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SUMMARY:Victoria Hoskins — Two proofs of the P=W conjecture
DTSTART:20231118T133000Z
DTEND:20231118T143000Z
DTSTAMP:20260505T192900Z
UID:indico-event-10794@indico.math.cnrs.fr
DESCRIPTION:The non-abelian Hodge theorem gives a diffeomorphism between t
 he moduli space of Higgs bundles on a smooth projective complex curve and 
 the character variety of (twisted) representations of its fundamental grou
 p. The P=W conjecture of de Cataldo\, Hausel and Migliorini predicts that 
 via the corresponding isomorphism on cohomology\, the perverse filtration 
 for the Hitchin fibration on the Higgs moduli space is identified with the
  weight filtration of the mixed Hodge structure on the character variety.W
 e will discuss two (recent) proofs of the P=W conjecture due to Maulik–S
 hen and Hausel–Mellit–Minets–Schiffmann. Since the cohomology of the
  Higgs moduli space is generated by tautological classes (Markman) and the
 ir weights on the character variety are known (Shende)\, the P=W conjectur
 e reduces to describing the interaction between the tautological classes a
 nd the perverse filtration. The proof of Maulik–Shen combines support th
 eorems for meromorphic Hitchin fibrations (after Ngô and Chaudouard–Lau
 mon)\, vanishing cycle techniques and Yun’s global Springer theory\, whi
 ch allows them to determine the strong perversity of tautological classes 
 by pulling back to a parabolic Higgs moduli space. The proof of Hausel–M
 ellit–Minets–Schiffmann shows the P- and W-filtrations both agree with
  a third representation-theoretic filtration for an sl_2-triple in a Lie a
 lgebra of polynomial Hamiltonian vector fields\, which acts on the cohomol
 ogy via Hecke operators and cup products by tautological classes.\n\nhttps
 ://indico.math.cnrs.fr/event/10794/
LOCATION:Hermite (IHP)
URL:https://indico.math.cnrs.fr/event/10794/
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