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, Volume 152, Issue 3–4, pp 533–537 | Cite as

Manifolds containing an ample \(\mathbb {P}^1\)-bundle

  • Daniel LittEmail author


Sommese has conjectured a classification of smooth projective varieties X containing, as an ample divisor, a \(\mathbb {P}^d\)-bundle Y over a smooth variety Z. This conjecture is known if \(d>1\), if \(\dim (X)\le 4\), or if Z admits a finite morphism to an Abelian variety. We confirm the conjecture if the Picard rank \(\rho (Z)=1\), or if Z is not uniruled. In general we reduce the conjecture to a conjectural characterization of projective space: namely that if W is a smooth projective variety, \(\mathscr {E}\) is an ample vector bundle on W, and \(\text {Hom}(\mathscr {E}, T_W)\not =0\), then \(W\simeq \mathbb {P}^n\).

Mathematics Subject Classification:

14J70 14J40 


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  1. 1.
    Andreatta, M., Wiśniewski, J.A.: On manifolds whose tangent bundle contains an ample subbundle. Invent. Math. 146(1), 209–217 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aprodu, M., Kebekus, S., Peternell, T.: Galois coverings and endomorphisms of projective varieties. Math. Z. 260(2), 431–449 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Beltrametti, M.C., Ionescu, P.: A view on extending morphisms from ample divisors. In: Interactions of Classical and Numerical Algebraic Geometry, vol. 496 of Contemp. Math., pp. 71–110. Am. Math. Soc., Providence, RI (2009)Google Scholar
  4. 4.
    Beltrametti, M.C., Sommese, A.J.: The Adjunction Theory of Complex Projective Varieties, vol. 16 of de Gruyter Expositions in Mathematics. Walter de Gruyter & Co., Berlin (1995)Google Scholar
  5. 5.
    Fania, M.L., Sato, E., Sommese, A.J.: On the structure of \(4\)-folds with a hyperplane section which is a \(\mathbb{P}^1\) bundle over a surface that fibres over a curve. Nagoya Math. J. 108, 1–14 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Kollár, J.: Rational Curves on Algebraic Varieties, vol. 32 of 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]. Springer, Berlin (1996)Google Scholar
  7. 7.
    Litt, D.: Non-abelian Lefschetz Hyperplane Theorems, Ph.D. Thesis, Stanford University. (2015)
  8. 8.
    Litt, D.: Non-abelian Lefschetz Hyperplane Theorems. arXiv:1601.07914 (2016)
  9. 9.
    Mourougane, C., Takayama, S.: Hodge metrics and positivity of direct images. J. Reine Angew. Math. 606, 167–178 (2007)MathSciNetzbMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Columbia UniversityNew YorkUSA

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