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Shafarevich-Tate groups of holomorphic Lagrangian fibrations II (2407.09178v3)
Published 12 Jul 2024 in math.AG and math.CV
Abstract: Let $X$ be a compact hyperk\"ahler manifold with a Lagrangian fibration $\pi\colon X\to B$. A Shafarevich-Tate twist of $X$ is a holomorphic symplectic manifold with a Lagrangian fibration $\pi\varphi\colon X\varphi\to B$ which is isomorphic to $\pi$ locally over the base. In particular, $\pi\varphi$ has the same fibers as $\pi$. A twist $X\varphi$ corresponds to an element $\varphi$ in the Shafarevich-Tate group of $X$. We show that $X\varphi$ is K\"ahler 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 K\"ahler manifold.