients are and the increasing-entry coefficients are . Up to the fixed nonzero normalization and the common weight sign, the Donaldson–Futaki numerator is
Therefore the hypothesis forces , and . The scalar coefficient formula uses the identical positive factor:
| (4.110) |
By (4.107), (4.108), and (4.102), we obtain
| (4.111) |
Both inequalities are equalities. Since the leading coefficients already agree, equality of the two coefficient expressions gives
| (4.112) |
We use (4.112) in Step 4 to prove the uniform bound in (4.90). Equality in (4.102) yields
| (4.113) |
Since is a nonnegative measure, it equals for some . This proves (4.84).
Step 4. In this step, we prove the uniform bound for . The exact expression for has the form
| (4.114) |
For the affine part of , expansion into powers of and the exact power-sum formulas give a polynomial in . Apply the kernel identity (4.78) with and . Since is a finite signed measure and , we obtain
| (4.115) |
The function is Lipschitz, and its corresponding sum is the integral term times with an remainder. After including the factors in (4.114), all remaining terms are , uniformly in the fractional part of . Hence
| (4.116) |
Combining (4.106), (4.107), (4.112), and (4.116) in (4.103) gives
| (4.117) |
Since all four are strictly positive on , put . After increasing the lower bound on , every factor in (4.104) is at least , uniformly for . Thus
| (4.118) |
Every is nonnegative, and hence
| (4.119) |
This proves (4.90). ∎
4.4. Constancy on rational slices
The summability estimate (4.90) and the fixed-pair inequality (4.67) force the asymptotic Chow invariant to vanish on every rational ray, after which a strictness theorem removes all variation inside each rational slice.
Theorem 4.14 (Strictness for filtrations, [Sze15, Proposition 11]).
Let be a polarized manifold admitting a cscK metric in , and assume that its polarized automorphism group is finite and that the complete section ring of is generated in degree one. A multiplicative linearly bounded filtration of this ring with positive -norm has strictly positive asymptotic Chow invariant.
A multiplicative linearly bounded filtration is brought to the normalization of [Sze15], in which the filtration is increasing and begins at level zero, by an entrywise shift linear in the degree; the shift changes neither the centered -norm nor any finite Chow weight (4.20).
Lemma 4.15 (Vanishing on every rational ray).
Proof.
We prove the assertion in three steps. First we identify the ray invariant with a liminf of finite Chow defects. We then choose finitely many fixed symmetric pairs in one affine interval of , and finally use the summability bound to make the total defect of one pair arbitrarily small.
Step 1. In this step, we identify the finite Chow normalization. Write
for the Hilbert function in ray degree . Let be the finitely generated approximation generated by the weighted ray-degree- space , let be the total central algebraic weight of in ray degree , and let be the leading central weight coefficient of in the original ray grading. In the ray variables, the finite Chow normalization (4.20) reads
| (4.121) |
The exact-ray ring is generated in ray degree one. Dilation from ray degree one to ray degree , together with the sign change from increasing entries to central algebraic weights, gives
| (4.122) |
It follows from (4.121) and (4.122) that
By (4.20), the intrinsic asymptotic Chow invariant is the full-sequence lower limit of these finite invariants. Consequently,
| (4.123) |
Step 2. In this step, we choose the rational symmetric pairs. Since is irrational, the rational number lies in one of the two open intervals on which is affine. Denote that interval by . Fix a positive integer , choose pairwise distinct positive rational numbers
and put
| (4.124) |
For each , choose a denominator divisible by and by the reduced denominators of . Lemma 4.11 applies to this fixed pair on the progression . Passing to the least common multiple of the finitely many gives one progression on which all inequalities hold, each with an error tending to zero.
Step 3. In this step, we use the summable defects. The input block indices are distinct. By (4.90), for every sufficiently large on the common progression, one pair satisfies
| (4.125) |
Pass to a subsequence on which the chosen index is fixed. Since is affine on ,
Subtracting from (4.67) gives
| (4.126) |
Since , the target block lies on the chosen exact ray. Together, (4.123) and (4.126) give
Letting proves (4.120). ∎
Lemma 4.16 (Constancy on rational slices).
Notation and conditions are as in Set-up 4.1. For every , the restriction of to is constant almost everywhere.
Proof.
Choose as in (4.25), and put . The product of constant-curvature metrics on the curve factors is cscK in . Every curve has genus greater than one, so the polarized automorphism group of is finite, and the section ring of is generated in degree one by Lemma 4.8. By Theorem 4.14, positive ray norm would imply strictly positive asymptotic Chow invariant. Lemma 4.15 therefore gives . Formula (4.27) shows that the slice variance of vanishes. Thus is constant almost everywhere. ∎
4.5. Propagation from rational slices
Lemma 4.17 extends constancy on rational slices to every interior slice of .
Lemma 4.17 (Propagation from rational slices).
Let be a finite convex function on . If is constant almost everywhere for every , then there exists a finite convex function such that
| (4.127) |
at every interior point of . The value is the normalized slice average of .
Proof.
A finite convex function is continuous on the interior of its domain. On a rational interior slice, almost-everywhere constancy and continuity imply constancy on the relative interior of that slice. Fix and two points in the relative interior of . For a rational sequence , the affine side lengths ensure that
for all sufficiently large . Hence , and interior continuity gives .
Let be this common value. For fixed , the map
is affine. Restricting to this path shows that is convex. The boundary of each rectangular slice has four-dimensional measure zero, so is also the normalized slice average. ∎
4.6. Exclusion of the irrational crease
Algebraicity now excludes the remaining possible crease at through the rational spectral structure of an algebraic Duistermaat–Heckman measure.
Theorem 4.18 (Volume from normalized blow-ups, [BHJ17, Lemma 5.1]).
Let be a normal projective variety of dimension , let be ample, and let be a graded subalgebra containing an ample series. For every sufficiently large positive integer , let be the base ideal of , let be its normalized blow-up, and write . Then
| (4.128) |
Proof.
This is [BHJ17, Lemma 5.1]. ∎
Theorem 4.19 (Piecewise polynomial density of Duistermaat–Heckman measures, [BHJ17, Theorem 5.10]).
Let be a normal projective variety of dimension , let be ample, and let be a finitely generated integer filtration. Then has linear growth, and the density of the absolutely continuous part of its limit measure is piecewise polynomial of degree at most .
Lemma 4.20 (The evaluation-ideal system).
Let be an integral projective variety, let be ample, and let be a finitely generated integer filtration. After an integral character shift and passage to one fixed Veronese, put . There exist a pointed rational polyhedral cone and a finitely generated saturated semigroup
| (4.129) |
with the following properties. For , let
| (4.130) |
where when the filtered piece is zero and . Then is a finitely generated -graded system of coherent ideals:
| (4.131) |
Proof.
Choose finitely many bihomogeneous generators of the filtered Rees algebra, including its parameter of section degree zero, and let be the semigroup generated by their bidegrees. We use the convention in which a section in has bidegree and the Rees parameter has bidegree . An integral character shift makes the second coordinate of every remaining generator nonnegative. Those generators have positive first coordinate, so their degrees together with generate a pointed cone. A vector in the intersection of this cone with its negative has first coordinate zero, so it lies on the nonnegative parameter ray; its negative lies on that ray only when the vector is zero. Since is integral, a monomial in nonzero Rees generators is nonzero. Thus is exactly the support of the filtered Rees algebra. We may choose the Veronese so that a nonzero degree-one piece exists. Its bidegree together with generates . Let be the rational polyhedral cone generated by . The saturation of in is . To see that it is finitely generated, choose integral generators of . If belongs to , write
The set on the right is bounded and contains only finitely many lattice points. Those points together with the generate .
The convention in (4.130) extends the system from its supported degrees to by zero ideals. Multiplicativity of the filtration and of the evaluation maps gives (4.131). Every filtered section is a sum of monomials in the chosen Rees generators. Its evaluation ideal is therefore a sum of products of the finitely many evaluation ideals of those generators. Consequently,
| (4.132) |
is a finitely generated -graded -algebra. The zero slots add no generators. This verifies the lattice, saturation, zero-piece, multiplicativity, and finite-generation hypotheses used below. ∎
Theorem 4.21 (Rational chamber decomposition for base ideals, [ELM+06, Proposition 4.7], with the corrected proof in [ELM+23, Proposition 1.1]).
Let be as in Theorem 4.19, and construct its saturated evaluation-ideal system as in Lemma 4.20. Then there exist a positive integer and a finite smooth rational fan with support and primitive integral ray generators . Thus every two adjacent ray generators span a unimodular cone. Their finite slopes are strictly increasing; a boundary ray of the full cone may be vertical. The integer may be enlarged so that belongs to the supported semigroup for every ray: since belongs to the saturation of the support semigroup, some positive multiple of belongs to that support, and the conclusion below remains valid after replacing by a positive multiple. Every integral vector in a chamber spanned by two adjacent generators has a unique expression
| (4.133) |
and
| (4.134) |
Theorem 4.22 (Rational spectral structure).
Let be a normal projective variety, let be ample, and let a finitely generated integer filtration of the full section ring be given. The density of the absolutely continuous part of its limit weight measure is polynomial with rational coefficients on each interval of a finite partition whose breakpoints are rational.
Proof.
We prove the rationality assertion in three steps. First, we place all base ideals on one birational model over a finite rational chamber decomposition. We then obtain a rational polynomial formula for the filtered volume on each chamber. Finally, we differentiate the tail distribution and identify its breakpoints.
