$L^2$-sheaves of holomorphic functions and $n$-forms on complex spaces with isolated singularities

TL;DR

This paper provides a natural resolution for the canonical sheaf of square-integrable holomorphic forms on complex spaces with isolated singularities, offering explicit models for $L^2$-$ar{ ext{d}}$-cohomology and related sheaves.

Contribution

It introduces a natural resolution for the canonical sheaf of $L^2$ holomorphic forms on complex spaces with isolated singularities, connecting sheaf theory with $L^2$-cohomology.

Findings

Explicit smooth model for $L^2$-$ar{ ext{d}}$-cohomology

Resolutions for sheaves of $ar{ ext{d}}$-closed $L^2$-functions

Natural resolution for the canonical sheaf

Abstract

Let $X$ be a Hermitian complex space of pure dimension $n$ with isolated singularities. In the present paper, we give a natural resolution for the canonical sheaf of square-integrable holomorphic $n$-forms with Dirichlet boundary condition on $X$. As application, we obtain an explicit smooth model for the $L^2$-$\bar{\partial}$-cohomology, including natural resolutions for sheaves of $\bar{\partial}$-closed (holomorphic) $L^2$-functions.