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

Abstract

AbstractIn 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.KeywordsCauchy-Riemann equations \(L^2\)-theorySerre dualityDolbeault cohomologyVanishing theoremsSingular complex spacesRational singularities