Step 1. In this step, we construct the chamber decomposition and a common model for its ray ideals. Put , and let be the base ideal of the filtered piece of at weight . Piecewise polynomiality follows from Theorem 4.19. We retain the multigraded base-ideal data in order to prove the rationality assertion. Lemma 4.20 supplies the lattice , the pointed rational cone , the saturated semigroup , and the finitely generated evaluation-ideal system, with zero ideals at unsupported indices. The preliminary changes in that lemma preserve the assertion that we are proving. An integral character shift translates normalized weights by an integer. If the fixed Veronese has index , its degree- spectrum is the degree- spectrum of the original filtration, divided by rather than by . Its limit measure is therefore the pushforward of the original one under ; this is also [BHJ17, Remark 5.6]. Integer translation and positive integral dilation preserve rational breakpoints and rational polynomial coefficients, in both directions. We may consequently prove the assertion after these changes, and retain the notation and for the resulting polarization and filtration.
By Theorem 4.21, there exist a positive integer and finitely many integral chamber generators
whose finite slopes are strictly increasing, with the following property. Every integral vector in the chamber spanned by adjacent vectors has a unique expression
and
| (4.135) |
Choose one normal projective birational model that dominates the normalized blow-ups of the finitely many ray ideals. Write
| (4.136) |
where every is an integral Cartier divisor. On , the product of the two ray ideals is invertible. The universal property therefore factors first through the ordinary blow-up. Since is normal, this factorization lifts through its normalization. Integral closure does not change a normalized blow-up, so (4.135) identifies that blow-up with the normalized blow-up of . Pulling back its exceptional Cartier divisor gives the explicit chain
| (4.137) | ||||
Step 2. We compute the filtered volume on one chamber and prove that it is a polynomial with rational coefficients in this step. Let be the maximal asymptotic weight of the modified filtration, and use the convention
For every , choose with . By [BHJ17, Theorem 5.3(i)], the threshold series contains an ample series. For all sufficiently large ,
The decreasing property of the filtration shows that, in every sufficiently large degree , the degree- piece of the graded series
contains the degree- piece of the -th Veronese of . Since containing an ample series is an asymptotic condition, the rounded series itself contains an ample series. Above its volume is zero; outside the compact support supplied by [BHJ17, Theorem 5.3(iii)], the tail is constant and its density vanishes. It therefore suffices to work on a finite-slope chamber with . Put . Dividing the two coordinates of the ray decomposition by shows that
are affine functions of with rational coefficients: they are obtained by inverting the integral two-by-two matrix with columns . For each integral vector in this chamber, the Cartier divisor
is the pullback of the moving divisor on the normalized blow-up of . It is globally generated and hence nef. Thus the intersection number in (4.138) is the moving self-intersection supplied by the base-ideal construction, not the volume of an arbitrary non-nef divisor. For this integral pair, set
| (4.138) |
For a real number in the interior of this chamber, put and define
| (4.139) |
where and are the rational-affine functions obtained from the ray decomposition. The spaces
| (4.140) |
form a graded linear series: multiplicativity of the filtration and give the required product inclusion. Its degree- base ideal is . Therefore Theorem 4.18, applied to the -Veronese and divided by , together with (4.137), gives
| (4.141) |
We now compare this rounded-ray series with the actual threshold at . Choose so that and remain in the interior of the chamber. For all sufficiently large ,
Since the filtration is decreasing, the corresponding inclusions are
Hence
| (4.142) | ||||
The function is a polynomial and is therefore continuous. Letting in (4.142) proves both the existence of the actual filtered-volume limit and its equality with . Thus
| (4.143) |
which is a polynomial in with rational coefficients, because the coefficients and are rational-affine and the intersections of integral Cartier divisors are integers.
Step 3. We pass from the chamber volume to the density and determine the breakpoints in this step. Let be the limit measure in the present grading. Restrict the filtration to the -th Veronese . Its limit measure is by [BHJ17, Remark 5.6]. Applying the tail formula [BHJ17, Theorem 5.3(iii), equation (5.4)] at the threshold , and using the limit proved in (4.142), gives
Integration of equation (5.4) from to then gives, away from an endpoint atom,
| (4.144) |
The first equality in (4.144) is the probability normalization in [BHJ17, equation (5.4)]; the second uses and the preceding volume identity. Thus passage to degrees divisible by computes the pushforward measure, not a different subsequential limit. The absolutely continuous density is therefore on the interior of each chamber and has rational polynomial coefficients there. Each finite chamber endpoint is a slope and is rational. This proves Theorem 4.22. The argument does not exclude a finite atomic part at the chamber boundaries. ∎
Remark 4.23 (Use for a fiber-scaling initial).
In the application in Lemma 4.24, finite generation is asserted only for the algebraic source test configuration after a sufficiently divisible Veronese. Either fiber-scaling initial filtration of Set-up 4.1 has weight multiplicities equal to those of the source in every degree, so its limit measure equals the source limit measure. It inherits Theorem 4.22 without any assertion that the initial filtration is finitely generated. Returning from the Veronese coordinate to actual degree divides the normalized weight coordinate by that positive integer and preserves rational breakpoints and rational coefficients.
Lemma 4.24 (Exclusion of the irrational crease).
Notation and conditions are as in Set-up 4.1. Assume that is the convex transform in actual degree of a fiber-scaling initial filtration of an algebraic test configuration. Let be its normalized slice average, and assume
| (4.145) |
Then there exist such that
| (4.146) |
at every interior point of .
Proof.
We prove Lemma 4.24 in three steps. The first removes transverse variation, the second records the rational spectral constraint inherited from the algebraic source, and the third treats all possible signs of the two slopes.
Step 1. In this step, we prove that agrees with . By Fubini,
for -almost every . Equality of the pushforward measures in (4.145) gives equality of their second moments. Therefore
| (4.147) | ||||
Thus almost everywhere. Convexity and interior continuity upgrade this equality to every interior point. Since is a nonnegative measure supported at one point,
| (4.148) |
for some and .
Step 2. In this step, we record the spectral restriction coming from the algebraic source. After passing to a sufficiently divisible Veronese, the filtration of the source test configuration is a finitely generated integer filtration. Theorem 4.22 shows that its absolutely continuous Duistermaat–Heckman density is polynomial with rational coefficients on a finite partition with rational breakpoints. By Lemma 4.8, the initial filtration has an entry multiset identical to that of the source in every degree, so the two filtrations have equal measures. Passing between exponent-one degree and actual degree only rescales the weight coordinate by the positive integer , and therefore preserves the rational spectral structure.
Suppose for a contradiction that , and put
The two slopes of are . We shall use the expansion
| (4.149) | ||||
Its two highest coefficients are nonzero, and on .
Step 3. In this step, we exclude the crease by considering the possible signs of and .
We first record the breakpoints used in the argument. A point at which the two one-sided polynomial density germs differ, or at which one branch appears with a nonzero one-sided jump, is an endpoint of the rational partition in Theorem 4.22 and is therefore rational. Since is positive on , every outer support endpoint is rational. A branch endpoint not shared by a second branch is also a rational breakpoint: its jump has size or , where and are the absolute branch slopes. If an explicit density formula agrees on a nonempty open interval with a polynomial from the rational partition, polynomial identity forces all of its coefficients to be rational. Redundant partition points therefore do not affect the argument.
Case 3.1. Assume that the two slopes share a weak sign. Then is monotone. On every nonconstant branch, its pushforward density is obtained from by an affine change of variable and division by the absolute value of the branch slope. If both slopes are nonzero, the leading coefficients of the two density germs at the image of are
where is the coefficient of in . Since in the monotone cases, this image is a genuine rational breakpoint. In a rational coordinate translated from an outer support endpoint of the left branch, its density is . The ratio of its coefficients of and is
where . Rationality of these coefficients forces . The distance from the image of to this endpoint is , which forces .
If , the left branch gives an atom. On the right branch, in the coordinate translated from its outer support endpoint, the constant coefficient is . If , the corresponding coefficient on the left branch is . In either case the nonzero slope is rational. The distance between the two rational support endpoints then forces the relevant distance to to be rational. Every monotone case contradicts the irrationality of .
Case 3.2. Assume . Put
| (4.150) |
Suppose first that . At the end of the shorter branch, that branch disappears with the nonzero jump or . The minimum and the longer outer endpoint are support endpoints, while the end of the shorter branch is a rational breakpoint. Hence all three points, and therefore both heights, are rational. If , the unmatched left tail has density in a rational coordinate translated from the outer endpoint of the left branch. Its two highest coefficients force , and then forces . If , the unmatched right tail has density . Its constant coefficient forces , after which also gives a contradiction.
It remains to consider . In this case the minimum and the common outer endpoint are rational support endpoints. Their difference is therefore a nonzero rational number. The constant coefficient of the two-branch density at the minimum is
| (4.151) |
Reduction of (4.149) modulo gives
| (4.152) |
This number is irrational, contradicting the rationality of the density coefficient in (4.151).
Both cases are impossible. Therefore , and (4.146) follows. ∎
4.7. The affine-transform theorem
We now combine rational-slice constancy, convex propagation, and the spectral exclusion to obtain the form used in the equality classification.
Proposition 4.25 (Affine opposite initials).
Notation and conditions are as in Set-up 4.1. For , let be the convex transform of in actual -degree normalization. There exist such that
| (4.153) |
at every interior point of .
The entry probability laws in actual degree of and are both
| (4.154) |
In the exponent-one grading of , their common law is
| (4.155) |
No equality between the two affine functions is asserted.
Proof.
Fix , write and , and let be the function defined by (4.81) and (4.82). We first remove the transverse variables. Lemma 4.13 gives
Lemmas 4.15 and 4.16 show that is constant almost everywhere on every rational interior slice. Lemma 4.17 then gives
| (4.156) |
at every interior point of .
5. Semistability and the integral affine comparator
This section proves semistability at every exponent and converts the affine initial data of Section 4 into an integral product comparator after a ramified base change and normalization.
5.1. Semistability at every exponent
We first combine the approximate cscK metrics of Proposition 3.12 with two analytic results on the centered norm of a test configuration.
Theorem 5.1 (Donaldson’s scalar-curvature lower bound, [Don05b, Theorem 2]).
