Huybrechts proved that if X and Y are birational HK manifolds they have to be deformation equivalent, but this implication is far from being an equivalence. It is then natural to ask which additional assumptions are needed to get that two HK manifolds of the same deformation type are birational. In the first part of the talk I will give a gentle introduction on HK manifolds and lattice theory for HK manifolds,  I will recall the definition and I will provide some examples. Then I will focus on HK manifolds of OG10 type recalling the geometric constructions that are known so far. In the second part of the talk I will expose a lattice-theoretic criterion to determine when two HK manifolds of OG10 type are birational, and I will show an application to the Li-Pertusi-Zhao variety. What I will present is the result of a joint work with C.Felisetti and F.Giovenzana. 

https://indico.math.cnrs.fr/event/10905/