Local Properties of Self-Dual Harmonic 2-forms on a 4-Manifold

TL;DR

This paper proves a Moser-type theorem for self-dual harmonic 2-forms on closed 4-manifolds, classifies local forms near singular circles, and analyzes the resulting contact structures on boundary components.

Contribution

It introduces a new classification of local forms of self-dual harmonic 2-forms near singular circles and explores the induced contact structures on boundary components.

Findings

Classification of local forms near singular circles

Identification of two possible contact forms on S^1×S^2

Construction of symplectic manifolds with contact boundary

Abstract

We will prove a Moser-type theorem for self-dual harmonic 2-forms on closed 4-manifolds, and use it to classify local forms on neighborhoods of singular circles on which the 2-form vanishes. Removing neighborhoods of the circles, we obtain a symplectic manifold with contact boundary - we show that the contact form on each S^1\times S^2, after a slight modification, must be one of two possibilities.