Let be a polarized smooth projective variety, and let be an exponent-one normal ample test configuration with positive Donaldson centered norm . Every Kähler metric satisfies
| (5.1) |
where is the average scalar curvature in . The inequality (5.1) follows from the cited theorem. We now translate the two algebraic quantities in that inequality into the conventions of this paper. If
| (5.2) |
and the increasing-entry total weight is
| (5.3) |
then
| (5.4) |
Thus (5.1) has the displayed sign, and the Donaldson and Boucksom–Hisamoto–Jonsson centered norms have identical zero loci.
Theorem 5.2 (Vanishing of the centered norm, [BHJ17, Corollary B and Lemma 2.10]).
Let be a polarized variety. An ample test configuration for has zero centered norm if and only if it is almost trivial. Equivalently, after twisting the linearization by a scalar character, its normalization is the trivial test configuration. The vertical term in [BHJ17, Lemma 2.10] is precisely this scalar-character twist after the exponent has been incorporated into . Thus a normal ample test configuration of zero norm is the trivial product up to such a scalar twist and has zero Donaldson–Futaki invariant.
The conclusion of Theorem 5.2 is narrower than the automorphism-induced polarized products allowed in Definition 2.2.
Proposition 5.3.
Let be a positive integer. Every normal ample algebraic test configuration whose generic polarized fiber is has nonnegative Donaldson–Futaki invariant.
Proof.
We first rescale the approximate metrics from to . We then use Theorems 5.1 and 5.2 according to whether the centered norm is positive or zero. We begin by proving that the infimum of the centered scalar-curvature norm in is zero. Proposition 3.12 gives Kähler metrics such that
The metrics belong to . Since has complex dimension five, we have
Consequently,
| (5.5) |
We now exclude a negative Donaldson–Futaki invariant for an arbitrary normal ample test configuration of . Let be such a test configuration, regarded as an exponent-one test configuration of the polarized pair . If , then (5.4), (5.1), and (5.5) exclude . If , then Theorem 5.2 gives . Thus in both cases. ∎
5.2. Common rational affine data
We next identify the two affine profiles from Proposition 4.25 and use algebraicity of the source test configuration to prove that their common coefficients are rational.
Lemma 5.4 (Moment rigidity).
Proof.
Corollary 5.5.
Notation and conditions are as in Set-up 4.1. There exist unique real numbers such that, after division by the actual degree, both opposite initial filtrations have convex transform
| (5.8) |
at every interior point of . The entry probability laws of , , and are all
| (5.9) |
In the exponent-one grading of , the common law is
| (5.10) |
Proof.
Proposition 4.25 gives affine transforms and . It also identifies both pushforwards of with the entry law of after division by the actual degree. Lemma 5.4 gives
The formulas after division by the actual degree and in exponent-one degree are the two conclusions of Proposition 4.25. Uniqueness follows from Lemma 5.4. ∎
Lemma 5.6 (Rational moments of algebraic Duistermaat–Heckman measures).
Let be an -dimensional polarized projective scheme, and let be an ample exponent-one algebraic test configuration. For all sufficiently large , write
| (5.11) |
for the decomposition into weight spaces on the central fiber, and set
| (5.12) |
If converges weakly to , then
| (5.13) |
for every nonnegative integer .
Proof.
We first express all weight moments by rational generating functions. We then recover the limiting moments from the rational leading coefficients of the resulting quasipolynomials.
Step 1. We construct a rational generating function for each unnormalized weight moment. Put
Properness of makes finite-dimensional over . Choose a homogeneous basis of the finite-dimensional -module , and choose homogeneous generators of as a -algebra. Write the bidegree of as , where . Give the variables of the corresponding bidegrees. The chosen basis of makes a finite multigraded -module. A finite multigraded free resolution over gives
| (5.14) |
where is a Laurent polynomial with integer coefficients.
Fix . Acting by on (5.14) and setting , we obtain a rational one-variable series
whose denominator is a product of powers of . Let be a common multiple of the integers . Partial fractions after splitting the denominator over the -th roots of unity show that is, for all sufficiently large , a quasipolynomial with rational coefficients and period dividing .
Step 2. We identify the limiting moment with a rational leading coefficient. Algebraicity gives a uniform linear bound on the central weights. Hence the supports of the measures in (5.12) lie in one compact interval. Weak convergence implies convergence of every fixed moment, and
On each residue class modulo , the limit is the coefficient of in a polynomial with rational coefficients, with value zero if the degree is smaller. The full sequence has one weak limit, so the residue-class limits agree. Their common value is the moment in (5.13). ∎
Corollary 5.7.
The coefficients and in Corollary 5.5 belong to .
Proof.
Lemma 5.6, applied to regarded as an exponent-one test configuration of , shows that the zeroth moment is the positive rational leading Hilbert coefficient. Dividing by this mass shows that every moment of the associated central-weight probability law is rational. The convention for increasing entries in Set-up 4.1 provides such that this law is the law of
| (5.15) |
Suppose that . Rationality of the second and third central moments of , together with (5.6), gives
| (5.16) |
Both coefficients multiplying and in (5.16) are nonzero rational numbers. Thus , and
The mean of is rational. Together, (5.6) and (5.15) then give . If , rationality of the mean gives directly. ∎
5.3. Normalization and the Donaldson–Futaki invariant
We record the normalization formula, including the codimension and Duistermaat–Heckman conclusions that will be used after base change.
Lemma 5.8 (Normalization comparison).
Let be a polarized normal projective complex variety of dimension . Let be an integral ample algebraic test configuration whose generic polarized fiber is , and let
| (5.17) |
be its normalization, where . Write
| (5.18) |
Write
| (5.19) | ||||
| (5.20) | ||||
| (5.21) |
using increasing-entry totals and . Put
| (5.22) |
For all sufficiently large , put
| (5.23) |
and let be the -length of . Then there exists such that
| (5.24) |
and
| (5.25) |
If the two Donaldson–Futaki invariants are equal, then
| (5.26) |
If the two Donaldson–Futaki invariants are equal, the total-variation distance between the normalized probability measures of the increasing entries in degree is ; equivalently, the reflected central-weight measures have this property. The two test configurations have equal Duistermaat–Heckman probability measures.
Proof.
We compare the two extension lattices, read their determinant-weight difference from Smith normal form, and then interpret the equality case by the Hilbert polynomial of the normalization quotient.
Step 1. We express the weight difference as the length of the normalization quotient. The normalization is an isomorphism away from the central fiber, so is a coherent sheaf supported on that fiber. Tensor the defining sequence of by . Relative Serre vanishing [Sta26, Tag 02O1] and the projection formula give, for all sufficiently large , the exact sequence of coherent -modules
| (5.27) |
The morphism is flat. The normal integral total space has no -torsion, and torsion-free modules over the principal ideal domain are flat; hence is flat as well. Cohomology and base change therefore make and locally free of equal rank for large . The module is finite and supported at , so its global sections have finite -length. Since is affine, taking global sections in (5.27) gives
| (5.28) |
The first two terms are free -modules of rank , and the last term is -torsion. Put the first map in Smith normal form, with diagonal factors
The increasing-entry convention gives
| (5.29) |
The determinant of the lattice inclusion is a unit times . The inclusion is equivariant, so comparison of the two equivariant determinant lines, with of increasing-entry weight one, gives the second equality in (5.29). This argument uses Smith normal form only for the underlying lattices and does not require a Smith basis compatible with the weight decompositions.
Step 2. We obtain the normalization formula from the leading coefficient of the length polynomial. Relative Serre vanishing identifies with the Euler characteristic, and hence with the Hilbert polynomial, of with respect to . Its degree is at most , because is supported on the -dimensional central fiber. Its coefficient at is nonnegative and is positive exactly when has an -dimensional support component. This proves (5.24). Comparing coefficients in (5.29) gives . Substitution into
gives (5.25).
Step 3. We prove the codimension and measure statements when the two Donaldson–Futaki invariants are equal. By (5.25), we have . Thus has support dimension at most , which proves (5.26), and .
Put and, in degree ,
Tensoring (5.28) with gives the exact sequence
| (5.30) |
The sequence is -equivariant, because (5.28) is equivariant and the ideal is invariant. Cohomology and base change identify the two middle terms with the degree- central-fiber section representations. Both have dimension . Since is supported at , its -length is its complex vector-space dimension. Hence
The equality of the dimensions of the two middle terms in (5.30) then gives
Let be the image of the middle arrow in (5.30). It is a common -representation: it is a quotient of and a subrepresentation of . Finite-dimensional -representations are semisimple, so the two weight multisets have the weights of in common and have at most unmatched weights on either side. Let denote total-variation distance. If and denote the empirical probability measures of the central weights divided by , then
Reflection under preserves total-variation distance. Thus the increasing-entry measures also have distance , and the two Duistermaat–Heckman probability measures coincide. ∎
5.4. The integral product comparator
We finally clear the rational affine coefficients by base change and use the normalization comparison to retain both the zero invariant and the Duistermaat–Heckman law.
Proposition 5.9 (Integral product comparator).
Notation and conditions are as in Set-up 4.1, and let be the coefficients in Corollary 5.5. There exists a positive integer with the following properties. Let be the ordinary base change by , and let be its normalization. Then is integral, while is a normal ample test configuration for with
| (5.31) |
Its central-weight Duistermaat–Heckman probability law is the -dilation of the central-weight law of .
The integers
| (5.32) |
define a polarized product test configuration . Its increasing jump on is
| (5.33) |
After division by the actual degree, both opposite initial filtrations of , as well as the product filtration of , have convex transform
| (5.34) |
Proof.
We clear the denominators of the affine coefficients, prove integrality of the ordinary base change, compare it with its normalization, and then construct the product configuration from the resulting integral affine profile.
Step 1. We choose the base-change order and prove that the ordinary base change is integral. Choose such that the two numbers in (5.32) are integers, and form
The base-changed family is flat over , so multiplication by is injective on every affine coordinate ring. After inverting , the marked family is the integral product . Suppose that a product of two elements in an affine coordinate ring vanishes. One factor vanishes after localization, so a power of annihilates that factor. Torsion-freeness over makes the factor zero. Thus every affine coordinate ring is a domain, and is integral.
