Gluing construction of compact complex surfaces with trivial canonical bundle
type: Item
instance of: scholarly article
title: Gluing construction of compact complex surfaces with trivial canonical bundle
zbMATH Open document ID: 1178.32013
author: Mamoru Doi
published in: Journal of the Mathematical Society of Japan
publication date: 2009-09-15T00:00:00Z
full work available at URL: article
review text: Let \(X\) (resp., \(X'\)) be a compact complex surface with a smooth irreducible anticanonical divisor \(D\) (resp., \(D'\)). Suppose there exists an isomorphism \(f:D \to D'\) such that the corresponding normal bundles \(N_{D/X}\) and \(N_{D'/X'}\) are dual to each other via \(f\). Then the author shows the existence of a pair of tubular neighborhoods \((W_1,W_2)\) for \(D\) in \(X\) (resp., \((W'_1,W'_2)\) for \(D'\) in \(X'\)) with \(\overline W_1 \subset W_2\), \(\overline W'_1 \subset W'_2\) and a diffeomorphism \(h:W_2 \setminus \overline W_1 \to W'_2 \setminus \overline W'_1\) such that the gluing of \( X \setminus \overline W_1 \) with \( X' \setminus \overline W'_1 \) using \(h\) yields a complex surface with trivial canonical bundle. Applications are given to new constructions of complex tori, Kodaira surfaces and K3 surfaces, as well as the construction of smoothings of some normal crossing surfaces.
Mathematics Subject Classification ID: 32J15
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 58J37
Mathematics Subject Classification ID: 53C56
zbMATH DE Number: 5603964
zbMATH Keywords: compact complex surface
zbMATH Keywords: gluing construction
zbMATH Keywords: anticanonical divisor
zbMATH Keywords: trivial canonical divisor
reviewed by: Alexandru Dimca
MaRDI profile type: Publication
OpenAlex ID: W2051232392
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DOI: 10.2969/JMSJ/06130853
description: scientific article; zbMATH DE number 5603964