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$P=W$ via $\mathcal{H}_2$

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arxiv 2209.05429 v2 pith:PPGEH3ZO submitted 2022-09-12 math.AG math.DGmath.RT

Tamas Hausel , Anton Mellit , Alexandre Minets , Olivier Schiffmann This is my paper · ORCID
classification math.AGmath.DGmath.RT
keywords mathcalbundlesdegreefiltrationhiggsmodulirankspace
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abstract

Let $\mathcal{H}_2$ be the Lie algebra of polynomial Hamiltonian vector fields on the symplectic plane. Let $X$ be the moduli space of stable Higgs bundles of fixed relatively prime rank and degree, or more generally the moduli space of stable parabolic Higgs bundles of arbitrary rank and degree for a generic stability condition. Let $H^*(X)$ be the cohomology with rational coefficients. Using the operations of cup-product by tautological classes and Hecke correspondences we construct an action of $\mathcal{H}_2$ on $H^*(X)[x,y]$, where $x$ and $y$ are formal variables. We show that the perverse filtration on $H^*(X)$ coincides with the filtration canonically associated to $\mathfrak{sl}_2\subset \mathcal{H}_2$ and deduce the $P=W$ conjecture of de Cataldo-Hausel-Migliorini.

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Cited by 6 Pith papers

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