Let
be the normalization. Normalization is finite for a finite-type complex scheme. Finite pullback preserves relative ampleness and projectivity, and the action and marking lift to . Its normal integral total space is torsion-free, hence flat, over the principal ideal domain . Therefore is a normal ample test configuration for .
Step 2. We prove that normalization preserves the zero invariant and the dilated Duistermaat–Heckman law in this situation. Ordinary -fold base change multiplies every central weight by , and hence
| (5.35) |
Lemma 5.8, applied to , gives the following comparison. If is the coefficient in (5.24) and is the leading Hilbert coefficient, then
Proposition 5.3 gives . Thus
The equality clause of Lemma 5.8 identifies the Duistermaat–Heckman laws of and . Ordinary base change dilates every weight by , so this common law is the -dilation of the law of .
Step 3. We identify the common affine profile after base change. Corollary 5.5 applied to gives a common affine profile for its two opposite initial filtrations. Reflecting the dilated central-weight law to the increasing-entry coordinate, their entry law after division by the actual degree is
Step 4. We construct the integral product comparator. In the convention for increasing entries in Definition 2.1, the integer is the label of the scalar character and is the label of fiber scaling. Let be the resulting polarized product test configuration. Its increasing jump on is , which is (5.33). Dividing by the actual degree gives
Thus the product filtration has the transform in (5.34). No degreewise equality between and is asserted at this stage. That equality is the conclusion of the marked Smith argument in Section 6. ∎
6. Classification of zero-invariant test configurations
In this section, we prove Theorem B.
We use the increasing filtration convention fixed in Set-up 4.1. In particular, the section ring of is the Veronese ring in (4.9), and its blocks for fiber scaling are those in (4.7).
6.1. Normal section algebras and marked rigidity
In this subsection, we prove that a sufficiently small marked relative Smith spectrum determines a normal ample test configuration.
Definition-Lemma 6.1 (Relative Smith spectrum).
Let be a discrete valuation ring with uniformizer and fraction field , and let be full -lattices in an -dimensional -vector space. There exist a -basis and uniquely determined integers , up to permutation, such that
| (6.1) |
We call the multiset the relative Smith spectrum of and .
Proof.
We choose -bases of the two lattices and let be the matrix whose columns are the coordinates of the second basis in the first. Choose an integer such that has entries in . For a nonzero matrix over , move an entry of minimum valuation to the upper-left corner. This entry divides every other entry because is a discrete valuation ring. Elementary row and column operations over therefore clear its column and row. Induction on , followed by absorbing units into the bases, gives
| (6.2) |
If the original bases are written as row vectors and , then is a basis of and is a basis of . Thus (6.1) holds with . For uniqueness, let be the fractional ideal generated by the -by- minors of . Left or right multiplication by an element of does not change this ideal. If the integers are ordered increasingly, the diagonal form gives
| (6.3) |
The successive differences of these intrinsic valuations determine the ordered list , and hence determine the multiset. ∎
Lemma 6.2 (Normal Veronese section algebras).
Let
| (6.4) |
be a flat projective morphism such that is normal and integral and . Let be a -ample -linearized line bundle, where the -action on covers the standard scaling action on , and put
| (6.5) |
Then is a normal finitely generated domain over . Moreover, there exists a positive integer such that
| (6.6) |
is a normal finitely generated domain generated over by , and the same holds for every positive multiple of . In particular, defines a relatively projectively normal equivariant embedding.
Let be the generic polarized fiber, and fix an equivariant product identification over . Define the decreasing extension-order filtration
| (6.7) |
Then this filtration has Rees algebra
| (6.8) |
with its section degree and -weight retained. If
| (6.9) |
denotes the corresponding increasing filtration, then the marked extension lattice in (6.8) can be written as
| (6.10) |
Proof.
We first prove finite generation of the full section algebra, then choose a Veronese generated in degree one. We next prove normality by divisorial valuations. Finally, we identify the equivariant section algebra with the Rees algebra of the extension-order filtration.
Step 1. In this step, we prove that is finitely generated over .
Choose such that is relatively very ample and gives a closed immersion
For , the module
is the section module of the coherent sheaf for this embedding. For a finite graded presentation of its pushforward to , the higher cohomology of the finitely many coherent relation sheaves vanishes after sufficiently large twists. Consequently, finitely many homogeneous sections generate all sufficiently large components over . After adjoining the finitely many lower components, each is finite over this polynomial ring. The finite direct sum of the modules is , so is a finitely generated -algebra.
Step 2. In this step, we choose a Veronese of generated in degree one.
Choose homogeneous algebra generators of positive degrees , and let be a common multiple of the integers . Inside , with its rescaled grading, put
Each is integral over . Hence , and therefore , is finite over . Choose homogeneous -module generators of , of rescaled degrees , and choose . We prove that is generated by .
Let and . Write
Since is standard graded, the multiplication map
is surjective. Every term is consequently a sum of products of an element of and an element of . Induction on proves that is generated by . Replace by a positive multiple for which is relatively very ample. The -th Veronese remains generated in degree one, which proves the generation assertion in (6.6).
Step 3. In this step, we prove that and are normal domains.
Choose a nonzero rational section of , and let be its Cartier divisor. If is the function field of , then identifies with the subring of formed by the terms such that
For every prime divisor of , this condition is
For every prime divisor , let be the homogeneous valuation subring defined by this inequality. Thus . Each ring in this intersection is integrally closed, and therefore is a normal domain.
The Veronese is the invariant subring for the finite cyclic action that multiplies by the -th power of a primitive -th root of unity. If an element of is integral over , then it is integral over , belongs to , and is fixed by the cyclic action. It belongs to , which proves the normality of the Veronese.
Step 4. In this step, we identify the Rees algebra after fixing the product marking over .
The filtration in (6.7) is multiplicative because products of extensions extend. The -action decomposes every into weight spaces. A section homogeneous for this weight restricts to a Laurent monomial , which extends precisely when . Taking weight components preserves the module of global sections. Hence extension holds coefficient by coefficient, and (6.8) follows. ∎
Lemma 6.3 (Marked affine module gluing).
Let , let , and let be a finite-dimensional -vector space. For , let be a finite torsion-free -module spanning . Suppose that and that
| (6.11) |
as submodules of . Then inside .
Proof.
Proposition 6.4 (Rigidity from the marked Smith spectrum).
Let be a connected normal -dimensional polarized complex projective variety. For , let
| (6.13) |
be a normal ample algebraic test configuration for . Equip the two configurations with a common -equivariant product trivialization over , including a common marking of the generic line bundle and a common linearization convention. Their marked relative section algebras
| (6.14) |
lie in the common generic graded algebra
| (6.15) |
Choose a common positive integer for which the -th Veronese of each is normal and generated in rescaled degree one, as in Lemma 6.2. Put
| (6.16) | ||||||||
Put . Let be the relative Smith elementary divisors of and in the convention of Definition-Lemma 6.1. Assume that
| (6.17) |
Then inside (6.15). Consequently, the two test configurations are isomorphic as marked polarized -equivariant test configurations.
Proof.
We prove Proposition 6.4 in three steps. First, we record a min–max criterion for the relative Smith integers. Second, a divisorial valuation turns any difference between the two localized Veronese algebras into a quadratic lower bound that contradicts (6.17). Third, normality removes the common Veronese and relative recovers the marked test configuration.
Put
By Lemma 6.2, each is a normal domain generated in rescaled degree one. Each localization is also a normal domain generated in rescaled degree one, and
inside the common marked generic algebra.
Step 1. In this step, we relate lattice-order separation on a subspace to the relative Smith spectrum.
For an -lattice in a finite-dimensional -vector space and a nonzero element , put
Order the Smith integers decreasingly. Let , and let be a -dimensional -subspace such that
for every nonzero . Choose a Smith basis with
For , we have
If fewer than Smith integers were at least , then would meet the span of the remaining basis vectors nontrivially, contradicting the order separation. Thus at least Smith integers are at least . If instead on a -dimensional subspace, the argument with the two lattices interchanged shows that at least Smith integers are at most .
Step 2. In this step, we prove the quantitative contrapositive to (6.17).
Assume that . Since both algebras are generated in degree one, their degree-one lattices differ. After interchanging the algebras if necessary, choose
Since is a normal Noetherian domain, it is the intersection of its localizations at the height-one prime ideals inside its fraction field by [Sta26, Tag 031T]. The element therefore has negative order along a height-one prime ; write its normalized valuation as
We may choose homogeneous. The finite set of negative prime components of the divisor of the homogeneous element is preserved by the connected section-grading torus, so each component is fixed. Moreover, . If , then is a unit in and , contradicting . Put .
Since , we have . The discrete valuation ring is Cohen–Macaulay and hence universally catenary by [Sta26, Tag 00NM]. Its finite-type algebra is therefore catenary by the definition of universal catenarity. Let
The generic fiber of is the section ring of the -fold , so its fraction field has transcendence degree over . The dimension formula [Sta26, Tag 02IJ] therefore gives . Catenarity of the chain gives . Since is a standard graded complex domain and is its irrelevant ideal, [Sta26, Tag 00P6] gives
The ideal is homogeneous and the degree-zero part of the quotient is . Thus is a standard graded complex domain of dimension . Its Hilbert function agrees in large degree with a polynomial of degree and positive leading coefficient. Hence
for a positive constant . Choose lifts
of a complex basis modulo . If are not all zero and , put . At least one has nonzero image in . The images of the form a complex basis modulo , so has nonzero image modulo . Factoring out therefore gives
In particular, the chosen lifts are -linearly independent.
