# Local Euler Obstructions of Reflective Projective Varieties

Xiping Zhang<EMAIL_ADDRESS>
###### Abstract.

In this note we introduce the concept of reflective projective varieties.
These are stratified projective varieties with certain dimension constraints
on their dual varieties. We prove that for such varieties, the Chern-Schwartz-
MacPherson classes of the strata completely determine the local Euler
obstructions and the polar degrees. We also propose an algorithm to compute
the local Euler obstructions when such varieties are formed by group orbits.
As examples we compute the local Euler obstructions of quadratic hypersurfaces
and ordinary determinantal varieties to illustrate our method.

MSC Classification: 14C17 32S25 32S05 32S50

##  1\. Introduction

The local geometric invariants around singular points are essential objects to
study in algebraic geometry and singularity theory. Among all sorts of
invariants, the local Euler obstructions are of central importance in the
study of singular spaces. Defined as the obstruction to extend the distance
$1$-form after lifting to the Nash transform; it is the key ingredient in
MacPherson’s proof of the existence and uniqueness of Chern class on singular
spaces, which was conjectured by Deligne and Grothendieck. For more about
local Euler obstructions we refer to the nice papers [2] and [5]. Such Chern
class is equivalent to a previous construction of Schwartz (Cf. [3][24][25]),
thus is called Chern-Schwartz-MacPherson class.

In [7] the authors proved that the local Euler obstruction is equal to the
local characteristics defined by Kashiwara in [14], in which he introduced the
famous index theorem for holonomic D-modules. As the key ingredients in both
singularity theory and Kashiwara’s index theorem, the local Euler obstructions
of stratified spaces have been deeply studied in singularity theory. In
particular when such stratification is induced by a group action, following
the work of Kazhdan-Luzstig [15] the local Euler obstructions are closely
related to the coefficients of the Kazhdan-Luzstig polynomials. Thus they are
also intensely studied in geometric representation theory. Despite the great
importance, the local Euler obstructions are in general very hard to compute.
Many authors have been working on formulas that make the computation easier.
We refer to [4] [6] [12] [13] [19] [27] for many different approaches.

In this note we consider a special type of projective varieties: the
reflective projective varieties. These are stratified varieties with certain
dimension constraints on the dual varieties of the strata. The precise
definition is given in Definition Definition. A common example of reflective
projective varieties is the projectivizations of orbital closures of a
$G$-representation $V$, where $G$ is a connected algebraic group acting on $V$
by finite orbits. We call such orbits reflective group orbits. We take
advantage of Aluffi’s duality result in [1], which relates the Chern-Mather
classes between dual varieties, to prove the main result of this paper:

######  Theorem (Theorem3.1, Corollary1).

Let $V$ be a vector space of dimension $N$. Let
$X=\sqcup_{i=1}^{n}X_{i}\subset\mathbb{P}(V)$ be a reflective projective
variety, with its dual variety $Y=X^{\vee}$ stratified by
$Y=\sqcup_{j=1}^{s}Y_{j}$. The Chern-Schwartz-MacPherson classes of the strata

$\\{c_{sm}^{X_{1}}(H),\cdots,c_{sm}^{X_{n}}(H);c_{sm}^{Y_{1}}(H),\cdots,c_{sm}^{Y_{s}}(H)\\}$

completely determine the local Euler obstruction functions $Eu_{X}$ and
$Eu_{Y}$, and thus the Chern-Mather classes $c_{M}^{X}(H)$, $c_{M}^{Y}(H)$, or
equivalently the polar degrees $\\{P_{k}(X),P_{r}(Y)|k,r\\}$.

As pointed out in [8][22], for certain group orbits the Chern-Schwartz-
MacPherson classes are precisely the solutions of some constraint equations
coming from localizations and specializations/restrictions of the group
action. These solutions are certain weight functions corresponding to the
$K$-theoretic stable envelops studied in [18]. This provides a way to obtain
the Chern-Schwartz-MacPherson classes directly from representation theory,
besides the classical intersection method. This allows more flexible on
computing in practice. Thus in Corollary 2 and Algorithm 1 we prove an
effective algorithm to compute the local Euler obstructions for reflective
group orbits, using the Chern-Schwartz-MacPherson classes. As applications we
consider quadratic hypersurfaces and (ordinary) determinantal varieties. They
are natural examples of reflective projective varieties and reflective group
orbits, and in Corollary 3 and Corollary 4 we show that how to apply this
duality method on effectively computing their local Euler obstructions. We
also compute the Milnor classes for quadratic hypersurfaces.

The paper is organized as follows. In §2 we briefly recall the definitions and
the tools we will use, namely the theory of Chern-Schwartz-MacPherson classes
and Aluffi’s projective duality. Then in §3 we state and prove the main result
of the paper. In sections §4 and §5 we show that our method can effectively
compute the local Euler obstructions for quadratic hypersurfaces and
(ordinary) determinantal varieties.

##  2\. Preliminary

###  2.1. Chern-Schwartz-MacPherson Classes

Let $X\subset\mathbb{P}^{N}$ be a projective variety. The group of
constructible function is defined as the abelian group generated by indicator
functions $\mathds{1}_{V}$ for all irreducible subvarieties $V\subset X$. We
define the pushforward for a proper morphism $f\colon X\to Y$ as follows. For
any closed subvariety $V\subset X$, the pushforward $F(f)(\mathds{1}_{V})(y)$
evaluates $\chi(f^{-1}(y)\cap V)$ for any $y\in Y$. This makes $F$ a functor
from projective complex varieties to the Abelian group category. The group
$F(X)$ has $\\{\mathds{1}_{V}\\}$ as a natural base.

When the base field is $\mathbb{C}$, in 1974 MacPherson defined a local
measurement for singularities and names it the local Euler obstruction. He
proved that the local Euler obstruction functions $\\{Eu_{V}|\text{V is a
subvariety of X}\\}$ also form a base for $F(X)$. He also constructed a
natural transformation $c_{*}$ by sending the local Euler obstruction function
$Eu_{V}$ to the Chern-Mather class $c_{M}(V)$. The precise definition is left
to the next subsection. The following theorem answered a conjecture by Deligne
and Grothendieck:

######  Theorem 2.1 ([17]).

The map $c_{*}$ defined above is the unique natural transform from $F$ to the
homology functor $H_{*}$ satisfying the following normalization property:
$c_{*}(\mathds{1}_{X})=c(TX)\cap[X]$ when $X$ is smooth.

######  Remark 1.

Another definition of Chern classes on singular varieties is due to M.-H.
Schwartz, who uses obstruction theory and radial frames to construct such
classes. Details of the construction can be found in [25] [24] [3]. Also in
[3] it is shown that these classes correspond, by Alexander isomorphism, to
the classes defined by MacPherson in the above theorem.

The local Euler obstruction was originally defined via obstruction theory over
$\mathbb{C}$, as the obstruction to extend the lift of the distance $1$ form
on the Nash transform. In [12] the author gave an algebraic formula using the
Nash blowup that works for arbitrary algebraically closed field. An equivalent
formula using conormal spaces was given by Sabbah in [23]. In 1990 Kennedy
modified Sabbah’s Lagrangian intersections and proved the following
generalization.

######  Theorem 2.2 ([16]).

Replace the homology functor by the Chow functor, MacPherson’s natural
transformation extends to arbitrary algebraically closed field of
characteristic $0$.

