Simone Billi

Integral lattices often appear in topology and complex/algebraic geometry, they usually raise from pairings on (co)homology of manifolds. For some classes of complex manifolds, an extremely fine invariant consists of an integral lattice which is naturally endowed with a Hodge structure. Our focus is dedicated to the example of cubic fourfolds, where the lattice we consider is the degree 4 (primitive) cohomology with the Poincaré pairing. The aim of the first part of the seminar is to motivate the study of integral lattices discussing in a friendly and non-technical manner the Torelli theorem for cubic fourfolds, in analogy to the much more popular and intelligible case of complex tori. The the second part of the seminar is devoted to a rigorous but down to earth introduction to integral lattices, with plenty of examples followed by a discussions on embeddings of integral lattices and gluing subgroups. Time permitting, a hint of the classification of algebraic lattices for cubic fourfolds with a prime-order automorphism obtained in collaboration with A. Grossi will be mentioned.