Analytic cycles in flip passages and in instanton moduli spaces over non-K\"ahlerian surfaces

TL;DR

This paper derives a formula for the boundary of analytic cycles in moduli spaces of bundles over non-Kähler surfaces, aiding the understanding of their geometric structure and incidence relations.

Contribution

It introduces a general boundary formula for cycles in blow-up moduli spaces and applies it to instanton moduli spaces, revealing new incidence relations.

Findings

Derived a homological boundary formula for cycles in moduli spaces.

Established incidence relations between closures of extension families.

Enhanced understanding of moduli space geometry beyond classical deformation theory.

Abstract

Let $\mathcal{M}^{\mathrm{st}}$ ($\mathcal{M}^{\mathrm{pst}}$) be a moduli space of stable (polystable) bundles with fixed determinant on a complex surface with $b_1=1$, $p_g=0$, and let $Z\subset \mathcal{M}^{\mathrm{st}}$ be a pure $k$-dimensional analytic set. We prove a general formula for the homological boundary $\delta[Z]^{BM}\in H_{2k-1}^{BM}(\partial\hat{\mathcal M}^{\mathrm{pst}},\mathbb{Z})$ of the Borel-Moore fundamental class of $Z$ in the boundary of the blow up moduli space $\hat {\mathcal M}^{\mathrm{pst}}$. The proof is based on the holomorphic model theorem (proved in a previous article), which identifies a neighborhood of a boundary component of $\hat {\mathcal M}^{\mathrm{pst}}$ with a neighborhood of the boundary of a "blow up flip passage". We then focus on a particular instanton moduli space which intervenes in our program for proving the existence of curves on class VII surfaces. Using our result, combined with general properties of the Donaldson cohomology classes, we prove incidence relations between the Zariski closures (in the considered moduli space) of certain families of extensions. These incidence relations are crucial for understanding the geometry of the moduli space, and cannot be obtained using classical complex geometric deformation theory.