Characterization of projective spaces and \(\mathbb{P}^{r}\)-bundles as ample divisors. (English) Zbl 1411.14054
The projective space is characterized by the ampleness of its tangent bundle. Several generalizations of this result has been proposed in terms of positive properties of the tangent bundle (see the Introduction of the paper under review and references therein). The main result in the paper under review proposed the following characterization of the projective spaces:
Theorem 1.1. Let \(X\) be a projective manifold of dimension \(n\) and suppose that its tangent bundle contains an ample subsheaf of positive rank; then \(X\) is the projective space of dimension \(n\) and the subsheaf is either the whole tangent bundle or a split vector bundle, direct sum of \(r\) copies of the hyperplane bundle.
In the particular case in which the subsheaf is coming from the image of an ample vector bundle one gets (see Corollary 2) that the existence of a nonzero map from an ample vector bundle to the tangent bundle characterizes the projective space (as conjectured in [D. Litt, Manuscr. Math. 152, No. 3–4, 533–537 (2017; Zbl 1359.14048)]. As an application (see Theorem 1.3), a complete classification of projective manifolds containing a projective bundle as an ample divisor is presented.
Theorem 1.1. Let \(X\) be a projective manifold of dimension \(n\) and suppose that its tangent bundle contains an ample subsheaf of positive rank; then \(X\) is the projective space of dimension \(n\) and the subsheaf is either the whole tangent bundle or a split vector bundle, direct sum of \(r\) copies of the hyperplane bundle.
In the particular case in which the subsheaf is coming from the image of an ample vector bundle one gets (see Corollary 2) that the existence of a nonzero map from an ample vector bundle to the tangent bundle characterizes the projective space (as conjectured in [D. Litt, Manuscr. Math. 152, No. 3–4, 533–537 (2017; Zbl 1359.14048)]. As an application (see Theorem 1.3), a complete classification of projective manifolds containing a projective bundle as an ample divisor is presented.
Reviewer: Roberto Munoz (Madrid)
MSC:
| 14M20 | Rational and unirational varieties |
| 14C20 | Divisors, linear systems, invertible sheaves |
| 37F75 | Dynamical aspects of holomorphic foliations and vector fields |
Citations:
Zbl 1359.14048References:
| [1] | M.Andreatta and J. A.Wiśniewski, On manifolds whose tangent bundle contains an ample subbundle, Invent. Math.146(1) (2001), 209-217.10.1007/PL00005808 · Zbl 1081.14060 · doi:10.1007/PL00005808 |
| [2] | M.Aprodu, S.Kebekus and T.Peternell, Galois coverings and endomorphisms of projective varieties, Math. Z.260(2) (2008), 431-449.10.1007/s00209-007-0282-5 · Zbl 1162.14010 · doi:10.1007/s00209-007-0282-5 |
| [3] | C.Araujo, Rational curves of minimal degree and characterizations of projective spaces, Math. Ann.335(4) (2006), 937-951.10.1007/s00208-006-0775-2 · Zbl 1109.14032 · doi:10.1007/s00208-006-0775-2 |
| [4] | C.Araujo and S.Druel, On Fano foliations, Adv. Math.238 (2013), 70-118.10.1016/j.aim.2013.02.003 · Zbl 1282.14085 · doi:10.1016/j.aim.2013.02.003 |
| [5] | C.Araujo, S.Druel and S. J.Kovács, Cohomological characterizations of projective spaces and hyperquadrics, Invent. Math.174(2) (2008), 233-253.10.1007/s00222-008-0130-1 · Zbl 1162.14037 · doi:10.1007/s00222-008-0130-1 |
| [6] | L.Bădescu, “Hyperplane sections and deformations”, in Algebraic Geometry, Bucharest 1982 (Bucharest, 1982), Lecture Notes in Mathematics 1056, Springer, Berlin, 1984, 1-33.10.1007/BFb0071766 · Zbl 0527.00002 · doi:10.1007/BFb0071766 |
| [7] | M. C.Beltrametti and P.Ionescu, “A view on extending morphisms from ample divisors”, in Interactions of Classical and Numerical Algebraic Geometry, Contemporary Mathematics 496, American Mathematical Society, Providence, RI, 2009, 71-110.10.1090/conm/496/09719 · Zbl 1188.14032 · doi:10.1090/conm/496/09719 |
