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Varieties with ample tangent sheaves

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This paper generalises Mori’s famous theorem about “Projective manifolds with ample tangent bundles” (Mori in Ann Math (2) 110(3):593–606, 1979) to normal projective varieties in the following way: a normal projective variety over \({\mathbb {C}}\) with ample tangent sheaf is isomorphic to the complex projective space.

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References

  1. Aprodu, M., Kebekus, S., Peternell, T.: Galois coverings and endomorphisms of projective varieties. Math. Z. 260(2), 431–449 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ancona, V.: Faisceaux amples sur les espaces analytiques. Trans. Am. Math. Soc. 274(1), 89–100 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  3. Andreatta, M., Wiśniewski, J.A.: On manifolds whose tangent bundle contains an ample subbundle. Invent. Math. 146(1), 209–217 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ballico, E.: On singular varieties with ample tangent bundle. Indag. Math. (N.S.) 4(1), 1–10 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cho, K., Miyaoka, Y., Shepherd-Barron, N.I.: Characterizations of projective space and applications to complex symplectic manifolds. Higher dimensional birational geometry (Kyoto, 1997), Adv. Stud. Pure Math., vol. 35, Math. Soc. Japan, Tokyo, 2002, pp. 1–88

  6. Debarre, O.: On coverings of simple abelian varieties. Bull. Soc. Math. Fr. 134(2), 253–260 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Graf, P., Kovács, S.J.: An optimal extension theorem for 1-forms and the Lipman–Zariski conjecture. Doc. Math. 19, 815–830 (2014)

    MathSciNet  MATH  Google Scholar 

  8. Greb, D., Kebekus, S., Kovács, S.J.: Extension theorems for differential forms and Bogomolov–Sommese vanishing on log canonical varieties. Compos. Math. 146(1), 193–219 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Grauert, H., Peternell, T., Remmert, R. (eds.): Several Complex Variables VII. Springer, Berlin (1994)

    MATH  Google Scholar 

  10. Kawamata, Y.: Minimal models and the Kodaira dimension of algebraic fiber spaces. J. Reine Angew. Math. 363, 1–46 (1985)

    MathSciNet  MATH  Google Scholar 

  11. Kebekus, S.: Characterizing the projective space after Cho, Miyaoka and Shepherd-Barron. In: Bauer, I., Catanese, F., Kawamata, Y., Peternell, T., Siu, Y.T. (eds.) Complex Geometry (Göttingen, 2000), pp. 147–155. Springer, Berlin (2002)

    Chapter  Google Scholar 

  12. Kollár, J.: Lectures on Resolution of Singularities, Annals of Mathematics Studies, vol. 166. Princeton University Press, Princeton, NJ (2007)

    MATH  Google Scholar 

  13. Kubota, K.: Ample sheaves. J. Fac. Sci. Univ. Tokyo Sect. I A Math. 17, 421–430 (1970)

    MathSciNet  MATH  Google Scholar 

  14. Lazarsfeld, R.: Positivity in Algebraic Geometry. II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 49. Springer, Berlin (2004). Positivity for vector bundles, and multiplier ideals

  15. Liu, J.: Characterization of projective spaces and \({\mathbb{P}}^{r}\)-bundles as ample divisors. Nagoya Math. J. arXiv:1611.05823 (2016)

  16. Miyaoka, Y., Mori, S.: A numerical criterion for uniruledness. Ann. Math. (2) 124(1), 65–69 (1986)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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Sieder, P. Varieties with ample tangent sheaves. manuscripta math. 157, 257–261 (2018). https://doi.org/10.1007/s00229-017-0979-7

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  • DOI: https://doi.org/10.1007/s00229-017-0979-7

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