Explicit Serre duality on complex spaces
type: Item
instance of: scholarly article
title: Explicit Serre duality on complex spaces
zbMATH Open document ID: 1359.32009
author: Jean Ruppenthal
author: Håkan Samuelsson Kalm
author: Elizabeth Wulcan
published in: Advances in Mathematics
publication date: 2016-11-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/1401.8093
Mathematics Subject Classification ID: 32C35
Mathematics Subject Classification ID: 32A26
Mathematics Subject Classification ID: 32A27
Mathematics Subject Classification ID: 32B15
Mathematics Subject Classification ID: 32C30
zbMATH DE Number: 6652666
zbMATH Keywords: singular complex space
zbMATH Keywords: dbar-cohomology
zbMATH Keywords: Serre duality
zbMATH Keywords: residue current
MaRDI profile type: Publication
OpenAlex ID: W1654399329
arXiv ID: 1401.8093
cites work: Integral representation with weights. I
cites work: Residue currents and ideals of holomorphic functions
cites work: Integral representation with weights. II: Division and interpolation
cites work: Coleff-Herrera currents, duality, and noetherian operators
cites work: Pseudomeromorphic currents on subvarieties
cites work: A Dolbeault-Grothendieck lemma on complex spaces via Koppelman formulas
cites work: Residue currents with prescribed annihilator ideals
cites work: Decomposition of residue currents
cites work: Direct images of semi-meromorphic currents
cites work: Q4405461
cites work: Q4183005
cites work: A formula for interpolation and division in \({\mathbb C}^n\).
cites work: Regularizations of residue currents
cites work: Les courants residuels associes à une forme méromorphe
cites work: Grothendieck duality and base change
cites work: Q4317713
cites work: Residues and duality. Lecture notes of a seminar on the work of A. Grothendieck, given at Havard 1963/64. Appendix: Cohomology with supports and the construction of the \(f^!\) functor by P. Deligne
cites work: Q4143433
cites work: Abelian differentials on singular varieties and variations on a theorem of Lie-Griffiths
cites work: Residues and principal values on complex spaces
cites work: Q3995750
cites work: Various approaches to products of residue currents
cites work: Residues, currents, and their relation to ideals of holomorphic functions.
cites work: The divisor class groups of some rings of holomorphic functions
cites work: Complexe dualisant et théorèmes de dualité en géométrie analytique complexe
cites work: Q4043341
cites work: Un théorème de dualité
DOI: 10.1016/J.AIM.2016.10.013
review text: 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.
description: scientific article; zbMATH DE number 6652666