Abstract: |
Let C be a smooth projective curve. The non-abelian Hodge theory of Simpson is a diffeomorphism between the character variety M_B of C and the moduli of (semi)stable Higgs bundles M_D on C. Since this diffeomorphism is not algebraic, it induces an isomorphism of cohomology rings, but does not preserve finer information, such as the weight filtration. Based on computations in small rank, de Cataldo-Hausel-Migliorini conjectured that the weight filtration on H^*(M_B) gets sent to the perverse filtration on H^*(M_D), associated to the Hitchin map. In this talk, I will explain a recent proof of this conjecture, which crucially uses the action of Hecke correspondences on H^*(M_D). If time permits, I will discuss the relation of this proof to cohomological Hall algebras. Based on joint work with T. Hausel, A. Mellit, O. Schiffmann. |