The Betti Numbers of a Determinantal Variety

TL;DR

This paper computes the Poincaré polynomial of the determinantal variety defined by matrices with zero determinant, revealing its topological structure in the context of algebraic geometry.

Contribution

It provides an explicit calculation of the Betti numbers for the determinantal variety, a significant advancement in understanding its topological invariants.

Findings

Explicit Poincaré polynomial derived for the determinantal variety

Betti numbers characterized for the variety of singular matrices

Topological structure of the determinantal variety clarified

Abstract

We determine the Poincar\'e polynomial of the determinantal variety $\{\det = 0\}$ in the projective space associated with the monoid of $n\times n$ matrices.

Equations

$$Y:=\mathbb{P}(Y^0)=(Y^0\setminus \{0\})/{\mathbb{C}}^{\ast },$$

$$H_k(Y)\cong \{\begin{array}{ll}0& \text{ if k is odd and k<n2−1};\\ \mathbb{Z}& \text{ if k is even and k<n2−1};\\ H_{k+1-(n^2-1)}({\text{PSU}}_n)& \text{ if k is odd and k≥n2−1}.\end{array}$$

$$P_Y(t)$$

$$B(t):=t^{n^2-1}{P}_{{\text{PSU}}_n}(t)=\sum _{i=n^2-1}^{2(n^2-1)}{b}_{i-(n^2-1)}{t}^i$$

$$A^k(X):=Z^k(X)/Ra{t}_{\dim X-k}(X)=A_{\dim X-k}(X).$$

$$f_{\ast }:A_k(X)\to A_k(Y).$$

$$f^{\ast }:A_k(Y)\to A_{k+n}(X).$$

$$A_k(Y)\stackrel{i_{\ast }}{⟶}{A}_k(X)\stackrel{j^{\ast }}{⟶}{A}_k(U)\to 0$$

$${\Delta }^k=\text{Spec}(\mathbb{Z}[t_1,\dots ,t_k]/(t_1+\cdots +t_k-1)).$$

$$C{H}^i(X,m):=H_m(Z^i(X,\bullet )).$$

$$A_p(X,k):=H_k(Z_p(X,\bullet )).$$

$$\cdots \to A_p(Y,k)\to$$

$$\cdots \to A_p(X-Y,1)\to$$

$${\bar{H}}_q(Y)=H^{n-q}({\mathbb{R}}^n,{\mathbb{R}}^n\setminus Y).$$

$$\cdots \to {\bar{H}}_{i+1}(U)\to {\bar{H}}_i(Y)\stackrel{i_{\ast }}{⟶}{\bar{H}}_i(X)\stackrel{j^{\ast }}{⟶}{\bar{H}}_i(U)\to {\bar{H}}_{i-1}(Y)\to \cdots$$

$$\begin{array}[]{lcl}M&:&\text{ the monoid of $n\times n$ matrices defined over ${\mathbb{C}}$;}\\ G&:&\text{ the general linear group of $n\times n$ matrices defined over ${\mathbb{C}}$;}\\ T&:&\text{ the maximal torus consisting of diagonal matrices in $G$;}\\ Z&:&\text{ the center of $G$;}\\ X&:&\text{ the projectivization of $M$, $X:=(M\setminus\{0\})/Z$;}\\ Y_{0}&:&\text{ the vanishing locus of the determinant in $M$;}\\ Y&:&\text{ the projectivization of $Y$, $Y:=Y^{0}/Z$;}\\ U&:&\text{ the projectivization of $G$, $U:=G/Z=\textrm{PGL}_{n}$.}\\ \end{array}$$

$$\cdots \to {\bar{H}}_q(Y)\to {\bar{H}}_q(X)\to {\bar{H}}_q(U)\to {\bar{H}}_{q-1}(Y)\to \dots ,$$

$$0\to {\bar{H}}_{2q+1}(U)\to H_{2q}(Y)\to H_{2q}(X)\to {\bar{H}}_{2q}(U)\to H_{2q-1}(Y)\to 0.$$

$${\bar{H}}_q(U)=H^{2(n^2-1)-q}(U)=H^{2(n^2-1)-q}({\text{PSL}}_n)=H^{2(n^2-1)-q}({\text{PSU}}_n).$$