| [8] | M. C.Beltrametti and A. J.Sommese, The Adjunction Theory of Complex Projective Varieties, de Gruyter Expositions in Mathematics 16, Walter de Gruyter & Co., Berlin, 1995.10.1515/9783110871746 · Zbl 0845.14003 · doi:10.1515/9783110871746 |
| [9] | S.Bosch, W.Lütkebohmert and M.Raynaud, Formal and rigid geometry. IV. The reduced fibre theorem, Invent. Math.119(2) (1995), 361-398.10.1007/BF01245187 · Zbl 0839.14014 · doi:10.1007/BF01245187 |
| [10] | F.Campana, Connexité rationnelle des variétés de Fano, Ann. Sci. Éc. Norm. Supér. (4)25(5) (1992), 539-545.10.24033/asens.1658 · Zbl 0783.14022 · doi:10.24033/asens.1658 |
| [11] | F.Campana, Orbifolds, special varieties and classification theory, Ann. Inst. Fourier (Grenoble)54(3) (2004), 499-630.10.5802/aif.2027 · Zbl 1062.14014 · doi:10.5802/aif.2027 |
| [12] | F.Campana and T.Peternell, Rational curves and ampleness properties of the tangent bundle of algebraic varieties, Manuscripta Math.97(1) (1998), 59-74.10.1007/s002290050085 · Zbl 0932.14024 · doi:10.1007/s002290050085 |
| [13] | M. L.Fania, E.-i.Sato and A. J.Sommese, On the structure of 4-folds with a hyperplane section which is a P^1 bundle over a surface that fibres over a curve, Nagoya Math. J.108 (1987), 1-14.10.1017/S0027763000002622 · Zbl 0602.14014 · doi:10.1017/S0027763000002622 |
| [14] | A.Grothendieck, Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes, Publ. Math. Inst. Hautes Études Sci.8 (1961), 222. |
| [15] | A.Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Publ. Math. Inst. Hautes Études Sci.24 (1965), 231. · Zbl 0135.39701 |
| [16] | A.Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Publ. Math. Inst. Hautes Études Sci.28 (1966), 255.10.1007/BF02684343 · Zbl 0144.19904 · doi:10.1007/BF02684343 |
| [17] | R.Hartshorne, Ample vector bundles, Publ. Math. Inst. Hautes Études Sci.29 (1966), 63-94. · Zbl 0173.49003 |
| [18] | R.Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics 52, Springer, New York, 1977.10.1007/978-1-4757-3849-0 · Zbl 0367.14001 · doi:10.1007/978-1-4757-3849-0 |
| [19] | J.Kollár, Rational Curves on Algebraic Varieties, 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] 32, Springer, Berlin, 1996.10.1007/978-3-662-03276-3 · Zbl 0877.14012 · doi:10.1007/978-3-662-03276-3 |
| [20] | J.Kollár, Singularities of the Minimal Model Program, Cambridge Tracts in Mathematics 200, Cambridge University Press, Cambridge, 2013, With a collaboration of Sándor Kovács.10.1017/CBO9781139547895 · Zbl 1282.14028 · doi:10.1017/CBO9781139547895 |
| [21] | J.Kollár, Y.Miyaoka and S.Mori., Rational connectedness and boundedness of Fano manifolds, J. Differential Geom.36(3) (1992), 765-779.10.4310/jdg/1214453188 · Zbl 0759.14032 · doi:10.4310/jdg/1214453188 |
| [22] | K.Kubota, Ample sheaves, J. Fac. Sci. Univ. Tokyo Sect. I A Math.17 (1970), 421-430. · Zbl 0212.26102 |
| [23] | D.Litt, Manifolds containing an ample P^1 -bundle, Manuscripta Math.152(3) (2017), 533-537.10.1007/s00229-016-0872-9 · Zbl 1359.14048 · doi:10.1007/s00229-016-0872-9 |
| [24] | Y.Miyaoka, “Deformations of a morphism along a foliation and applications”, in Algebraic Geometry, Bowdoin 1985 (Brunswick, Maine, 1985), Proceedings of Symposia in Pure Mathematics 46, American Mathematical Society, Providence, RI, 1987, 245-268.10.1090/pspum/046.1/927960 · Zbl 0659.14008 · doi:10.1090/pspum/046.1/927960 |
| [25] | S.Mori, Projective manifolds with ample tangent bundles, Ann. of Math. (2)110(3) (1979), 593-606.10.2307/1971241 · Zbl 0423.14006 · doi:10.2307/1971241 |
| [26] | A. J.Sommese, On manifolds that cannot be ample divisors, Math. Ann.221(1) (1976), 55-72.10.1007/BF01434964 · Zbl 0306.14006 · doi:10.1007/BF01434964 |
| [27] | J. M.Wahl, A cohomological characterization of P^n, Invent. Math.72(2) (1983), 315-322.10.1007/BF01389326 · Zbl 0544.14013 · doi:10.1007/BF01389326 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
