On the topology of determinantal links

Let $X\hookrightarrow U\subset\mathbb{C}^{n}$ be the closed embedding of an equidimensional reduced complex analytic variety of dimension $d$ in some open set $U$ of $\mathbb{C}^{n}$ and suppose $\{V^{\alpha}\}_{\alpha\in A}$ is a complex analytic Whitney stratification for $X$. Such stratifications always exist; a construction of a unique minimal stratification has been described in [TT81] and if not specified further, we will in the following always assume $X$ to be endowed with its minimal Whitney stratification. Moreover, we may suppose that every stratum is connected for if this was not the case, we could replace it by its distinct irreducible components.

Recall the notions of the real and complex links of $X$ along its strata. These can be defined as follows: Let $V^{\alpha}$ be a stratum of $X$ and $x\in V^{\alpha}\subset X$ an arbitrary point. For simplicity, we may assume $x=0\in\mathbb{C}^{n}$ to be the origin. Choose a normal slice to $V^{\alpha}$ through $x$, i.e. a submanifold $(N_{x},x)\subset(\mathbb{C}^{n},x)$ of complementary dimension which meets $V^{\alpha}$ transversally in $x$. Let $B_{\varepsilon}(x)$ be the closed ball of radius $\varepsilon$ centered at $x$. Then the real link of $X$ along the stratum $V^{\alpha}$ is

(5)

$$\mathcal{K}(X,V^{\alpha})=X\cap N_{x}\cap\partial B_{\varepsilon}(x)$$

for $1\gg\varepsilon>0$ sufficiently small. Similarly, the complex link is

(6)

$$\mathcal{L}(X,V^{\alpha})=X\cap N_{x}\cap B_{\varepsilon}(x)\cap l^{-1}(\{\delta\})$$

for a sufficiently general linear form $l$ and $1\gg\varepsilon\gg|\delta|>0$. For a rigorous definition of these objects see for example [GM88, Part I, Chapter 1.4] and [GM88, Part II, Chapter 2.2]. There, one can also find a proof of the fact that the real and complex links are independent of the choices involved in their definition: They are invariants of the particular stratification of $X$ and unique up to non-unique homeomorphism.

In this note we shall also be concerned with the real and complex links of higher codimension for a germ $(X,0)\subset(\mathbb{C}^{n},0)$ at the point $0$.

By “sufficiently general” we mean that $D_{i}$ has to belong to a Zariski open subset $U_{i}\subset\operatorname{Grass}(n-i,n)$ in the Grassmannian. This set consists of planes for which variation of $D_{i}$ results in a Whitney equisingular family in $X\cap D_{i}$ and we shall briefly discuss the existence of such a set. From this it will follow immediately that the real and complex links of higher codimension are invariants of the germ $(X,0)$ itself and unique up to non-unique homeomorphism.

Consider the Grassmann modification of $X$:

$$G_{i}X=\{(x,D)\in X\times\operatorname{Grass}(n-i,n):x\in D\}$$

where by $\operatorname{Grass}(n-i,n)$ we denote the Grassmannian of codimension $i$ planes in the ambient space $\mathbb{C}^{n}$ through the point $0$. It comes naturally with two projections

(7)

where by construction $\rho(\pi^{-1}(\{D\}))=X\cap D$ for any plane $D\in\operatorname{Grass}(n-i,n)$. The Grassmannian itself has a natural embedding into $G_{i}X$ as the zero section $E_{0}$ of $\pi$ and we may think of $E_{0}$ as parametrizing the intersections of $X$ with planes of codimension $i$. For $i\leq d=\dim X$ the intersection of any plane $D$ with $(X,0)$ will be of positive dimension at $0$ and therefore $E_{0}$ is necessarily contained in the closure of its complement.

Note that for an arbitrary point $p\in X$ the fiber $\rho^{-1}(\{p\})$ over $p$ in $G_{i}X$ consists of all planes $D\in\operatorname{Grass}(n-i,n)$ containing both $p$ and the origin. It follows that the restriction

$$\rho\colon G_{i}X\setminus E_{0}\to X\setminus\{0\}$$

is not an isomorphism, but nevertheless a holomorphic fiber bundle with smooth fibers. In particular, any given Whitney stratification on $X$ determines a unique Whitney stratification of $G_{i}X\setminus E_{0}$ by pullback of the strata.