$${\bar{H}}_q(U)=\{\begin{array}{ll}0& \text{ if }q<n^2-1,\\ H_{q-(n^2-1)}({\text{PSU}}_n)& \text{ if }q\ge n^2-1.\end{array}$$

$$H_q(Y)=\{\begin{array}{ll}0& \text{ if q is odd and q<n2−1},\\ \mathbb{Z}& \text{ if q is even and q<n2−1}.\end{array}$$

$${\bar{H}}_{2q}(Y)/{\bar{H}}_{2q+1}(U)\cong \mathbb{Z}\text{ and }{\bar{H}}_{2q-1}(Y)\cong {\bar{H}}_{2q}(U).$$

Full text

**Keywords: Determinantal variety, Betti numbers, Chow groups, Borel-Moore homology

MSC: 14M12, 20M32, 14C15**

1. Introduction

In this note, we look closely at the homology groups of a classical variety. Let $Y^{0}$ denote the semigroup defined by the vanishing of the determinant polynomial in $n\times n$ matrices. More precisely, we set $Y^{0}:=M\setminus G$, where $M$ is the monoid of all linear operators on an $n$ dimensional complex vector space $V$, and $G=\textrm{GL}(V)$. The purpose of this note is to describe the Poincaré polynomial of the quotient,

[TABLE]

where ${\mathbb{C}}^{*}$ is the center of $\textrm{GL}(V)$.

It is not difficult to see that if $\dim V=2$, then $Y$ is isomorphic to the quadric surface ${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$ in ${\mathbb{P}}^{3}$. In particular, the Poincaré polynomial of $Y$ is given by $1+2t^{2}+t^{4}$. However, in general, $Y$ has a large singular locus, which is given by the projectivization of a $G\times G$-orbit closure, $(\overline{GeG}\setminus\{0\})/{\mathbb{C}}^{*}$, where $e$ is an idempotent of rank $n-2$ in $M$. It is natural question to ask for a description of the Poincaré polynomial of $Y$ for $n:=\dim V>2$. It turns out that the degrees as well as the coefficients of monomials in $P_{Y}(t)$ have interesting patterns, although $P_{Y}(t)$ is neither symmetric nor unimodal.

Our main result and its corollary are the following statements.

Theorem 1.2**.**

Let $Y$ denote the determinantal variety defined as in (1.1). If $V$ is $n$ dimensional, then the homology groups of $Y$ satisfy the following isomorphisms:

[TABLE]

Finally, if $k$ is even and $k\geq n^{2}-1$, then we have $H_{k}(Y)/H_{k+1-(n^{2}-1)}(\textrm{PSU}_{n})\cong{\mathbb{Z}}$. Here, $\textrm{PSU}_{n}$ denotes the projective special unitary group.

Let us denote by $P_{\textrm{PSU}_{n}}(t)$ the polynomial $\prod_{i=1}^{n-1}(1+t^{2i+1})$. In other words, $P_{\textrm{PSU}_{n}}(t)$ is the Poincaré polynomial of $\textrm{PSU}_{n}$. It is easy to check that, starting from $n=5$ the polynomial $P_{\textrm{PSU}_{n}}(t)$ is no longer unimodal. On the other hand, as a product of palindromic polynomials, $P_{\textrm{PSU}_{n}}(t)$ is palindromic for all $n$. Let us write $P_{\textrm{PSU}_{n}}(t)$ in the form $P_{\textrm{PSU}_{n}}(t)=\sum_{i=0}^{n^{2}-1}b_{i}t^{i}$ with $b_{i}\in{\mathbb{Z}}_{\geq 0}$.

Corollary 1.3**.**

Let $Y$ denote the determinantal variety defined as in (1.1). If $V$ is $n$ dimensional, then the Poincaré polynomial of $Y$ is expressible as a sum of two polynomials,

[TABLE]

where $A(t)=1+t^{2}+\cdots+t^{2\lfloor\frac{n^{2}-1}{2}\rfloor}$, and $\widetilde{B}(t)$ is the polynomial that is obtained from

[TABLE]

by adding 1 to the coefficients of the terms $b_{i-(n^{2}-1)}t^{i}$ with $i$ odd.