The degree-one lattices are commensurable. Choose such that
Generation in degree one gives
For positive integers , put , and let be the -subspace spanned by
It has dimension . For a nonzero element
with , the generators lie in , and hence
On the other hand,
If , then . Therefore
and
| (6.18) |
Choose a positive rational number such that , with any positive allowed when . Take infinitely many positive pairs with . There exists such that the right-hand side of (6.18) is at least . The integer is a fixed positive proportion of , so for a positive constant . The -dimensional subspace used to obtain (6.18) therefore forces at least relative Smith integers to have absolute value at least . Hence
along an infinite sequence. This contradicts (6.17), and we conclude that .
Step 3. In this step, we remove the common Veronese and recover the marked test configurations.
We first recover the two Veronese algebras over . Equality and finite generation allow us to clear the finitely many denominators in homogeneous generating sets. Hence there exists such that
Over , the fixed punctured marking gives . Since , the open sets and cover . The two identifications are restrictions of the identity inside (6.15). If and
| (6.19) |
then and are finite torsion-free -submodules of the common space . Applying Lemma 6.3 degree by degree gives
| (6.20) |
and shows that the two sides agree for . Hence
Lemma 6.2 states that each full section algebra is normal. If is homogeneous in , then . Thus is integral over . It also belongs to , and hence to . Normality gives . Symmetry gives inside the marked generic algebra. For each , the canonical morphism
| (6.21) |
is an isomorphism by [Sta26, Tag 0C6J]. Under this isomorphism, the canonical evaluation map identifies with by [Sta26, Tag 01QI]. Equality of the full graded algebras therefore recovers the original polarization, not only its th power. The grading, common generic marking, and linearization are preserved, so the resulting isomorphism is an isomorphism of marked polarized equivariant test configurations. ∎
6.2. Residual filtrations and the oriented Smith spectrum
In this subsection, we identify the relative Smith spectrum with the jumps of the oriented initial and prove quadratic decay after subtracting the integral product profile.
Theorem 6.5 (Filtered jump measures, [BC11, Theorem 1.11 and Remark 1.12(i)]).
Let be a projective variety, let be an ample line bundle on , and let carry a multiplicative filtration. Assume that, in the decreasing convention, the filtration is pointwise left bounded and linearly right bounded. Fix a full-rank valuation with one-dimensional leaves, and let be the associated concave transform on the Newton–Okounkov body. If and are the filtration jumps in degree , counted with multiplicity, then
| (6.22) |
weakly, where is the Newton–Okounkov body of .
Lemma 6.6 (Quadratic decay of the residual jumps).
Notation and conditions are as in Proposition 5.9. Let be an integral increasing filtration of that preserves every block . Assume that is multiplicative and two-sided linearly bounded, and that its convex transform after division by the actual -degree is (5.34). If are its jumps, where , define the residual filtration by
| (6.23) |
Its jumps satisfy
| (6.24) |
where is the integral product jump in (5.33). Then is an integral, multiplicative, two-sided linearly bounded filtration of the full ring , and its convex transform is zero. If
| (6.25) |
then
| (6.26) |
weakly, and
| (6.27) |
Proof.
We prove Lemma 6.6 in three steps. First, the inverse product profile gives an integral multiplicative filtration, and its blockwise sum with is . Second, the convex transforms cancel, so the filtered jump measures converge to . Third, the common linear bound converts this weak convergence into the quadratic estimate.
Step 1. In this step, we prove the algebraic properties of the residual filtration.
The negative product profile assigns the scalar jump to . By (5.33), it is additive under multiplication:
Moreover, and (5.32) give
Choose in each block a basis whose vectors of jump at most span the -th filtered piece of . Formula (6.23) shifts every basis jump in by . If and , multiplicativity of and additivity of the scalar profile give
Thus is multiplicative. The filtration is integral and two-sided linearly bounded, while is integral and satisfies . Hence is integral and two-sided linearly bounded.
Step 2. In this step, we determine the weak limit of the residual jump measures.
The convex transform of is (5.34), and (5.33) has precisely that affine transform. Their difference is the zero transform. After passing from increasing jumps to the decreasing real extension, satisfies the hypotheses of Theorem 6.5 for the complete series of the ample line bundle . The zero transform gives (6.26).
Step 3. In this step, we deduce the quadratic decay from (6.26).
The two-sided linear bound places the supports of all measures in (6.26) in one compact interval. Choose a bounded continuous function that agrees with on this interval. Weak convergence gives
Since has dimension five, there exists a constant such that
Multiplying the normalized quadratic sum by proves (6.27). ∎
Definition 6.7 (Initial filtrations oriented by the weights).
Let carry an increasing filtration . Put
| (6.28) |
and
| (6.29) |
We call these the initials obtained by taking the highest and lowest weight components, respectively. For an integer , the -oriented initial is if , is if , and may be either one if .
Lemma 6.8 (Smith spectrum of an oriented initial).
Let
| (6.30) |
be a finite-dimensional complex vector space with finitely many nonzero summands, and let be an exhaustive separated increasing integer filtration of . Fix integers , and let be the split filtration whose jump on is
| (6.31) |
Put and . Associate to an increasing filtration the lattice
| (6.32) |
Put . Let be the relative Smith elementary divisors of and in the convention of Definition-Lemma 6.1, and let be the -oriented initial in Definition 6.7. There exists a homogeneous basis , with , and integers such that is spanned by the vectors with . For this basis,
| (6.33) |
as multisets. In particular,
| (6.34) |
Proof.
We first construct a basis adapted to the filtration and to the weight flag. We then gauge the product lattice to the standard lattice and read the Smith integers from the gauged filtered basis. It is enough to prove the case . The case follows after replacing every weight by , and the case follows from .
Step 1. In this step, we construct a homogeneous basis for the initial obtained by taking the highest weight components.
Put
For each pair , choose a complement of
inside , omitting repeated filtration levels. Choose a basis of each complement. Induction on the finite ordered grid of pairs shows that the vectors chosen up to span . Their union is therefore a basis of .
For each , let and be the indices of the complement containing . Then is the -jump of , the vector belongs to , and its image
is nonzero. For fixed , the vectors with and form a basis of
Thus is a homogeneous basis for , and the jump of is .
Step 2. In this step, we diagonalize the relative lattice after gauging the product filtration.
Define a -linear automorphism by
Then . Write
We have
Put
Relative to the complex basis , the matrix with columns has constant term equal to the identity. It belongs to . Since the basis is adapted to , we obtain
The relative Smith spectrum is invariant under the common gauge transformation . In the -basis , the two lattices are diagonal with exponents . This proves (6.33) and (6.34). ∎
Remark 6.9 (Dependence on the orientation).
Let , with comparator jumps on and on . Let be given by
| (6.35) |
The filtration in (6.29) has residual jumps . The relative lattices are
| (6.36) |
In the -basis of , the transition matrix is
| (6.37) |
Its minimum entry valuation is and its determinant has valuation , so the relative Smith multiset is and its square sum is . Thus the opposite orientation does not compute the relative Smith spectrum in this example, which explains the sign convention in Lemma 6.8.
6.3. Descent from the integral comparator
In this subsection, we eliminate the rounding introduced by deck descent and prove Theorem B.
Lemma 6.10 (Elimination of the ceiling error).
Let be a positive integer and let . For every positive integer and , put
| (6.38) |
Assume that there exists a constant such that
| (6.39) |
for all sufficiently large . Then
| (6.40) |
Consequently,
| (6.41) |
for every positive integer and every .
Proof.
Write in lowest terms, with . If , then on any consecutive values of , the fractional parts of form a translate of the -grid. The sum of their ceiling errors is at least
The left-hand side of (6.39) would grow linearly with , which is a contradiction. Thus , and .
Proof of Theorem B.
The nonnegativity assertion is Proposition 5.3. Assume that . Corollary 5.7 gives the rational coefficients of the common affine transform. Proposition 5.9 gives a positive integer , the normalized base change , and the integral product comparator . They satisfy (5.32), (5.33), and (5.34), and .
We prove the equality assertion in four steps. First, the oriented initial of Definition 6.7 identifies the relative Smith spectrum of and with the residual jumps. Second, quadratic decay and marked rigidity identify the normalized base change with the product comparator. Third, deck invariants descend the equality of the marked extension lattices to a filtration that is scalar on each block and has ceiling jumps. Finally, the uniform bound in (4.90) removes the ceiling error.
Step 1. In this step, we prove that the relative Smith spectrum of and has quadratic sum .
Choose from (6.28) if , choose from (6.29) if , and choose either filtration if . Denote its jumps on by
Let be the relative Smith integers of the marked extension lattices of and . In section degree , Lemma 6.8, with and , gives
| (6.42) |
The projected initial just chosen is the Grassmannian initial used in Set-up 4.1. To prove this, fix , let be the degree- filtration of , and let act on by . For the highest-weight flag, choose by successive complements on the finite grid of pairs a basis simultaneously adapted to and to the flag . Write , where and . If is the -jump of , then the with span , whereas the corresponding span . The normalized frames converge to the linearly independent vectors as . Hence in the Grassmannian. Replacing every weight by gives as . Since and in (4.10), this proves and . Thus the selected initial is when , is when , and may be either one when .
Lemma 4.8, applied to , now shows that the selected initial is block preserving, integral, multiplicative, and two-sided linearly bounded on . By Proposition 5.9, its convex transform is (5.34). Hence Lemma 6.6 applies and gives
Together with (6.42), this proves
| (6.43) |
Step 2. In this step, we identify the normalized base change with the product comparator.
The test configurations and carry a common generic marking. On , this is the base change of the original marking under . The pulled-back polarization has its canonical deck linearization: the deck group acts trivially on the marked generic line-bundle factor and sends to . This action lifts functorially through normalization. The product comparator is placed in the common marked Laurent algebra with this linearization. Twisting by an additional -character would define a different descent datum and is not part of either marked base change.
Choose a common Veronese in section degree as in Lemma 6.2. The estimate (6.43) remains valid after this fixed rescaling of the section degree. Since , the exponent in (6.17) is . Therefore Proposition 6.4 gives a marked polarized equivariant isomorphism
Equivalently, if is the marked degree- extension lattice, then
| (6.44) |
inside the marked Laurent extension. The equality in (6.44) is equality of submodules of that fixed ambient algebra, not an abstract isomorphism chosen afterward. The deck action on its right-hand side is consequently the restriction of the canonical ambient action.