Given such a stratification on the complement of $E_{0}$, there exists some maximal Zariski open subset $U_{i}\subset E_{0}\cong\operatorname{Grass}(n-i,n)$ such that all the strata of $G_{i}X\setminus E_{0}$ satisfy Whitney’s conditions (a) and (b) along $U_{i}$. We may extend the stratification on the open subset $G_{i}X\setminus E_{0}$ to a Whitney stratification of $G_{i}X$, containing $U_{i}$ as a stratum. Since the Grassmannian $\operatorname{Grass}(n-i,n)$ is irreducible and complex analytic subsets have real codimension $\geq 2$, this stratum is necessarily dense and connected.

A plane $D$ of codimension $i$ is sufficiently general in the sense of Definition 2.1 if it belongs to $U_{i}$. The projection $\pi$ provides us with a canonical choice of normal slices along $U_{i}$ and by construction, we may therefore identify the real and complex links of $G_{i}X$ along $U_{i}$ with the real and complex links of the germ $(X\cap D,0)\subset(\mathbb{C}^{n},0)$ for any $D\in U_{i}$. Given that the classical real and complex links are invariants of the strata in a Whitney stratification, it follows immediately that the real and complex links of higher codimension for a germ $(X,0)$ are well defined invariants of the germ itself and unique up to non-unique homeomorphism.

While it was already established in the previous discussion that the real and complex links of higher codimension are invariants of the germ $(X,0)\subset(\mathbb{C}^{n},0)$ itself, a more specific setup will be required in the following. Let $\underline{l}=l_{1},l_{2},\dots,l_{d}\in\operatorname{Hom}(\mathbb{C}^{n},\mathbb{C})$ be a sequence of linear forms defining a flag of subspaces

$$\mathcal{D}:\mathbb{C}^{n}=D_{0}\supset D_{1}\supset D_{2}\supset\dots\supset D_{d}\supset\{0\}$$

in $\mathbb{C}^{n}$ via $D_{i+1}=D_{i}\cap\ker l_{i+1}$.

This particular definition is chosen with a view towards the fibration theorems and inductive arguments used in Section 4.1. However, we shall also need the results on polar varieties by Lê and Teissier from [TT81]. To this end, we recall the necessary definitions and explain how Definition 2.2 fits into the context of their paper.

The geometric setup for the treatment of limiting tangent spaces is the Nash modification. Let $X\subset U$ be a suitable representative of $(X,0)$ in some open neighborhood $U$ of the origin. The Gauss map is defined on the regular locus $X_{\mathrm{reg}}$ via

$$X_{\mathrm{reg}}\to\operatorname{Grass}(d,n),\quad p\mapsto[T_{p}X_{\mathrm{reg}}\subset T_{p}\mathbb{C}^{n}].$$

Then the Nash blowup $\tilde{X}$ of $X$ is defined as the closure of the graph of the Gauss map. It comes with two morphisms

where $\nu$ is the projection to $X$ and $\gamma$ the natural prolongation of the Gauss map. We denote by $\tilde{T}$ the pullback of the tautological bundle on $\operatorname{Grass}(d,n)$ along $\gamma$ and by $\tilde{\Omega}^{1}$ the dual of $\tilde{T}$. By construction, the fiber $\nu^{-1}(\{p\})$ over a point $p\in X$ consists of pairs $(p,E)\in\tilde{X}\subset\mathbb{C}^{n}\times\operatorname{Grass}(d,n)$ where $E$ is a limiting tangent space to $X$ at $p$. In particular, such a limiting tangent space is unique at regular points $p\in X_{\mathrm{reg}}$ and $\nu\colon\nu^{-1}(X_{\mathrm{reg}})\to X_{\mathrm{reg}}$ is an isomorphism identifying the restriction of $\tilde{T}$ with the tangent bundle of $X_{\mathrm{reg}}$.

