Moduli spaces and Fredholm theory for pseudoholomorphic subvarieties associated to self-dual, harmonic \(2\)-forms
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 constructs a moduli space for a certain subset of pseudoholomorphic subvarieties. To describe this subset, digress momentarily to note that the aforementioned, prescribed structure for the metric and 2-form near the latter's zero set guarantees that this zero set is a disjoint union of embedded circles with a distinguished 2-plane subbundle in its normal bundle. With the digression now ended, the subset of pseudoholomorphic subvarieties in question consists of those which approach the zero set of the self-dual form with one tangent asymptotic to the distinguished 2-plane subbundle. The following theorem summaries the basic picture: Theorem. The set of pseudoholomorphic subvarieties has a natural topology for which the subset, \(M_b\), of boundary regular subvarieties is open. Moreover, the latter set is a countable, nested union \(M_b= M_{b,0}\supset M_{b,1}\supset \dots \), where, for each \(k\geq 0, M_{b,k}\supset M_{b,k+1}\) is closed and each \(C \in M_{b,k} - M_{b,k+1}\) has an open neighborhood in \( M_{b,k}\) which is homeomorphic to the zero set of a smooth map between balls about the origin in the kernel and cokernel of a Fredholm operator''.
scientific article; zbMATH DE number 1531020