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Vector bundles on Fano threefolds and K3 surfaces

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Abstract

Let X be a Fano threefold, and let \(S\subset X\) be a K3 surface. Any moduli space \({\mathscr {M}}_{S}\) of simple vector bundles on S carries a holomorphic symplectic structure. Following an idea of Tyurin, we show that in some cases, those vector bundles which come from X form a Lagrangian subvariety of \({\mathscr {M}}_{S}\). We illustrate this with a number of concrete examples.

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References

  1. Arbarello, E., Cornalba, M., Griffiths, P.: Geometry of Algebraic Curves, Volume II. Grund. der Math. Wiss., vol. 268. Springer, Heidelberg (2011)

    Book  Google Scholar 

  2. Altman, A., Kleiman, S.: Compactifying the Picard scheme. Adv. Math. 35(1), 50–112 (1980)

    Article  MathSciNet  Google Scholar 

  3. Beauville, A.: Variétés de Prym et jacobiennes intermédiaires. Ann. Sci. Éc. Norm. Sup. 10, 309–391 (1977)

    Article  Google Scholar 

  4. Beauville, A.: Determinantal hypersurfaces. Mich. Math. J. 48, 39–64 (2000)

    Article  MathSciNet  Google Scholar 

  5. Beauville, A.: Vector bundles on the cubic threefold. Contemp. Math. 312, 71–86 (2002)

    Article  MathSciNet  Google Scholar 

  6. Beauville, A.: An introduction to Ulrich bundles. Eur. J. Math. 4(1), 26–36 (2018)

    Article  MathSciNet  Google Scholar 

  7. Brambilla, M.-C., Faenzi, D.: Moduli spaces of rank-2 ACM bundles on prime Fano threefolds. Mich. Math. J. 60(1), 113–148 (2011)

    Article  MathSciNet  Google Scholar 

  8. Hartshorne, R.: Stable vector bundles of rank 2 on \(P^3\). Math. Ann. 238(3), 229–280 (1978)

    Article  MathSciNet  Google Scholar 

  9. Hassett, B.: Special cubic fourfolds. Compos. Math. 120(1), 1–23 (2000)

    Article  MathSciNet  Google Scholar 

  10. Huybrechts, D., Lehn, M.: The Geometry of Moduli Spaces of Sheaves. Cambridge University Press, Cambridge (2010)

    Book  Google Scholar 

  11. Mukai, S.: Symplectic structure of the moduli space of sheaves on an abelian or K3 surface. Invent. Math. 77(1), 101–116 (1984)

    Article  MathSciNet  Google Scholar 

  12. Mukai, S.: On the Moduli Space of Bundles on K3 Surfaces, I. Vector Bundles on Algebraic Varieties (Bombay, 1984), vol. 11, pp. 341–413. Tata Institute of Fundamental Research Studies in Mathematics, Bombay (1987)

  13. Saccà, G.: Birational geometry of the intermediate Jacobian fibration of a cubic fourfold. Preprint. arXiv:2002.01420

  14. Tyurin, A.: The moduli spaces of vector bundles on threefolds, surfaces and curves, I. In: Vector Bundles. Collected Works, vol. I, pp. 176–213. Universitätsverlag Göttingen, Göttingen (2008)

  15. Voisin, C.: Hodge Theory and Complex Algebraic Geometry, II. Cambridge Studies in Advanced Mathematics, vol. 77. Cambridge University Press, Cambridge (2007)

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Acknowledgements

I am grateful to Daniele Faenzi for pointing out the reference [7], which has allowed me to extend and simplify the results of Sect. 8.

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  1. CNRS-Laboratoire J.-A. Dieudonné, Université Côte d’Azur, Parc Valrose, 06108, Nice Cedex 2, France

    Arnaud Beauville

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  1. Arnaud Beauville
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Correspondence to Arnaud Beauville.

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To Fabrizio, on his 70th birthday.

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Beauville, A. Vector bundles on Fano threefolds and K3 surfaces. Boll Unione Mat Ital 15, 43–55 (2022). https://doi.org/10.1007/s40574-020-00266-1

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  • DOI: https://doi.org/10.1007/s40574-020-00266-1

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