Among Fano varieties of K3 type (or FK3 for short) the cubic fourfold stands out for its historical significance. Indeed, long before the terminology FK3 was born there were already examples of a relation between this famous cubic hypersurface and irreducible holomorphic symplectic (or IHS for short) manifolds. One method to associate a smooth cubic fourfold with an IHS manifold involves the Fano variety of lines on it. This is, as proven by Beauville–Donagi, an IHS manifold of type K3[2] . This relation becomes even more intriguing when considering mildly singular cubic fourfolds, e.g. cubic fourfolds Y that are triple covering of P4 branched over a singular cubic threefold. In this case we have that F (Y ), the Fano variety of lines on Y , is birational to an IHS manifold of type K3[2] . This fact has been used by Boissière–Camere–Sarti and by me to study some compactification of the moduli spaces of irreducible holomorphic symplectic manifolds with an order three non-symplectic automorphism. In order to achieve this result the authors do not consider the rich geometry of F (Y ). I will present recent results obtained in collaboration with Samuel Boissière and Paola Comparin that explain how the geometry of F (Y ) gives us a better understanding of the deep relation between cyclic cubic fourfolds and IHS manifolds of type K3[2] with a non-symplectic automorphism of order three.
Thursday, 11.01.2024, 16:00-17:00, room B312 in the main building of Leibniz Universität Hannover
