The automorphism and cohomology of cyclic coverings

par Lyu Renjie

vendredi 31 mai 2019, 10:30 → 11:30 Europe/Paris

112 (ICJ)

The “strong” Torelli problem asks whether a given isomorphism of Hodge structures between two compact Kähler manifolds X and Y is induced by a unique isomorphism between X and Y. The answer would be true if the action of the automorphism of X on its cohomology group is faithful. In this talk, we concerns the finite cyclic cover over a projective space branched along a smooth hypersurface. We will show that the automorphisms of cyclic coverings over complex numbers faithfully act on its cohomology group except a few cases. The proof is based on the deformation of automorphisms and the infinitesimal Torelli theorem for the cyclic coverings. In positive characteristic, the faithfulness problem can be reduced to characteristic zero. The reduction argument relies on the degeneration of the relative Hodge-de Rham spectral sequence for a smooth family of cyclic coverings.