The $\bar{\partial}$-equation, duality, and holomorphic forms on a reduced complex space

TL;DR

This paper develops solutions to the $ar{ ext{-}} ext{d}$-equation on reduced complex spaces, establishes an explicit Serre duality, and provides conditions for global solvability, advancing understanding of holomorphic forms on singular spaces.

Contribution

It introduces concrete sheaves of currents to explicitly realize Serre duality and extends $ar{ ext{-}} ext{d}$-equation solutions to higher forms on singular complex spaces.

Findings

Local solutions to the $ar{ ext{-}} ext{d}$-equation on reduced complex spaces.

An explicit version of Serre duality using concrete sheaves.

Conditions for global solvability of the $ar{ ext{-}} ext{d}$-equation.

Abstract

We solve the $\bar{\partial}$-equation for $(p,q)$-forms locally on any reduced pure-dimensional complex space and we prove an explicit version of Serre duality by introducing suitable concrete fine sheaves of certain $(p,q)$-currents. In particular this gives a precise condition for the $\bar{\partial}$-equation to be globally solvable. Our results extend results for $(0,q)$-forms and give information about holomorphic $p$-forms on singular spaces.