We describe the cohomological structure of the moduli space of stable SL_n Higgs bundles on a curve following the topological mirror symmetry conjecture of Hausel-Thaddeus. For the approach, we establish a connection between:
(a) the moduli space of twisted Higgs bundles by an effective divisor of degree greater than 2g-2, and
(b) the moduli space of K_C-Higgs bundles,
using vanishing cycle functors. This allows us to apply Ngô's support theorem, which has a simpler form in the case (a) (by Ngô, Chaudouard-Laumon, de Cataldo), to the case (b) which concerns hyper-Kähler geometries. In particular, this gives a new proof of the Hausel-Thaddeus conjecture proven previously by Groechenig-Wyss-Ziegler via p-adic integration. We will also discuss connections to the P=W conjecture if time permits. Based on joint work with Davesh Maulik.
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