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Fano n-folds with nef tangent bundle and Picard number greater than \(n-5\)

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We prove that Fano n-folds with nef tangent bundle and Picard number greater than \(n-5\) are rational homogeneous manifolds.

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References

  1. Ancona, V., Peternell, T., Wiśniewski, J.A.: Fano bundles and splitting theorems on projective spaces and quadrics. Pac. J. Math. 163(1), 17–42 (1994). http://projecteuclid.org/euclid.pjm/1102622627

  2. Araujo, C.: Identifying quadric bundle structures on complex projective varieties. Geom. Dedic. 139, 289–297 (2009). doi:10.1007/s10711-008-9324-3

    Article  MathSciNet  MATH  Google Scholar 

  3. Bonavero, L., Casagrande, C., Debarre, O., Druel, S.: Sur une conjecture de Mukai. Comment. Math. Helv. 78(3), 601–626 (2003). doi:10.1007/s00014-003-0765-x

    Article  MathSciNet  MATH  Google Scholar 

  4. Campana, F., Peternell, T.: Projective manifolds whose tangent bundles are numerically effective. Math. Ann. 289(1), 169–187 (1991). doi:10.1007/BF01446566

    Article  MathSciNet  MATH  Google Scholar 

  5. Campana, F., Peternell, T.: \(4\)-folds with numerically effective tangent bundles and second Betti numbers greater than one. Manuscr. Math. 79(3–4), 225–238 (1993). doi:10.1007/BF02568341

    Article  MathSciNet  MATH  Google Scholar 

  6. Cho, K., Miyaoka, Y., Shepherd-Barron, N.I.: Characterizations of projective space and applications to complex symplectic manifolds. In: Mori, S., Miyaoka, Y. (eds.) Higher Dimensional Birational Geometry (Kyoto, 1997), vol. 35, pp. 1-88. Mathematical Society of Japan, Tokyo (2002)

  7. Demailly, J.P., Peternell, T., Schneider, M.: Compact complex manifolds with numerically effective tangent bundles. J. Algebr. Geom. 3(2), 295–345 (1994)

    MathSciNet  MATH  Google Scholar 

  8. Hwang, J.M.: Rigidity of rational homogeneous spaces. In: Sanz-Solé, M., Soria, J., Varona, J.L., Verdera, J. (eds.) International Congress of Mathematicians, vol. II, pp. 613–626. European Mathematical Society, Zürich (2006)

  9. Kanemitsu, A.: Fano \(5\)-Folds with nef Tangent Bundles. (2015). arXiv:1503.04579v1

  10. Kawamata, Y., Matsuda, K., Matsuki, K.: Introduction to the minimal model problem. In: Oda, T. (ed.) Algebraic Geometry, Sendai, 1985, vol. 10, pp. 283–360. North-Holland, Amsterdam (1987)

  11. Kollár, J., Miyaoka, Y., Mori, S.: Rational connectedness and boundedness of Fano manifolds. J. Differ. Geom. 36(3), 765–779 (1992). http://projecteuclid.org/euclid.jdg/1214453188

  12. Kollár, J., Mori, S.: Birational geometry of algebraic varieties. In: Cambridge Tracts in Mathematics, vol. 134. Cambridge University Press, Cambridge (1998). doi:10.1017/CBO9780511662560 (with the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original)

  13. Miyaoka, Y.: Numerical characterisations of hyperquadrics. In: Miyajima, K., Furushima, M., Kazama, H., Kodama, A., Noguchi, J., Ohsawa, T., Tsuji, H., Ueda, T. (eds.) Complex Analysis in Several Variables–Memorial Conference of Kiyoshi Oka’s Centennial Birthday, vol. 42, pp. 209–235. Mathematical Society of Japan, Tokyo (2004)

  14. Mok, N.: On Fano manifolds with nef tangent bundles admitting 1-dimensional varieties of minimal rational tangents. Trans. Am. Math. Soc. 354(7), 2639–2658 (2002). doi:10.1090/S0002-9947-02-02953-7

    Article  MathSciNet  MATH  Google Scholar 

  15. Muñoz, R., Occhetta, G., Solá Conde, L.E.: On rank 2 vector bundles on Fano manifolds. Kyoto J. Math. 54(1), 167–197 (2014). doi:10.1215/21562261-2400310

