Related papers: Poincar\'e-Hopf Theorem for Isolated Determinantal…

Let $M$ be a non-compact connected manifold with a cocompact and properly discontinuous action of a discrete group $G$. We establish a Poincar\'{e}-Hopf theorem for a bounded vector field on $M$ satisfying a mild condition on zeros. As an…

In this paper, we use Conley index theory to examine the Poincare index of an isolated invariant set. We obtain some limiting conditions on a critical point of a planar vector field to be an isolated invariant set. As a result we show the…

If (V,0) is an isolated complete intersection singularity and X a holomorphic vector field tangent to V one can define an index of X, the so called GSV index, which generalizes the Poincare-Hopf index. We prove that the GSV index coincides…

We define two types of local indices of a vector field at an isolated zero on the boundary, and prove Poincare-Hopf-type index theorems for certain vector fields on a compact smooth manifold which have only isolated zeros.

The goal of this work is to generalize the Gauss-Bonnet and Poincar\'{e}-Hopf Theorems to the case of orbifolds with boundary. We present two such generalizations, the first in the spirit of Satake. In this case, the local data (i.e.…

Braid Floer homology is an invariant of proper relative braid classes. Closed integral curves of 1-periodic Hamiltonian vector fields on the 2-disc may be regarded as braids. If the Braid Floer homology of associated proper relative braid…

We propose singular variants of the Singer-Hopf conjecture, formulated in terms of the Euler-Mather characteristic, intersection homology Euler characteristic and, resp., virtual Euler characteristic of a closed irreducible subvariety of an…

Let $X\subseteq \mathbb{P}^N$ be a non-degenerate normal projective variety of codimension $e$ and degree $d$ with isolated $\mathbb{Q}$-Gorenstein singularities. We prove that the Castelnuovo-Mumford regularity…

The main objects of this paper include some degenerate and nonlocal elliptic operators which naturally arise in the conformal invariant theory of Poincar\'e-Einstein manifolds. These operators generally reflect the correspondence between…

Let $D$ be a smooth divisor on a closed K\"ahler manifold $X$. Suppose that $Aut_0(D)=\{Id\}$. We prove that the Poincar\'e type extremal K\"ahler metric with a cusp singularity at $D$ is unique up to a holomorphic transformation on $X$…

A degenerate dynamical system is characterized by a state-dependent multiplier of the time derivative of the state in the time evolution equation. It can give rise to Hamiltonian systems whose symplectic structure possesses a non-constant…

This article is about 1-forms on complex analytic varieties and it is particularly relevant when the variety has non-isolated singularities. We first show how the radial extension technique of M.-H. Schwartz can be adapted to 1-forms,…

We extend the circle of ideas from a previous paper on hypersurfaces to functions $f \colon (\mathbb C^n, 0) \to (\mathbb C^k, 0)$ with an isolated singularity in a stratified sense on an arbitrary, but fixed complex analytic germ $(X, 0)$.…

We develop versions of the Poincar\'e-Birkhoff-Witt and Cartier-Milnor-Moore theorems in the setting of braided Hopf algebras. To do so, we introduce new analogues of a Lie algebra in the setting of a braided monoidal category, using the…

We study codimension two determinantal varieties with isolated singularities. These singularities admit a unique smoothing, thus we can define their Milnor number as the middle Betti number of their generic fiber. For surfaces in C^4, we…

Equivariant versions of the radial index and of the GSV-index of a vector field or a 1-form on a singular variety with an action of a finite group are defined. They have values in the Burnside ring of the group. Poincar\'e-Hopf type…

In this paper, We develop the stratified de Rham theory on singular spaces using modern tools including derived geometry and stratified structures. This work unifies and extends the de Rham theory, Hodge theory, and deformation theory of…

Motivated by the Pierce-Birkhoff conjecture, we launch an extension program for single variable expansivity theory. We study this notion under tuples of polynomials in the ring $\mathbb{R}[x_1,x_2,\ldots,x_n]$. As an application, we develop…

Let X be a normal variety such that $K_X$ is Q-Cartier, and let $f: X \rightarrow X$ be a finite surjective morphism of degree at least two. We establish a close relation between the irreducible components of the locus of singularities that…

It is shown that if a finite generically smooth morphism $f\,:\,Y\,\longrightarrow\, X$ of smooth projective varieties induces an isomorphism of the \'etale fundamental groups, then the induced map of the stratified fundamental groups…