On the moduli spaces of parabolic Higgs bundles on a curve
Speaker: Do Viêt Cuong (University of Science, Vietnam National University)

Time: 14h00 Friday, 1/7/2022

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Meeting ID: 896 0321 0669

Passcode: 340252

Abstract: Let $C$ be a projective curve. The moduli space of Higgs bundles on $C$, introduced by Hitchin, is an interesting object of study in geometry. If $C$ is defined over the complex numbers, the moduli space of Higgs bundles is diffeomorphic to the space of representations of the fundamental group of the curve. If $C$ is defined over finite fields, the adelic description of the stack of Higgs bundles on $C$ is closely related to spaces occurring in the study of the trace formula. It is a start point to Ngo's proof for the fundamental lemma for Lie algebras.

A natural generalization of the Higgs bundles is the parabolic Higgs bundles (that we shall equip each bundle of a parabolic structure, i.e the choice of flags in the fibers over certain marked points, and some compatible conditions). Simpson proved that there is analogous relation between the space of representations of the fundamental group of a punctured curve (the marked points are the points that are took out from the curve) with the moduli space of parabolic Higgs bundles.

Despite their good applications, the cohomology of the moduli space of (parabolic) Higgs bundles has not yet been determined. In this talk, I shall explain an algorithm to calculate the (virtual) motive (i.e in a suitable Grothendieck group) of the moduli spaces of (parabolic) Higgs bundles. In the case when the moduli space is quasi-projective, the virtual motive allows us to read off the dimensions of its cohomology spaces.

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