Langlands Duality in the Generic Fibres of Classical Hitchin Fibrations

We study L-twisted endomorphisms of holomorphic vector bundles over a compact Riemann surface. In particular, we generalise Hitchin's computation of the generic bres of the Hitchin bration to the more general L-twisted case, which presents new challenges since we lose properties intrinsic to the canonical bundle. However, we impose a basepoint-free condition on L to use classical theorems, such as Bertini's theorem, to ensure the generic spectral curves are either smooth or have mild singularities. Moreover, we compute the generic bres for each classical simple Lie group and GLn. Akin to Hitchin's result, the bres are torsors of abelian varieties, and we demonstrate Langlands duality in the Hitchin bration by proving the generic bres corresponding to Langland's dual groups are indeed dual abelian varieties. The thesis consists of two parts. In Part I, we review the necessary background of complex algebraic geometry. Chapter 1 provides an account of the rudiments of divisors and holomorphic vector bundles. Chapter 2 extends the work of Chapter 1 and restricts to the case of compact Riemann surfaces, which lays the foundation for Higgs bundles. Chapter 3 reviews the necessary de nitions and details surrounding complex abelian varieties, which we need to study the generic bres of the Hitchin bration and the duality. In Part II, we compute the generic bres of the Hitchin brations. Chapter 4 computes GLn and type An Hitchin brations, and demonstrates the self-duality in GLn, and duality between SLn and PGLn: Chapter 5 computes the generic bres for Sp2n and SO2n+1 and demonstrates the Langlands duality in the generic bres. Chapter 6 computes the generic bres for SO2n and demonstrates the self-duality. In the SO2n+1 and SO2n cases, generic spectral curves are singular. We resort to normalising the spectral curve in the SO2n case, and in the SO2n+1 case, we use explicit local calculations and Hecke modi cations. As far as the author is aware, this is a new approach to the SO2n+1 computation, as other methods resort to extension classes. In Appendix A, we recount some results in classical Lie theory for complex simple Lie groups and provide a computation of the Langlands dual group in each classical case. Appendix B consists of a self-contained proof of a property speci c to abelian varieties that one could take as a given throughout the project.

Author dc:creator

• Klingner, Tyson

Advisors dc:contributor.advisor

• Baraglia, David
• Vilonen, Kari

• Higgs bundles
• Hitchin fibration
• Langlands duality

Language dc:language.iso

en

Handle dc:identifier.uri

https://hdl.handle.net/2440/137508

OAI identifier oai:identifier

oai:digital.library.adelaide.edu.au:2440/137508