Universal spaces of parameters for complex Grassmann manifolds $G_{q+1,2}$.

Any element $L\in G_{q+1,2}$ may be represented by matrix $A_{L}$, which columns forms a basis in $L$:

This matrix is not unique, but any other such matrix $B_{L}$ may be obtained from $A_{L}$ by right action of $GL(2,\mathbb{C})$.

Let $P_{ij}=a_{i}b_{j}-a_{j}b_{i}$.

For the proof see [9] or [10].

This is a classical result, that Grassman manifold $G_{q+1,2}$ of two-planes in $\mathbb{C}^{q+1}$ may be viewed as homogeneous space of groups $U(n)$ and $GL(q+1,\mathbb{C})$. Really, both groups acts transitivilly on the set of 2-planes in $\mathbb{C}^{n}$. In the case of $U(n)$ the stabilizer of $span_{\mathbb{C}}\{e_{1},e_{2}\}$ is equal to $U(2)\times U(n-2)$ (here $e_{1},e_{2}$ are vectors from standard basis of $\mathbb{C}^{n}$). So, we have

From this point of view one can see that the torus $T^{q+1}\subset U(q)$ acts on $G_{q+1,2}$. This action is an example of Hamiltonian action from symplectic geometry. The easiest way to see it is follows: as we mentioned before, there is the Plucker embedding $\Phi:G_{q+1,2}\rightarrow\mathbb{C}P^{\binom{q+1}{2}-1}$ is $T^{q+1}$-equivariant. Hence, the pull-back $\Phi^{*}\omega_{FS}$ of Fubini-Studi form is Kahler form on $G_{q+1,2}$. It is not hard to show that the action of $T^{q+1}$ is actually Hamiltonian. Moreover, one can show, that this metric is also $U(q+1)$-invariant.

There is a $T^{q+1}$-equivariant map $\mu:G_{q+1,2}\rightarrow\Delta_{q+1,2}=\{x\in\mathbb{R}^{n}\penalty\ |\penalty\ 0\leq x_{i}\leq 1,\penalty\ x_{1}+\dots+x_{n}=2\}$. We are going to write the formula for $\mu$. Denote by $e_{i}$ standard basis of $\mathbb{R}^{q+1}$. Let $e_{ij}:=e_{i}+e_{j}$ and $P(L):=\sum_{i<j}{\lvert P_{ij}\rvert}^{2}$. In these definitions the formula for $\mu$ is very simple:

It’s not hard to see that the map is $T^{n}$-equivariant and maps $G_{q+1,2}$ onto $\Delta_{q+1,2}$. One could show that $\mu$ is moment map for Kahler metric $\Phi^{*}\omega_{FS}$, which was mentioned above.

There is another way to represent $G_{q+1,2}$ as a homogeneous space. As we noticed before, $GL(n,\mathbb{C})$ acts transitively on 2-planes in $\mathbb{C}^{n}$. The stabilizer (it is parabolic subgroup of $GL(n,\mathbb{C})$) consists of matrices

Here $A_{0}\in GL(2,\mathbb{C})$, $A_{2}\in GL(n-2,\mathbb{C})$ and $A_{1}$ is an arbitrary complex matrix with two rows and $(n-2)$ columns. So, the complex torus $H:=(\mathbb{C}^{*})^{n}\subset GL(n,\mathbb{C})$ acts on $G_{q+1,2}$.

Here we describe a coordinate way. We can represent any $L\in G_{q+1,2}$ by $2\times n$ matrix:

Let $t=(t_{1},\dots,t_{n})\in H$. The action of $H$ describes as follows:

Denote $\tau_{ij}=t_{i}t_{j}$. It is easy to see that we have an equality: $P_{ij}(\tau L)=\tau_{ij}P_{ij}(L)$. Note, that $\frac{\tau_{ij}}{\tau_{ik}}=\frac{t_{j}}{t_{k}}$. Hence, we can reconstruct all $t_{j}$ by $\tau_{ij}$ up to a common factor. Since the common factor is just an element of diagonal of $H$, we can completely rebuild torus action up to action of diagonal (which acts trivially). One could notice that an element $t\in H$ fixes $L$ iff $t_{i}t_{j}P_{ij}(L)=\lambda P_{ij}(L)$ for some $\lambda\in\mathbb{C}^{*}$. Without loss of generality we can assume that $\lambda=1$.

In this section we define the notion of strata on $G_{q+1,2}$. We give two definitions of stratification and show their equivalence.

Let $K=\{I\subset 2^{\{1,\dots,n\}}|\penalty\ |I|=2\}$. Suppose $Y_{I}=G_{n,k}\setminus M_{I}\penalty\ \forall I\in K$. Here $M_{I}$ – standard coordinate chart on $G_{q+1,2}$. Suppose also $\sigma=\{I_{1},\dots,I_{l}\}$ and all $I_{j}\in K$.