    Article  MathSciNet  MATH  Google Scholar 

  16. Muñoz, R., Occhetta, G., Solá Conde, L.E.: Rational curves, Dynkin diagrams and Fano manifolds with nef tangent bundle. Math. Ann. 361(3–4), 583–609 (2015). doi:10.1007/s00208-014-1083-x

    Article  MathSciNet  MATH  Google Scholar 

  17. Muñoz, R., Occhetta, G., Solá Conde, L.E., Watanabe, K., Wiśniewski, J.A.: A survey on the Campana–Peternell conjecture. Rend. Istit. Mat. Univ. Trieste 47, 127–185 (2015). doi:10.13137/0049-4704/11223

    MathSciNet  MATH  Google Scholar 

  18. Novelli, C., Occhetta, G.: Ruled Fano fivefolds of index two. Indiana Univ. Math. J. 56(1), 207–241 (2007). doi:10.1512/iumj.2007.56.2905

    Article  MathSciNet  MATH  Google Scholar 

  19. Occhetta, G., Solá Conde, L.E., Watanabe, K., Wiśniewski, J.A.: Fano Manifolds Whose Elementary Contractions are Smooth \({\mathbb{P}}^1\)-Fibrations: A Geometric Characterization of Flag Varieties. To apprear in Ann. Sc. Norm. Super. Pisa Cl. Sci. doi:10.2422/2036-2145.201508_007

  20. Okonek, C., Schneider, M., Spindler, H.: Vector bundles on complex projective spaces. In: Coates, J., Helgason, S. (eds.) Progress in Mathematics, vol. 3. Birkhäuser, Boston (1980)

  21. Ottaviani, G.: Spinor bundles on quadrics. Trans. Am. Math. Soc. 307(1), 301–316 (1988). doi:10.2307/2000764

    Article  MathSciNet  MATH  Google Scholar 

  22. Ottaviani, G.: On Cayley bundles on the five-dimensional quadric. Boll. Un. Mat. Ital. A (7) 4(1), 87–100 (1990)

    MathSciNet  MATH  Google Scholar 

  23. Solá Conde, L.E., Wiśniewski, J.A.: On manifolds whose tangent bundle is big and 1-ample. Proc. Lond. Math. Soc. (3) 89(2), 273–290 (2004). doi:10.1112/S0024611504014856

    Article  MathSciNet  MATH  Google Scholar 

  24. Watanabe, K.: Fano 5-folds with nef tangent bundles and Picard numbers greater than one. Math. Z. 276(1–2), 39–49 (2014). doi:10.1007/s00209-013-1185-2

    Article  MathSciNet  MATH  Google Scholar 

  25. Watanabe, K.: \(\mathbb{P}^1\)-bundles admitting another smooth morphism of relative dimension one. J. Algebra 414, 105–119 (2014). doi:10.1016/j.jalgebra.2014.05.026

    Article  MathSciNet  MATH  Google Scholar 

  26. Watanabe, K.: Fano manifolds with nef tangent bundle and large Picard number. Proc. Jpn. Acad. Ser. A Math. Sci. 91(6), 89–94 (2015). doi:10.3792/pjaa.91.89

    Article  MathSciNet  MATH  Google Scholar 

  27. Wiśniewski, J.A.: On deformation of nef values. Duke Math. J. 64(2), 325–332 (1991). doi:10.1215/S0012-7094-91-06415-X

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The author wishes to express his gratitude to Professor Yoichi Miyaoka, his supervisor, for his encouragement, comments and suggestions. The author is also grateful to Professor Kiwamu Watanabe for his helpful comments and suggestions, and for sending us his preprint of [26]. The author also wishes to thank Professor Hiromichi Takagi and Doctor Fumiaki Suzuki for their careful reading of the manuscript and for helpful comments. The author is a JSPS Research Fellow and he is supported by the Grant-in-Aid for JSPS fellows (JSPS KAKENHI Grant Number 15J07608). This work was supported by the Program for Leading Graduate Schools, MEXT, Japan.

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  1. Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan

    Akihiro Kanemitsu

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Kanemitsu, A. Fano n-folds with nef tangent bundle and Picard number greater than \(n-5\) . Math. Z. 284, 195–208 (2016). https://doi.org/10.1007/s00209-016-1652-7

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