Instanton moduli spaces on non-K\"ahlerian surfaces. Holomorphic models around the reduction loci

TL;DR

This paper investigates the local holomorphic structure of moduli spaces of rank 2 bundles on Gauduchon surfaces near reduction loci, introducing a model called a 'blowup flip passage' to describe neighborhoods around these loci.

Contribution

It introduces the concept of a 'blowup flip passage' as a local model for moduli spaces near reduction loci on non-Kählerian surfaces, providing a detailed geometric and complex-analytic description.

Findings

Neighborhoods of the boundary of the blown-up moduli space are holomorphically equivalent to neighborhoods of a flip passage.

The model space is a manifold with boundary, with a boundary that is a projective fibration.

The interior of the model space has a natural complex structure.

Abstract

Let $\mathcal{M}$ be a moduli space of polystable rank 2-bundles bundles with fixed determinant (a moduli space of $\mathrm{PU}(2)$-instantons) on a Gauduchon surface with $p_g=0$ and $b_1=1$. We study the holomorphic structure of $\mathcal{M}$ around a circle $\mathcal{T}$ of regular reductions. Our model space is a "blowup flip passage", which is a manifold with boundary whose boundary is a projective fibration, and whose interior comes with a natural complex structure. We prove that a neighborhood of the boundary of the blowup $\hat{\mathcal{M}}_{\mathcal{T}}$ of $\mathcal{M}$ at $\mathcal{T}$ can be smoothly identified with a neighborhood of the boundary of a "flip passage" $\hat Q$, the identification being holomorphic on $\mathrm{int}(\hat Q)$.