A Dolbeault-Grothendieck lemma on complex spaces via Koppelman formulas

TL;DR

This paper introduces a new resolution of the structure sheaf on complex spaces using fine sheaves of currents and explicit Koppelman formulas, extending Dolbeault-Grothendieck theory.

Contribution

It constructs a Dolbeault resolution on complex spaces via intrinsic Koppelman formulas, generalizing classical results to singular settings.

Findings

Defines fine sheaves of currents on complex spaces.

Provides an explicit semi-global Koppelman formula.

Establishes a resolution of the structure sheaf.

Abstract

Let $X$ be a complex space of pure dimension. We introduce fine sheaves $\A^X_q$ of $(0,q)$-currents, which coincides with the sheaves of smooth forms on the regular part of $X$, so that the associated Dolbeault complex yields a resolution of the structure sheaf $\hol^X$. Our construction is based on intrinsic and quite explicit semi-global Koppelman formulas.