Step 3. In this step, we descend (6.44) by taking deck invariants.
The deck group acts on the normalized base change. On an affine normal chart of , the ordinary base-change ring is
If is its normalization, then
| (6.45) |
Proposition 5.9 shows that is a domain. Localizing this domain at gives
which is again a domain; hence is irreducible over . Monicity gives the free -basis . Thus the extension in (6.46) has degree , and its displayed -action is the full Galois group. The fraction fields satisfy
| (6.46) |
An invariant element therefore belongs to and is integral over . Normality of places it in . Every element of maps to a -invariant element of , which gives the reverse inclusion. Thus .
Write and , and let be the quotient morphism. The canonical base-change linearization is carried by . The projection formula and (6.45) give
| (6.47) |
Taking global sections in (6.47) identifies the downstairs degree- section lattice with the invariant part of the upstairs degree- lattice.
Under the pulled-back generic marking, acts trivially on every and sends to for . This marking is the pullback of the original marking under , so the deck group acts only on the parameter ; explicitly,
| (6.48) |
For every integer , the invariant block is
| (6.49) |
A section downstairs pulls back to an invariant section upstairs, and -lattice is the invariant lattice by (6.47). Deck invariance requires the -exponent to be divisible by . On , the least admissible invariant exponent is
This identity is valid without a sign restriction on . Consequently, the increasing jump of the original filtration on is
| (6.50) |
Step 4. In this step, we remove the ceiling error and identify the original test configuration with a product.
The filtration in (6.50) is scalar on every block . Fiber scaling acts by a scalar on each block and therefore preserves every filtered subspace. Both opposite initials equal the descended filtration. In the actual degree , its normalized jump is
The difference from is less than , so the convex profile is . By (4.89), the block-average gap for the descended filtration is exactly
| (6.51) |
Lemma 4.13, applied to this zero-invariant initial, and (4.90) give a constant such that
for all sufficiently large supported degrees .
Lemma 6.10 now gives
Substitution in (6.50) gives the jump in (1.7) on every block . The marked section algebra is the product Rees algebra in which fiber scaling contributes and the scalar character contributes to the increasing entry on , in the convention of Definition 2.1. Relative recovers as the polarized product test configuration described in Theorem B. ∎
7. Proof of Theorem A
Proof of Theorem A.
We verify the three assertions in Theorem A in the order in which they are stated. We retain the notation of Construction 3.7 throughout the proof. Lemma 3.3 gives curves with the genera in (1.1) and the vanishing in (1.2). With these curves, Construction 3.7 defines from the degrees in (1.3). Proposition 3.8 shows that is a smooth projective fivefold, that is ample, and that , where the action is fiber scaling.
We next prove K-polystability at every positive exponent. Fix a positive integer , and let be a normal ample algebraic test configuration whose generic polarized fiber is . Proposition 5.3 gives
| (7.1) |
If equality holds, Theorem B identifies with the polarized product induced by an integral one-parameter subgroup of fiber scaling together with a scalar character on the polarization. Since and were arbitrary, this proves the second assertion of Theorem A.
7.1. Scheme-theoretic consequences
This subsection proves Corollary 1.5 for arbitrary ample scheme-theoretic test configurations and then proves Corollary 1.6 for the unrepaired equality convention.
Proof of Corollary 1.5.
Let be the normalization, and put . We first verify that this is a normal ample test configuration, then use the normalization comparison, and finally determine the equality case.
Step 1. In this step, we show that is a normal ample test configuration for . The generic polarized fiber is the marked product with the smooth connected variety , and is therefore integral. If is affine, flatness over makes the homomorphism
| (7.2) |
injective. The target is the coordinate ring of an affine open subset of and is a domain. Hence is a domain, so is integral.
The normalization is finite and inherits the -action and the marking over . Its coordinate rings are torsion free over the principal ideal domain , and hence the normalization remains flat over . Finite pullback preserves relative ampleness. Consequently, is a normal ample algebraic test configuration for .
Step 2. In this step, we prove the nonnegativity of the Donaldson–Futaki invariant. Put
| (7.3) |
Lemma 5.8 gives a number and the identity
| (7.4) |
Theorem B gives
| (7.5) |
Step 3. We determine the equality case in this step. If , then both terms on the right-hand side of (7.4) vanish. Theorem B identifies the normalization with the polarized product induced by an integral one-parameter subgroup of fiber scaling together with a scalar character. Since , Lemma 5.8 shows that is supported in total-space codimension at least two. The finite morphism is an isomorphism away from this support.
Conversely, suppose that the normalization is the asserted polarized product and that is an isomorphism away from a closed subset of total-space codimension at least two. The product has zero Donaldson–Futaki invariant: inverting both the one-parameter subgroup and the scalar character yields another normal ample product configuration whose invariant is the negative of the first, and Proposition 5.3 makes both nonnegative. The support condition gives in (7.4). Hence . This completes the proof of Corollary 1.5. ∎
Proof of Corollary 1.6.
We construct an explicit exponent-six test configuration of , compute its Donaldson–Futaki invariant, and compare it with the Fubini–Study metric.
Step 1. In this step, we construct a nonnormal scheme-theoretic test configuration with trivial normalization. Inside , let
| (7.6) |
Give these generators the grading
| (7.7) |
The algebra is a finitely generated domain, and its degree- piece is the free -module
| (7.8) |
Thus
| (7.9) |
is flat over . The standard presentation of the sixth Veronese is generated in degree one, so is a relatively ample line bundle. It follows that is an algebraic test configuration for , and hence is an exponent-six test configuration for .
The fraction fields of and agree because , and is integral over because it satisfies . Since is integrally closed, it is the normalization of . Therefore the normalization of is the product . After inverting , the identity gives , while after inverting , the identity gives . The normalization quotient is therefore supported on . The relations and show that its support is the codimension-two locus . In relative Proj this is the single point defined by
| (7.10) |
Thus the normalization is an isomorphism away from that point.
The central fiber has the presentation
| (7.11) |
The basis (7.8) shows that the class of is nonzero. Its annihilator in (7.11) is , whereas the minimal prime is . Hence is an embedded associated prime. In particular, the central fiber is nonreduced and is not the product test configuration.
Step 2. In this step, we compute the Donaldson–Futaki invariant of . Let act with weight one on and , and with weight zero on . This action covers the standard action on and becomes the trivial product action on the normalization. In polarization degree , set . The basis in (7.8) gives
| (7.12) |
Thus both functions in (7.12) are computed from the free pieces of the sixth Veronese, which is the section algebra of the chosen polarization . Thus the leading and subleading coefficients of the total weight are both zero, and
| (7.13) |
Step 3. We compare the test configuration with the cscK metric and conclude the proof in this step. The Fubini–Study metric is a cscK metric in . On the other hand, (7.13) holds and is not a product because its central fiber in (7.11) is nonreduced. These facts violate the unmodified equality condition requiring an actual product. This is the codimension-two phenomenon isolated by Stoppa as being trivial in codimension two; see [Sto11, p. 1 and Definition 1]. Hence the literal scheme-theoretic equality convention is not necessary for cscK existence. This completes the proof of Corollary 1.6. ∎
8. Vanishing of the reduced Donaldson–Futaki quotient
The goal of this section is to construct the test configurations in Theorem C and compute their reduced Donaldson–Futaki quotient.
Lemma 8.1 (Sufficiently divisible Veronese gradings).
Let be a finitely generated graded algebra over the Noetherian ring . There exists a positive integer such that, for every positive multiple of , the Veronese algebra
| (8.1) |
is generated over by .
Proof.
Choose homogeneous algebra generators of positive degrees , and let be a common multiple of the . The algebra , with its rescaled grading, is finite over the algebra
| (8.2) |
which is generated in degree one. Choose homogeneous -module generators of , of degrees , and choose .
Let and write an element of as
| (8.3) |
Since is generated in degree one, multiplication from onto is surjective. Each term in (8.3) is therefore a sum of products of an element of and an element of . Induction on shows that is generated in degree one. Put . If is a positive multiple of , then is a Veronese of the degree-one-generated algebra and is again generated in degree one. ∎
Theorem 8.2 (Openness of relative ampleness, [Sta26, Tag 0D2N]).
Let be a proper morphism of schemes with Noetherian, let be a line bundle on , and let . If is ample, then there exists an open neighborhood of such that
| (8.4) |
Proof.
This is [Sta26, Tag 0D2N]. ∎
Theorem 8.3 (Relative ampleness and projectivity, [Sta26, Tags 01VJ and 0B45]).
Let be a quasi-compact morphism, and let be a line bundle on . If has an affine open covering such that is ample for every , then is -ample. If, in addition, has an ample line bundle and is proper, then is projective.
Proof.
Proposition 8.4 (Rational single creases).
Let , where and are coprime positive integers, and let be a positive integer such that . Put
| (8.5) |
For every sufficiently divisible positive integer , the saturated epigraph of , with its -th Veronese polarization in section degree, defines a normal nonproduct -equivariant algebraic test configuration of exponent for . For every , its increasing filtration entry on is
| (8.6) |
Its Donaldson–Futaki invariant is
| (8.7) |
The limiting distribution of the normalized fiber coordinate is the probability measure
| (8.8) |
The product test configuration generated by fiber scaling satisfies
| (8.9) |
Proof.
We first construct the test configuration from its saturated epigraph and then compute its invariant from the exact dimensions of the blocks in large degree. We use the notation of Set-up 3.9.
