Related papers: Geometric Invariants of Recursive Group Orbit Stra…

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…

Let $k$ be a field, let $G$ be a reductive algebraic group over $k$, and let $V$ be a linear representation of $G$. Geometric invariant theory involves the study of the $k$-algebra of $G$-invariant polynomials on $V$, and the relation…

Equivariant localization expresses global invariants in terms of local invariants, and many of them appearing in equivariant index theory, (holomorphic) Morse theory, geometric quantization and supersymmetric localization can be…

The purpose of this paper is to study finite dimensional equivariant moduli problems from the viewpoint of stratification theory. We show that there exists a stratified obstruction system for a finite dimensional equivariant moduli problem.…

For $m\geq n$, Let $K$ be an algebraic closed base field, and define $\tau_{m,n,k}$ to be the set of $m\times n$ matrices over $K$ with kernel dimension $\geq k$. This is a projective subvariety of $\mathbb{P}^{mn-1}$, and is usually called…

Integrals of the Pfaffian form over the nonsingular part of a projective variety compute information closely related to the Mather-Chern class of the variety and to other invariants such as the local Euler obstruction along strata of its…

Several authors have proved Lefschetz type formulae for the local Euler obstruction. In particular, a result of this type is proved in [BLS].The formula proved in that paper turns out to be equivalent to saying that the local Euler…

We unify problems about the equivariant geometry of symmetric quiver representation varieties, in the finite type setting, with the corresponding problems for symmetric varieties $GL(n)/K$ where $K$ is an orthogonal or symplectic group. In…

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…

This paper announces results on the behavior of some important algebraic and topological invariants --- Euler characteristic, arithmetic genus, and their intersection homology analogues; the signature, etc. --- and their associated…

We introduce notions of finiteness obstruction, Euler characteristic, L^2-Euler characteristic, and M\"obius inversion for wide classes of categories. The finiteness obstruction of a category Gamma of type (FP) is a class in the projective…

Symplectic invariants introduced in math-ph/0702045 can be computed for an arbitrary spectral curve. For some examples of spectral curves, those invariants can solve loop equations of matrix integrals, and many problems of enumerative…

Consider a finite-dimensional real vector space equipped with a finite group acting unitarily on it. We address the general problem of constructing Euclidean stable embeddings of the quotient space of orbits. Our approach is based on…

Here we review background in differential topology related to the calculation of an euler characteristic, and background on localization in equivariant cohomology. We then outline Gromov-Witten invariants in algebraic geometry and give…

This work deals with relations between a bounded cohomological invariant and the geometry of Hermitian symmetric spaces of noncompact type. The invariant, obtained from the K\"ahler class, is used to define and characterize a special class…

Let $G = GL(n)$ and $K = GL(p) \times GL(q)$ with $p+q=n$, where the groups are taken over $\C$. In this paper we study a certain family of $K$-orbit closures on the flag variety $X$ of $G$. The geometry of these orbit closures plays a…

Left invariant metrics induced by the p-norms of the trace in the matrix algebra are studied on the general lineal group. By means of the Euler-Lagrange equations, existence and uniqueness of extremal paths for the length functional are…

We study character varieties arising as moduli of representations of an orientable surface group into a reductive group $G$. We first show that if $G/Z$ acts freely on the representation variety, then both the representation variety and the…

We approach the well-studied problem of supervised group invariant and equivariant machine learning from the point of view of geometric topology. We propose a novel approach using a pre-processing step, which involves projecting the input…

Consider a smooth projective curve and a given embedding into projective space via a sufficiently positive line bundle. We can form the secant variety of $k$-planes through the curve. These are singular varieties, with each secant variety…