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 $\mathscr{ A}^{ p,q}(X)$.
Categories: math.CV