Canonical sheaves at isolated canonical Gorenstein singularities

TL;DR

This paper studies the behavior of canonical sheaves of holomorphic forms with boundary conditions at isolated canonical Gorenstein singularities, advancing the $L^2$-theory for the $ar{ ext{d}}$-operator on singular complex spaces.

Contribution

It classifies and describes the behavior of the sheaf of square-integrable forms with boundary conditions at isolated canonical Gorenstein singularities, providing new insights for $L^2$-theory.

Findings

Classification of $ ext{K}_X^s$ behavior at singularities

Applications to $L^2$-theory for $ar{ ext{d}}$-operator

Enhanced understanding of canonical sheaves in singular spaces

Abstract

It is well known that the Grauert-Riemenschneider canonical sheaf $\mathcal{K}_X$ of holomorphic square-integrable $n$-forms is a central tool in $L^2$-theory for the $\overline\partial$-operator on a singular complex space $X$ of pure dimension $n$. It was shown a few years ago that a comprehensive $L^2$-theory requires also the study of the sheaf $\mathcal{K}_X^s$ of holomorphic square-integrable $n$-forms with a Dirichlet boundary condition at the singular set of $X$. In the present paper, we describe and classify the behaviour of $\mathcal{K}_X^s$ in isolated canonical Gorenstein singularities, and give applications to the $L^2$-theory for the $\overline\partial$-operator on such spaces.