Explicit Serre duality on complex spaces

TL;DR

This paper provides an explicit analytic realization and proof of Serre duality on reduced pure complex spaces using residue currents and integral formulas, introducing concrete sheaves that form a dualizing complex.

Contribution

It introduces concrete sheaves of currents that realize Serre duality explicitly and provides a new analytic proof applicable to all reduced pure complex spaces.

Findings

Constructs explicit fine sheaves of currents on complex spaces.

Shows the Dolbeault complex of these sheaves forms a dualizing complex.

Specializes to Cohen-Macaulay spaces where it resolves the dualizing sheaf.

Abstract

In this paper we use recently developed calculus of residue currents together with integral formulas to give a new explicit analytic realization, as well as a new analytic proof of Serre duality on any reduced pure $n$-dimensional paracompact complex space $X$. At the core of the paper is the introduction of concrete fine sheaves $\mathscr{B}_X^{n,q}$ of certain currents on $X$ of bidegree $(n,q)$, such that the Dolbeault complex $(\mathscr{B}_X^{n,\bullet},\,\bar{\partial})$ becomes, in a certain sense, a dualizing complex. In particular, if $X$ is Cohen-Macaulay (e.g., Gorenstein or a complete intersection) then $(\mathscr{B}_X^{n,\bullet},\,\bar{\partial})$ is an explicit fine resolution of the Grothendieck dualizing sheaf.