Symplectic surfaces in symplectic 4-manifolds
type: Item
instance of: scholarly article
title: Symplectic surfaces in symplectic 4-manifolds
zbMATH Open document ID: 1032.57018
author: Mi Sung Cho
author: Yong Seung Cho
published in: Taiwanese Journal of Mathematics
publication date: 2004-03-09T00:00:00Z
review text: Let \(X\) be a closed, minimal, symplectic 4-manifold with symplectic form \(\omega,\) and \(F\) a symplectic surface in \(X\) satisfying \(c_1(TX)[F] > 0.\) D. McDuff asked whether it follows that \(X\) must be rational or ruled. Partial results have been obtained by \textit{D. McDuff} in [J. Am. Math. Soc. 3, 679-712 (1990; Zbl 0723.53019); J. Am. Math. Soc. 5, 987-988 (1992; Zbl 0799.53039)]. In this paper the authors use McDuff's results and Seiberg-Witten and Gromov invariants to solve a restricted form of the problem, proving: Let \(X\) be a closed, connected, minimal symplectic 4-manifold containing a symplectic surface \(F\) with genus \(g(F) = g \geq 1\) and \(c_1(TX)[F] > g.\) Then \(X\) is rational or ruled.
Mathematics Subject Classification ID: 57N13
Mathematics Subject Classification ID: 53D35
Mathematics Subject Classification ID: 37J05
Mathematics Subject Classification ID: 57R57
zbMATH DE Number: 1926031
zbMATH Keywords: symplectic surface
zbMATH Keywords: Gromov invariant
zbMATH Keywords: rational surface
zbMATH Keywords: ruled surface
zbMATH Keywords: Seiberg-Witten invariant
zbMATH Keywords: symplectic 4-manifold
reviewed by: Terry Lawson
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.11650/twjm/1500407518
OpenAlex ID: W4245291661
DOI: 10.11650/TWJM/1500407518
description: scientific article; zbMATH DE number 1926031