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Seminar of the Department of Algebra and of the Department of Algebraic Geometry (Shafarevich Seminar)
December 12, 2017 15:00, Moscow, Steklov Mathematical Institute, room 540 (Gubkina 8)

Asymptotic behaviour of certain families of harmonic bundles on Riemann surfaces I

T. Mochizuki

Abstract: Let $(E,\overline{\partial}_E,\theta)$ be a stable Higgs bundle of degree 0 on a compact connected Riemann surface. Once we fix a flat metric $h_{\det(E)}$ on the determinant of $E$, we have the harmonic metrics $h_t (t>0)$ for the stable Higgs bundles $(E,\overline{\partial}_E,t\theta)$ such that $\det(h_t)=h_{\det(E)}$.
In this series of talks, we will discuss two results on the behaviour of $h_t$ when $t$ goes to $\infty$. First, we show that the Hitchin equation is asymptotically decoupled under some assumption for the Higgs field. We apply it to the study of the so called Hitchin WKB-problem. Second, we discuss the convergence of the sequence $(E,\overline{\partial}_E,\theta,h_t)$ in the case where the rank of $E$ is 2. We explain a rule to describe the parabolic weights of a “limiting configuration”, and we show the convergence of the sequence to the limiting configuration in an appropriate sense.
In the talk I, we will give an overview. In the talks II and III, we will give more details without assuming that the audience have listened to the talk I.

Language: English

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