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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. We construct a generalized spectral curve, an (almost) Lagrangian subvariety of the complexified cotangent space of the surface. 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.
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Submitted on : Wednesday, September 30, 2020 - 11:58:11 AM
Last modification on : Thursday, October 1, 2020 - 7:27:39 AM


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  • HAL Id : tel-02615338, version 2



Alexander Thomas. Higher complex structures and higher Teichmüller theory. Group Theory [math.GR]. Université de Strasbourg, 2020. English. ⟨NNT : 2020STRAD006⟩. ⟨tel-02615338v2⟩



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