The stacky P=W conjecture

Ben Davison (University of Edinburgh)

Tuesday 28th November, 2023 16:00-17:00 Maths 110

Abstract

Given a smooth complex projective curve C, nonabelian Hodge theory establishes a diffeomorphism between the coarse moduli spaces of semistable Higgs bundles on the one hand (the Dolbeault side), and representations of the fundamental group of the underlying real surface, on the other hand (the Betti side).  The P=W conjecture, states that the weight filtration on the cohomology on the Betti side gets sent to the perverse filtration on the Dolbeault side.  This is a striking result: the weight filtration has to do with Deligne's theory of mixed Hodge structures on cohomology of complex varieties, while the perverse filtration is a byproduct of the BBDG decomposition theorem for the Hitchin map, and these are two rather different bits of maths.

For smooth, fine moduli schemes, the P=W conjecture is now a theorem, proved by Maulik and Shen (twice) and independently by Hausel, Mellit, Minets and Schiffmann.  This talk will concern what happens when the moduli scheme is not smooth, and also the related conjecture for moduli stacks.  A crucial part of the story is the definition of a nonabelian Hodge isomorphism for Borel-Moore homology of the relevant moduli stacks: defining what all this means and how it is established will be the main object of the talk.  With this stacky version of the nonabelian Hodge correspondence in place, it is possible to formulate a version of the P=W conjecture, and in certain cases prove it.

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