Step 1. In this step, we construct and prove its normality and nonproductness. The function has integral slopes and intercepts. Put
| (8.10) |
and put . This is the positively homogeneous extension of . Its epigraph semigroup is
| (8.11) |
Triangulate the rational polyhedral cone given by the epigraph into rational simplicial cones and choose an integral generator on each ray. Every lattice point in one of these cones is the sum of a lattice point in the bounded fundamental parallelepiped and nonnegative integral multiples of its ray generators. Only finitely many lattice points lie in these parallelepipeds, so is finitely generated. It is saturated by its definition as the full set of lattice points in the cone. Hence the semigroup algebra , graded by and with degree-zero part , is a finitely generated normal graded -algebra; here corresponds to .
Enlarge the divisibility condition on so that . By Lemma 8.1, the algebra is generated over by its degree-one part. Its semigroup is saturated in its group, so the Veronese algebra is normal. Each homogeneous piece is a finite free -module, and the coordinate ring of every standard affine chart of its relative projective spectrum is torsion free over and hence flat. After is inverted, localization removes the lower bound on the third semigroup coordinate and identifies the relative projective spectrum with . Thus
| (8.12) |
is a normal flat toric degeneration of with an invertible relatively ample polarization .
We now globalize this family over . Define a graded -algebra by
| (8.13) |
To make the gluing explicit, choose an affine cover that trivializes and , and denote their transition functions by and . On the -summand of (8.13), the transition multiplier is
| (8.14) |
The exponents are additive, so these multipliers preserve multiplication and every binomial relation of the semigroup algebra. The cocycle law is inherited from those of and . Moreover, is the fiber-torus transition and glues the degree-one polarization. Hence
| (8.15) |
is obtained on every from . Thus the local families, the test action, the fiber action, and their polarizations glue to a normal flat -equivariant family with a proper morphism
| (8.16) |
On each trivializing open subset , (8.12) shows that the restriction of over is projective and that is ample relative to this restriction. Properness and relative ampleness are local on the target, so is proper and is -ample. Since is projective, the composite is proper. After inverting , its polarized fiber is .
We verify ampleness relative to this composite. Put . The quotient has the boundary-monomial basis
| (8.17) |
because a monomial whose third coordinate is strictly larger than is divisible by . The product of two boundary monomials survives modulo exactly when their exponent vectors lie in a common linearity cone of . For vectors in the interiors of the two different cones, the strict inequality
makes their product divisible by .
Let and be the two lower-face semigroups. Retaining the monomials on and killing the other boundary monomials defines a homomorphism to . The two homomorphisms give an homomorphism
| (8.18) |
Within either face, the retained boundary monomials map to distinct monomials and are therefore linearly independent. Since every boundary monomial belongs to at least one face, the two kernels have zero intersection. Thus the homomorphism in (8.18) is injective. The two kernels are incomparable prime ideals. Thus the local central fiber is reduced, has no embedded component, and has exactly the two components determined by the linearity intervals. The transition multipliers in (8.14) preserve both face semigroups, so this description glues over .
Put , , and . The two irreducible components of the reduced central fiber correspond to the two linearity intervals and . On the component corresponding to , the associated fiber over is a projective line bundle whose two fixed sections correspond to the fiber weights and . If this component is denoted by and its projection by , put . After removing the affine -character, the graded face algebra of this component is
| (8.19) |
This is the -th Veronese of the symmetric algebra of , twisted in degree by . Consequently,
| (8.20) |
The restrictions of to the two fixed sections of are
| (8.21) |
For and , the common face has algebra . Hence the two components meet along their common fixed section , and both restrictions of to this intersection equal . For every and every curve factor , the degree of is
| (8.22) |
by (3.27). Lemma 3.4, applied with and , shows that is ample on each component of the reduced central fiber. Lemma 3.6 then shows that it is ample on the central fiber. Theorem 8.2 gives a neighborhood of over which is relatively ample. Over , the polarized family is the product with and is relatively ample there. These two open subsets cover , so Theorem 8.3 shows that is ample relative to and that the proper morphism to is projective. This polarized family is .
It remains to identify the induced filtration. Put . Since is generated in degree one, is invertible and the punctured cone
| (8.23) |
is the associated -torsor. The group generated by the semigroup of has rank three, whereas . Thus has codimension two. Normality and the codimension statement give
| (8.24) |
A regular function on defines an element of lying in for every height-one prime , since no such prime belongs to . A normal noetherian domain is the intersection of its height-one localizations inside its fraction field by [Sta26, Tag 031T], so this element belongs to ; the reverse inclusion is immediate. Taking the weight- part on the torsor in (8.23) gives, for every ,
| (8.25) |
This identity is compatible with the transition multipliers in (8.14); it therefore glues to
| (8.26) |
Taking sections over gives the exact section lattice
| (8.27) |
inside the marked generic algebra. This proves (8.6). Finally, the crease is interior, so the entries are not affine in . Since every one-parameter subgroup of lies in , the resulting test configuration is not a product.
Step 2. In this step, we compute the Hilbert and weight coefficients and obtain the formula given by point evaluation. For all sufficiently large , every line bundle appearing in the summands on the four curve factors has degree greater than and therefore has dimension equal to its degree plus . The tensor-product decomposition gives, uniformly for ,
| (8.28) |
For a polynomial , expansion into monomials and the exact power-sum identities give
| (8.29) |
Applying (8.29) to the two polynomials in (8.28) gives
| (8.30) |
For a rational convex piecewise-linear function , define
| (8.31) |
For , the integer is a lattice point. Apply (8.29) separately on the two sides of this point to the polynomial pieces of and . The two endpoint terms with coefficient at the crease, followed by the subtraction of the duplicated lattice value, cancel because is continuous. Thus only the endpoints and contribute to the second coefficient, and the total increasing entry in test-configuration degree is
| (8.32) |
The exponent factors cancel in the Donaldson–Futaki normalization:
| (8.33) |
Since and , (8.33) gives
| (8.34) |
Here we used the four boundary identities , , and from Lemma 3.10. For , the measure is the unit point mass at . This proves (8.7).
Proof of Theorem C.
We argue in four steps. First, we construct the Fibonacci test configurations and prove convergence of their crease parameters. We then compute their Donaldson–Futaki invariants, compute their reduced non-Archimedean -functionals, and finally compare these quantities with the relative non-Archimedean Mabuchi functional. We write and throughout.
Step 1. In this step, we construct and prove the convergence in (1.12). The Fibonacci recurrence gives , so is in lowest terms. For , Proposition 8.4 applied with
| (8.36) |
gives a normal nonproduct -equivariant test configuration of exponent whose block entry in test-configuration degree is
| (8.37) |
Since , the inequalities and give
| (8.38) |
The recurrence gives
| (8.39) |
After division by , every limit point of in is a root of . The only such root in this interval is
| (8.40) |
This proves the asserted convergence.
Step 3. In this step, we compute the reduced non-Archimedean -functional and obtain its uniform lower bound. We use the conventions of Nitta–Saito [NS21, Section 3.2 and Definitions 3.6.1 and 3.6.3]. By (8.8), the limiting fiber-coordinate measure is
| (8.44) |
In test-configuration degree , the Duistermaat–Heckman normalization divides the entries by the exponent times the degree, namely . Formula (8.6) therefore gives the normalized entry , where . A real twist by the fiber-scaling torus adds a real multiple of ; after division by , we denote that multiple by . The Duistermaat–Heckman measure in the definition of is the central-weight measure. Since central weights are the negatives of increasing entries by Definition 2.1, is the entry mean minus the minimum entry. Therefore
| (8.45) |
Put
| (8.46) |
so that . The convex piecewise-linear function has slopes and on the two sides of . Consequently, the objective in (8.45) is
| (8.47) |
The first branch is minimized at , the third at , and the middle branch is affine. Its minimum is therefore attained at one of those two endpoints. Hence
| (8.48) |
Write
| (8.49) |
We have on , and hence . Suppose that . On we have and , while on we have and . It follows that
| (8.50) | ||||
| (8.51) |
Since , formula (8.48) gives
| (8.52) |
Taking and , using , and combining (8.43) with (8.52), we obtain
| (8.53) |
This proves (1.15).
Step 4. We compare the ratio in (8.53) with the relative non-Archimedean Mabuchi functional and conclude the proof in this step. The product configuration generated by fiber scaling has zero Donaldson–Futaki invariant by (8.9). For a product configuration this invariant is a positive multiple of the classical Futaki invariant of the generating vector field [Don02, Section 2.2]; since is one-dimensional by Proposition 3.8, the Futaki character vanishes identically. The nondegenerate Futaki–Mabuchi pairing therefore makes the extremal vector zero; see [NS21, Definitions 2.3.2 and 2.3.4]. The extremal correction in the relative non-Archimedean Mabuchi functional vanishes by [NS21, Definition 3.3.5(2)], so
| (8.54) |
The comparison formula in [SD19, Definition 3.4] gives
| (8.55) |
Together with (8.53), this yields
| (8.56) |
Thus no can satisfy
| (8.57) |
for every normal -equivariant test configuration . By [NS21, Definitions 3.3.5(2) and 3.7.1(4)], this is precisely the failure of uniform relative K-polystability asserted in Theorem C. This completes the proof of Theorem C. ∎
Appendix A Use of generative AI
by Bin Dong11 1 Beijing International Center for Mathematical Research & Center for Machine Learning Research, Peking University, No. 5 Yiheyuan Road, Haidian District, Beijing 100871, China. Email: dongbin@math.pku.edu.cn, Guoxiong Gao22 2 School of Mathematical Sciences, Peking University, No. 5 Yiheyuan Road, Haidian District, Beijing 100871, China. Email: samggx@stu.pku.edu.cn, and Jihao Liu
Artificial intelligence has lately proven strikingly good at producing counterexamples, and this paper too disproves a conjecture by exhibiting an explicit variety. It belongs, however, to a different genre than most counterexamples found by AI so far. The statement that an object is a counterexample can be of two kinds. It may reduce to a finite certificate, checkable by a finite mechanical computation once the object is written down: the recent counterexample to the Jacobian conjecture [Alp26] is typical, an explicit polynomial map whose Jacobian determinant is a nonzero constant by direct expansion and whose non-injectivity is witnessed by two points with the same image. Or it may admit no finite certificate even in principle, because it is itself a theorem, typically universally quantified: the recent construction of a non-sofic group [OAI26] is of this kind, since non-soficity asserts the nonexistence of approximate embeddings into finite symmetric groups of any size and can be witnessed by no computation with the group itself. A counterexample of the second kind is, in substance, a proof.
