Blow ups and symplectic embeddings in dimension 4
type: Item
instance of: scholarly article
title: Blow ups and symplectic embeddings in dimension 4
zbMATH Open document ID: 0731.53035
DOI: 10.1016/0040-9383(91)90021-U
author: Dusa McDuff
published in: Topology
publication date: 1991-01-01T00:00:00Z
review text: The purpose of this paper is to prove the following main results: (1) Let \(B_\lambda\) of the closed ball of radius \(\lambda\) in \(\mathbb R^4\). Then the space of symplectic embeddings of \(B_\lambda\) into the open unit ball in \(\mathbb R^4\) is connected. (2) Blow up the complex projective plane \(\mathbb{CP}^2\) at a point \(x\) and denote the resultant manifold by \(\mathbb{CP}^2\#\overline{\mathbb{CP}}^2\). Let \(S_L\) be the image in \(\mathbb{CP}^2\#\overline{\mathbb{CP}}^2\) of a complex line in \(\mathbb{CP}^2\) which does not meet \(x\). Then a symplectic form on \(\mathbb{CP}^2\#\overline{\mathbb{CP}}^2\) which restricts to a nonvanishing form on \(S_L\) is characterized by its cohomology class up to a diffeomorphism which preserves \(S_L\). By a blowing up process he proves that (1) is equivalent to the uniqueness up to diffeomorphisms of some symplectic structures on \(\mathbb{CP}^2\#\overline{\mathbb{CP}}^2\). Further, Gromov's theory of pseudoholomorphic curves leads us to prove (2).
reviewed by: Manuel de León
Mathematics Subject Classification ID: 53D05
Mathematics Subject Classification ID: 57R17
Mathematics Subject Classification ID: 57R40
Mathematics Subject Classification ID: 32Q65
zbMATH DE Number: 4209003
zbMATH Keywords: symplectic embeddings
zbMATH Keywords: symplectic form
zbMATH Keywords: blowing up process
zbMATH Keywords: pseudoholomorphic curves
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4209003