2026 Theses Doctoral

Shafarevich-Tate twists of holomorphic Lagrangian fibrations

Abasheva, Anna

In this thesis we introduce the notion of Shafarevich–Tate twists of Lagrangian fibrations on holomorphic symplectic manifolds. The twists are also holomorphic symplectic manifolds admitting a Lagrangian fibration but their geometry can be very different. Our results go towards answering two questions.

First, when are Shafarevich–Tate twists of a Lagrangian fibration Kähler and when are they projective? We find a necessary and sufficient condition for a Shafarevich–Tate twist to be projective and prove that Kählerness is preserved under Shafarevich–Tate deformations.

Second, how are cohomology groups of Shafarevich–Tate twists of a manifold different from cohomology groups of the manifold itself? We compute the second Betti number of all Shafarevich–Tate twists of a manifold and use this result to give a necessary condition for a twist not to be bimeromorphic to a Kähler manifold. Our results are general and work for all deformation classes of irreducible holomorphic symplectic manifolds.