Shafarevich-Tate groups of holomorphic Lagrangian fibrations II

TL;DR

The paper develops a comprehensive framework for Shafarevich-Tate twists of Lagrangian fibrations on irreducible holomorphic symplectic (hyperkähler) manifolds. It constructs and analyzes Sha twists X^φ, proving Kählerness when a multiple of φ lies in Sha^0, and establishing a period-map Torelli-based strategy to compare twisted and untwisted manifolds. It proves that projective twists correspond to torsion Sha classes and that degenerate/twistor-type deformations realize twists, with a detailed cohomological/topological description of how H^2 and higher pushforwards behave under twisting. The results yield criteria for Kählerness and Fujiki class C, describe the discrete vs. continuous components of Sha, and connect twist geometry to the Lefschetz-type structure of the base via the BBF form and period data, offering a robust toolkit for understanding the global geometry of holomorphic symplectic fibrations under twists.

Abstract

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

Theorems & Definitions (111)

• Definition 1.1.1
• Definition 1.1.2
• Definition 1.1.3
• Definition 1.1.4
• Definition 1.1.5
• Definition 1.1.6
• Theorem A: \ref{['subsub proof of kahlerness']}, Theorem \ref{['theorem kahler substitute']}
• Remark 1.2.1
• Theorem B: \ref{['subsub proof of theorem b']}
• Theorem C: \ref{['subsub proof of theorem c']}