######  Remark 2.

Let $X$ be a complete variety with constant map $k\colon X\to\\{p\\}$. Then
the covariance property of $c_{*}$ shows that

$\displaystyle\int_{X}c_{sm}(Y)=$ $\displaystyle\leavevmode\nobreak\
\int_{\\{p\\}}Afc_{*}(\mathds{1}_{Y})=\int_{\\{p\\}}c_{*}Ff(\mathds{1}_{Y})$
$\displaystyle=$ $\displaystyle\leavevmode\nobreak\
\int_{\\{p\\}}\chi(Y)c_{*}(\mathds{1}_{\\{p\\}})=\chi(Y).$

This observation gives a generalization of the classical Poincaré-Hopf Theorem
to possibly singular varieties.

###  2.2. Local Euler Obstruction

In this subsection we briefly review the definition of local Euler obstruction
and Chern-Mather class, which play central role in the construction of
$c_{*}$. The following algebraic definition is due to González-Sprinberg [12].

Let $\iota\colon X\to\mathbb{P}^{N}$ be a (closed) projective subvariety of
dimension $d$, we define the Nash transform of $X$ to be

$\hat{X}:=\text{closure of }\\{(x,T_{x}X)|x\text{ is a smooth point}\\}\subset
G_{d}(T\mathbb{P}^{N})$

The projection map $p\colon\hat{X}\to X$ is birational and is isomorphic over
the smooth locus $X_{sm}$. The universal sub-bundle $S$ of
$G_{d}(T\mathbb{P}^{N})$ restricts to a rank $d$ vector bundle on $\hat{X}$,
denoted by $\mathcal{T}_{X}$.

######  Definition.

The local Euler obstruction of $X$ is defined as follows: for any $x\in X$ we
define

$Eu_{X}(x):=\int_{p^{-1}(x)}c(\mathcal{T}_{X})\cap s(p^{-1}(x),\hat{X})\/.$

Here $s(A,B)$ is Fulton’s Segre class in [9]. The Chern-Mather class of $X$ is
defined as

$c_{M}^{X}:=\iota_{*}p_{*}(c(\mathcal{T}_{X})\cap[\hat{X}])\/.$

Recall that the Chow group(ring) of $\mathbb{P}^{N}$ is
$\mathbb{Z}[H]/H^{N+1}$, where $H$ here is the hyperplane class
$c_{1}(\mathcal{O}(1))\cap[\mathbb{P}^{N}]$.

######  Definition.

The Chern-Schwartz-MacPherson class of $X$, denoted by $c_{sm}^{X}(H)$ is
defined as the pushforward $\iota_{*}c_{*}(\mathds{1}_{X})$ in
$A_{*}(\mathbb{P}^{N})$.

Notice that when $X$ is smooth, the Chern-Mather class and the Chern-Schwartz-
MacPherson class all equal to the total Chern class $i_{*}(c(TX)\cap[X])$.

######  Proposition 2.3.

We have the following properties.

  1. (1)

$Eu_{X}(x)$ is a local invariant, thus only depends on an open neighborhood of
$x$ in $X$.

  2. (2)

If $x\notin X$, then $Eu_{X}(x)=0$.

  3. (3)

If $x\in X$ is a smooth point, then $Eu_{X}(x)=1$. But notice that
$Eu_{X}(x)=1$ does NOT imply that $x$ is smooth (Cf. [20],[19] and [30]).

  4. (4)

The Euler obstruction has the product property, i.e., $Eu_{X\times Y}(x\times
y)=Eu_{X}(x)\times Eu_{Y}(y)$.

######  Proposition 2.4.

Let $X\subset\mathbb{P}(V)$ be a projective variety, and let $\Sigma\subset V$
be its affine cone. Let $q\in V\setminus\\{0\\}$, and let $\bar{q}$ be its
image in $\mathbb{P}(V)$. Then we have

$Eu_{X}(\bar{q})=Eu_{\Sigma}(q)\/.$

###### Proof.

This is because that locally around $\bar{q}$, the affine cone $\Sigma$ is
isomorphic to the product $X\times\mathbb{C}^{*}$. This result follows from
the product property mentioned above. ∎

###  2.3. Involutions of Chern Classes

In this subsection we introduce the key ingredient we will use in the paper:
the projective duality involution introduced by Aluffi in [1]. Let
$\mathbb{Z}_{\leq d}[x]$ be the vector space of degree $\leq d$ polynomials.
For any $d\geq 0$ we define the following transformations
$\mathcal{I}_{d}\colon\mathbb{Z}_{\leq d}[x]\to\mathbb{Z}_{\leq d}[x]$by

$\displaystyle\mathcal{I}_{d}\colon$ $\displaystyle f(x)\mapsto
f(-1-x)-f(-1)((1+x)^{d+1}-x^{d+1})$

######  Proposition 2.5.

The transformations $\mathcal{I}_{d}$ has the following properties:

  1. (1)

For any polynomial $f$ with no constant term,
$\mathcal{I}_{d}(\mathcal{I}_{d}(f))=f$. Thus $\mathcal{I}_{d}$ is an
involution on the set of polynomials.

  2. (2)

The involution $\mathcal{I}_{d}$ is linear, i.e.,
$\mathcal{I}_{d}(af+bg)=a\mathcal{I}_{d}(f)+b\mathcal{I}_{d}(g)$.

Let $X\subset\mathbb{P}(V)$ be a projective variety of dimension $n$, the dual
variety $X^{\vee}$ is a projective variety in $\mathbb{P}(V^{*})$, defined as

$X^{\vee}:=\overline{\\{H\in\mathbb{P}(V^{*})|T_{x}X\subset H\text{ for some
smooth point }x\in X_{sm}\\}}$

######  Theorem 2.6 (Projective Duality [1]).

The involution $\mathcal{I}_{\dim V-1}$ takes the signed Chern-Mather class of
$X$ to the signed Chern-Mather class of the dual variety $X^{\vee}$. More
precisely, we have

$(-1)^{\dim X^{\vee}}c_{M}^{X^{\vee}}(H)=(-1)^{\dim X}\mathcal{I}_{\dim
V-1}(c_{M}^{X}(H))\/.$

For more details about projective dual varieties we refer to [9][26]. We will
use the following result.

######  Proposition 2.7 ([26], §2.2).

Let $G$ be a connected algebraic group, and let $V$ be a $G$-representation.
Let $V^{*}$ be the dual representation of $G$. If there are only finitely
orbits, and all the orbits are cones, then the projectivized orbits in
$\mathbb{P}(V)$ and $\mathbb{P}(V^{*})$ are projective dual to each other,

For the rest of the paper we consider projective varieties over algebraically
closed field $k$ of characteristic $0$.

##  3\. Local Euler Obstructions of Reflective Projective Varieties

In this section we consider a special type of projective varieties: the
reflective varieties.

###  3.1. Main Result

######  Definition.

Let $V$ be a vector space of dimension $N$. We say a projective variety
$X\subset\mathbb{P}(V)$ is reflective if $X$ admits a finite stratification
$X=\cup_{i=1}^{n}X_{i}$, such that

  1. (1)

$X_{n}=\bar{X}_{n}\subset\bar{X}_{n-1}\subset\cdots\bar{X}_{2}\subset\bar{X}_{1}=X$;

  2. (2)

The dimension of the dual varieties $\bar{X}_{i}^{\vee}$ satisfy the following
inequalities: $\dim
X^{\vee}=\dim\bar{X}^{\vee}_{0}<\dim\bar{X}^{\vee}_{1}<\cdots<\dim\bar{X}^{\vee}_{n}\/.$

  3. (3)

$Y=X^{\vee}=\cup_{j=1}^{m}Y_{j}$ form a finite stratification.

