The \(L_ 2-{\bar \partial}\)-cohomology of an algebraic surface
type: Item
instance of: scholarly article
title: The \(L_ 2-{\bar \partial}\)-cohomology of an algebraic surface
zbMATH Open document ID: 0682.32024
DOI: 10.1016/0040-9383(89)90019-0
author: William Pardon
published in: Topology
publication date: 1989-01-01T00:00:00Z
review text: Let S be a complex projective surface. Denote by \(H_{(2)}^{p,q}(S- Sing S)\) the corresponding \(L_ 2-{\bar \partial}\)-cohomology groups (defined via the Riemannian metric induced by a projective imbedding). Let \(\chi_ 2(S)=\Sigma (-1)^ i \dim H_{(2)}^{0,i}(S-Sing S)\) be the \(L_ 2\)-arithmetic genus of S (whenever \(H_{(2)}^{0,i}\) are finite-dimensional over \({\mathbb{C}}).\) The author proves the following results: 1. If S is normal, then each \(H_{(2)}^{0,i}(S\)-Sing S) is finite-dimensional. 2. Let \(\rho\) : \(X\to S\) be a desingularization such that \(Z=\rho^{-1}(Sing S)\) is a divisor with normal crossings. Then \(\chi\) (\({\mathcal O}_ X(Z-| Z|))\) is a birational invariant of S, \[ \chi ({\mathcal O}_ X)\geq \chi ({\mathcal O}_ X(Z-| Z|))\geq \chi_ 2(S), \] and the first inequality is in general strict. In such a way the author gives (in dimension 2) contra-examples to an older conjecture of MacPherson (stating that \(\chi\) (\({\mathcal O}_ X)=\chi_ 2(S)\), S projective variety and X a desingularization of it). Comments and results about revised conjectures are also given.
Mathematics Subject Classification ID: 32J15
Mathematics Subject Classification ID: 14J17
zbMATH DE Number: 4116938
zbMATH Keywords: \(L_ 2\)-cohomology
zbMATH Keywords: arithmetic genus of a surface
zbMATH Keywords: projective surface
MaRDI profile type: Publication
full work available at URL: 0040-9383(89)90019-0
OpenAlex ID: W2020673297
Wikidata QID: Q125590427
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description: scientific article; zbMATH DE number 4116938