A flag $\mathcal{D}$ as above determines a set of degeneraci loci in the Grassmannian as follows. Every linear form $l\in\operatorname{Hom}(\mathbb{C}^{n},\mathbb{C})$ can be pulled back to a global section $\nu^{*}l$ in the dual of the tautological bundle of $\operatorname{Grass}(d,n)$: On a fiber $E$ of the tautological bundle the section $\nu^{*}l$ is defined to be merely the restriction of $l$ to $E$ regarded as a linear subspace of $\mathbb{C}^{n}$. This generalizes in the obvious way for the linear maps

$$\varphi_{k}:=l_{1}\oplus l_{2}\oplus\dots\oplus l_{d-k+1}\colon\mathbb{C}^{n}\to\mathbb{C}^{d-k+1}$$

for every $0\leq k\leq d$. Adapting the notation from [TT81] we now have

	$\displaystyle c_{k}(\mathcal{D})$	$\displaystyle:=$	$\displaystyle\{E\in\operatorname{Grass}(d,n):\dim E\cap D_{d-k+1}\geq k\}$
		$\displaystyle=$	$\displaystyle\{E\in\operatorname{Grass}(d,n):\operatorname{rank}\nu^{*}\varphi_{k}<d-k+1\}.$

Note that by construction $c_{0}(\mathcal{D})=\operatorname{Grass}(d,n)$ is the whole Grassmannian. We set $\gamma^{-1}(c_{k}(\mathcal{D}))\subset\tilde{X}$ to be the corresponding degeneraci locus on the Nash modification and

$$P_{k}(\mathcal{D}):=\nu(\gamma^{-1}(c_{k}(\mathcal{D})))$$

its image in $X$. By construction, a point $p\in X$ belongs to $P_{k}(\mathcal{D})$ if and only if there exists a limiting tangent space $E$ to $X_{\mathrm{reg}}$ at $p$ such that the restriction of $\varphi_{k}$ to $E$ does not have full rank.

The $k$-th polar multiplicity of $(X,0)$ is then defined for $0\leq k<d$ to be

(8)

$$m_{k}(X,0):=m_{0}(P_{k}(\mathcal{D})),$$

the multiplicity of the $k$-th polar variety. We shall see below that for $k=d$ the variety $P_{d}(\mathcal{D})$ is empty for a generic flag so that a definition of $m_{d}(X,0)$ does not make sense. In the other extreme case where $k=0$, the polar multiplicity $m_{0}(X,0)$ is simply the multiplicity of $(X,0)$ itself.

The previous lemma provides the link of Definition 2.2 with the “Théorème de Bertini idéaliste” by Lê and Teissier [TT81, Théorème 4.1.3]. They establish the existence of Zariski open subsets $U^{\prime}_{i}\subset\operatorname{Grass}(n-i,n)$ with certain good properties concerning the variety $\gamma^{-1}(c_{d-i+1}(D_{i}))$ for $D_{i}\in U^{\prime}_{i}$. A posteriori, they discuss in [TT81, Proposition 4.1.5] that if the whole flag $\mathcal{D}$ has been chosen such that $D_{i}\subset U^{\prime}_{i}$ for all $i$, then also (9) is in fact satisfied for all $i$. Thus we obtain the following

We will henceforth assume that the sequence of linear forms $l_{i}$ has been chosen such that the associated flag $\mathcal{D}$ has $D_{i}\in U^{\prime}_{i}\cap U_{i}$ where $U^{\prime}_{i}$ is the Zariski open subset of Lê’s and Teissiers’ “Théorème de Bertini idéaliste” and $U_{i}$ the Zariski open subset of Whitney equisingular sections of codimension $i$ from the discussion of Definition 2.1.

Lê and Teissier have described a method to compute the Euler characteristic of complex links from the polar multiplicities in [TT81, Proposition 6.1.8]. We briefly sketch how to use their results inductively in our setup from Definition 2.2.