######  Example 1.

Let $X_{A}\subset\mathbb{P}(V)$ be a quadratic hypersurface defined by
$x^{t}Ax=0$ for some symmetric $n\times n$ matrix $A$. The hypersurface
$X_{A}$ is smooth if and only if $|A|\neq 0$, and the dual variety
$X_{A}^{\vee}$ is also a smooth hypersurface. When $A$ is singular of rank
$k<n$, as shown in [11], the dual variety $X_{A}^{\vee}$ is a smooth quadratic
hypersurface in $\mathbb{P}^{r-1}\subset\mathbb{P}(V)$. The singularity locus
$S_{A}$ of $X_{A}$ is given by the solution of the linear system $AX=0$, thus
is a dimension $n-r-1$ linear space. The dual space of $S_{A}$ is isomorphic
to $\mathbb{P}^{r}$. This shows that the singular quadratic hypersurfaces are
reflective.

######  Theorem 3.1.

We denote $I=I_{N-1}$ to be Aluffi’s projective duality involution. Assume
that there are not-all-zero integers $\\{\alpha_{i}|i=1,\cdots,n\\}$ and
$\\{\beta_{j}|j=0,\cdots,m\\}$ such that the following involution equality
holds.

$I\left(\sum_{r=1}^{n}\alpha_{r}\cdot
c_{sm}^{X_{r}}\right)=\sum_{s=1}^{m}\beta_{s}\cdot
c_{sm}^{Y_{s}};\quad\alpha_{1}=\beta_{1}=1\/.$

Then we have

$\alpha_{i}=Eu_{X}(X_{i});\quad\beta_{i}=Eu_{Y}(Y_{i})\/.$

###### Proof.

We define $\hat{c}_{M}^{\bar{X}_{i}}(H):=(-1)^{\dim
X_{i}}c_{M}^{\bar{X}_{i}}(H)$ to be the sighed Chern-Mather polynomials. Since
the local Euler obstructions form a basis for the constructible function
group, there are unique integers $\\{E_{r,i}\\}$ and $\\{D_{r,j}\\}$ such that
$E_{r,r}=(-1)^{\dim X_{r}}$, $D_{s,s}=(-1)^{\dim Y_{s}}$ and

$c_{sm}^{X_{r}}(H)=\sum_{i=r}^{n}E_{r,i}\cdot\hat{c}_{M}^{\bar{X}_{i}}(H);\quad
c_{sm}^{Y_{s}}(H)=\sum_{j=s}^{m}D_{s,j}\cdot\hat{c}_{M}^{\bar{Y}_{j}}(H)\/.$

The involution equality can then be written as

$\displaystyle\sum_{j=1}^{m}\left(\sum_{s=1}^{j}\beta_{s}D_{s,j}\right)\hat{c}_{M}^{\bar{Y}_{j}}$
$\displaystyle=\sum_{s=1}^{m}\sum_{j=s}^{m}\beta_{s}D_{s,j}\cdot\hat{c}_{M}^{\bar{Y}_{j}}=\sum_{s\geq
m}\beta_{s}\cdot c_{sm}^{Y_{s}}$ $\displaystyle=$ $\displaystyle
I\left(\sum_{r=1}^{n}\alpha_{r}\cdot
c_{sm}^{X_{r}}\right)=I\left(\sum_{r=1}^{n}\sum_{i=r}^{n}\alpha_{r}E_{r,i}\cdot\hat{c}_{M}^{\bar{X}_{i}}\right)$
$\displaystyle=$
$\displaystyle\sum_{r=1}^{n}\sum_{i=r}^{n}\alpha_{r}E_{r,i}\cdot
I(\hat{c}_{M}^{\bar{X}_{i}})=\sum_{r=1}^{n}\sum_{i=r}^{n}\alpha_{r}E_{r,i}\cdot\hat{c}_{M}^{\bar{X}^{\vee}_{i}}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\left(\sum_{r=1}^{i}\alpha_{r}E_{r,i}\right)\hat{c}_{M}^{\bar{X}^{\vee}_{i}}$

Thus we obtain the following equality

$\sum_{j=1}^{m}\left(\sum_{s=1}^{j}\beta_{s}D_{s,j}\right)\hat{c}_{M}^{\bar{Y}_{j}}=\sum_{i=1}^{n}\left(\sum_{r=1}^{i}\alpha_{r}E_{r,i}\right)\hat{c}_{M}^{\bar{X}^{\vee}_{i}}\/.$

Recall that we have an increasing chain of dimensions from the definition of
reflective projective variety:

$\dim\bar{Y}_{m}<\dim\bar{Y}_{m-1}<\cdots<\dim\bar{Y}_{1}=\dim\bar{X}^{\vee}_{1}<\dim\bar{X}^{\vee}_{2}<\cdots<\dim\bar{X}^{\vee}_{n}$

First we look at the highest dimension term $\bar{Y}_{m}$. Among all the
polynomials
$\\{\hat{c}_{M}^{\bar{X}^{\vee}_{i}};\hat{c}_{M}^{\bar{Y}_{j}}\\}$, the
polynomial $\hat{c}_{M}^{\bar{Y}_{m}}$ is the only polynomial that contains
$[\mathbb{P}^{\dim Y_{m}}]$. Thus we have $\sum_{s=1}^{m}\beta_{s}D_{s,m}=0$.
Since the coefficient of $\hat{c}_{M}^{\bar{Y}_{m}}$ is $0$, then the
polynomial $\hat{c}_{M}^{\bar{Y}_{m-1}}$ is the only polynomial that contains
$[\mathbb{P}^{\dim Y_{m-1}}]$. This forces
$\sum_{s=1}^{m-1}\beta_{s}D_{s,m-1}=0$.

Continue this process until we meet $\bar{Y}_{1}=\bar{X}^{\vee}_{1}$. At this
level we obtain $\beta_{1}D_{1,1}=\beta_{1}=\alpha_{1}E_{1,1}=\alpha_{1}$.
Then we repeat the above procedure on the dimension of $X_{r}$’s, eventually
we obtain the following linear system:

$\displaystyle\begin{cases}\sum_{j=1}^{s}\beta_{j}D_{j,r}=0&s=2,3,\cdots,m\\\
\beta_{1}D_{1,1}=\alpha_{1}E_{1,1}=1&\\\
\sum_{i=1}^{r}\alpha_{i}E_{i,r}=0&r=2,3,\cdots,m\end{cases}$

