Local Euler Obstruction Research Articles

Euler obstructions for the Lagrangian Grassmannian

We prove a case of a positivity conjecture of Mihalcea-Singh, concerned with the local Euler obstructions associated to the Schubert stratification of the Lagrangian Grassmannian LG(n,2n). Combined with work of Aluffi-Mihalcea-Sch\urmann-Su, this further implies the positivity of the Mather classes for Schubert varieties in LG(n,2n), which Mihalcea-Singh had verified for the other cominuscule spaces of classical Lie type. Building on the work of Boe and Fu, we give a positive recursion for the local Euler obstructions, and use it to show that they provide a positive count of admissible labelings of certain trees, analogous to the ones describing Kazhdan-Lusztig polynomials. Unlike in the case of the Grassmannians in types A and D, for LG(n,2n) the Euler obstructions e_{y,w} may vanish for certain pairs (y,w) with y <= w in the Bruhat order. Our combinatorial description allows us to classify all the pairs (y,w) for which e_{y,w}=0. Restricting to the big opposite cell in LG(n,2n), which is naturally identified with the space of n x n symmetric matrices, we recover the formulas for the local Euler obstructions associated with the matrix rank stratification.

The geometrical information encoded by the Euler obstruction of a map

In this work, we investigate the topological information captured by the Euler obstruction of a map-germ, [Formula: see text], where (X, 0) denotes a germ of a complex d-equidimensional singular space, with d &gt; 2, and its relation with the local Euler obstruction of the coordinate functions and consequently, with the Brasselet number. Moreover, under some technical conditions on the domain, we relate the Chern number of a special collection related to the map-germ f at the origin with the number of cusps of a generic perturbation of f on a stabilization of (X, f).

Local Euler obstruction, old and new, III

Local Euler obstruction, old and new, III

Local Euler obstructions for determinantal varieties

The goal of this note is to explain a derivation of the formulas for the local Euler obstructions of determinantal varieties of general, symmetric and skew-symmetric matrices, by studying the invariant de Rham complex and using character formulas for simple equivariant D-modules. These calculations are then combined with standard arguments involving Kashiwara's local index formula and the description of characteristic cycles of simple equivariant D-modules. The formulas are implicit in the work of Boe and Fu, and in the case of general matrices they have also been obtained recently by Gaffney–Grulha–Ruas, for skew-symmetric matrices by Promtapan and Rimányi, and for all cases by Zhang.

Local Euler obstructions and Chern–Mather classes of determinantal varieties

The local Euler obstructions are key ingredients in the study of both singularity theory and geometric representation theory. In this note, we consider matrix rank stratification over algebraically closed field of arbitrary characteristic. We compute the local Euler obstructions using a direct intersection-theoretic approach. The algebraic formula we prove are based on explicitly calculating the integrations of certain Chern classes of the universal bundles over the Grassmannians, and we proceed such calculations using formal Chern roots argument. Our formula generalizes the result over computed in [8] via topological methods. Based on our integration formula we propose an explanation to the Pascal triangle pattern observed by the authors in [8]. Then, we generalize the formula of the Chern–Mather class of in [18, theorem 10] to arbitrary algebraically closed base field. In particular, when k = 1 and we explicitly compute the Chern–Mather classes and prove some interesting symmetric patterns on the coefficients.

Euclidean Distance Degree of Projective Varieties

Abstract We give a positive answer to a conjecture of Aluffi–Harris on the computation of the Euclidean distance degree of a possibly singular projective variety in terms of the local Euler obstruction function.

Alternative algorithms for computing generic μ∗-sequences and local Euler obstructions of isolated hypersurface singularities

A new algorithm is introduced for computing [Formula: see text]-sequences of isolated hypersurface singularities. It is shown that the new algorithm results in better performance, compared to our previous algorithm that utilizes parametric local cohomology systems, in computation speed. Furthermore, it can be used to compute local Euler obstruction of a hypersurface with an isolated singularity. The key idea of the new algorithm is computing standard bases in a local ring over a field of rational functions.

