Classical deformations of noncompact surfaces and their moduli of instantons

TL;DR

This paper studies deformations of certain noncompact surfaces and shows that instanton moduli spaces vanish on nontrivial deformations, contrasting with their known structure on the original surfaces.

Contribution

It provides a detailed description of semiuniversal deformation spaces for noncompact surfaces and demonstrates that instanton moduli spaces are empty on nontrivial deformations.

Findings

Deformation spaces for surfaces $Z_k$ are semiuniversal.

Nontrivial deformations $Z_k( au)$ are affine.

Instanton moduli spaces are empty on deformed surfaces.

Abstract

We describe semiuniversal deformation spaces for the noncompact surfaces $Z_k := \operatorname{Tot} (\mathcal O_{\mathbb P^1} (-k))$ and prove that any nontrivial deformation $Z_k (\tau)$ of $Z_k$ is affine. It is known that the moduli spaces of instantons of charge $j$ on $Z_k$ are quasi-projective varieties of dimension $2j-k-2$. In contrast, our results imply that the moduli spaces of instantons on any nontrivial deformation $Z_k (\tau)$ are empty.