Topology of projective Tate-Shafarevich twists

Abstract

A Tate-Shafarevich twist $X^{\varphi }\to B$ of a fibration $X\to B$ modifies it by a $1$-cocycle of flows of vector fields relative to the base, locally in the analytic topology. Saccà conjectured that the total spaces of two projective Lagrangian fibrations related by such a twist are deformation-equivalent. Assuming that the class of the twist is torsion (which is often equivalent to the twist being realizable in the étale topology), we show that there is an isomorphism $H^{\ast }(X;\mathbb{Q})\cong H^{\ast }(X^{\varphi };\mathbb{Q})$ of graded vector spaces that respects (1) the Hodge structures and (2) the Hodge-Riemann pairing. Consequently, the rational Beauville-Bogomolov-Fujiki lattices of these two spaces are Hodge-similar. Assuming further that $B$ is smooth, and both the original fibration and its twist admit $C^{\infty }$-sections, we show Saccà's conjecture using the theory of degenerate twistor deformations.

20 pages. v2: added some references; modified the statement of Theorem 0.6; removed dependence on [10]