Moduli spaces and Fredholm theory for pseudoholomorphic subvarieties associated to self-dual, harmonic \(2\)-forms
type: Item
instance of: scholarly article
title: Moduli spaces and Fredholm theory for pseudoholomorphic subvarieties associated to self-dual, harmonic \(2\)-forms
zbMATH Open document ID: 0972.53055
published in: The Asian Journal of Mathematics
publication date: 2000-11-15T00:00:00Z
review text: From the author's preface: ``A compact, oriented Riemannian 4-manifold whose intersection form is not negative definite has a non-trivial closed, self-dual 2-form. Such a form defines a symplectic structure away from its zero set, and the metric can be used to define a compatible almost complex structure. Although this almost complex structure is singular across the zero set of the given 2-form, one can none-the-less study the associated pseudoholomorphic subvarieties in the complement of the zero set. In this regard, it is natural to restrict attention to those which have the following three properties: First, they are closed subsets of the complement of the zero set of the given self-dual form. Second, they are submanifolds except at a set which can be empty, but is at worst countable and non-accumulating. Third, the given self-dual 2-form has finite integral over the subvariety. With regard to the second point, remark that the condition of pseudoholomorphicity is no more nor less than the requirement that the almost complex structure preserve the tangent space at all manifold points. With regard to the third point, remember that a pseudoholomorphic variety is naturally oriented at its manifold points by the restriction of the symplectic form. The main theorem in [T1, \textit{C. H. Taubes}, Geom. Topol. 3, 167-210 (1999)] asserts that such pseudoholomorphic subvarieties exist when the given 4-manifold has a non-zero Seiberg-Witten invariant. Meanwhile, [T2, \textit{C. H. Taubes}, Geom. Topol. 2, 221-332 (1998; Zbl 0908.53013)] discusses aspects of the regularity theory of pseudoholomorphic subvarieties when the metric and self-dual 2-form have a certain prescribed structure near the zero set of the 2-form. This structure is also described below as it is assumed here as well. In the regard, note that there are no obstructions to finding metrics and the forms which obey the constraints from [T2]. This article serves as a sequel of sorts to [T1] and [T2] as it constr…
reviewed by: Valery Borisovich Marenich
Mathematics Subject Classification ID: 53D45
Mathematics Subject Classification ID: 58B15
Mathematics Subject Classification ID: 58D27
Mathematics Subject Classification ID: 53C07
Mathematics Subject Classification ID: 57R57
zbMATH DE Number: 1531020
zbMATH Keywords: symplectic structure
zbMATH Keywords: self-dual form
zbMATH Keywords: pseudoholomorphic variety
zbMATH Keywords: Seiberg-Witten invariant
author: Clifford Henry Taubes
MaRDI profile type: Publication
DOI: 10.4310/AJM.1999.V3.N1.A11
description: scientific article; zbMATH DE number 1531020