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Fano 5-folds with nef tangent bundles and Picard numbers greater than one

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Abstract

We prove that smooth Fano 5-folds with nef tangent bundles and Picard numbers greater than one 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)

    Article  MATH  Google Scholar 

  2. Borel, A.: Cohomologie Des Espaces Localement Compacts D’aprs J. Leray. Lecture Notes in Mathematics, vol. 2. Springer, Berlin (1964)

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

    Article  MATH  MathSciNet  Google Scholar 

  4. Casagrande, C.: Quasi-elementary contractions of Fano manifolds. Compos. Math. 144(6), 1429–1460 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  5. Campana, F., Peternell, T.: Projective manifolds whose tangent bundles are numerically effective. Math. Ann. 289, 169–187 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  6. 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)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cho, K., Miyaoka, Y., Shepherd-Barron, N.I.: Characterizations of projective space and applications to complex symplectic manifolds. In: Higher Dimensional Birational Geometry, Kyoto, 1997, Advanced Studies in Pure Mathematics, vol. 35, pp. 1–88. Mathematical Society of Japan, Tokyo (2002)

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

    MATH  MathSciNet  Google Scholar 

  9. Grothendieck, A.: Le groupe de Brauer I, II, III. Exemples et compléments. In: Dix Exposés Sur La Coho- Mologie Des Schémas. North-Holland, Amsterdam, pp. 46–188 (1968)

  10. Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, No. 52. Springer, New York (1977).

  11. Hwang, J.M.: Rigidity of rational homogeneous spaces. In: Proceedings of ICM. 2006 Madrid, vol. II, pp. 613–626. European Mathematical Society (2006)

  12. Kollár, J.: Rational curves on algebraic varieties. In: Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 32. Springer, Berlin (1996)

  13. Kollár, J., Miyaoka, Y., Mori, S.: Rational connectedness and boundedness of Fano manifolds. J. Differ. Geom. 36(3), 765–779 (1992)

    MATH  Google Scholar 

  14. Miyaoka, Y.: Numerical characterisations of hyperquadrics. In: Complex Analysis in Several variables-Memorial Conference of Kiyoshi Oka’s Centennial Birthday, Advanced Studies of Pure Mathematics, vol. 42, pp. 209–235. Mathematical Society of Japan, Tokyo (2004)

  15. 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)

    Article  MATH  MathSciNet  Google Scholar 

  16. Mori, S.: Projective manifolds with ample tangent bundles. Ann. Math. (2) 110(3), 593–606 (1979)

    Article  MATH  Google Scholar 

  17. Novelli, C., Occhetta, G.: Ruled Fano fivefolds of index two. Indiana Univ. Math. J. 56(1), 207–241 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  18. Occhetta, G., Wiśniewski, J.A.: On Euler–Jaczewski sequence and Remmert–Van de Ven problem for toric varieties. Math. Z. 241, 35–44 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  19. Okonek, C., Schneider, M., Spindler, H.: Vector Bundles Over Complex Projective Space, Progress in Math., vol. 3, Birkhäuser, Boston (1980)

  20. Ottaviani, G.: Spinor bundles on quadrics. Trans. Am. Math. Soc. 307(1), 301–316 (1988)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  22. Paranjape, K.H., Srinivas, V.: Self-maps of homogeneous spaces. Invent. Math. 98(2), 425–444 (1989)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

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Acknowledgments

The author would like to thank Dr. Kazunori Yasutake for reading this paper and his comments. He also would like to express his gratitude to referees for their careful reading of the text and useful suggestions and comments. The author is partially supported by the Grant-in-Aid for Research Activity Start-up \(\sharp 24840008\) from the Japan Society for the Promotion of Science.

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  1. Course of Mathematics, Programs in Mathematics, Electronics and Informatics, Graduate School of Science and Engineering, Saitama University, Shimo-Okubo 255, Sakura-ku, Saitama-shi, 338-8570, Japan

    Kiwamu Watanabe

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  1. Kiwamu Watanabe
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Correspondence to Kiwamu Watanabe.

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

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