Local properties of self-dual harmonic 2-forms on a 4-manifold. (English) Zbl 1065.53066

Honda, Ko

J. Reine Angew. Math. 577, 105-116 (2004).

Let \(M\) be a closed, oriented 4-manifold and let \(\omega\) be a self-dual harmonic 2-form on \(M\), having the zero set \(C\) – a union of circles – as an obstruction to the existence of a symplectic structure on \(M\). Such a 2-form is called also a singular symplectic form. In section 2, the author defines an almost complex structure \(J\) on \(M\setminus C\) which is naturally associated to \(\omega\), and section 3 is devoted to a discussion of a version of Moser’s theorem (Theorem 2) which applies to the considered singular symplectic forms. In section 4 the author uses the Moser-type theorem to classify local normal forms for the singular symplectic forms near \(S^1\), and in the last section he discusses the induced contact structures on the boundaries of a neighborhood \(N(S^1)\) of \(S^1\) (theorems 8 and 10).

Reviewer: Neculai Papaghiuc (Iaşi)

MSC:

53D35	Global theory of symplectic and contact manifolds
58A10	Differential forms in global analysis

Keywords:

Moser-type theorem; self-dual harmonic 2-forms