As before, let $(X,0)\subset(\mathbb{C}^{n},0)$ be a reduced, equidimensional complex analytic germ of dimension $d$, endowed with a Whitney stratification $\{V^{\alpha}\}_{\alpha\in A}$. We will assume that $V^{0}=\{0\}$ is a stratum and write $X^{\alpha}=\overline{V^{\alpha}}$ for the closure of any other stratum $V^{\alpha}$ of $X$. Throughout this section, we will assume that an admissible sequence of linear forms $l_{1},\dots,l_{d}$ and the associated flag $\mathcal{D}$ have been chosen as in Corollary 2.4 for all germs $(X^{\alpha},0)$ at once. This flag being fixed, we will in the following suppress it from our notation and simply write $P_{k}(X^{\alpha},0)$ for the polar varieties of the germ $(X^{\alpha},0)$ with $\mathcal{D}$ being understood.

Denote by $\mathcal{L}(X,V^{\alpha})$ the classical complex links of $X$ along the stratum $V^{\alpha}$ and by $\mathcal{L}^{i}$ the complex link of codimension $i$ of $X$ at the origin. Then by [TT81, Théorème 6.1.9]

(10)

$$\chi\left(\mathcal{L}^{0}\right)-\chi\left(\mathcal{L}^{1}\right)=\sum_{\alpha\neq 0}m_{0}\left(P_{d(\alpha)-1}\left(X^{\alpha},0\right)\right)\cdot(-1)^{d(\alpha)-1}\left(1-\chi(\mathcal{L}(X,V^{\alpha}))\right).$$

For our specific setup we may interpret this formula in the context of stratified Morse theory. To this end, note that the restriction of $l_{2}$ to the complex link $\mathcal{L}^{0}$

$$l_{2}\colon X\cap B_{\varepsilon}(0)\cap l_{1}^{-1}(\{\delta_{1}\})\to\mathbb{C}$$

is a stratified Morse function with critical points on the interior of the strata $V^{\alpha}\cap\mathcal{L}^{0}$ of $\mathcal{L}^{0}$ precisely at the intersection points

$$\mathcal{L}^{0}\cap P_{d(\alpha)-1}(X^{\alpha},0)=\{q^{\alpha}_{1},\dots,q^{\alpha}_{m_{0}(P(\alpha)_{d-1}(X^{\alpha},0))}\},$$

cf. [TT81, Corollaire 4.1.6] and [TT81, Corollaire 4.1.9]. The complex link of codimension $2$ can be identified with the general fiber

$$\mathcal{L}^{1}\cong l_{2}^{-1}(\{\delta_{2}\})\cap\mathcal{L}^{0}$$

for some regular value $\delta_{2}$ off the discriminant and we can use the function $\lambda=|l_{2}-\delta_{2}|^{2}$ as a Morse function in order to reconstruct $\mathcal{L}^{0}$ from $\mathcal{L}^{1}$. It can be shown that for suitable choices of the represenatives involved, the critical points of $\lambda$ on the boundary of $\mathcal{L}^{0}$ are “outward pointing” and hence do not contribute to changes in topology; see for instance [Zac17] or [PnZ18] for a discussion. For an interior critical point $q^{\alpha}_{j}\in V^{\alpha}\cap\mathcal{L}^{0}$ we have the product of the tangential and the normal Morse data

$$\left(D^{d(\alpha)-1},\partial D^{d(\alpha)-1}\right)\times\left(C(\mathcal{L}(X,V^{\alpha})),\mathcal{L}(X,V^{\alpha})\right)$$

where $D^{d(\alpha)}$ is the disc of real dimension $d(\alpha)=\dim_{\mathbb{C}}V^{\alpha}$ and $C(\mathcal{L}(X,V^{\alpha}))$ the cone over the complex link of $X$ along $V^{\alpha}$. It is a straighforward calculation that the Euler characteristic changes precisely by $(-1)^{d(\alpha)-1}(1-\chi(\mathcal{L}(X,V^{\alpha})))$ for the attachement of this cell at any of the critical points $q_{j}^{\alpha}$. Summation over all these points on all relevant strata therefore gives us back the Formula (10) by Lê and Teissier.

