Abstract. We prove that the mirror symmetry of Berglund-Hübsch-ChiodoRuan, applied to K3 surfaces with a non-symplectic involution, coincides with the mirror symmetry described by Dolgachev and Voisin.

We give a classification of all non-symplectic automorphisms of prime order p acting on irreducible holomorphic symplectic fourfolds deformation equivalent to the Hilbert scheme of two points on a K3 surface, for p = 2, 3 and 7 ≤ p ≤ 19. Our classification relates the isometry classes of two natural lattices associated to the action of the automorphism on the second cohomology group with integer coefficients with some invariants of the fixed locus and we provide explicit examples. As an application, we find new examples of non-natural non-symplectic automorphisms.

We define Enriques varieties as a higher dimensional generalization of Enriques surfaces and construct examples by using fixed point free automorphisms on generalized Kummer varieties. We also classify all automorphisms of generalized Kummer varieties that come from an automorphism of the underlying abelian surface. Higher dimensional Enriques varieties2.1. Irreducible holomorphic symplectic manifolds. A complex, compact, Kähler manifold X is called irreducible symplectic if X is simply connected and

Résumé. Onétablit quelques résultats généraux relatifsà la taille du groupe d'automorphismes de l'espace de Douady de points sur une surface, puis onétudie quelques propriétés des automorphismes provenant d'un automorphisme de la surface, en particulier leur action sur la cohomologie et la classification de leurs points fixes.