1. Home
  2. events
  3. Cohomology of the moduli of Higgs bundles,Ìýthe Hausel-Thaddeus conjecture, and the P=W conjecture

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

Algebraic Geometry Seminar

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

-
Junliang Shen, Massachusetts Institute of Technology

Zoom link:ÌýÌý

WeÌýdescribeÌýthe cohomological structure of theÌýmoduli space of stable SL_n Higgs bundles on a curveÌýfollowingÌýthe topological mirror symmetryÌýconjecture of Hausel-Thaddeus. For the approach, we establishÌýa 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,ÌýusingÌývanishing 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 caseÌý(b) which concerns hyper-Kähler geometries. In particular, thisÌýgives 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.ÌýBased on joint workÌýwith Davesh Maulik.

Ìý