The Dolbeault complex with weights according to normal crossings

TL;DR

This paper introduces a weighted Dolbeault complex tailored for singular complex spaces with normal crossings, providing a new tool for solving the d-bar-equation through local exactness and integral operators.

Contribution

It defines a novel weighted Dolbeault complex for normal crossings, proving its local exactness using Cauchy's Integral Formula, aiding analysis on singular spaces.

Findings

Constructed a local d-bar-solution operator using Cauchy's Integral Formula.

Proved the local exactness of the weighted Dolbeault complex.

Facilitated the study of the d-bar-equation on singular complex spaces.

Abstract

In this paper, we define a Dolbeault complex with weights according to normal crossings, which is a useful tool for studying the d-bar-equation on singular complex spaces by resolution of singularities (where normal crossings appear naturally). The major difficulty is to prove that this complex is locally exact. We do that by constructing a local d-bar-solution operator which involves only Cauchy's Integral Formula (in one complex variable) and behaves well for L^p-forms with weights according to normal crossings.