$$L^2$$ -Serre Duality on Singular Complex Spaces and Applications

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 ∂-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 ∂-equation in the L 2 -sense. More precisely: a singular point is rational if and only if the appropriate L 2 -∂-complex is exact in this point. So, we obtain an L 2 -∂-resolution of the structure sheaf in rational singular points.