It is evident that the above procedure can be applied inductively, cf. [TT81, Remarque 6.1.10]. This allows to reconstruct the codimension $i$ complex link $\mathcal{L}^{i}$ of $(X,0)$ from its hyperplane sections

$$\mathcal{L}^{i}\supset\mathcal{L}^{i+1}\supset\dots\supset\mathcal{L}^{d-1}=\{x_{1},\dots,x_{m_{0}(X,0)}\}$$

starting with $\mathcal{L}^{d-1}$ which is just a set of points whose number is equal to the multiplicity $m_{0}(X,0)$ of $(X,0)$ at the origin. We leave it to the reader to verify the formula

(11)

$$\chi(\mathcal{L}^{i})=\sum_{\alpha\in A}\left(\sum_{j=i+1}^{d(\alpha)}(-1)^{d(\alpha)-j}m_{0}\left(P_{d(\alpha)-j}(X^{\alpha},0)\right)\right)\cdot\left(1-\chi(\mathcal{L}(X,V^{\alpha})\right).$$

The coefficients appearing in this formula are nothing but the local Euler obstruction of $(X^{\alpha}\cap D_{i-1},0)$ at the origin:

(12)

$$\mathrm{Eu}(X^{\alpha}\cap D_{i-1},0)=\sum_{j=i}^{d(\alpha)}(-1)^{d(\alpha)-j}m_{0}\left(P_{d(\alpha)-j}(X^{\alpha},0)\right)$$

cf. [TT81, Corollaire 5.1.2].

We now turn towards the study of the generic determinantal varieties $M_{m,n}^{s}\subset\mathbb{C}^{m\times n}$. These are equipped with the rank stratification, i.e. the decomposition

$$M_{m,n}^{s}=\bigcup_{r<s}V_{m,n}^{r},\quad V_{m,n}^{r}=\{\varphi\in\mathbb{C}^{m\times n}:\operatorname{rank}\varphi=r\}.$$

Due to its local analytic triviality, this stratification is easily seen to satisfy both Whitney’s conditions (a) and (b).

The reduced Euler characteristics of the classical complex links have been computed by Ebeling and Gusein-Zade in [EGZ09, Proposition 3]:

(13)

$$1-\chi(\mathcal{L}(M_{m,n}^{s},0))=(-1)^{s-1}{m-1\choose s-1},$$

where, without loss of generality, it is assumed that $m\leq n$. The generic determinantal varieties admit a recursive pattern in the following sense. For $r<s\leq m\leq n$, a normal slice to the stratum $V_{m,n}^{r}\subset M_{m,n}^{s}$ through the point $\mathbf{1}_{m,n}^{r}$ is given by the set of matrices of the form

$$N_{m,n}^{r}=\mathbf{1}^{r}\oplus\mathbb{C}^{(m-r)\times(n-r)}=\begin{pmatrix}\mathbf{1}^{r}&0\\ 0&\mathbb{C}^{(m-r)\times(n-r)}\end{pmatrix}\subset\mathbb{C}^{m\times n}.$$

It follows immediately that $\mathcal{L}(M_{m,n}^{s},V_{m,n}^{r})\cong\mathcal{L}(M_{m-r,n-r}^{s-r},0)$ and hence

(14)

$$1-\chi(\mathcal{L}(M_{m,n}^{s},V_{m,n}^{r}))=(-1)^{s-r-1}{m-r-1\choose s-r-1}.$$

In order to determine the topological Euler characteristic of the complex links of higher codimension $\mathcal{L}^{i}(M_{m,n}^{s},0)$ of the generic determinantal varieties by means of the previous section, we need to know all the relevant polar multiplicities

(15)

$$e_{m,n}^{r,k}:=m_{k}(M_{m,n}^{r+1},0)=m_{0}\left(P_{k}(M_{m,n}^{r+1},0)\right).$$

There are several methods to achieve this. For instance, one could simply choose random linear forms $l_{i}$ and compute the resulting multiplicity with the aid of a computer algebra system using e.g. Serre’s intersection formula. However, this approach provides very little insight and the results a priori depend on the choice of the linear forms. Recently, X. Zhang has computed the polar multiplicities in [Zha17] using Chern class calculus which is an exact computation not depending on any particular choices. His formulas, however, are very complicated since they appear as byproducts of the study of the Chern-Mather classes of determinantal varieties. In this section we will follow a more direct approach to the computation of the polar multiplicities using Chern classes.

