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Fano manifolds with many free divisors

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Abstract

We classify Fano manifolds admitting many nontrivial free divisors. Our classification is motivated by a conjecture posed by Mukai.

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References

  1. Andreatta, M., Ballico, E., Wiśniewski, J.A.: Two theorems on elementary contractions. Math. Ann. 297(2), 191–198 (1993)

    Article  MathSciNet  Google Scholar 

  2. Andreatta, M., Chierici, E., Occhetta, G.: Generalized Mukai conjecture for special Fano varieties. Cent. Eur. J. Math. 2(2), 272–293 (2004)

    Article  MathSciNet  Google Scholar 

  3. Andreatta, M., Wiśniewski, J.A.: A note on nonvanishing and applications. Duke Math. J. 72(3), 739–755 (1993)

    Article  MathSciNet  Google Scholar 

  4. Bonavero, L., Casagrande, C., Debarre, O., Druel, S.: Sur une conjecture de Mukai. Comment. Math. Helv. 78(3), 601–626 (2003)

    Article  MathSciNet  Google Scholar 

  5. Casagrande, C.: On the Picard number of divisors in Fano manifolds. Ann. Sci. Éc. Norm. Supér. 45(3), 363–403 (2012)

    Article  MathSciNet  Google Scholar 

  6. Fujita, K.: Simple normal crossing Fano varieties and log Fano manifolds. Nagoya Math. J. 214, 95–123 (2014)

    Article  MathSciNet  Google Scholar 

  7. Fujita, K.: The Mukai conjecture for log Fano manifolds. Cent. Eur. J. Math. 12(1), 14–27 (2014)

    MathSciNet  MATH  Google Scholar 

  8. Fujita, K.: Around the Mukai conjecture for Fano manifolds. Eur. J. Math. 2(1), 120–139 (2016)

    Article  MathSciNet  Google Scholar 

  9. Fujita, T.: On polarized manifolds whose adjoint bundles are not semipositive. In: Oda, T. (ed.) Algebraic Geometry, Sendai, 1985. Advanced Studies in Pure Mathematics, vol. 10, pp. 167–178. North-Holland, Amdterdam (1987)

  10. Kobayashi, S., Ochiai, T.: Characterizations of complex projective spaces and hyperquadrics. J. Math. Kyoto Univ. 13, 31–47 (1973)

    MathSciNet  MATH  Google Scholar 

  11. Kollár, J., Mori, S.: Birational Geometry of Algebraic Varieties. Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge (1998)

    Book  Google Scholar 

  12. Maeda, H.: Classification of logarithmic Fano threefolds. Compositio Math. 57(1), 81–125 (1986)

    MathSciNet  MATH  Google Scholar 

  13. Mori, S.: Threefolds whose canonical bundles are not numerically effective. Ann. Math. 116(1), 133–176 (1982)

    Article  MathSciNet  Google Scholar 

  14. Mukai, S.: Problems on characterization of the complex projective space. In: Birational Geometry of Algebraic Varieties. Open Problems. Proceedings of the 23rd International Symposium of the Taniguchi Foundation at Katata, pp. 57–60. Katata (1988)

  15. Novelli, C., Occhetta, G.: Rational curves and bounds on the Picard number of Fano manifolds. Geom. Dedicata. 147, 207–217 (2010)

    Article  MathSciNet  Google Scholar 

  16. Wiśniewski, J.A.: On a conjecture of Mukai. Manuscripta Math. 68(2), 135–141 (1990)

    Article  MathSciNet  Google Scholar 

  17. Wiśniewski, J.A.: On Fano manifolds of large index. Manuscripta Math. 70(2), 145–152 (1991)

    Article  MathSciNet  Google Scholar 

  18. Wiśniewski, J.A.: On contractions of extremal rays of Fano manifolds. J. Reine Angew. Math. 417, 141–157 (1991)

    MathSciNet  MATH  Google Scholar 

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Correspondence to Kento Fujita.

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This work was supported by JSPS KAKENHI Grant Number 22K03269.

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Fujita, K. Fano manifolds with many free divisors. European Journal of Mathematics 8, 909–931 (2022). https://doi.org/10.1007/s40879-022-00559-z

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  • DOI: https://doi.org/10.1007/s40879-022-00559-z

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