The distinction is older than AI. Proposed disproofs of the Hodge conjecture have never been short of candidate classes; they founder on proving that the candidate is the class of no algebraic cycle whatsoever. Rationally connected varieties are expected not to be unirational in general, and plausible candidates abound—a very general hypersurface of bidegree in may well be one—but no known technique can prove any rationally connected variety non-unirational, so no candidate can be certified.
This paper is of the second kind. Candidate manifolds of the present shape have been available since [ACG+08], and producing candidates is precisely what contemporary AI does well; what had been missing, and what constitutes the mathematical content of this paper, is the proof that the mechanism works—that every normal ample algebraic test configuration of the fivefold has nonnegative Donaldson–Futaki invariant, with equality only for products, a statement quantified over all degenerations and established by the classification of Sections 4–7. This, we believe, is the axis along which AI-assisted mathematics should be judged: not whether the conclusion is a counterexample, but whether the assertion that it is one is a finite certificate or a proof.
In the course of exploring a number of open problems with Claude Code, the author found initial signs of a possible breakthrough on the problem studied here, and then had Claude Code (Fable 5), Codex (GPT-5.6-sol), and Danus [Liu+26] work on it in collaboration; the three systems together produced the counterexample and its proof. Human input was crucial at one point: the author realized that the example constructed in Theorem A is either a counterexample to the Codogni--Stoppa conjecture or a counterexample to the cscK Yau--Tian--Donaldson conjecture---either case would be striking---and asked the agents to go all-in on this particular example and determine which conjecture is false. To understand what the AI systems had in fact contributed, the problem was then attempted afresh by an improved version of Danus33 3 This improved version of Danus will be open-sourced shortly. working alone. Given only the original problem and none of the earlier findings, it settled the problem 5 hours and 29 minutes into its run, in less time than the collaboration had taken and without the human input that had been key to the first attempt, and produced a counterexample and a complete proof of this paper. The text of the main part of the paper then went through several rounds of discussion between the author and Danus to improve its presentation, none of which altered the mathematics, and a final round of human polishing and checking.
Danus is a mathematical agent built on top of Rethlas [Ju+26] by the same team and designed for long-horizon reasoning. A main agent orchestrates Rethlas agents as workers on a single problem: the workers produce facts, which accumulate into a fact graph whose edges record their dependencies, and the run stops only when the target statement itself appears in the graph as a fact. The design is described in [Liu+26] and the accompanying open-source code.44 4 https://github.com/frenzymath/Danus The improved version differs from it in several respects. The Rethlas workers now run on GPT-5.6-sol, and the main agent moved to a Codex-based implementation using the same model as well, so the whole system runs on a single API. With models of this strength, a design in which the main agent does no mathematics itself and defers high-level planning to a strategic consultation of GPT-5.5 Pro leaves their capability underused; the consultation was removed, and the main agent now does the mathematical thinking and directs the global strategy, dispatching Codex subagents—three in the present work—to extend its mathematical reach and to relieve the context burden of a large fact graph and memory. Its global strategic reflection was strengthened, so that it does not stall in a dead end and lose sight of the need to change direction, and fact granularity was retuned so that workers produce longer, more complex local results. Together these changes let Danus make fuller use of the current generation of models.
The authors also tested how other agent systems perform on this problem, posed to them as a request to prove or disprove the conjecture: QED [An+26], ProofCouncil [Sch+26], MechMath [Cao+26], Codex (GPT-5.6-sol), Claude Code (Fable 5), and GPT-5.6-sol Pro asked directly through its web interface. The first three all run on GPT-5.6-sol, with the effort level of their Codex components or GPT API set to “xhigh”; Codex and Claude Code were set to “max” effort. Given the original problem, the first three ran for twelve hours without producing a solution; Codex produced a proof, rejected it in its own verification, and gave no valid result within the time limit; Claude Code reported that this is a well-known conjecture and offered some possible approaches but no solution, and GPT-5.6-sol Pro, after 133 minutes of thinking, reached much the same assessment and likewise claimed no solution. On the easier task of being given the counterexample of this paper and asked only for a proof that it is one, none of the six produced a complete proof under the same twelve-hour limit and settings. Verifying this counterexample is therefore not trivial, as the second kind of counterexample described above requires.
The original run, the one this paper comes from, had eight Rethlas workers, with the Codex effort level set to “xhigh” for four of them and “high” for the other four. Its final fact graph contains 616 verified facts, and its global memory contains 353 conclusions, 143 identified obstacles, 96 directions, 33 proof attempts, 26 plans, 17 counterexamples, 2 recorded dead ends, and 924 verification records. Table 1 records how the 616 facts divide: 88 lie in the supporting closure of the final main-theorem fact, computed from the dependencies recorded with each fact, and the remaining 528 lie outside it, unused by the paper.
| Count | Share of all facts | |
|---|---|---|
| Main-theorem closure | 88 | 14% |
| Outside the closure, unused by the paper | 528 | 86% |
| Total | 616 | 100% |
Table 2 describes what the 88 in-closure facts do. The decomposition is approximate: a fact often serves several purposes at once, and each is placed by the role of its final clause. Six of the 88 are theorems from the literature, restated by a worker so that other facts could cite them; the remaining 82 the swarm proved itself. Only a minority appear in the paper as statements in their own right; the rest are internal steps beneath the argument the paper presents.
| Role | Count | Function in the proof |
|---|---|---|
| Veronese gradings and rays | 20 | Fix the gradings under which a test configuration is read as a filtration, then work one rational ray at a time: sufficiently divisible Veronese exponents with normal, finitely generated section algebras, multiplication along blocks and rays, and the Chow weight carried by each ray. |
| The scalar profile | 14 | Expand the block dimensions and total weights in the degree, average them over the fiber index into a one-variable convex profile, evaluate the Donaldson–Futaki invariant of that profile against the boundary data, and bound the accumulated blockwise gaps by summability estimates. |
| Newton–Okounkov convex transforms | 10 | Compute the product-box Newton–Okounkov body of the split bundle and control the Boucksom–Chen transforms on it: convexity and rigidity of the slice averages, and the theorem that both opposite initials of a zero-invariant configuration have affine transforms. |
| Duistermaat–Heckman measures | 10 | Constrain the limit weight measure: invariance under specialization, rational breakpoints, coefficients and moments, and the exclusion, by rationality alone, of a crease at the irrational double zero. |
| Semistability and polystability | 8 | Signs and equality cases quantified over whole classes of degenerations of , from nonnegativity for every equivariant filtration up to K-semistability at every exponent, the classification of the zero-invariant case as products, and the capstone. |
| Nonexistence of extremal metrics | 7 | The analytic half: admissible metrics whose scalar-curvature error tends to zero, the automorphism and maximal-compactness input that makes the admissible ansatz exhaustive, and the conclusion that contains no extremal metric. |
| Comparator and Smith spectrum | 7 | Prepare a zero-invariant configuration for comparison against an integral product comparator: normalization priced against the invariant, ramified base change realized as a weight dilation, and the relative Smith elementary divisors identified with the residual jumps. |
| The admissible boundary polynomial | 6 | Fix the curve datum with its genera and degrees, produce the admissible boundary polynomial with the required endpoint data, and certify its nonnegativity on the interval with a single interior double zero at an irrational point. |
| Recorded external results | 6 | Published theorems restated in internal notation with no derived content, among them the admissible-extremal criterion of [ACG+08] and standard identifications of the Duistermaat–Heckman measure. |
Table 3 summarizes the 528 facts outside the closure. Unused here means unused by the final proof, not mathematically meaningless or unhelpful to the search: some were proved for arbitrary data rather than for the fivefold, and the proof needs only particular instances of them; some prove the Donaldson–Futaki inequality one class of degenerations at a time, later absorbed by the general classification; some record that a step fails, and so kept workers from retrying it; others rule out simpler candidates, and so fixed the shape of the example.
| Cluster | Count | Content |
|---|---|---|
| Superseded duplicates | 146 | Earlier or duplicate forms of conclusions the graph later carries in better shape, including the endgame theorem itself, which several workers wrote out independently. Only one instance of each conclusion can enter the closure. |
| Lemmas beyond the fivefold | 113 | Statements proved for arbitrary graded algebras, filtrations, Newton–Okounkov bodies and products of arbitrarily many curves, where the paper instantiates a handful of them once each. Staying general is what let one lemma serve every worker and every exponent. |
| Abandoned torus-specialization route | 64 | Structural analysis of a hypothetical zero-invariant non-product configuration along the Codogni–Stoppa specialization strategy. The final proof reaches the same conclusion through the affine transforms and the oriented Smith spectrum instead. |
| Restricted-class nonnegativity | 61 | Sign results for the fivefold proved one named class of degenerations at a time, blow-ups, monomial and flag filtrations among them, each a restricted case of Theorem A(2) that the final classification later absorbs. |
| Undischarged hypotheses | 58 | Positive results resting on an antecedent the run never verified, chiefly finite generation. The most consequential is the criterion a boundary polynomial would have to meet, which the explicit four-factor datum was then built to satisfy. |
| Refuted proof steps | 56 | Facts whose whole content is that a proposed step fails: an implication that does not hold, an estimate that cannot exist, an ansatz that is the flat specialization of no test configuration. Each closes a branch. |
| Excluded candidate families | 24 | Negative existence results over whole families of candidates, showing that split line bundles over one, two or three curves satisfy the conjecture. This is why the base is a product of four curves. |
| Off-path external results | 6 | Published results transcribed into the run’s own notation so that workers could apply them inside the graph, on paths the final proof does not take. |
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