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

Abstract

We previously developed an approach to Bialynicki-Birula theory for holomorphic ${\mathbb{C}}^{\ast }$ 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.

arXiv comment366+xx pages, 1 figure. Supporting background material drawn from our monograph arXiv:math/0203047. This is a preprint version. The final corrected version is to appear as a book in the series Mathematical Monographs and Surveys, published by the American Mathematical Society

References in corpus (24)

• Stable pairs on curves and surfaces
• Monopoles and Four-Manifolds
• Virtual neighborhoods and pseudo-holomorphic curves
• Orbifold Riemann Surfaces and the Yang-Mills-Higgs Equations
• Quaternionic Monopoles
• Superconformal invariance and the geography of four-manifolds
• Non-abelian monopoles and vortices
• Global existence and convergence of solutions to gradient systems and applications to Yang-Mills gradient flow
• An SO(3)-monopole cobordism formula relating Donaldson and Seiberg-Witten invariants
• Moduli spaces of PU(2)-monopoles
• SO(3) Monopoles, Level-One Seiberg-Witten Moduli Spaces, and Witten's Conjecture in Low Degrees
• PU(2) monopoles and links of top-level Seiberg-Witten moduli spaces
• PU(2) monopoles. II: Top-level Seiberg-Witten moduli spaces and Witten's conjecture in low degrees
• PU(2) monopoles and a conjecture of Marino, Moore, and Peradze
• A continuity theorem for families of sheaves on complex surfaces
• Geography of symplectic 4-manifolds with Kodaira dimension one
• PU(2) monopoles. III: Existence of gluing and obstruction maps
• Superconformal simple type and Witten's conjecture
• Non-abelian Seiberg-Witten theory and projectively stable pairs
• The SO(3) monopole cobordism and superconformal simple type
• Moduli spaces of PU(2)-monopoles (revised version)
• Morse Theory, Higgs fields and Yang-Mills-Higgs functionals
• Gluing in geometric analysis via maps of Banach manifolds with corners and applications to gauge theory
• On the Donaldson-Uhlenbeck compactification of instanton moduli spaces on class VII surfaces

Cited by in corpus (1)

• The SO(3) Vortex Equations over Orbifold Riemann Surfaces