Higher Complex Structures and Higher Teichm\"uller Theory

TL;DR

This thesis introduces a new geometric framework called higher complex structures on surfaces, linking them to Hitchin components and generalizing classical structures using advanced algebraic geometry tools.

Contribution

It constructs higher complex structures via the punctual Hilbert scheme, explores their deformation to flat connections, and proposes a conjectural equivalence with Hitchin components, extending to simple Lie algebras.

Findings

Constructed higher complex structures on surfaces.

Established a conjectural diffeomorphism with Hitchin's component.

Generalized constructions to simple Lie algebras.

Abstract

In this PhD thesis, we give a new geometric approach to higher Teichm\"uller theory. In particular we construct a geometric structure on surfaces, generalizing the complex structure, and we explore its link to Hitchin components. The construction of this structure, called higher complex structure, uses the punctual Hilbert scheme of the plane. Its moduli space admits similar properties to Hitchin's component. Given a higher complex structure, we try to canonically deform it to a flat connection. The space of such connections, called "parabolic", is obtained by imitating the Atiyah--Bott reduction. It is a space of pairs of commuting differential operators. Under some conjecture, we establish a canonical diffeomorphism between our moduli space and Hitchin's component. Finally, we generalize certain constructions, like the punctual Hilbert scheme and the higher complex structure, to the case of a simple Lie algebra.