PhD in Mathematics

Higgs bundles and non-abelian Hodge correspondence

Higgs bundles and non-abelian Hodge correspondence

Academic year 2025/2026

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Course objectives

After an overview of the course and some historical remarks on non-abelian Hodge correspondence, we will introduce basic concepts from Riemannian geometry (real vector bundles, smooth sections, holonomy), and then move on to differential complex geometry (holomorphic vector bundles, Hermitian metrics, and Chern curvature). We will discuss the classification problem for holomorphic vector bundles over Riemann surfaces of genus g, the introduction of the notion of stability, and the Narasimhan–Seshadri theorem. We will then turn to Donaldson’s proof of this theorem, which for the first time translates the algebraic notion of stability into analytic terms and represents a special case of the non-abelian Hodge correspondence. Subsequently, we will introduce Higgs bundles, their notion of stability, Hitchin’s equations, and explain the proof strategy of the correspondence in as much detail as possible, making use of harmonic maps and harmonic metrics. In the final part of the course, one may either discuss the generalization of the correspondence to real Lie groups and explain its applications in the context of higher Teichmüller theory, or consider the case in which the Riemann surface of genus g is replaced by a compact Kähler manifold.

Program

The aim of this course is to introduce the analytic and holomorphic aspects of Higgs bundles defined over a compact Riemann surface, and to explore their connection with the representation theory of the fundamental group of the surface into SL(n, ℂ). We will begin by reviewing key concepts from complex differential geometry, such as holomorphic vector bundles, Hermitian metrics, and Chern connections. The core of the course will be devoted to a proof of the correspondence between polystable Higgs bundles and reductive representations—known as the non-abelian Hodge correspondence—based on the theory of harmonic maps from a Riemann surface into a non-compact symmetric space. Depending on students’ interests, we may delve deeper into the notion of stability for vector bundles, possibly extending the discussion to the setting where the Riemann surface is replaced by a compact Kähler manifold. Alternatively, we may explore how more general versions of this correspondence underpin current developments in higher Teichmüller theory, which studies representations of surface groups into real semisimple Lie groups.

Details:

• Hours: 20
• Teaching period: Week of January 19 - last week of March 2026

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