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jeunes en arithmétique et variétés algébriques

HODGE THEORY OF CHARACTER VARIETIES AND THE P=W CONJECTURE

program by Luca Migliorini
tatihou island (normandy), 23-27 July 2018

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.


ARBEITSGEMEINSCHAFT
À LA FRANÇAISE

We plan to revive the tradition of an annual series of conferences in arithmetic geometry based on the model of Oberwolfach's Arbeitsgemeinschaft. They were previously organized by Jean-Benoît Bost and François Loeser from 1995 to 2002 at Luminy. The first editions covered topics such as Euler systems, higher class field theory or modular forms and Galois representations.

The public we have in mind consists mainly of PhD students and early postdocs, with the aim of offering a friendly ambience to learn mathematical subjects that do not necessarily belong to one's own research area. At the end of each edition, the topic for the next one will be voted, and a scientific program will be written in close collaboration with a leading expert. About six months before the conference, participants will apply and the talks will be distributed among them.


ORGANIZERS

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

SCIENTIFIC COMMITTEE

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