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Holomorphic Lagrangian subvarieties in holomorphic symplectic manifolds with Lagrangian fibrations and special Kähler geometry

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Abstract

Let M be a holomorphic symplectic Kähler manifold equipped with a Lagrangian fibration \(\pi \) with compact fibers. The base of this manifold is equipped with a special Kähler structure, that is, a Kähler structure \((I, g, \omega )\) and a symplectic flat connection \(\nabla \) such that the metric g is locally the Hessian of a function. We prove that any Lagrangian subvariety \(Z\subset M\) which intersects smooth fibers of \(\pi \) and smoothly projects to \(\pi (Z)\) is a torus fibration over its image \(\pi (Z)\) in B, and this image is also special Kähler. This answers a question of Nigel Hitchin related to Kapustin–Witten BBB/BAA duality.

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Acknowledgements

Great many thanks to Richard Thomas for his comments, questions and examples. We are indebted to Nigel Hitchin for his interest and inspiration, and to Laura Schaposnik for her comments and expertise. We express our gratitude to Stony Brook University and to the SCGP, where this paper was prepared, and for their hospitality. We thank the referees for their suggestions. Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

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Authors and Affiliations

  1. Department of Mathematics, 3-115, Stony Brook University, Stony Brook, NY, 11794-3651, USA

    Ljudmila Kamenova

  2. Instituto Nacional de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorina, 110, Jardim Botânico, Rio de Janeiro, 22460-320, Brazil

    Misha Verbitsky

  3. National Research University Higher School of Economics, Laboratory of Algebraic Geometry, 9 Usacheva Str., Moscow, Russia, 119048

    Misha Verbitsky

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  1. Ljudmila Kamenova
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Correspondence to Ljudmila Kamenova.

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Ljudmila Kamenova was partially supported by a grant from the Simons Foundation/SFARI (522730, LK). Misha Verbitsky was partially supported by the Russian Academic Excellence Project ‘5-100’, FAPERJ E-26/202.912/2018 and CNPq—Process 313608/2017-2.

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Kamenova, L., Verbitsky, M. Holomorphic Lagrangian subvarieties in holomorphic symplectic manifolds with Lagrangian fibrations and special Kähler geometry. European Journal of Mathematics 8, 514–522 (2022). https://doi.org/10.1007/s40879-021-00488-3

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  • DOI: https://doi.org/10.1007/s40879-021-00488-3

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