Skip to main content
Log in

Cohomological characterizations of projective spaces and hyperquadrics

  • Published:
Inventiones mathematicae Aims and scope

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

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

    Article  MATH  MathSciNet  Google Scholar 

  2. Araujo, C.: Rational curves of minimal degree and characterizations of projective spaces. Math. Ann. 335(4), 937–951 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Araujo, C., Kollár, J.: Rational curves on varieties. In: Higher Dimensional Varieties and Rational Points (Budapest 2001). Bolyai Soc. Math. Stud., vol. 12, pp. 13–68. Springer, Berlin (2003)

    Google Scholar 

  4. Beauville, A.: Symplectic singularities. Invent. Math. 139(3), 541–549 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bogomolov, F., McQuillan, M.: Rational curves on foliated varieties. IHES (2001, Preprint)

  6. Bonavero, L., Casagrande, C., Druel, S.: On covering and quasi-unsplit families of rational curves. J. Eur. Math. Soc. (JEMS) 9(1), 45–76 (2007)

    MATH  MathSciNet  Google Scholar 

  7. Campana, F.: Connexité rationnelle des variétés de Fano. Ann. Sci. Éc. Norm. Supér., IV. Sér. 25(5), 539–545 (1992)

    MATH  MathSciNet  Google Scholar 

  8. Campana, F.: Orbifolds, special varieties and classification theory. Ann. Inst. Fourier (Grenoble) 54(3), 499–630 (2004)

    MATH  MathSciNet  Google Scholar 

  9. Campana, F., Peternell, T.: Rational curves and ampleness properties of the tangent bundle of algebraic varieties. Manuscr. Math. 97(1), 59–74 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  10. Debarre, O.: Higher-Dimensional Algebraic Geometry. Universitext. Springer, New York (2001)

    Google Scholar 

  11. Eisenbud, D.: Commutative Algebra. Grad. Texts Math., vol. 150. Springer, New York (1995). With a view toward algebraic geometry

    MATH  Google Scholar 

  12. Fujita, T.: On the structure of polarized varieties with Δ-genera zero. J. Fac. Sci., Univ. Tokyo, Sect. IA Math. 22, 103–115 (1975)

    MATH  MathSciNet  Google Scholar 

  13. Graber, T., Harris, J., Starr, J.: Families of rationally connected varieties. J. Am. Math. Soc. 16(1), 57–67 (2003) (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  14. Harder, G., Narasimhan, M.S.: On the cohomology groups of moduli spaces of vector bundles on curves. Math. Ann. 212, 215–248 (1975). MR0364254 (51 ♯509)

    Article  MATH  MathSciNet  Google Scholar 

  15. Huybrechts, D., Lehn, M.: The Geometry of Moduli Spaces of Sheaves. Aspects Math., vol. E31. Friedr. Vieweg & Sohn, Braunschweig (1997)

    MATH  Google Scholar 

  16. Hwang, J.-M.: Geometry of minimal rational curves on Fano manifolds. In: School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000). ICTP Lect. Notes, vol. 6, pp. 335–393. Abdus Salam Int. Cent. Theoret. Phys., Trieste (2001). MR1919462 (2003g:14054)

    Google Scholar 

  17. Hwang, J.-M.: Deformation of holomorphic maps onto Fano manifolds of second and fourth Betti numbers 1. Ann. Inst. Fourier 57(3), 815–823 (2007)

    MATH  MathSciNet  Google Scholar 

  18. Hwang, J.-M., Mok, N.: Birationality of the tangent map for minimal rational curves. Asian J. Math. 8(1), 51–64 (2004)

    MATH  MathSciNet  Google Scholar 

  19. Källström, R.: Liftable derivations for generically separably algebraic morphisms of schemes. Trans. Am. Math. Soc. (2008). Article electronically published on June 26, 2008. DOI: 10.1090/S0002-9947-08-04534-0

  20. Kebekus, S.: Families of singular rational curves. J. Algebr. Geom. 11(2), 245–256 (2002)

    MATH  MathSciNet  Google Scholar 

  21. Kebekus, S., Solá Conde, L., Toma, M.: Rationally connected foliations after Bogomolov and McQuillan. J. Algebr. Geom. 16(1), 65–81 (2007). MR2257320

    MATH  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  23. Kollár, J.: Rational Curves on Algebraic Varieties. Ergeb. Math. Grenzgeb., vol. 32. Springer, Berlin (1996)

    Google Scholar 

  24. Matsumura, H.: Commutative Algebra, 2nd edn. Math. Lecture Note Ser., vol. 56. Benjamin/Cummings Publishing Co., Inc., Reading, Mass. (1980)

    MATH  Google Scholar 

  25. Mehta, V.B., Ramanathan, A.: Semistable sheaves on projective varieties and their restriction to curves. Math. Ann. 258(3), 213–224 (1981/82)

    Google Scholar 

  26. Miyaoka, Y.: Deformations of a morphism along a foliation and applications. In: Algebraic Geometry, Bowdoin, 1985 (Brunswick, Maine, 1985). Proc. Sympos. Pure Math., vol. 46, pp. 245–268. Am. Math. Soc., Providence, RI (1987)

    Google Scholar 

  27. Miyaoka, Y.: Relative deformations of morphisms and applications to fibre spaces. Comment. Math. Univ. St. Pauli 42(1), 1–7 (1993)

    MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  29. Seidenberg, A.: Derivations and integral closure. Pac. J. Math. 16, 167–173 (1966)

    MATH  MathSciNet  Google Scholar 

  30. Siu, Y.T., Yau, S.T.: Compact Kähler manifolds of positive bisectional curvature. Invent. Math. 59(2), 189–204 (1980). MR577360 (81h:58029)

    Article  MATH  MathSciNet  Google Scholar 

  31. Wahl, J.M.: A cohomological characterization of P n. Invent. Math. 72(2), 315–322 (1983)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Sándor J. Kovács.

Additional information

Mathematics Subject Classification (2000)

14M20

Rights and permissions

Reprints and permissions

About this article

Cite this article

Araujo, C., Druel, S. & Kovács, S. Cohomological characterizations of projective spaces and hyperquadrics. Invent. math. 174, 233–253 (2008). https://doi.org/10.1007/s00222-008-0130-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-008-0130-1

Keywords

Navigation