Note that a complete description of the (torsion in the) cohomology ring of $\textrm{PSU}_{n}$ has recently been given by Haibao Duan in [3].

2. Preliminaries

We start with reviewing some well known facts about the Chow groups and Borel-Moore homology groups. We follow the presentation in [5, Chapter 19]; if $X$ is a topological space, then $\bar{H}_{*}(X)$ stands for the Borel-Moore homology group with integer coefficients.

2.1.

Let $k$ be a nonnegative integer, and let $X$ be a scheme. The free abelian group generated by all $k$ dimensional subvarieties of $X$ is denoted by $Z_{k}X$. The elements of $Z_{k}X$ are called $k$-cycles. A $k$-cycle $\alpha$ is said to be rationally equivalent to 0, and written $\alpha\sim 0$ if there are a finite number of $k+1$ dimensional subvarieties $Y_{1},\dots,Y_{s}$ and rational functions

$f_{i}\in{\mathbb{C}}(Y_{i})^{*}$ ($i=1,\dots,s$) such that $\alpha=\sum[\text{div}(f_{i})]$. The set of $k$-cycles which are rationally equivalent to 0 is a subgroup of $Z_{k}(X)$, denoted by $Rat_{k}(X)$. The quotient group $A_{k}(X):=Z_{k}(X)/Rat_{k}(X)$ is called the group of $k$-cycles modulo rational equivalence, or the $k$-th Chow group. The total Chow group $A_{*}(X):=\bigoplus_{k=0}^{\dim X}A_{k}(X)$ is a graded abelian group; if $X$ is equidimensional, then $A_{\dim X}(X)$ is freely generated by the classes of irreducible components of $X$.

If $X$ is an equidimensional scheme, by replacing $Z_{k}(X)$ with $Z^{k}(X)$, that is the group of $k$-codimensional cycles, we have the Chow group

[TABLE]

We set $A^{*}(X):=\oplus A^{i}(X)$. If $X$ is smooth, then there is an intersection pairing on $A^{*}(X)$, and hence, $A^{*}(X)$ becomes a ring.

Let $s$ be an element of ${\mathbb{Z}}_{\geq 0}\cup\{*\}$. We will denote the vector spaces $A^{s}(X)\otimes{\mathbb{Q}}$ and $A_{s}(X)\otimes{\mathbb{Q}}$ by $A^{s}(X)_{\mathbb{Q}}$ and $A_{s}(X)_{\mathbb{Q}}$, respectively.

Chow groups behave nicely with respect to certain classes of morphisms.

1. (1)

If $f:X\rightarrow Y$ is a proper morphism, then there is a (covariant) homomorphism

[TABLE] 2. (2)

If $f:X\rightarrow Y$ is flat morphism of relative dimension $n$, then there is a (contravariant) homomorphism

[TABLE]

Let $i:Y\hookrightarrow X$ be an inclusion of a closed subscheme $Y$ into a scheme $X$. Let $U$ denote the complement $X-Y$ and let $j:U\rightarrow X$ denote the inclusion. Then there is an exact sequence

[TABLE]

for all $k$. To understand the image of $i_{*}$ in $A_{k}(X)$ we need to consider Edidin and Graham’s version of Bloch’s higher Chow groups.

Let $X$ be a quasi-projective scheme, and let $\Delta^{k}$ denote the algebraic version of the regular $k$-simplex:

[TABLE]

A face of $X\times\Delta^{k}$ is the subscheme of the form $X\times\Delta^{m}$, where the second factor $\Delta^{m}$ is the image of an injective canonical morphism $\rho:\Delta^{m}\rightarrow\Delta^{k}$. We denote by $Z^{i}(X,\bullet)$ the complex whose $k$-th term is the group of cycles of codimension $i$ in $X\times\Delta^{k}$ which intersect properly all of the faces in $X\times\Delta^{k}$. In [1], Bloch considered the following higher Chow groups:

[TABLE]

Let $Z_{p}(X,\bullet)$ denote the complex whose $k$-th term is the group of cycles of dimension $p+k$ in $X\times\Delta^{k}$ intersecting the faces properly.