In [TT81, Théorème 5.1.1], Lê and Teissier give the following formula for the polar multiplicities of a germ $(X,0)\subset(\mathbb{C}^{n},0)$:

(16)

$$m_{0}\left(P_{k}(\mathcal{D})\right)(-1)^{d-1}\int_{\mathfrak{Y}}c_{k}(\tilde{T})\cdot c_{1}(\mathcal{O}(1))^{d-k-1}.$$

Here the integral is taken over the exceptional divisor $\mathfrak{Y}$ of the blowup $\mathfrak{X}$ of the Nash modification $\tilde{X}$ along the pullback of the maximal ideal at the origin for $(X,0)\subset(\mathbb{C}^{n},0)$ and $\mathcal{O}(1)$ denotes the dual of the tautological bundle for that blowup. By construction, these spaces can be arranged in a commutative diagram

(17)

where $\mathrm{Bl}_{0}X$ denotes the usual blowup of $X$ at the origin. We will now describe this diagram for the particular case where $(X,0)=(M_{m,n}^{s},0)\subset(\mathbb{C}^{m\times n},0)$ is a generic determinantal variety and deduce our particular formula (21) from that.

The Nash blowup of $(M_{m,n}^{s},0)\subset(\mathbb{C}^{m\times n},0)$ has been studied by Ebeling and Gusein-Zade in [EGZ09] and we briefly review their discussion. Let $r=s-1$ be the rank of the matrices in the open stratum $V_{m,n}^{r}\subset M_{m,n}^{s}$. The tangent space to $V_{m,n}^{r}$ at a point $\varphi$ is known to be

(18)

$$T_{\varphi}V_{m,n}^{r}=\{\psi\in\operatorname{Hom}(\mathbb{C}^{n},\mathbb{C}^{m}):\psi(\ker\varphi)\subset\mathrm{im}\,\varphi\}.$$

This fact can be exploited to replace the Grassmannian for the Nash blowup of $M_{m,n}^{s}$ by a product: Let $\operatorname{Grass}(r,m)$ be the Grassmannian of $r$-planes in $\mathbb{C}^{m}$ and $\operatorname{Grass}(r,n)$ the Grassmannian of $r$-planes in $(\mathbb{C}^{n})^{\vee}$, the dual of $\mathbb{C}^{n}$. Then the Gauss map factors through

$$\hat{\gamma}:V_{m,n}^{r}\to\operatorname{Grass}(r,n)\times\operatorname{Grass}(r,m),\quad\varphi\mapsto\left((\ker\varphi)^{\perp},\mathrm{im}\,\varphi\right),$$

where by $(\ker\varphi)^{\perp}$ we mean the linear forms in $(\mathbb{C}^{n})^{\vee}$ vanishing on $\ker M\subset\mathbb{C}^{n}$. We denote this double Grassmannian by $G=\operatorname{Grass}(r,n)\times\operatorname{Grass}(r,m)$. On $G$ we have the two tautological exact sequences $0\to S_{1}\overset{i_{1}}{\longrightarrow}\mathcal{O}^{n}\overset{\pi_{1}}{\longrightarrow}Q_{1}\to 0$ and $0\to S_{2}\overset{i_{2}}{\longrightarrow}\mathcal{O}^{m}\overset{\pi_{2}}{\longrightarrow}Q_{2}\to 0$ coming from the two factors with $S_{2}$ corresponding to the images and $Q_{1}^{\vee}$ to the kernels of the matrices $\varphi$ under the modified Gauss map $\hat{\gamma}$. With this notation, the space of matrices

$$\operatorname{Hom}(\mathbb{C}^{n},\mathbb{C}^{m})\cong(\mathbb{C}^{n})^{\vee}\otimes\mathbb{C}^{m}$$

pulls back to the trivial bundle $\mathcal{O}^{n}\otimes\mathcal{O}^{m}$ from which we may project to the product

$$\pi=\pi_{1}\otimes\pi_{2}\colon\mathcal{O}^{n}\otimes\mathcal{O}^{m}\to Q_{1}\otimes Q_{2}.$$

Then the condition on $\psi$ in (18) becomes $\pi(\psi)=0$ and consequently the Nash bundle on $\tilde{M}_{m,n}^{s}$ is given by

(19)

$$\tilde{T}=\hat{\gamma}^{*}\left(\ker\pi\right).$$