Markman--Markushevich principle and Lagrangian fibrations on O'Grady six-manifolds
Abstract
To any compact holomorphically symplectic manifold and a nef BBF-isotropic class we associate a quasi-projective variety called the Markman--Markushevich space which parametrises the Lagrangian fibrations with this isotropic class modulo Tate--Shafarevich twists. We give a systematic interpretation of known results about Tate--Shafarevich twists of Lagrangian fibrations using this space, employ it to obtain a conjectural result for the last deformation type for which such a result is not yet proposed, and make some first steps towards its proof. As a byproduct, we generalise the results of Kumar on existence Shioda--Inose partners for abelian surfaces to a slightly more general class of K3 surfaces, and associate a canonical K3 surface with transcendental lattice isometric to U2⊕⟨−2⟩2 to a smooth cubic surface with a choice of a Schläfli double-six. Finally, we construct a noncompact six-manifold with an abelian fibration associated to a cubic surface with a Schläfli double-six, and give its interpretation in terms of the K3 surfaces we introduced.
Related
- Lagrangian fibrations on hyperkahler manifolds have no multiple fibers in codimension one
- Smoothness criterion for Fano orbifolds and bases of Lagrangian fibrations
- Gurtas Lefschetz Fibrations, Rational Blowdowns, and Exotic Symplectic Four-Manifolds
- Lagrangian Foliations on compact Kähler manifolds
- Comparison Principle and Semi-Lagrangian Methods for a Hamilton--Jacobi--Bellman System with Recursive Utility
BibTeX
@misc{deev2026markmanmarkushevich,
title = {{Markman--Markushevich principle and Lagrangian fibrations on O'Grady six-manifolds}},
author = {Rodion Déev},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20044},
eprint = {2610.20044},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20044},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Markman--Markushevich principle and Lagrangian fibrations on O'Grady six-manifolds
AU - Rodion Déev
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.20044
UR - https://arxiv.org/abs/2610.20044
AB - To any compact holomorphically symplectic manifold and a nef BBF-isotropic class we associate a quasi-projective variety called the Markman--Markushevich space which parametrises the Lagrangian fibrations with this isotropic class modulo Tate--Shafarevich twists. We give a systematic interpretation of known results about Tate--Shafarevich twists of Lagrangian fibrations using this space, employ it to obtain a conjectural result for the last deformation type for which such a result is not yet proposed, and make some first steps towards its proof. As a byproduct, we generalise the results of Kumar on existence Shioda--Inose partners for abelian surfaces to a slightly more general class of K3 surfaces, and associate a canonical K3 surface with transcendental lattice isometric to U^2⊕⟨−2⟩^2 to a smooth cubic surface with a choice of a Schläfli double-six. Finally, we construct a noncompact six-manifold with an abelian fibration associated to a cubic surface with a Schläfli double-six, and give its interpretation in terms of the K3 surfaces we introduced.
ER -