Definition 2.2**.**

The $(p,k)$-th higher Chow group of a quasi-projective scheme $X$ is defined by

[TABLE]

The point of this definition is that $X$ does not need to be equidimensional. If $X$ is equidimensional of dimension $n$, then it is easy to see that $A_{p}(X,k)=CH^{n-p}(X,k)$.

Now we state the localization long exact sequence for higher Chow groups.

Lemma 2.4**.**

Let $Y$ be a closed, not necessarily equidimensional subscheme of an equidimensional scheme $X$. Then there is a long exact sequence of higher Chow groups;

[TABLE]

Proof.

See [4, Lemma 4]. ∎

Remark 1**.**

It is not clear if the localization long exact sequence terminates for an arbitrary scheme.

2.2.

The Borel-Moore homology groups of a space are defined by using ordinary cohomology groups as follows. Let $Y$ be a topological space that is embedded as a closed subspace of ${\mathbb{R}}^{n}$ for some positive integer $n$. Then the $q$th Borel-Moore homology of $Y$, denoted by $\bar{H}_{q}(Y)$ is defined by

[TABLE]

1. (1)

If $f:Y\rightarrow X$ is a proper morphism of complex schemes, then there are covariant homomorphisms $f_{*}:\bar{H}_{i}(Y)\rightarrow\bar{H}_{i}(X)$. 2. (2)

If $j:U\hookrightarrow Y$ is an open imbedding, then there are contravariant restriction homomorphisms $j^{*}:\bar{H}_{i}(Y)\rightarrow\bar{H}_{i}(U)$. 3. (3)

If $Y$ is the complement of $U$ in $X$ and $i:Y\rightarrow X$ is the closed imbedding, then there is a long exact sequence

[TABLE] 4. (4)

If $X$ is a disjoint union of a finite number of spaces, $X=X_{1}\cup\cdots\cup X_{n}$, then $\bar{H}_{i}(X)=\oplus\bar{H}_{i}(X_{j})$. 5. (5)

There is a Künneth formula for Borel-Moore homology. 6. (6)

If $X$ is an $n$-dimensional complex scheme, then $\bar{H}_{i}(X)=0$ for all $i>2n$, and $\bar{H}_{2n}(X)$ is a free abelian group with one generator for each irreducible component $X_{i}$ of $X$. The generator corresponding to $X_{i}$ will be denoted by $cl(X_{i})$. More generally, we will use the following notation: If $V$ is a $k$-dimensional closed subscheme of $X$, and $i:V\hookrightarrow X$ is the closed imbedding, then $cl_{X}(V)$ stands for $i_{*}cl(V)$, which lives in $\bar{H}_{2k}(X)$. If confusion is unlikely, we will omit the subscript $X$ from the notation. 7. (7)

If $f:V\rightarrow W$ is a proper, surjective morphism of varieties, then $f_{*}cl(V)=\text{deg}(V/W)\cdot cl(W)$. Since we do not need it for our purposes, we will not define $\text{deg}(V/W)$ here; see [5, Section 1.4] for its definition. 8. (8)

For any complex scheme $X$, there is a homomorphism from algebraic $k$-cycles on $X$ to the $k$-th Borel-Moore homology, $cl:Z_{k}(X)\rightarrow\bar{H}_{2k}(X)$, defined by $cl(\sum n_{i}[V_{i}])=\sum n_{i}cl_{X}(V_{i})$. This homomorphism factors through the “algebraic equivalence” (which we didn’t introduce), hence, by composition, it induces a homomorphism from the $k$-th Chow group of $X$ onto the $2k$-th Borel-Moore homology. We will denote the resulting homomorphism by $cl$ also, and call it the cycle map. 9. (9)

If a complex scheme $X$ has a cellular decomposition, then the cycle map $cl_{X}:A_{k}(X)\rightarrow\bar{H}_{2k}(X)$ is an isomorphism (see [5, Section 19.1.11]). 10. (10)

Finally, let us mention that if $X$ is an $n$-dimensional oriented manifold, then $\bar{H}_{k}(X)\cong H^{n-k}(X)$.

