Morse-Bott theory on singular analytic spaces and applications to the topology of symplectic four-manifolds

We describe two extensions, called the virtual Morse-Bott index and circle-equivariant virtual Morse-Bott index, of the classical Morse-Bott index of a Morse-Bott function on a smooth manifold to the setting of (a) suitably defined analytic functions on singular analytic spaces and (b) suitably defined Hamiltonian functions on almost symplectic, singular analytic spaces equipped with circle actions. Almost symplectic, singular analytic spaces with circle actions are pervasive in gauge theory and key examples include the moduli spaces of Higgs pairs over Riemann surfaces, moduli spaces of projective vortices over complex Kaehler manifolds, and moduli spaces of non-Abelian monopoles over smooth Riemannian four-manifolds. We explain how the concept of circle-equivariant virtual Morse-Bott index can potentially be used to answer questions arising in the geography of smooth four-manifolds, such as whether constraints on the topology of compact complex surfaces of general type (the Bogomolov-Miyaoka-Yau inequality) continue to hold for symplectic four-manifolds or even for smooth four-manifolds of Seiberg-Witten simple type.