Higher Complex Structures and Higher Teichmüller Theory

Abstract

In this PhD thesis, we give a new geometric approach to higher Teichmüller 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.

arXiv commentPhD thesis, 147 pages

References in corpus (7)

• Mirror symmetry, Langlands duality, and the Hitchin system
• Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces
• Principal Nilpotent pairs in a semisimple Lie algebra, I
• Hyperkaehler structures on total spaces of holomorphic cotangent bundles
• Cyclic Higgs bundles and the affine Toda equations
• Higher Complex Structures and Flat Connections
• Generalized Punctual Hilbert Schemes and $\mathfrak{g}$-complex structures