Lagrangian spheres, symplectic surfaces and the symplectic mapping class group
type: Item
instance of: scholarly article
title: Lagrangian spheres, symplectic surfaces and the symplectic mapping class group
zbMATH Open document ID: 1253.53073
author: Tian-Jun Li
author: Weiwei Wu
published in: Geometry \& Topology
publication date: 2012-08-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/1012.4146
reviewed by: Akira Asada
Mathematics Subject Classification ID: 53D05
Mathematics Subject Classification ID: 53D12
Mathematics Subject Classification ID: 53D42
zbMATH DE Number: 6068622
zbMATH Keywords: Lagrangian sphere
zbMATH Keywords: symplectomorphism group
zbMATH Keywords: Kodaira dimension
MaRDI profile type: Publication
OpenAlex ID: W1998272992
arXiv ID: 1012.4146
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DOI: 10.2140/GT.2012.16.1121
review text: It is known that a Lagrangian sphere \(L\) in a symplectic Del Pezzo surface with Euler number at most 7 can be displaced from certain symplectic spheres with positive Chern number up to Hamiltonian isotopy [\textit{J. D. Evans}, J. Topol. 3, No. 1, 181--227 (2010; Zbl 1235.53084)]. This paper generalizes this result in the following two ways.NEWLINENEWLINE Theorem 1.1. Let \(L\) be a Lagrangian spere in a symplectic 4-manifold \((M,\omega)\), and \(A\in H_2(M;\mathbb{Z})\) with \(A^2\geq -1\). Suppose \(A\) is represented by a symplectic sphere \(C\). Then \(C\) can be isotoped symplectically to another representative of \(A\) which intersects \(L\) minimally.NEWLINENEWLINE Theorem 1.2. Suppose \((M,\omega)\) is a symplectic 4-manifold with \(b^+= 1\) and \(L\) is a Lagrangian sphere. Assume \(A\in H^2(M,\mathbb{Z})\) satisfies \(\omega(A)> 0\), \(A^2> 0\) and \(A\cdot E\geq 0\) for all \(\omega\)-exceptional \(E\). Then there exists a symplectic surface in the class \(nA\) intersecting \(L\) minimally for large \(n\in\mathbb{N}\).NEWLINENEWLINE Here, two closed surfaces are said to intersect minimally if they intersect transversally at \(|k|\) points, where \(k\) is the homological intersection number, and \(E\in H_2(M,\mathbb{Z})\) is said to be of \(\omega\)-exceptional class if \(E\) is represented as an \(\omega\)-symplectic \((-1)\) sphere.NEWLINENEWLINE To show these theorems, first embedded connected pseudoholomorphic submanifolds for a class of compatible almost complex structures suitable for applying symplectic field theory are produced by using symplectic Seiberg-Witten theory and Taubes' \(\text{SW}\Rightarrow\text{GT}\) [\textit{C. H. Taubes}, J. Differ. Geom. 52, No. 3, 453--609 (1999; Zbl 1040.53096), Prop. 3.5]. Then, via neck stretching (explained in \S2.2), the symplectic field theory (SFT; a brief review is given in \S2), especially a compactness result in the SFT [\textit{F. Bourgeois} et al., Geom. Topol. 7, 799--888 (2003; Zbl 1131.53312), Th…
description: scientific article; zbMATH DE number 6068622