Topology of projective Tate-Shafarevich twists
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910121894871040 |
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| author | Bai, David Zhiyuan |
| author_facet | Bai, David Zhiyuan |
| contents | A Tate-Shafarevich twist $X^ϕ\to B$ 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 $H^\ast(X;\mathbb Q)\cong H^\ast(X^ϕ;\mathbb Q)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_21554 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Topology of projective Tate-Shafarevich twists Bai, David Zhiyuan Algebraic Geometry Differential Geometry A Tate-Shafarevich twist $X^ϕ\to B$ 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 $H^\ast(X;\mathbb Q)\cong H^\ast(X^ϕ;\mathbb Q)$ 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. |
| title | Topology of projective Tate-Shafarevich twists |
| topic | Algebraic Geometry Differential Geometry |
| url | https://arxiv.org/abs/2602.21554 |