This definition of stratum was given by Buchstaber and Terzic in[1] and [2] for so-called $(2p;q)$ manifolds.

One can notice that $G_{n,k}=\cup W_{\sigma}$ and $W_{\sigma}\bigcap W_{\eta}=\emptyset$. We also notice, that all strata are invariant under $T^{n}$ and $H$ action.

Now we give another definotion of stratification. This definition was given by Gelfand and MacPherson [7]. This definition may be found in the paper of Kapranov [5].

This stratification is also $H$ and $T^{n}$ invariant. So, we have a question about connection between these two stratifications.

Further, we will use notions of Buchstaber and Terzic.

We also need definition of a spaces of parameters.

This definition was given in [1]. In this article Buchstaber and Terzic also showed that for any stratum there is a homeomorphism $h_{\sigma}:W_{\sigma}/T^{n}\rightarrow\mathring{P}_{\sigma}\times F_{\sigma}$, defined by maps $\mu:W_{\sigma}/T^{n}\rightarrow\mathring{P}_{\sigma}$ and by canonical projection $p_{\sigma}:W_{\sigma}/T^{q+1}\rightarrow W_{\sigma}/H$.

As we mentioned before, the hypersimplex $\Delta_{q+1,2}$ is the moment polytope for $G_{q+1,2}$. It has a very simple description:

We need to understand which subpolytopes in $\Delta_{q+1,2}$ may be an admissible.

This is an obvious and we omit the proof.

The polytopes in boundary of $\Delta_{q+1,2}$ has a very nice description: it’s hypersimplices, defined by equations $x_{i}=0$ or $x_{i}=1$.

There is a polytopes of codimension one in $\Delta_{q+1,2}$, defined by equation $l_{I}(x)=x_{i_{1}}+\dots+x_{i_{k}}=1$ for some $k\geq 2$ and $I=\{i_{1},\dots,i_{k}\}$. As Kapranov proved in [5], all admissible polytopes of codimension 1 are either lies in $\partial\Delta_{q+1,2}$ or defined by the form $l_{I}$ for some $I$.

Now suppose that a plane $L\in G_{q+1,2}$ does not lie in any $L_{i}$ for any $i$ and $L$ has non-trivial stabilizer in $(\mathbb{C}^{*})^{q+1}$.

Here is an obvious proposition.

Now we can construct an equivalence relation on $[q+1]$, which needs to describe the moment polytope for $H.L$.

Note that this theorem is actually true not only for some $L$, but for whole stratum, which consists $L$.

In this section we enlist $6$ axioms of $(2p,q)$-manifolds. These axioms were given in [2].

Let $M$ is a compact oriented simple-sonnectd manifold, $\dim_{\mathbb{R}}(M)=2n$. Suppose we have an action $\theta$ of torus $T^{q}$. Suppose also we have an equivariant map $\mu:M\rightarrow\mathbb{R}^{q}$, with trivial $T^{q}$ action on $\mathbb{R}^{q}$ and the image of $\mu$ is a convex polytope $P^{q}$. Triple $(M,\theta,\mu)$ called $(2n,k)$-manifold, if it satisfies next $6$ axioms.

Next, we will use definition of stratum from the previous section.

Let $\Sigma$ be set of all admissible sets (set called admissible if corresponding stratum is non-empty). define a map $s:\Sigma\rightarrow S(P)$, which maps each admissible set from $\sigma$ into polytope $P_{\sigma}=conv\langle\mu(x_{i_{1}});\dots;\mu(x_{i_{l}})\rangle$. We will call such polytopes an admissible polytopes (there is no contradictions with previous section).

Now we can formulate next axiom.

Denote $T^{\sigma}:=T^{k}/\chi(W_{\sigma})$.

Notice, that $\mu:M\rightarrow P^{k}$ induce a map $\hat{\mu}:M/T^{k}\rightarrow P^{k}$.

Examples of $(2n,k)$-manifolds is spheres (type of $(2n,1)$) and quasitoric manifolds (type of $(2n,n)$). Also, Grassman Manifolds $G_{4,2}$ and $G_{5,2}$ are also such manifolds of types $(8;3)$ and $(12;4)$ respectfully. It was showed by Buchstaber and Terzic in [3] and [1]. For other $G_{q+1,2}$ with $q\geq 5$ it was unknown, because there is no proof of veracity of axiom 6 in this case.

Here we briefly describe the notion of Chow factor of a projective variety $X$ which possesses an algebraic group $G$. One can find further details and references in [5].

