Virtual Morse-Bott index, moduli spaces of pairs, and applications to topology of smooth four-manifolds

TL;DR

This paper develops a method using virtual Morse-Bott indices to analyze the topology of moduli spaces of non-Abelian monopoles on complex Kähler surfaces, with applications to four-manifold topology and inequalities.

Contribution

It extends Bialynicki-Birula theory and virtual Morse-Bott index concepts to the study of non-Abelian monopole moduli spaces, linking to four-manifold topology.

Findings

Computed virtual Morse-Bott indices for critical strata.

Proved positivity of indices in a setting related to the Bogomolov-Miyaoka-Yau inequality.

Supported conjectures on the topology of smooth four-manifolds.

Abstract

We previously developed an approach to Bialynicki-Birula theory for holomorphic $\mathbb{C}^*$ actions on complex analytic spaces and the concept of virtual Morse-Bott indices for singular critical points of Hamiltonian functions for the induced circle actions (see Feehan, arXiv:2206.14710). For Hamiltonian functions of circle actions on closed, complex Kaehler manifolds, the virtual Morse-Bott index coincides with the classical Morse-Bott index due to Bott (1954) and Frankel (1959). A key principle in our approach is that positivity of the virtual Morse-Bott index at a critical point of the Hamiltonian function implies that the critical point cannot be a local minimum even when that critical point is a singular point in the moduli space. In this monograph, we consider our method in the context of the moduli space of non-Abelian monopoles over a closed, complex, Kaehler surface. We use the Hirzebruch-Riemann-Roch Theorem to compute virtual Morse-Bott indices of all critical strata (Seiberg-Witten moduli subspaces) and we prove that these indices are positive in a setting motivated by the conjecture that all closed, smooth four-manifolds of Seiberg-Witten simple type obey the Bogomolov-Miyaoka-Yau inequality.