Written into matrix form we have

$\displaystyle\begin{bmatrix}D_{1,m}&D_{2,m}&\cdots&D_{m-1,m}&D_{m,m}\\\
D_{1,m-1}&D_{2,2}&\cdots&D_{m-1,m-1}&0\\\
\cdots&\cdots&\cdots&\cdots&\cdots\\\
D_{1,1}&0&\cdots&0&0\end{bmatrix}\begin{bmatrix}\beta_{1}\\\ \beta_{2}\\\
\cdots\\\ \beta_{m-1}\\\ \beta_{m}\end{bmatrix}=$
$\displaystyle\begin{bmatrix}0\\\ 0\\\ \cdots\\\ 0\\\ 1\end{bmatrix}$
$\displaystyle\begin{bmatrix}E_{1,n}&E_{2,n}&\cdots&E_{n-1,n}&E_{n,n}\\\
E_{1,n-1}&E_{2,2}&\cdots&D_{n-1,n-1}&0\\\
\cdots&\cdots&\cdots&\cdots&\cdots\\\
E_{1,1}&0&\cdots&0&0\end{bmatrix}\begin{bmatrix}\alpha_{1}\\\ \alpha_{2}\\\
\cdots\\\ \alpha_{m-1}\\\ \alpha_{m}\end{bmatrix}=$
$\displaystyle\begin{bmatrix}0\\\ 0\\\ \cdots\\\ 0\\\ 1\end{bmatrix}$

The two matrices are both of full rank, since that have diagonal entries
$D_{s,s}=E_{r,r}=1$ for any $r,s$, thus the solution of
$\\{\alpha_{1},\cdots,\alpha_{n},\beta_{1},\cdots,\beta_{m}\\}$ is unique.
Meanwhile, notice that
$\\{\alpha_{i}=Eu_{X}(X_{i});\beta_{j}=Eu_{Y}(Y_{j})\\}$ is indeed a solution:
the involution does interchanges the signed Chern-Mather classes between dual
varieties. Thus by the uniqueness of the solution we completes the proof. ∎

This theorem shows that for reflective projective varieties, the Chern-
Schwartz-MacPherson classes of the strata completely determine the local Euler
obstructions. Since the Chern-Mather classes are linear sums of Chern-
Schwartz-MacPherson classes with weights being local Euler obstructions, they
also determine the Chern-Mather classes. Moreover, as shown in [21], the
coefficients of the Chern-Mather classes are equivalent to the polar degrees,
i.e., degrees of the polar varieties. This gives us the following.

######  Corollary 1.

For a reflective projective variety $X=\sqcup X_{i}$ with dual variety
$X^{\vee}=Y=\sqcup Y_{j}$, the Chern-Schwartz-MacPherson classes of the strata
$\\{X_{i}\\}$ and $\\{Y_{j}\\}$ completely determine the local Euler
obstructions $Eu_{X}(X_{i})$ and $Eu_{Y}(Y_{j})$. Moreover, they completely
determine the Chern-Mather classes $c_{M}^{X}$ and $c_{M}^{Y}$, and thus the
polar degrees $P_{k}(X)$ and $P_{k}(Y)$.

###  3.2. On Reflective Group Orbits

A common example of reflective projective varieties comes from group actions.
Let $G$ be a connected algebraic group and $V$ be a linear $G$ representation
of dimension $N$ with finite orbits $\mathcal{O}_{1},\cdots,\mathcal{O}_{s}$.
From Proposition 2.7 we can label the dual orbits by the duality
correspondence: $\mathbb{P}(\bar{\mathcal{O}}_{k})$ is dual to
$\mathbb{P}(\bar{\mathcal{O}}^{\prime}_{s-k})$.

######  Definition.

We call such representation $V$ reflective if the following assumption is
satisfied.

  1. (1)

The $G$ action contains the scalar multiplication: the orbits are $k^{*}$
cones.

  2. (2)

For any $i$ we have $\bar{\mathcal{O}}_{i}=\cup_{j\geq i}\mathcal{O}_{j}$ and
$\bar{\mathcal{O}}^{\prime}_{i}=\cup_{j\geq i}\mathcal{O}^{\prime}_{j}$.

  3. (3)

All the orbits $\mathcal{O}_{i}$ and $\mathcal{O}^{\prime}_{i}$ are purely
dimensional.

The assumptions forces $\mathcal{O}_{s}=\mathcal{O}^{\prime}_{s}=\\{0\\}$. We
call such group orbits Reflective Orbits.

######  Corollary 2.

For any $i$ we denote $S_{i}=\mathbb{P}(\mathcal{O}_{i})$ and
$S^{\prime}_{i}=\mathbb{P}(\mathcal{O}^{\prime}_{i})$ to be the projectivized
orbits in $\mathbb{P}(V)$ and $\mathbb{P}(V^{*})$. Let $I=I_{N-1}$ be Aluffi’s
projective duiality involution. Let $c_{sm}^{S_{i}}=c_{sm}^{S_{i}}(H)$ and
$c_{sm}^{S^{\prime}_{i}}=c_{sm}^{S^{\prime}_{i}}(H)$ be the Chern-Schwartz-
MacPherson polynomials in $H$. For any $r$, assume that there are not-all-
zeros integers $\\{\alpha^{r}_{i}|i=r,\cdots,s-1\\}$ and
$\\{\beta^{r}_{j}|j=s-r,\cdots,s-1\\}$ such that

$I_{N-1}\left(\sum_{i\geq r}\alpha^{r}_{i}\cdot
c_{sm}^{S_{i}}\right)=\sum_{k\geq s-r}\beta^{r}_{i}\cdot
c_{sm}^{S^{\prime}_{k}};\quad\alpha^{r}_{r}=\beta^{r}_{s-r}=1\/.$

Then we have

$\alpha^{r}_{i}=Eu_{\bar{S}_{r}}(S_{i});\quad\beta^{r}_{i}=Eu_{\bar{S}^{\prime}_{s-r}}(S^{\prime}_{i})\/.$

###### Proof.

Denote $m_{i}$ and $m^{\prime}_{j}$ to be the sighed Chern-Mather class
polynomials: $m_{i}:=(-1)^{\dim S_{i}}c_{M}^{\bar{S}_{i}}(H)$ and
$m^{\prime}_{j}:=(-1)^{\dim S^{\prime}_{j}}c_{M}^{\bar{S}^{\prime}_{j}}(H)$.
For any $r$ there are unique integers $E_{r,i}$ and $E^{\prime}_{s-r,j}$ such
that $E_{r,r}=(-1)^{\dim S_{r}}$, $E^{\prime}_{s-r,s-r}=(-1)^{\dim
S^{\prime}_{s-r}}$ and

$\displaystyle c_{sm}^{S_{r}}(H)=$
$\displaystyle\sum_{i=r}^{s-1}E_{r,i}\cdot(-1)^{\dim
S_{i}}c_{M}^{\bar{S}_{i}}(H)=\sum_{i=r}^{s-1}E_{r,i}m_{i};$ $\displaystyle
c_{sm}^{S^{\prime}_{s-r}}(H)=$
$\displaystyle\sum_{j=s-r}^{s-1}E^{\prime}_{s-r,j}\cdot(-1)^{\dim
S^{\prime}_{i}}c_{M}^{\bar{S}^{\prime}_{i}}(H)=\sum_{j=s-r}^{s-1}E^{\prime}_{r,i}m^{\prime}_{j}\/.$

Since $\bar{S}_{k}$ is dual to $\bar{S}^{\prime}_{s-k}$, recall from
Proposition 2.3 that the involution $I_{N-1}$ then takes $m_{k}$ to
$m^{\prime}_{s-k}$. Thus the involution equality can be written as

