**Hélène Esnault** (Freie Universität Berlin)

*Cohomologically rigid complex local systems with finite determinant and quasi-unipotent monodromies at infinity are integral.*

14:30—15:30 : *Miscellaneous on companions*

16:00—17:00 : *The Betti to ℓ-adic proof.*

Joint work with Michael Groechenig.

If we dropped ‘Cohomologically’ from the title, this would be a complete (positive) answer to Simpson’s conjecture.

We prove the conjecture under the (perhaps ?) stronger assumption that the local systems are cohomologically rigid. Our proof can’t be extended to the case of rigid connections, and I shall explain why.

Initially, in the projective case, our proof consisted in going to the de Rham side, showing that the restriction of the connection on the p-adic varieties is a Frobenius isocrystal, descend mod p, considering the ℓ-adic companions (the existence of which has been proven recently by Abe and myself) and going back to the complex numbers by showing that the induced local systems stemming from the companions are still cohomologically rigid. This last step requires the study of weights and the L-functions of the companions.

The new proof is purely Betti-ℓ-adic. To produce ℓ-adic companions, one has to be able to descend the representation completed at an integral place to the arithmetic fundamental group of the mod p-variety. This requires a variant of a classical argument of Simpson. In doing so one has to make sure that one keeps the conditions at infinity. The companions one uses are now ℓ-adic and have been constructed a few years ago by Drinfeld. One then proves that the formation of companions in our situation preserves the conditions at infinity (this relies on a theorem of Deligne) and one proves, again using weights, that the induced complex local systems are cohomologically rigid as well.

This proof does not request the projective geometry one needs to prove the F-isocrystal theorem mentioned above. For this reason it can be performed in this generality.

We thank Pierre Deligne for putting us under pressure so we do not restrict our proof to the projective case. His understanding of intersection cohomology of the local systems considered plays an important role in the proof.

For lack of time, I won’t lecture on the F-isocrystals

17:30—18:30 : **Adrian Langer** (Uniwersytet Warszawski)

*Rigid representations of projective fundamental groups*

I will talk about some results related to Simpson’s conjecture : rigid irreducible representation of the fundamental group of a smooth complex variety comes from some family of smooth projective varieties. I will sketch the proof of this conjecture in case of rank 3 representations. This is a joint work with Carlos Simpson.