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

Abstract

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