$\displaystyle\sum_{k\geq s-r}\beta^{r}_{k}\cdot
c_{sm}^{S_{k}}(H)=\sum_{k=s-r}^{s-1}\sum_{j=s-r}^{k}\beta^{r}_{i}E^{\prime}_{j,k}m^{\prime}_{k}$
$\displaystyle=$ $\displaystyle I_{N-1}\left(\sum_{k\geq r}\alpha^{r}_{k}\cdot
c_{sm}^{S_{k}}(H)\right)=I_{N-1}\left(\sum_{k=r}^{s-1}\alpha^{r}_{k}\cdot\sum_{i=k}^{s-1}E_{k,i}m_{i}\right)$
$\displaystyle=$
$\displaystyle\sum_{k=r}^{s-1}\alpha^{r}_{k}\cdot\sum_{i=k}^{s-1}E_{k,i}m^{\prime}_{s-i}=\sum_{k=1}^{s-r}\sum_{i=r}^{s-k}\alpha^{r}_{i}E_{i,s-k}m^{\prime}_{k}$

Since $\alpha^{r}_{r}=\beta^{r}_{s-r}=1$, we have

$m^{\prime}_{s-r}+\sum_{k=s-r+1}^{s}\sum_{i=s-r}^{k}\beta^{r}_{j}E^{\prime}_{j,k}m^{\prime}_{k}=\sum_{k=1}^{s-r-1}\sum_{i=r}^{s-k}\alpha^{r}_{i}E_{i,s-k}m^{\prime}_{k}+m^{\prime}_{s-r}\/.$

The corollary then follows from the same argument used in the proof of Theorem
3.1. ∎

An advantage of reflective group orbits is that, as pointed out in [8][22],
besides the classical approach from intersection theory the Chern-Schwartz-
MacPherson classes of such group orbits can also be obtained from
representation theory. This provides more choices in practice. In this case we
have the following algorithm to compute the local Euler obstructions:

######  Algorithm 1 (Algorithm for Reflective Group Orbits).

The following algorithm computed the local Euler obstructions for Reflective
Group Orbits.

  1. Step 0

For each pairs of orbits $\mathcal{O}_{i}$ and $\mathcal{O}_{j}$, notice that
$Eu_{\bar{\mathcal{O}}_{i}}(\mathcal{O}_{j})=Eu_{\bar{S}_{i}}(S_{j})$ passes
to the projective setting for $j\leq s$.

  2. Step 1

For each projectivized orbit $S_{i}$ we compute its Chern-Schwartz-MacPherson
class $c_{sm}^{S_{i}}(H)$.

  3. Step 2

For each projectivized orbit $S^{\prime}_{i}$ we compute its Chern-Schwartz-
MacPherson class $c_{sm}^{S^{\prime}_{i}}(H)$.

  4. Step 3

For each $r$ we set up the following linear system

$\displaystyle\sum_{k=r}^{s-1}x^{r}_{k}c_{sm}^{S_{k}}(H)-\sum_{k=s-r}^{s-1}y^{r}_{k}c_{sm}^{S^{\prime}_{k}}(H)=0;x^{r}_{r}=y^{r}_{s-r}=1\/.$

This is a linear system since $c_{sm}^{S_{k}}(H)$ and
$c_{sm}^{S^{\prime}_{k}}(H)$ are polynomials in $H$, and the equality gives at
least $s+1$ linear equations concerning the coefficients of powers of $H$. As
proved in the previous Proposition, the solution $\\{x^{r}_{k},y^{r}_{k}\\}$
are the local Euler obstructions of $\bar{\mathcal{O}}_{r}$ and
$\bar{\mathcal{O}}^{\prime}_{s-r}$ at each stratum.

  5. Step 4

The local Euler obstruction of $\bar{\mathcal{O}}_{r}$ at
$\mathcal{O}_{s}=\\{0\\}$ come from the algebraic Brasselet-Lê-Seade type
formula proved in [30].

$Eu_{\bar{\mathcal{O}}_{r}}(0)=\sum_{k\geq r}(-1)^{\dim
S_{k}}c_{sm}^{S_{k}}(-1)\cdot Eu_{\bar{\mathcal{O}}_{r}}(\mathcal{O}_{k})\/.$

Same argument applies to $Eu_{\bar{\mathcal{O}}^{\prime}_{s-r}}(0)$.

######  Remark 3.

More precisely, in [8][22] they proved that the equivariant Chern-Schwartz-
MacPherson classes are characterized by certain Axioms. The equivariant
classes are expressed as polynomials in the weights of the $G$ action, the
axioms then provide constraint equations the Chern polynomials have to satisfy
by studying localizations and specializations/restrictions. The solutions are
certain weight functions from representation theory. In fact, in [8] they
proved that for such orbits the solutions correspond to the $K$-theoretic
stable envelops studied in [18]. Thus this algorithm provides a way to study
the singularities of orbits closures directly from representation theory.

######  Example 2.

We use the following example to illustrate the algorithm. We consider the
symmetric rank stratification $M^{S}_{3}=\cup_{i=0}^{3}\Sigma^{S\circ}_{3,i}$.
From the computation in [29] we have

$\displaystyle c_{sm}^{\tau^{S\circ}_{3,0}}=$ $\displaystyle
3H^{4}+6H^{3}+6H^{2}+3H+1$ $\displaystyle c_{sm}^{\tau^{S\circ}_{3,1}}=$
$\displaystyle 3H^{5}+6H^{4}+10H^{3}+9H^{2}+3H;\quad
c_{sm}^{\tau^{S\circ}_{3,1}}(-1)=-1$ $\displaystyle c_{sm}^{\tau^{S}_{3,2}}=$
$\displaystyle 3H^{5}+6H^{4}+4H^{3};\quad c_{sm}^{\tau^{S}_{3,2}}(-1)=-1$

The orbits $\tau^{S}_{3,1}$ is dual to $\tau^{S}_{3,2}$, with dimension $7$
and $4$ respectively. Then we have

$c_{sm}^{\tau^{S\circ}_{3,1}}(-1-H)+((1+H)^{6}-H^{6})+x\cdot
c_{sm}^{\tau^{S}_{3,2}}(-1-H)+x\cdot((1+H)^{6}-H^{6})=c_{sm}^{\tau^{S}_{3,2}}(H)\/.$

Expand the polynomials we get

$3H^{5}+6H^{4}+4H^{3}+x\cdot(3H^{5}+6H^{4}+10H^{3}+9H^{2}+3H)=3H^{5}+6H^{4}+4H^{3}\/.$

There is a unique solution $x=0$. This shows that
$Eu_{\Sigma^{S}_{3,1}}(\Sigma^{S\circ}_{3,2})=0$. The Brasselet-Lê-Seade
formula then gives

$Eu_{\Sigma^{S}_{3,1}}(0)=Eu_{\Sigma^{S}_{3,1}}(\Sigma^{S\circ}_{3,1})\cdot
c_{sm}^{\tau^{S\circ}_{3,1}}(-1)+Eu_{\Sigma^{S}_{3,1}}(\Sigma^{S\circ}_{3,2})\cdot
c_{sm}^{\tau^{S}_{3,2}}(-1)=(-1)\cdot(-1)+(-1)\cdot 0=1\/.$

##  4\. Application I: Quadratic Hypersurfaces

Let $X_{A}\subset\mathbb{P}(V)$ be a quadratic hypersurface defined by
$x^{t}Ax=0$ for some symmetric $n+1\times n+1$ matrix $A$. Here $\dim V=n+1$.
The hypersurface $X_{A}\subset\mathbb{P}(V^{*})$ is smooth if and only if
$|A|\neq 0$, and the dual variety $X_{A}^{\vee}$ is also a smooth
hypersurface. When $A$ is singular of rank $r\leq n$, the singular locus
$S_{A}$ is the projective subspace $\mathbb{P}(K)=\mathbb{P}^{n-r}$, where
$K=\ker A\subset V$.

