Cohomology of the moduli of Higgs bundles, the Hausel-Thaddeus conjecture, and the P=W conjecture

Cohomology of the moduli of Higgs bundles, the Hausel-Thaddeus conjecture, and the P=W conjecture

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Junliang Shen, Massachusetts Institute of Technology

Zoom link:  https://princeton.zoom.us/j/91248028438

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 Gröchenig-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.