The Anti-Self-Dual Deformation Complex and a conjecture of Singer

TL;DR

This paper investigates conditions under which anti-self-dual four-manifolds are unobstructed, providing conformally invariant criteria that influence the structure of their moduli spaces and construction methods.

Contribution

It introduces conformally invariant conditions ensuring unobstructedness of ASD manifolds of positive Yamabe type, advancing understanding of their deformation theory.

Findings

Conformally invariant conditions for unobstructedness.

Characterization of unobstructed ASD manifolds via cohomology.

Implications for moduli space structure and construction methods.

Abstract

Let $(M^4,g)$ be a smooth, closed, oriented anti-self-dual (ASD) four-manifold. $(M^4,g)$ is said to be unobstructed if the cokernel of the linearization of the self-dual Weyl tensor is trivial. This condition can also be characterized as the vanishing of the second cohomology group of the ASD deformation complex, and is central to understanding the local structure of the moduli space of ASD conformal structures. It also arises in construction of ASD manifolds by twistor and gluing methods. In this article we give conformally invariant conditions which imply an ASD manifold of positive Yamabe type is unobstructed.