Mathematisches Seminar

Dr. Jan Swoboda (Universität München): “ Limits and degenerations of Higgs bundle moduli spaces I und II”

30.09.2015 von 16:15 bis 17:15

LMS 4 - Raum 424 - Kleiner Hörsaal

Auf Einladung von Prof. Dr. Weiß.


Abstract:

Moduli spaces of Higgs bundles arise naturally in several rather different contexts: as generalizations of the concept of stable bundles in algebraic geometry, as solution spaces of certain elliptic equations over a Riemann surface, as representation varieties of surface groups as well as phase spaces of integrable systems. Being complete, noncompact hyperkähler manifolds, these are also of some interest in Riemannian geometry.

In the first part of this lecture series I shall give an introduction to the basic theory of Higgs bundles over a Riemann surface. I will focus on the construction of its moduli space as the set of gauge equivalence classes of solutions to Hitchin's self-duality equation, discuss concepts such as stability and the Hitchin fibration, and describe its realization as a hyperkähler quotient.

In the second part, I plan to report about recent joint work with Mazzeo, Weiß and Witt on the asymptotic behaviour of solutions in the limit of large Higgs fields. In a rather different direction, I shall focus on the behaviour of the moduli space under degeneration of the underlying Riemann surface, which can be understood using gluing methods from geometric analysis.

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