When $r=1$, the defining equation of $X_{A}$ is given by $l^{2}$ for some
linear form $l$. Thus the reduced structure of $X_{A}$ is smooth: it is a
hyperplane. When $r=2$, the defining equation of $X_{A}$ is given by
$l_{1}\cdot l_{2}$ for two distinct linear forms $l_{1},l_{2}$. Thus the
hypersurface $X_{A}$ is the union of two hyperplanes intersecting transversely
at $\mathbb{P}^{n-2}$. After taking normal slice this case is the nothing but
the line arrangement $V(xy=0)\subset\mathbb{P}^{2}$. So we will assume $r\geq
3$ in this section.

Let $K^{\perp}\subset V^{*}$ be the space of linear forms that vanish on $K$.
As shown in [11], the dual variety $X_{A}^{\vee}$ is a smooth quadratic
hypersurface in the linear subspace $\mathbb{P}(K^{\perp})=\mathbb{P}^{r-1}$.
The dual variety $S_{A}^{\vee}$ is isomorphic to the linear subspace
$\mathbb{P}^{r-1}$.

######  Remark 4.

In general, let $A$ be a $m\times n$ matrix with $m\geq n$, we may consider
the hypersurface in $\mathbb{P}^{m+n-1}$ of the form $X^{t}AY=0$. This
hypersurface is exactly the hypersurface $X_{B}$, where $B$ is the
$(m+n)\times(m+n)$ symmetric matrix $\begin{bmatrix}0&A\\\
A^{t}&0\end{bmatrix}\/.$

######  Corollary 3.

Let $r=r(A)$ be the rank of $A$. We assume that $r\geq 3$.

  1. (1)

Let $\Sigma_{A}\subset V$ be the affine cone of $X_{A}$. The Euler
characteristic of the complex link space of $\Sigma_{A}$ at $0$ is $-2$.

  2. (2)

The Milnor number of $X_{A}$ at any point of $S_{A}$ equals $(-1)^{n+r}$.

  3. (3)

The Milnor class of $X_{A}$ is given by

$\mathcal{M}(X_{A})=(-1)^{n+r}\sum_{k=0}^{n-r}\left(\sum_{j=0}^{k}\binom{n-r+1}{k-j}(-2)^{j}\right)\cdot
H^{k+r}$

  4. (4)

We denote $X_{A}^{\circ}:=X_{A}\setminus S_{A}$. The local Euler obstruction
of $X_{A}$ then equals:

$Eu_{X_{A}}=\mathds{1}_{X_{A}^{\circ}}+((-1)^{r}+1)\mathds{1}_{S_{A}}\/.$

###### Proof.

The Milnor class of $X_{A}$ can be uniquely written as

(1)
$\mathcal{M}(X_{A})=(-1)^{n-1}\left(\frac{2H((1+H)^{n+1}-H^{n+1})}{1+2H}-c_{sm}^{X_{A}}(H)\right)=\frac{\mu\cdot
c_{sm}^{S_{A}}}{1+2H}\/.$

Let $e=Eu_{X_{A}}(S_{A})$ be the local Euler obstruction of $X_{A}$ at
$S_{A}$, and we can uniquely write

$c_{sm}^{X_{A}}=c_{M}^{X_{A}}-(e-1)c_{M}^{S_{A}}=c_{M}^{X_{A}}-(e-1)c_{sm}^{S_{A}}\/.$

We denote $\alpha:=e-1$, $\lambda:=c_{sm}^{X_{A}}(-1)$ and
$B_{n}:=(1+H)^{n+1}-H^{n+1}$. Then we consider the following polynomial:

$P:=(-1)^{n-1}\left(2HB_{n}-(1+2H)(c_{M}^{X_{A}}-\alpha\cdot
c_{M}^{S_{A}})\right)-\mu c_{M}^{S_{A}}\/.$

From (1) we know that $P=0$. Let $\mathcal{I}_{n}$ be the duality involution,
then we have $\mathcal{I}_{n}(P)=0$.

On the other hand, we can directly compute $I_{n}(P)$. From Proposition 2.5 we
have:

$\mathcal{I}_{n}\left(2H(c_{M}^{X_{A}}-\alpha\cdot
c_{M}^{S_{A}})\right)=(-2-2H)\mathcal{I}_{n}(c_{M}^{X_{A}}-\alpha\cdot
c_{M}^{S_{A}})-\lambda HB_{n}\/.$

Meanwhile, by a direct computation we have
$\mathcal{I}_{n}(HB_{n})=(-1)^{n+1}HB_{n}\/.$ Thus we have the following

$\displaystyle\mathcal{I}_{n}(P)=$
$\displaystyle(-1)^{n-1}\mathcal{I}_{n}(2HB_{n})-\mathcal{I}_{n}(c_{M}^{X_{A}}-\alpha
c_{M}^{S_{A}})-\mathcal{I}_{n}(2H(c_{M}^{X_{A}}-\alpha
c_{M}^{S_{A}}))-\mathcal{I}_{n}(\mu c_{M}^{S_{A}})$ $\displaystyle=$
$\displaystyle
2HB_{n}-(-1)^{n-1}\left((-1)^{n+r-1}c_{M}^{X^{\vee}_{A}}-(-1)^{n-1}\alpha
c_{M}^{S^{\vee}_{A}}\right)+(-1)^{n-1}\lambda HB_{n}$ $\displaystyle+$
$\displaystyle(-1)^{n-1}(2+2H)\left((-1)^{n+r-1}c_{M}^{X^{\vee}_{A}}-(-1)^{n-1}\alpha
c_{M}^{S^{\vee}_{A}}\right)-(-1)^{n-1}\mu c_{M}^{S^{\vee}_{A}}$
$\displaystyle=$
$\displaystyle(2+(-1)^{n-1}\lambda)HB_{n}+(-1)^{r}(1+2H)c_{M}^{X^{\vee}_{A}}-(1+2H)\alpha
c_{M}^{S^{\vee}_{A}}-(-1)^{n-1}\mu c_{M}^{S^{\vee}_{A}}\ =$ $\displaystyle 0$

Notice that $X_{A}^{\vee}$ is a smooth quadratic hypersurface in
$\mathbb{P}^{r-1}$, and $S_{A}^{\vee}$ is isomorphic to $\mathbb{P}^{r-1}$.
Thus we have

$c_{M}(X_{A}^{\vee})=c_{sm}^{X_{A}^{\vee}}=\frac{2H^{n-r+2}(1+H)^{r}}{1+2H};\quad
c_{M}(S_{A}^{\vee})=c_{sm}^{S_{A}^{\vee}}=H^{n-r+1}(1+H)^{r}\/.$

Thus we have

$\displaystyle 0=$
$\displaystyle(2+(-1)^{n-1}\lambda)H\left((1+H)^{n+1}-H^{n+1}\right)-(-1)^{n-1}\mu
H^{n-r+1}(1+H)^{r}$ $\displaystyle+(-1)^{r}2H^{n-r+2}(1+H)^{r}-(1+2H)\alpha
H^{n-r+1}(1+H)^{r}\/.$

This gives the solution

$\lambda=(-1)^{n}\cdot 2;\quad e=\alpha+1=(-1)^{r}+1;\quad\mu=(-1)^{n+r}\/.$

∎

##  5\. Application II: Generic Determinantal Varieties

The (generic) determinantal varieties are fundamental examples of non-isolated
singularities, and are of great interest in singularity theory. For ordinary
determinantal varieties the local Euler obtructions were computed in [10]
using topological method, and in [28] for arbitrary algebraically closed field
using intersection theory. In this section we use Corollary 2 to give a very
short proof, from an observation on the Chern-Schwartz-MacPherson class
formula.

