Anti-self-dual blowups

TL;DR

This paper proves the existence of Riemannian metrics on certain four-manifolds where the dual cohomology class to an embedded sphere with self-intersection -1 is represented by an anti-self-dual harmonic form, even with multiple such spheres.

Contribution

It demonstrates the construction of metrics with anti-self-dual harmonic forms dual to embedded (-1)-spheres in four-manifolds with specific topological constraints.

Findings

Existence of such metrics under given conditions.

Construction of metrics with multiple disjoint (-1)-spheres.

Extension to cases with several embedded (-1)-spheres.

Abstract

Let $X$ be a closed, oriented four-manifold containing an embedded sphere with self-intersection number $(-1)$. Suppose that $b_2^+(X) \leq 3$. We show that there exists a Riemannian metric on $X$ such that the cohomology class dual to this sphere is represented by an anti-self-dual harmonic form. Furthermore, such a metric can be constructed even when there are multiple disjoint embedded $(-1)$-spheres.