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Rational curves on primitive symplectic varieties of OG\(_6^s\)-type

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Abstract

We prove that any ample class on a primitive symplectic variety that is a locally trivial deformation of O’Grady’s singular 6-dimensional example is proportional to the first Chern class of a uniruled divisor. This result answers a question of Lehn-Mongardi-Pacienza (Lehn et al. 2021, Remark 4.7) extending their result (Lehn et al. 2021, Theorem 1.3) for primitive symplectic varieties of this deformation type. In order to get our result we produce examples of positive uniruled divisors on singular moduli spaces of sheaves that are locally trivial deformations of O’Grady’s example. We show that all possible square and divisibility of polarizations arise on such moduli spaces, hence we conclude by maximality of the monodromy group of this deformation class of singular symplectic varieties. Finally we provide some considerations on the smooth case, motivating why our techniques fail on the smooth setting and pointing out what information is needed to conclude the existence of positive uniruled divisors on smooth O’Grady’s sixfolds starting from our result.

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Notes

  1. This is the reference for the same result in the smooth setting, i.e. the case of ihs manifods.

  2. For a complex compact variety X the integration over its singular top cohomology group \(\textrm{H}^{2n}(X,\mathbb Z)\) is defined as \(\int _X\alpha :=[X]\cap \alpha \), where \([X]\in \textrm{H}_{2n}(X,\mathbb Z)\) is the fundamental class. Integration commutes with pullbacks by bimeromorphic morphisms.

  3. This is the reference for the same result in the smooth setting, i.e. the case of ihs manifods.

  4. Note that the terminology used in [29] is different from the one we have chosen, as he refers to primitive symplectic varieties as irreducible symplectic varieties, cfr. Definition 1.(3) therein.

  5. A divisor is positive if so is its first Chern class.

  6. [26, Theorem 1.7] computes the Fujiki constant for any singular moduli space on a surface with trivial canonical bundle, but in this particular case, i.e. when the singular moduli space admits a symplectic resolution given by an ihs manifold, the computation is easier as the Fujiki constant equals the one of its resolution.

  7. This is the reference in the smooth setting, i.e. for ihs manifolds.

  8. For the definition of transvection, or Eichler transvection, we refer to [9, Lemma 3.1]

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Acknowledgements

We wish to thank Christian Lehn, Giovanni Mongardi, Claudio Onorati and Gianluca Pacienza for their hints and suggestions.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Genova (DIMA), Genoa, Italy

    Valeria Bertini

  2. Université Paris-Saclay, CNRS, Laboratoire de Mathématiques d’Orsay, Rue Michel Magat, Bât. 307, 91405, Orsay, France

    Annalisa Grossi

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  1. Valeria Bertini
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  2. Annalisa Grossi
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Correspondence to Valeria Bertini.

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The authors have been partially supported by the DFG through the research grant Le 3093/2-2. Valeria Bertini has been partially funded by Portuguese national funds through FCT (Fundação para a Ciência e a Tecnologia) through the project EXPL/MAT-PUR/1162/2021 and through CMUP (Centro de Matemática da Universidade do Porto) under the project UIDB/00144/2020, and by the PRIN Project 2020 “Curves, Ricci flat varieties and their interactions”. Annalisa Grossi is supported by ERC Synergy Grant ERC-2020-SyG-854361-HyperK.

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Bertini, V., Grossi, A. Rational curves on primitive symplectic varieties of OG\(_6^s\)-type. Math. Z. 304, 36 (2023). https://doi.org/10.1007/s00209-023-03296-5

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