Note on MacPherson’s Local Euler Obstruction

This is a note on MacPherson’s local Euler obstruction, which plays an important role recently in the Donaldson–Thomas theory by the work of Behrend. We introduce MacPherson’s original definition and prove that it is equivalent to the algebraic definition used by Behrend, following the method of González-Sprinberg. We also give a formula of the local Euler obstruction in terms of Lagrangian intersections. As an application, we consider a scheme or DM stack X admitting a symmetric obstruction theory. Furthermore, we assume that there is a C∗ action on X that makes the obstruction theory C∗-equivariant. The C∗-action on the obstruction theory naturally gives rise to a cosection map in the Kiem–Li sense. We prove that Behrend’s weighted Euler characteristic of X is the same as the Kiem–Li localized invariant of X by the C∗-action.

The local Euler obstruction and topology of the stabilization of associated determinantal varieties

This work has two complementary parts, in the first part we compute the local Euler obstruction of generic determinantal varieties and apply this result to compute the Chern–Schwartz–MacPherson class of such varieties. In the second part we compute the Euler characteristic of the stabilization of an essentially isolated determinantal singularity (EIDS). The formula is given in terms of the local Euler obstruction and Gaffney’s $$m_{d}$$ multiplicity.

An Implementation of the Lê–Teissier Method for Computing Local Euler Obstructions

Local Euler obstruction of a hypersurface with possibly positive dimensional singularities, a constructible function on the hypersurface, is considered in the context of symbolic computation. It is shown that the method due to Le and Teissier (Ann Math 114:457–491, 1981) that requires genericity conditions for computing local Euler obstructions can be realized as an algorithm in a deterministic way. The key ideas of the proposed algorithm are the use of comprehensive Grobner systems and of parametric local cohomology systems.

Projective duality and a Chern-Mather involution

We observe that linear relations among Chern-Mather classes of projective varieties are preserved by projective duality. We deduce the existence of an explicit involution on a part of the Chow group of projective space, encoding the effect of duality on Chern-Mather classes. Applications include Plücker formulae, constraints on self-dual varieties, generalizations to singular varieties of classical formulas for the degree of the dual and the dual defect, formulas for the Euclidean distance degree, and computations of Chern-Mather classes and local Euler obstructions for cones.

Chern classes and characteristic cycles of determinantal varieties

Let K be an algebraically closed field of characteristic 0. For m≥n, we define τm,n,k to be the set of m×n matrices over K with kernel dimension ≥k. This is a projective subvariety of Pmn−1, and is called the (generic) determinantal variety. In most cases τm,n,k is singular with singular locus τm,n,k+1. In this paper we give explicit formulas computing the Chern–Mather class (cM) and the Chern–Schwartz–MacPherson class (cSM) of τm,n,k, as classes in the projective space. We also obtain formulas for the conormal cycles and the characteristic cycles of these varieties, and for their generic Euclidean Distance degree. Further, when K=C, we prove that the characteristic cycle of the intersection cohomology sheaf of a determinantal variety agrees with its conormal cycle (and hence is irreducible).Our formulas are based on calculations of degrees of certain Chern classes of the universal bundles over the Grassmannian. For some small values of m,n,k, we use Macaulay2 to exhibit examples of the Chern–Mather classes, the Chern–Schwartz–MacPherson classes and the classes of characteristic cycles of τm,n,k.On the basis of explicit computations in low dimensions, we formulate conjectures concerning the effectivity of the classes and the vanishing of specific terms in the Chern–Schwartz–MacPherson classes of the largest strata τm,n,k∖τm,n,k+1.The irreducibility of the characteristic cycle of the intersection cohomology sheaf follows from the Kashiwara–Dubson's microlocal index theorem, a study of the ‘Tjurina transform’ of τm,n,k, and the recent computation of the local Euler obstruction of τm,n,k.

Local Euler obstructions of toric varieties

We use Matsui and Takeuchi's formula for toric A-discriminants to give algorithms for computing local Euler obstructions and dual degrees of toric surfaces and 3-folds. In particular, we consider weighted projective spaces. As an application we give counterexamples to a conjecture by Matsui and Takeuchi. As another application we recover the well-known fact that the only defective normal toric surfaces are cones.

