Explicit Serre duality on complex spaces (Q340449)

Let \(F\to X\) be a complex vector bundle over an \(n\)-dimensional complex manifold \(X\). Serre duality gives a pairing between the cohomology groups of the space of smooth \(F\)-valued \((0,q)\)-forms on the one hand, and of smooth compactly supported \((n,q)\)-forms with values in the dual bundle \(F^*\) on the other hand, by NEWLINE\[NEWLINE ([\varphi]_{\bar\partial}, [\psi]_{\bar\partial})\mapsto\int_X \varphi\wedge\psi.NEWLINE\]NEWLINE This does not work in general in the case of singular spaces. Using a recently developed calculus of residue currents [\textit{M. Andersson} and \textit{H. Samuelsson}, Invent. Math. 190, No. 2, 261--297 (2012; Zbl 1271.32009)], the authors replace the sheaves of smooth forms by certain fine sheaves of currents and get a new explicit analytic realization of Serre duality on reduced pure dimensional paracompact complex spaces.