L2 d-bar cohomology groups of some singular complex spaces

TL;DR

This paper investigates the L2-d-bar cohomology groups of certain singular complex spaces near isolated singularities, relating them to resolution data and providing new insights into their structure.

Contribution

It expresses L2-(0,q)-d-bar cohomology groups in terms of resolution data for complex spaces with isolated singularities, extending understanding of their cohomological properties.

Findings

Cohomology groups are expressed via resolution data.

Identifications are provided for smooth points of the space.

Results apply to both compact and open complex analytic subsets.

Abstract

Let X be a pure n-dimensional (n>1) complex analytic set in C^N with an isolated singularity at 0. In this paper we express the L2-(0,q)-d-bar-cohomology groups for all q with 0<q<n+1, of a sufficiently small deleted neighborhood of the singular point, in terms of resolution data. We also obtain identifications of the L2-(0,q)-d-bar-cohomology groups of the smooth points of X in terms of resolution data, when X is either compact or an open relatively compact complex analytic subset of a reduced complex space with finitely many isolated singularities.