####  5.0.1. $q$-Polynomial of Determinantal Varieties

Let $K$ be a characteristic $0$ algebraically closed field. Let
$M_{m,n}=Hom(K^{n},K^{n})$ be the space of $n\times n$ ordinary matrices . The
group $GL_{n}(K)\times GL_{n}(K)$ acts on $M_{n,n}$ by $(P,X,Q)\mapsto
PXQ^{-1}$. The orbits consist of matrices of fixed rank. For $0\leq k\leq n-1$
we denote them by

$\Sigma^{\circ}_{n,k}:=\\{X|\dim\ker X=k\\};\quad\Sigma_{n,k}:=\\{X|\dim\ker
X\geq k\\}\/.$

They are algebraic varieties of dimension $n^{2}-k^{2}$. We denote the
projections $\mathbb{P}(\Sigma^{\circ}_{n,k})$ and $\mathbb{P}(\Sigma_{n,k})$
by $\tau^{\circ}_{n,k}$ and $\tau_{n,k}$ respectively.

As shown in [29], we define the $Q$ classes in the Grassmannian $G(r,n)$:

$q_{n,r}(d):=\left(\sum_{k=0}^{n(n-r)}(1+d)^{n(n-r)-k}c_{k}(Q^{\vee
n})\right)\left(\sum_{k=0}^{nr}d^{nr-k}c_{k}(S^{\vee n})\right)\/;$

and the $q$ polynomial in variable $q_{n,r}(H)$ by setting

$q_{n,r}(d):=\int_{G(r,n)}c(S^{\vee}\otimes Q)\cdot
q_{n,r}(d)\cap[G(r,n)]-d^{n^{2}}\binom{n}{r}$

for integers $d$. The Chern-Schwartz-MacPherson classes
$c_{sm}^{\tau^{\circ}_{n,k}}$ are then given by

$c_{sm}^{\tau_{n,k}^{\circ}}(H)=\sum_{r=k}^{n-1}(-1)^{r-k}\binom{r}{k}\cdot
q_{n,r}(H)\/.$

####  5.0.2. Local Euler Obstruction of Determinantal Varieties

Despite the complicated form, as we will show later, the $q_{n,r}(d)$, enjoys
very nice symmetric property: it is symmetric with the $d\mapsto-1-d$
substitution. This induces the following interesting application:

######  Corollary 4.

The local Euler obstruction function of determinantal varieties are given by

$Eu_{\Sigma_{n,k}}(\Sigma^{\circ}_{n,r})=\binom{r}{k}\/.$

Thus the polynomials $q_{n,r}(H)$ equal the Chern-Mather classes:
$q_{n,r}(H)=c_{M}^{\tau_{n,k}}(H)\/.$

###### Proof.

Substituting $d$ by $-1-d$ in the expression of $q_{n,r}(d)$ we have the
following:

$\displaystyle q_{n,r}(-1-d)+(-1-d)^{n^{2}}\binom{n}{r}$
$\displaystyle=\int_{G(r,n)}c(S^{\vee}\otimes
Q)\left(\sum_{k=0}^{n(n-r)}(-d)^{n(n-r)-k}c_{k}(Q^{\vee
n})\right)\left(\sum_{k=0}^{nr}(-1-d)^{nr-k}c_{k}(S^{\vee n})\right)$
$\displaystyle=\int_{G(n-r,n)}c(S^{\vee}\otimes
Q)\left(\sum_{k=0}^{n(n-r)}(-d)^{n(n-r)-k}c_{k}(S^{n})\right)\left(\sum_{k=0}^{nr}(-1-d)^{nr-k}c_{k}(Q^{n})\right)$
$\displaystyle=(-1)^{n^{2}}\int_{G(n-r,n)}c(S^{\vee}\otimes
Q)\left(\sum_{k=0}^{n(n-r)}d^{n(n-r)-k}c_{k}(S^{\vee
n})\right)\left(\sum_{k=0}^{nr}(1+d)^{nr-k}c_{k}(Q^{\vee n})\right)$
$\displaystyle=$
$\displaystyle(-1)^{n^{2}}q_{n,n-r}(d)+(-1)^{n^{2}}d^{n^{2}}\binom{n}{n-r}$

Recall that the dimension of $\tau_{n,r}$ is $n^{2}-r^{2}-1$. Since above
computation holds for any integer $d$, thus as polynomial we have

$\hat{q}_{n,r}(-1-H)-\hat{q}_{n,r}(-1)\left((1+H)^{n^{2}}-H^{n^{2}}\right)=\hat{q}_{n,n-r}(H)\/.$

Here $\hat{q}_{n,r}(H)=(-1)^{\dim\tau_{n,r}}\cdot q_{n,r}(H)$ is the signed
polynomial. Recall from §2.3 that the projective duality operation
$I:=I_{n^{2}-1}$ takes the signed Chern-Mather class of $\tau_{n,r}$ to the
signed Chern-Mather class of $\tau_{n,n-r}$: the dual variety of $\tau_{n,k}$
is exactly $\tau_{n,n-k}$. Let $c_{k}:=(-1)^{\dim\tau_{n,k}}\cdot
c_{M}^{\tau_{n,k}}$ be the Chern-Mather polynomials, this says that
$I(c_{i})=c_{n-i}$ for $i=1,\cdots,n-1$.

Proved in [29], the $q$ polynomial $q_{n,r}(H)$ equals $c_{*}(\phi_{n,r})$ for
some constructible function $\phi_{n,r}\in F(\tau_{n,r})$, defined by
$\phi_{n,r}(p)=\binom{r}{l}$ for $p\in\tau^{\circ}_{n,l}$. $($In fact,
$\phi_{n,r}$ is the pushforward $p_{*}(\mathds{1}_{\hat{\tau}_{n,r}})$, where
$p\colon\hat{\tau}_{n,r}\to\tau_{n,r}$ is the Tjurina transform.$)$ Since the
local Euler obstructions is a basis for $F(\tau_{n,r})$, there are unique
integers $\beta^{r}_{k}$ such that
$q_{n,r}(H)=\sum_{k=r}^{n-1}\beta^{r}_{k}c_{M}^{\tau_{n,k}}(H)$,
$\beta_{r}^{r}=1$. Then the involution equality shows that

$I(\hat{q}_{n,r}(H))=I(\sum_{k=r}^{n-1}\beta^{r}_{k}c_{k})=\sum_{k=r}^{n-1}\beta^{r}_{k}c_{n-k}=\hat{q}_{n,n-r}(H)=\sum_{k=n-r}^{n-1}\beta^{r}_{k}c_{k}\/.$

