Home /  GT Program Seminar: Morse Theory on Moduli Spaces of Pairs and the Bogomolov-Miyaoka-Yau Inequality

Seminar

GT Program Seminar: Morse Theory on Moduli Spaces of Pairs and the Bogomolov-Miyaoka-Yau Inequality November 08, 2022 (11:00 AM PST - 12:00 PM PST)
Parent Program:
Location: SLMath: Eisenbud Auditorium, Online/Virtual
Speaker(s) Paul Feehan (Rutgers University)
Description No Description
Keywords and Mathematics Subject Classification (MSC)
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC
Video

GT Program Seminar: Morse Theory On Moduli Spaces Of Pairs And The Bogomolov-Miyaoka-Yau Inequality

Abstract/Media

To participate in this seminar, please register HERE.

We describe an approach to Bialynicki-Birula theory for holomorphic C^* actions on complex analytic spaces and Morse-Bott theory for Hamiltonian functions for the induced circle actions. A key principle is that positivity of a suitably defined "virtual Morse-Bott index" at a critical point of the Hamiltonian function implies that the critical point cannot be a local minimum. Inspired by Hitchin’s 1987 study of the moduli space of Higgs monopoles over Riemann surfaces, we apply our method in the context of the moduli space of non-Abelian monopoles or, equivalently, stable holomorphic pairs 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 show that these indices are positive in a setting motivated by a conjecture that all closed, smooth four-manifolds of Seiberg-Witten simple type (including symplectic four-manifolds) obey the Bogomolov-Miyaoka-Yau inequality.

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GT Program Seminar: Morse Theory On Moduli Spaces Of Pairs And The Bogomolov-Miyaoka-Yau Inequality