The $\bar\partial$-equation for $(p,q)$-forms on a non-reduced analytic space

TL;DR

This paper develops a Dolbeault complex framework for smooth forms and currents on possibly non-reduced analytic spaces, establishing local solvability of the $ar{ ext{d}}$-equation and duality relations.

Contribution

It introduces sheaves of smooth forms and currents on non-reduced spaces, proving the exactness of the Dolbeault complex and duality between cohomologies.

Findings

Sheaves of smooth forms $\

$ar{ ext{d}}$-equation is locally solvable in the constructed sheaves.

Duality between Dolbeault cohomology of forms and currents on non-reduced spaces.

Abstract

On any pure $n$-dimensional, possibly non-reduced, analytic space $X$ we introduce the sheaves $\mathscr{E}_X^{p,q}$ of smooth $(p,q)$-forms and certain extensions $\mathscr{A}_X^{p,q}$ of them such that the corresponding Dolbeault complex is exact, i.e., the $\bar\partial$-equation is locally solvable in $\mathscr{A}_X$. The sheaves $\mathscr{A}_X^{p,q}$ are modules over the smooth forms, in particular, they are fine sheaves. We also introduce certain sheaves $\mathscr{B}_X^{n-p,n-q}$ of currents on $X$ that are dual to $\mathscr{A}_X^{p,q}$ in the sense of Serre duality. More precisely, we show that the compactly supported Dolbeault cohomology of $\mathscr{B}^{n-p,n-q}(X)$ in a natural way is the dual of the Dolbeault cohomology of $\mathscr{A}^{p,q}(X)$.