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jeunes en arithmétique et variétés algébriques

H O D G E   T H E O R Y   O F   C H A R A C T E R   V A R I E T I E S
A N D   T H E   P = W   C O N J E C T U R E

p r o g r a m   b y   l u c a   m i g l i o r i n i
t a t i h o u   i s l a n d   ( n o r m a n d y )   2 3 - 2 7   j u l y   2 0 1 8

The starting point is the non-abelian Hodge correspondence (due to Corlette, Donaldson, Hitchin, and Simpson), according to which the moduli spaces of flat connections $M_{\mathrm{dR}}$, Higgs bundles $M_\mathrm{Higgs}$, and local systems $M_{\mathrm{Betti}}$ (also known as character variety) on a smooth projective variety over the complex numbers are all diffeomorphic.

Despite the fact that these three spaces are algebraic varieties, the natural maps given by the correspondence are not algebraic, as one already sees in the case of rank one objects. As a manifestation of this phenomenon, the Hodge structures on the singular cohomology are very different: $H^\ast(M_\mathrm{Higgs})$ is pure whereas $H^\ast(M_{\mathrm{Betti}})$ is in general mixed. This raises the following question: what does the weight filtration induce on the cohomology of $M_\mathrm{Higgs}$?

A conjectural answer, the so called $P = W$ conjecture, was proposed by de Cataldo, Hausel and Migliorini around 2010: the weight should correspond to the filtration coming from the perverse Leray spectral sequence associated to the Hitchin fibration, which is the proper map from $M_\mathrm{Higgs}$ to an affine space sending a Higgs bundle to the coefficients of its characteristic polynomial. In a seminal paper (Annals of Math., 2012), the aforementioned authors proved that this is indeed the case for rank two objects on curves.


Giuseppe Ancona (Université de Strasbourg) /
Javier Fresán (École Polytechnique) /
Marco Maculan (Institut Mathématique de Jussieu) /

S C I E N T I F I C   C O M M I T T E E

Anna Cadoret (Institut Mathématique de Jussieu)
François Charles (Université Paris-Sud XI)
Luca Migliorini (Università di Bologna)
Jérôme Poineau (Université de Caen)