Aula D'Antoni, Dip.Matematica, Università "Tor Vergata", Roma

Abstract.

Given a compact complex smooth hyperkaehler manifold (HK) X with an automorphism $\alpha$, it is known that if $\alpha$ is symplectic and $dim(X)>2$, in general $X/\alpha$ does not admit a symplectic
resolution. On the other hand, if the automorphism is non symplectic is still possible that it preserves the volume form and in this case one can ask if the quotient of $X/\alpha$ admits a crepant resolution.
If it is so, one obtains a Calabi--Yau manifold (CY). In the talk we discuss the properties required to the $\alpha$ in order to obtain a CY, and we observe that among the known explicit examples of pairs
(X,\alpha), the unique possibility is that the dimension of $X$ is 4 and the order of $\alpha$ is 2. In this case one is able to compute the Hodge numbers of the Calabi--Yau fourfold $Y$, desingularization
of $X/\alpha$, and to discuss several geometric properties if $X$ is the Hilbert scheme of 2 points of a K3 surface S and $\alpha$ is an involution induced on $X$ by an involution on $S$. Moreover, we relate
$Y$ with another CY fourfold, the Borcea--Voisin of $S$, and we discuss the problem of finding a mirror CY for Y.
This is a joint work with Chiara Camere and Giovanni Mongardi.