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On Fano manifolds of Picard number one

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Abstract

Küchle classified the Fano fourfolds that can be obtained as zero loci of global sections of homogeneous vector bundles on Grassmannians. Surprisingly, his classification exhibits two families of fourfolds with the same discrete invariants. Kuznetsov asked whether these two types of fourfolds are deformation equivalent. We show that the answer is positive in a very strong sense, since the two families are in fact the same! This phenomenon happens in higher dimension as well.

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References

  1. Iliev, A., Manivel, L.: Fano manifolds of Calabi-Yau Hodge type. J. Pure Appl. Algebra 219, 2225-2244 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  2. Iskovskih V., Prokhorov Y.: Fano varieties. In: Algebraic Geometry V, pp. 1-247, Encycl. Math. Sci. 47, Springer (1999)

  3. Kuznetsov A.: On Küchle manifolds with Picard number greater than 1, arXiv.math/1501.03299

  4. Küchle, O.: On Fano 4-folds of index 1 and homogeneous vector bundles over Grassmannians. Math. Z. 218, 563-575 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  5. Mukai S.: Fano 3-folds. In: Complex Projective Geometry (Trieste, 1989/Bergen, 1989), pp. 255-263, London Math. Soc. Lecture Note Ser. 179, Cambridge University Press (1992)

  6. Snow, D.: Cohomology of twisted holomorphic forms on Grassmann manifolds and quadric hypersurfaces. Math. Ann. 276, 159-176 (1986)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

I thank A. Kuznetsov for his interesting comments.

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  1. UMR 7373 CNRS/Aix-Marseille Université, Technopôle Château-Gombert, Institut de Mathématiques de Marseille, 39 rue Frêdêric Joliot-Curie, 13453, Marseille Cedex 13, France

    Laurent Manivel

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  1. Laurent Manivel
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Correspondence to Laurent Manivel.

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Manivel, L. On Fano manifolds of Picard number one. Math. Z. 281, 1129–1135 (2015). https://doi.org/10.1007/s00209-015-1523-7

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  • DOI: https://doi.org/10.1007/s00209-015-1523-7

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