Almost complex structure and the quotient four-manifold by an anti-symplectic involution
type: Item
instance of: scholarly article
title: Almost complex structure and the quotient four-manifold by an anti-symplectic involution
zbMATH Open document ID: 1200.57016
author: Yong Seung Cho
author: Yoon Hi Hong
published in: Topology and its Applications
publication date: 2009-12-15T00:00:00Z
review text: The authors study the quotient of an oriented, closed, symplectic four-manifold \(X\) under an anti-symplectic involution \(\sigma\) with two-dimensional compact fixed point submanifold \(X^\sigma\) (not necessarily orientable or connected). They show that if \(b_2^+(X) \not \equiv b_1(X) + 3 \mod 4,\) then the quotient \(X/\sigma\) does not admit any almost complex structure. They prove a converse holds if we restrict to simply connected \(X\) or \(X\) Kähler and \(\sigma\) anti-holomorphic. Important tools in the authors' arguments are the fundamental classification theorems on quadratic forms of smooth 4-manifolds due to \textit{V. A. Rokhlin} [Dokl. Akad. Nauk SSSR, n. Ser. 84, 221--224 (1952; Zbl 0046.40702)], \textit{S. K. Donaldson} [J. Differ. Geom. 18, 279-315 (1983; Zbl 0507.57010)] and \textit{J.-H. Kim} [Math. Ann. 329, No.~1, 31--47 (2004; Zbl 1070.57021)] as well as the criteria of \textit{F. Hirzebruch} and \textit{H. Hopf} [Math. Ann. 136, 1566--172 (1958; Zbl 0088.39403)] for the existence of an almost complex structure. An important motivation for the work in the paper is an open conjecture of Akbulut that the quotient is completely decomposable, i.e. \(X/\sigma \simeq \#r \mathbb C \mathbb P^2 \# s\overline{\mathbb C \mathbb P}^2\) or \(\# n (S^2 \times S^2)\) for some \(r,s,n\). \textit{S. Akbulut} [J. Reine Angew. Math. 447, 83--90 (1994; Zbl 0861.57027)] proved this with the additional hypotheses that \(X\) is a complex algebraic surface and \(\sigma\) is the complex conjugation with a real algebraic surface as a fixed point set.
reviewed by: Terry Lawson
Mathematics Subject Classification ID: 57R17
Mathematics Subject Classification ID: 14J27
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 53D05
Mathematics Subject Classification ID: 57R22
Mathematics Subject Classification ID: 57R57
zbMATH DE Number: 5647959
zbMATH Keywords: anti-symplectic involution
zbMATH Keywords: symplectic four-manifold
zbMATH Keywords: almost complex structure
zbMATH Keywords: Lagrangian surface
zbMATH Keywords: quotient manifold
MaRDI profile type: Publication
full work available at URL: j.topol.2009.09.007
OpenAlex ID: W2081666345
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DOI: 10.1016/J.TOPOL.2009.09.007
description: scientific article; zbMATH DE number 5647959