$L^2$-Serre duality on singular complex spaces and applications

TL;DR

This survey develops an $L^2$-Serre duality framework for singular complex spaces, leading to vanishing theorems, extension results, and a characterization of rational singularities via $L^2$-$ar{ ext{d}}$-complexes.

Contribution

It introduces a topological $L^2$-Serre duality for singular spaces and applies it to derive vanishing theorems, extension theorems, and a new characterization of rational singularities.

Findings

Proves Hartogs' extension theorem for $(n-1)$-complete spaces.

Characterizes rational singularities via $L^2$-$ar{ ext{d}}$-complex exactness.

Provides an $L^2$-$ar{ ext{d}}$-resolution of the structure sheaf at rational singular points.

Abstract

In this survey, we explain a version of topological $L^2$-Serre duality for singular complex spaces with arbitrary singularities. This duality can be used to deduce various $L^2$-vanishing theorems for the $\overline\partial$-equation on singular spaces. As one application, we prove Hartogs' extension theorem for $(n-1)$-complete spaces. Another application is the characterization of rational singularities. It is shown that complex spaces with rational singularities behave quite tame with respect to some $\overline\partial$-equation in the $L^2$-sense. More precisely: a singular point is rational if and only if the appropriate $L^2$-$\overline\partial$-complex is exact in this point. So, we obtain an $L^2$-$\overline\partial$-resolution of the structure sheaf in rational singular points.