Suppose we have a projective variety $X$ over $\mathbb{C}$ and an algebraic group $G$,which acts on $X$. Suppose also, that there is an open subset $U\subset X$, such $G$ acts freely on $U$. Then for any $x\in U$ the closure of it’s $G$-orbit $\overline{G.x}$ (suppose that dimension of such cycle is equal $r$) is an algebraic cycle. Moreover, all cycles, obtained this way has the same degree (as algebraic subvariety in $X$) and represent the same homology class $\delta$ in $H_{2r}(X,\mathbb{C})$. Obviously, all these cycles are parametrized by $U/G$. Hence we have an embedding of $U/G$ into $\mathcal{C}_{r}(X;\delta)$ – the set of all cycles in $X$, which has fixed dimension $r$ and represents homological class $\delta$. $\mathcal{C}_{r}(X;\delta)$ has a structure of complex (even projective) manifold (look [5], chapter 0.1 and Barlet paper [11]), что $\mathcal{C}_{r}(X;\delta)$.

So, we have an important definition.

In the case of $X=G_{q+1,2}$ and $G=H$ there is an important theorem, proved by Kapranov ([5], theorem 4.1.8):

In this section we define and describe some objects, named cross-ratios. Cross-ratios are a crucial element of our construction, so it is necessary to study they before proving the main theorem.

It is not so hard to check that total number of cross-ratios on $G_{q+1,2}$ is equal $\binom{q+1}{4}$.

There is some useful formulas for cross-ratios.

Now we briefly describe Gelfand-MacPherson correspondence for arbitrary Grassmann manifolds. This correspondence connect points of $G_{q+1,k}$ and configuration of points on $\mathbb{C}P^{k-1}$. For more details see [5] and [7].

Denote by $M_{q+1,k}(\mathbb{C})$ the space of all comples $(q+1)\times k$ (matrices with $n$ rows and $k$ columns). One could notice that $M_{q+1,k}(\mathbb{C})$ possesses actions of two complex Lie groups: left action of $H=(\mathbb{C}^{*})^{q+1}$ and right action of $GL(k;\mathbb{C})$. There is an open set $M^{max}_{q+1,k}(\mathbb{C})$ of all matrices in $M_{q+1,k}(\mathbb{C})$. $M^{max}_{q+1,k}(\mathbb{C})$ consists of all matrices, such each $k\times k$ minor is non-zero. One could see that $M^{max}_{q+1,k}(\mathbb{C})$ is invariant under action of $H$ and $GL(k,\mathbb{C})$. Moreover, we have next proposition.

The proof of this proposition is straightforward.

Now we can look at $(\mathbb{C}P^{k-1})^{n}$. We can choose a subset of pairwise distinct points $(\mathbb{C}P^{k-1})^{n}_{max}:=\{(x_{1},\dots,x_{q+1})\in(\mathbb{C}P^{k-1})^{n}\lvert x_{i}\neq x_{j}\}$. $(\mathbb{C}P^{k-1})^{q+1}_{max}$ has a very simple description.

The proof is also straightforward and obvious.

Now we can combine two previous propositions and obtain an important corollary.

This is exactly Gelfand-MacPherson correspondence. One could observe that $(\mathbb{C}P^{k-1})^{q+1}_{max}/PGL(k;\mathbb{C})$ is set of all configurations of pairwise distinct points up to automorphism of $\mathbb{C}P^{k-1}$. In the case $k=2$ we have that $F$ consists of all such configurations on $\mathbb{C}P^{1}$. This observation is very crucial for us. The observation is also explains the definiton of "cross-ratios".

For the sake of completeness, we should mention the strongest version of this corresponedce.

In this section we describe connection between torus action on $G_{q+1,2}$ and cross-ratios.

Fix some point $L\in W$. It’s Plucker coordinates are $P_{ij}(L)\neq 0$ for an arbitrary $i,j\in\{1,\dots,n\}$. Denote orbit of $L$ under action of $H$ as $H.L$. The Plucker coordinates $P_{ij}(\tau.L)$ of the element $\tau.L\in H.L$ are equal $\tau_{i}\tau_{j}P_{ij}(L)$. One could see the next identity:

Now, for some $I=\{i,j,k,l\}\subset\{1,\dots,n\}$ and for $L\in W$ denote $c^{\prime}_{I}(L):=P_{ik}(L)P_{jl}(L)$ and $c_{I}(L):=P_{il}(L)P_{jk}(L)$. Now, we can write previous equality in the next form:

or

So, orbit of $L$ lies in the set of common zeroes of such polynomials. Also, we should notice, that $c_{I}\neq c^{\prime}_{I}$. Really, if $c_{I}=c^{\prime}_{I}=c\neq 0$, we can see, that

But by Plucker identities, we have, that

Hence, we have, that either $P_{ij}(\tau.L)=0$ or $P_{kl}(\tau.L)=0$ and we have a contradiction that $L\in W$.

Now, we can prove the next proposition.