Anti-self-dual blowups II

TL;DR

This paper proves the existence of Riemannian metrics on certain four-manifolds where specific embedded spheres are represented by anti-self-dual harmonic forms, extending previous results to include (-2)-spheres using contact topology techniques.

Contribution

It extends earlier work by constructing anti-self-dual harmonic forms representing (-2)-spheres on four-manifolds, utilizing Eliashberg's h-principle for overtwisted contact structures.

Findings

Existence of metrics with anti-self-dual forms representing (-2)-spheres

Application of Eliashberg's h-principle to four-orbifolds

Extension of previous results from (-1)-spheres to (-2)-spheres

Abstract

Let $X$ be a closed, oriented four-manifold with $b_2^+ \leq 3$, and suppose $X$ contains a collection of pairwise disjoint embedded $(-2)$-spheres. We prove that there is a Riemannian metric on $X$ such that the Poincare dual of each of these spheres is represented by an anti-self-dual harmonic form. This extends our earlier result for $(-1)$-spheres. The main new ingredient is an application of Eliashberg's $h$-principle for overtwisted contact structures, which we use to construct self-dual harmonic forms on four-orbifolds with prescribed local behaviour near the orbifold singular set.