L²-Serre duality on singular complex spaces and applications

2014/09/04 by Ruppenthal, Jean
#Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1409.1382

Abstract

In this survey, we explain a version of topological L²-Serre duality for singular complex spaces with arbitrary singularities. This duality can be used to deduce various L²-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 L²-sense. More precisely: a singular point is rational if and only if the appropriate L²-∂-complex is exact in this point. So, we obtain an L²-∂-resolution of the structure sheaf in rational singular points.

Related

• L²-Serre duality on singular complex spaces and rational singularities
• L² Serre Duality on Domains in Complex Manifolds and Applications
• L²-theory for the ∂-operator on complex spaces with isolated singularities
• L²-Dolbeault resolutions and Nadel vanishing on weakly pseudoconvex complex spaces with singular Hermitian metrics
• Griffiths and Nakano positivity and Nakano-Nadel vanishing theorems for singular Hermitian metrics on complex spaces