Speaker
Abstract
In this series of lectures I will discuss the tamely
ramified geometric Langlands correspondence over the complex
numbers. I will use the case of $GL(2)$ local systems on the
projective line with tame ramification at five points to illustrate a
general method for constructing Hecke eigensheaves by combining
Fourier-Mukai duality with non-abelian Hodge theory. I will explain
how the program converts the construction problem into a purely
algebraic geometric question which can be solved explicitly by a
higher dimensional version of the spectral cover construction. The
focus will be on the projective geometry of the moduli spaces
involved, and on the singularities and geometric subtleties needed
for the correct formulation of the correspondence.
Tentatively the four lectures will cover the following topics:
1) Moduli spaces of $GL(2)$ parabolic bundles and parabolic Higgs bundles on
$\mathbb{P}^{1}$ with tame ramification at five points. Symplectic
leaves and Hitchin fibers.
2) Wobbly bundles and the modular spectral cover. Geometry of the
parabolic Hecke correspondence. Formulation of the parabolic Hecke
eigensheaf problem for Higgs bundles.
3) Abelianization and the geometry of the abelianized Hecke
correspondence. Standard divisors and the parabolic Picard groups of
relevant moduli spaces.
4) Pullbacks, pushforwards, and tensor products of tame parabolic
Higgs bundles. Parabolic Hecke eigensheaf construction and
verification of all non-abelian Hodge theory and Hecke eigensheaf
constraints.
Some references:
Ron Donagi, Tony Pantev "Langlands duality for Hitchin systems",
Invent. math. (2012) 189: 653.
https://arxiv.org/abs/math/0604617
Ron Donagi, Tony Pantev "Geometric Langlands and non-Abelian Hodge
theory", Surveys in Differential Geometry Volume 13 (2008), 85 – 116.
R. Donagi, T. Pantev, C. Simpson "Direct Images in Non Abelian Hodge
Theory".
https://arxiv.org/abs/1612.06388.