For further details of this useful homology theory, see [2].

3. Proof

We will use the following notation in the sequel:

[TABLE]

Since $Y$ is a projective variety, we have $\bar{H}_{q}(Y)=H_{q}(Y)$ for $q\in\{0,\dots,\dim Y\}$. Of course, similar equalities hold true for $X\cong{\mathbb{P}}^{n^{2}-1}$ as well. Both of the spaces $X$ and $Y$ are path connected, therefore, we have $H_{0}(X)=H_{0}(Y)={\mathbb{Z}}$. The complement of $Y$ in $X$ is given by the group $U$. Since $U$ is open in $X$, there is a long exact sequence for their Borel-Moore homology,

[TABLE]

As complex projective spaces have zero odd homology, the long exact sequence in (3.1) breaks up into short exact sequences. More precisely, for $q=1,\dots,n^{2}-1$, we have

[TABLE]

We will identify $U=\textrm{PGL}_{n}$ with the (complex) projective special linear group, $\textrm{PSL}_{n}$. In turn, as a real manifold, $\textrm{PSL}_{n}$ has the (Cartan-Malcev-Iwasawa) decomposition $\textrm{PSL}_{n}\cong\textrm{PSU}_{n}\times{\mathbb{R}}^{s}$, where $\textrm{PSU}_{n}$ is the projective special unitary group, and $s=n^{2}-1$. Note that, as a (real) Lie group, $\textrm{PGL}_{n}$ is an oriented $2(n^{2}-1)$-dimensional manifold, therefore, its Borel-Moore homology groups are actually cohomology groups,

[TABLE]

The unitary groups are compact. Since $\textrm{PSU}_{n}$ is a $(n^{2}-1)$-manifold, by Poincaré duality, we see the following fact.

Lemma 3.3**.**

The homology groups of $U=\textrm{PGL}_{n}$ are given by

[TABLE]

By using (3.4) and the short exact sequence in (3.2), we determine the homology groups of $Y$ in lower degrees.

Lemma 3.5**.**

The homology groups $H_{q}(Y)$ for $q<n^{2}-1$ are given by

[TABLE]

Remark 2**.**

Since $Y$ is an irreducible hypersurface in $X$, the knowledge of the lower degree homology groups as in (3.6) could also be obtained by using the Lefschetz hyperplane theorem, see [7, Corollary 1.24].

We are now ready to state and prove our main result that is stated in the introduction.

Proof of Theorem 1.2.

For $q\in\{1,\dots,n^{2}-2\}$, we have the commuting diagram of Chow groups and Betti numbers as in Figure 3.1.

We have two remarks in order:

1. (1)

Since $X$ has a cellular decomposition, the vertical map $cl_{X}$ is an isomorphism. 2. (2)

Secondly, as a result of a deep result Totaro, we know that the Chow groups of $U$ are almost always zero, except at the degree $n^{2}-1$, where it is ${\mathbb{Z}}$. Indeed, by [6, Theorem 16.6], we know that the Chow ring $A^{*}(\textrm{GL}_{n}/{\mathbb{C}}^{*})$, which is Poincaré dual to $A_{*}(U)$, is isomorphic to ${\mathbb{Z}}$. In particular, $A_{q}(U)=0$ for $q\in\{0,\dots,n^{2}-3\}$, and $A_{0}(U)\cong{\mathbb{Z}}$.

As a consequence of these two remarks, we see that, for $q\in\{1,\dots,n^{2}-3\}$, the map $i^{*}$ in the top row of diagram in (3.1) is zero, hence, the top $j^{*}$ is surjective. It follows that the bottom $j^{*}$ is surjective as well. But then, by the exactness of the bottom row, the kernel of the bottom $i^{*}$ is equal to $\bar{H}_{2q}(X)$, hence it is the zero map. In other words, we have

[TABLE]

Thus, combining these isomorphisms with Lemma 3.5, we finish the proof of our theorem.

∎

It is now easy to verify that the Poincaré polynomial of $Y$ is as given in Corollary 1.3.