[Seminar] "Towards a Bogomolov-Miyaoka-Yau inequality for symplectic 4-manifolds, II" by Paul Feehan

Title:  Towards a Bogomolov-Miyaoka-Yau inequality for symplectic 4-manifolds, II

Abstract: The Bogomolov-Miyaoka-Yau inequality for minimal compact complex surfaces of general type was proved in 1977 independently by Miyaoka, using methods of algebraic geometry, and by Yau, as an outgrowth of his proof of the Calabi conjectures. In this talk, we describe progress in our ongoing program to prove the conjecture that symplectic 4-manifolds obey the Bogomolov-Miyaoka-Yau inequality. Our program uses Morse theory on the gauge theoretic moduli space of non-Abelian monopoles, where the Morse function is a Hamiltonian for a natural circle action and natural two-form. We shall describe generalizations of Donaldson’s symplectic subspace criterion (1996) from finite to infinite dimensions and generalizations of Taubes' perturbation analysis from the Seiberg-Witten to non-Abelian monopole equations. These methods have application to the problem of showing that the fundamental two-form is non-degenerate and thus an almost symplectic form on the moduli space of non-Abelian monopoles. This talk is based on joint work with Tom Leness and the monographs https://arxiv.org/abs/2010.15789 (to appear in AMS Mathematical Surveys and Monographs),  https://arxiv.org/abs/2206.14710  and https://arxiv.org/abs/2410.13809.