Since the lowest degree terms all the polynomials $c_{i}$ have different
degrees, expand the above equalities for $r=1,2,\cdots,n-1$ one can see that
there is a unique solution

$\displaystyle\beta_{B}^{A}=$ $\displaystyle\begin{cases}1&A=B\\\ 0&A\neq
B\end{cases}$

This shows that $\phi_{n,r}=Eu_{\tau_{n,r}}$, and thus
$q_{n,r}(H)=c_{M}^{\tau_{n,r}}$ is the Chern-Mather polynomial. ∎

######  Remark 5.

However we fail to obtain such an easy proof for the Chern-Mather classes of
symmetric and skew-symmetric determinantal varieties. This is due to the term
$\left(\sum_{k=0}^{e}d^{e-k}c_{k}(S^{\vee}\otimes Q^{\vee})\right)$, when
substituting $d$ by $-1-d$ we lose the symmetry. The fact that the involution
still interchanges the signed Chern-Mather classes indicates that there should
be more symmetry and vanishing patterns hide behind the integrations. It would
be interesting to use Schubert calculus to give a similar proof to the
symmetric and skew-symmetric determinantal varieties.

## References

  * [1] Paolo Aluffi.  Projective duality and a Chern-Mather involution.  Trans. Amer. Math. Soc., 370(3):1803–1822, 2018.
  * [2] J. P. Brasselet.  Local Euler obstruction, old and new.  In XI Brazilian Topology Meeting (Rio Claro, 1998), pages 140–147. World Sci. Publ., River Edge, NJ, 2000.
  * [3] J.-P. Brasselet and M.-H. Schwartz.  Sur les classes de Chern d’un ensemble analytique complexe.  In The Euler-Poincaré characteristic (French), volume 82 of Astérisque, pages 93–147. Soc. Math. France, Paris, 1981\.
  * [4] J.-P. Brasselet, Lê D ung Tráng, and J. Seade.  Euler obstruction and indices of vector fields.  Topology, 39(6):1193–1208, 2000.
  * [5] Jean-Paul Brasselet and Nivaldo G. Grulha, Jr.  Local Euler obstruction, old and new, II.  In Real and complex singularities, volume 380 of London Math. Soc. Lecture Note Ser., pages 23–45. Cambridge Univ. Press, Cambridge, 2010.
  * [6] P. Bressler, M. Finkelberg, and V. Lunts.  Vanishing cycles on Grassmannians.  Duke Math. J., 61(3):763–777, 1990.
  * [7] Jean-Luc Brylinski, Alberto S. Dubson, and Masaki Kashiwara.  Formule de l’indice pour modules holonomes et obstruction d’Euler locale.  C. R. Acad. Sci. Paris Sér. I Math., 293(12):573–576, 1981\.
  * [8] Laszlo M. Feher, Richard Rimanyi, and Andrzej Weber.  Motivic Chern classes and K-theoretic stable envelopes.  arXiv e-prints, page arXiv:1802.01503, February 2018.
  * [9] William Fulton.  Intersection theory.  Springer-Verlag, Berlin, 1984.
  * [10] Terence Gaffney, Nivaldo G. Grulha, Jr., and Maria A. S. Ruas.  The local Euler obstruction and topology of the stabilization of associated determinantal varieties.  Math. Z., 291(3-4):905–930, 2019.
  * [11] I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky.  Discriminants, resultants and multidimensional determinants.  Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, 2008.  Reprint of the 1994 edition.
  * [12] Gerardo González-Sprinberg.  L’obstruction locale d’Euler et le théorème de MacPherson.  In The Euler-Poincaré characteristic (French), volume 83 of Astérisque, pages 7–32. Soc. Math. France, Paris, 1981\.
  * [13] Jose Israel Rodriguez and Botong Wang.  Computing Euler obstruction functions using maximum likelihood degrees.  arXiv e-prints, page arXiv:1710.04310, October 2017.
  * [14] Masaki Kashiwara.  Index theorem for a maximally overdetermined system of linear differential equations.  Proc. Japan Acad., 49:803–804, 1973.
  * [15] David Kazhdan and George Lusztig.  Schubert varieties and Poincaré duality.  In Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), Proc. Sympos. Pure Math., XXXVI, pages 185–203. Amer. Math. Soc., Providence, R.I., 1980.
  * [16] Gary Kennedy.  MacPherson’s Chern classes of singular algebraic varieties.  Comm. Algebra, 18(9):2821–2839, 1990.
  * [17] R. D. MacPherson.  Chern classes for singular algebraic varieties.  Ann. of Math. (2), 100:423–432, 1974.
  * [18] Davesh Maulik and Andrei Okounkov.  Quantum groups and quantum cohomology.  Astérisque, (408):ix+209, 2019.
  * [19] Leonardo C. Mihalcea and Rahul Singh.  Mather classes and conormal spaces of Schubert varieties in cominuscule spaces.  arXiv e-prints, page arXiv:2006.04842, June 2020.
  * [20] Bernt Ivar Utstø l Nø dland.  Local Euler obstructions of toric varieties.  J. Pure Appl. Algebra, 222(3):508–533, 2018.
  * [21] Ragni Piene.  Polar varieties revisited.  In Computer algebra and polynomials, volume 8942 of Lecture Notes in Comput. Sci., pages 139–150. Springer, Cham, 2015.
  * [22] Richárd Rimányi and Alexander Varchenko.  Equivariant Chern-Schwartz-MacPherson classes in partial flag varieties: interpolation and formulae.  In Schubert varieties, equivariant cohomology and characteristic classes—IMPANGA 15, EMS Ser. Congr. Rep., pages 225–235. Eur. Math. Soc., Zürich, 2018.
  * [23] C. Sabbah.  Quelques remarques sur la géométrie des espaces conormaux.  Number 130, pages 161–192. 1985.  Differential systems and singularities (Luminy, 1983).
  * [24] Marie-Hélène Schwartz.  Classes caractéristiques définies par une stratification d’une variété analytique complexe. I.  C. R. Acad. Sci. Paris, 260:3262–3264, 1965.
  * [25] Marie-Hélène Schwartz.  Classes caractéristiques définies par une stratification d’une variété analytique complexe. II.  C. R. Acad. Sci. Paris, 260:3535–3537, 1965.
  * [26] E. A. Tevelev.  Projective duality and homogeneous spaces, volume 133 of Encyclopaedia of Mathematical Sciences.  Springer-Verlag, Berlin, 2005.  Invariant Theory and Algebraic Transformation Groups, IV.
  * [27] Lê D ung Tráng and Bernard Teissier.  Variétés polaires locales et classes de Chern des variétés singulières.  Ann. of Math. (2), 114(3):457–491, 1981.
  * [28] Xiping Zhang.  Local Euler Obstruction and Chern-Mather classes of Determinantal Varieties.  arXiv e-prints, page arXiv:1706.02032, June 2017.
  * [29] Xiping Zhang.  Characteristic Classes of Homogeneous Essential Isolated Determinantal Varieties.  arXiv e-prints, page arXiv:2011.12578, November 2020.
  * [30] Xiping Zhang.  Local Geometric Invariants of Recursive Group Orbit Stratification.  arXiv e-prints, page arXiv:2009.09362, September 2020.