Shafarevich-Tate groups of holomorphic Lagrangian fibrations II

2024/07/12 by Abasheva, Anna · 2 citations
#14F25 #14J42 #32J25 #32Q15 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2407.09178

Abstract

Let X be a compact hyperkähler manifold with a Lagrangian fibration π\colon X→ B. A Shafarevich-Tate twist of X is a holomorphic symplectic manifold with a Lagrangian fibration π^φ\colon X^φ→ B which is isomorphic to π locally over the base. In particular, π^φ has the same fibers as π. A twist X^φ corresponds to an element φ in the Shafarevich-Tate group of X. We show that X^φ is Kähler when a multiple of φ lies in the connected component of unity of the Shafarevich-Tate group and give a necessary condition for X^φ to be bimeromorphic to a Kähler manifold.

Cited by

• Sections of Jacobian fibrations over lines
• The Tate-Shafarevich group of a polarised K3 surface

Related

• Shafarevich-Tate groups of holomorphic Lagrangian fibrations
• Sections of Lagrangian fibrations on holomorphic symplectic manifolds
• Twists of intermediate Jacobian fibrations
• Holomorphic Lagrangian fibrations of toric hyperkahler manifolds
• Topology of Lagrangian fibrations and Hodge theory of hyper-Kähler manifolds