An Equivariant Version of the Euler Obstruction

For a complex analytic variety with an action of a finite group and for an invariant 1-form on it, we give an equivariant version (with values in the Burnside ring of the group) of the local Euler obstruction of the 1-form and describe its relation with the equivariant radial index defined earlier. This leads to equivariant versions of the local Euler obstruction of a complex analytic space and of the global Euler obstruction.

Minimal Whitney stratification and Euler obstruction of discriminants

In this work we show how to obtain the minimal Whitney stratification of the discriminant of finitely determined map germs from \((\mathbb {C}^m,0)\) to \((\mathbb {C}^p,0)\), of corank one if \(n<p\), and only with \(A_k\) singularities, when \(m = n+p\) with \( n \ge 0\). We apply the theory developed by Gaffney which shows how to compute a Whitney stratification of discriminants of any finitely determined holomorphic map germ in the nice dimensions of Mather, or in its boundary. For the pairs cited above we show that both stratifications coincide. We also compute the local Euler obstruction at 0 in a class of discriminants of finitely determined map germs from \(\mathbb {C}^{n+p}\) to \(\mathbb {C}^p\) with \(n\ge 0\) and only with \(A_k\) singularities.

The Euler obstruction of a function on a determinantal variety and on a curve

Given an analytic function germ f: (X, 0) → ℂ on an isolated determinantal singularity or on a reduced curve, we present formulas relating the local Euler obstruction of f to the vanishing Euler characteristic of the fiber X ∩ f-1(0) and to the Milnor number of f. Restricting ourselves to the case where X is a complete intersection, we obtain an easy way to calculate the local Euler obstruction of f as the difference between the dimension of two algebras.

Unfolding plane curves with cusps and nodes

Given an irreducible surface germ (X, 0) ⊂ (ℂ3, 0) with a one-dimensional singular set Σ, we denote by δ1 (X, 0) the delta invariant of a transverse slice. We show that δ1 (X, 0) ≥ m0 (Σ, 0), with equality if and only if (X, 0) admits a corank 1 parametrization f :(ℂ2, 0) → (ℂ3, 0) whose only singularities outside the origin are transverse double points and semi-cubic cuspidal edges. We then use the local Euler obstruction Eu(X, 0) in order to characterize those surfaces that have finite codimension with respect to -equivalence or as a frontal-type singularity.

Index formula for MacPherson cycles of affine algebraic varieties

We give explicit MacPherson cycles for the Chern-MacPherson class of a closed affine algebraic variety $X$ and for any constructible function $\alpha$ with respect to a complex algebraic Whitney stratification of $X$. We define generalized degrees of the global polar varieties and of the MacPherson cycles and we prove a global index formula for the Euler characteristic of $\alpha$. Whenever $\alpha$ is the Euler obstruction of $X$, this index formula specializes to the Seade-Tibar-Verjovsky global counterpart of the Le-Teissier formula for the local Euler obstruction.

Singular Chern Classes of Schubert Varieties via Small Resolution

We discuss a method for calculating the Chern-Schwartz-MacPherson (CSM) class of a Schubert variety in the Grassmannian using small resolutions introduced by Zelevinsky. As a consequence, we show how to compute the Chern-Mather class and local Euler obstructions using small resolutions instead of the Nash blowup. The algorithm obtained for CSM classes also allows us to prove new cases of a positivity conjecture of Aluffi and Mihalcea.

L’obstruction d’Euler locale d’une application

Our objective is to present a generalization for the local Euler obstruction of a holomorphic function singular at the origin to the case of a holomorphic map f:(V,0)→(ℂ k ,0), where (V, 0) is a germ of complex analytic variety, equidimensional of dimension n≥k. The principal result (Theorem 6.1) is a formula which computes the local Euler obstruction, defined for k-frames by Brasselet, Seade, Suwa, in terms of the local Euler obstruction of f.