$L^2$-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 $\overline{\partial }$-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.

8 pages; survey submitted to the KSCV10 Symposium Proceedings