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Birational geometry of irreducible holomorphic symplectic tenfolds of O’Grady type

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Abstract

In this paper, we analyse the birational geometry of O’Grady ten dimensional manifolds, giving a characterization of Kähler classes and lagrangian fibrations. Moreover, we study symplectic compactifications of intermediate jacobian fibrations of smooth cubic fourfolds.

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Notes

  1. These birational models have a canonical isometry of their second cohomology with that of X, due to the fact that birational maps are well defined in codimension one as the canonical classes are nef.

  2. Notice that without a section the lagrangian fibration does not give a family of abelian varieties.

  3. The same construction already appeared in [36, Section 4.5] but there was an initial step to prove a wrong result, namely the existence of a twisted theta divisor of divisibility 1.

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Acknowledgements

We would like to thank Valeria Bertini, Antonio Rapagnetta, Ulrike Rieß  and Giulia Saccà for useful conversations. We are also very grateful to the referee for comments and suggestions. Both authors were partially supported by “Progetto di ricerca INdAM per giovani ricercatori: Pursuit of IHS”. GM is member of the INDAM-GNSAGA and received support from it. CO thanks the Research Council of Norway project no. 250104 for financial support.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta San Donato 5, 40126, Bologna, Italy

    Giovanni Mongardi

  2. Dipartimento di Matematica, Università degli studi di Roma Tor Vergata, via della ricerca scientifica 1, 00133, Roma, Italy

    Claudio Onorati

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  1. Giovanni Mongardi
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Correspondence to Giovanni Mongardi.

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To Olivier Debarre, whose work on IHS manifolds (and more) has been an inspiration for us.

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Mongardi, G., Onorati, C. Birational geometry of irreducible holomorphic symplectic tenfolds of O’Grady type. Math. Z. 300, 3497–3526 (2022). https://doi.org/10.1007/s00209-021-02966-6

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