×

zbMATH — the first resource for mathematics

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.

MSC:
14M20 Rational and unirational varieties
14C20 Divisors, linear systems, invertible sheaves
37F75 Dynamical aspects of holomorphic foliations and vector fields
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Andreatta, M.; Wiśniewski, J. A., On manifolds whose tangent bundle contains an ample subbundle, Invent. Math., 146, 1, 209-217, (2001) · Zbl 1081.14060
[2] Aprodu, M.; Kebekus, S.; Peternell, T., Galois coverings and endomorphisms of projective varieties, Math. Z., 260, 2, 431-449, (2008) · Zbl 1162.14010
[3] Araujo, C., Rational curves of minimal degree and characterizations of projective spaces, Math. Ann., 335, 4, 937-951, (2006) · Zbl 1109.14032
[4] Araujo, C.; Druel, S., On Fano foliations, Adv. Math., 238, 70-118, (2013) · Zbl 1282.14085
[5] Araujo, C.; Druel, S.; Kovács, S. J., Cohomological characterizations of projective spaces and hyperquadrics, Invent. Math., 174, 2, 233-253, (2008) · Zbl 1162.14037
[6] Bădescu, L., Hyperplane sections and deformations, Algebraic Geometry, Bucharest 1982 (Bucharest, 1982), 1-33, (1984), Springer: Springer, Berlin · Zbl 0527.00002
[7] Beltrametti, M. C.; Ionescu, P., A view on extending morphisms from ample divisors, Interactions of Classical and Numerical Algebraic Geometry, 71-110, (2009), American Mathematical Society: American Mathematical Society, Providence, RI · Zbl 1188.14032
[8] Beltrametti, M. C.; Sommese, A. J., The Adjunction Theory of Complex Projective Varieties, (1995), Walter de Gruyter & Co.: Walter de Gruyter & Co., Berlin · Zbl 0845.14003
[9] Bosch, S.; Lütkebohmert, W.; Raynaud, M., Formal and rigid geometry. IV. The reduced fibre theorem, Invent. Math., 119, 2, 361-398, (1995) · Zbl 0839.14014
[10] Campana, F., Connexité rationnelle des variétés de Fano, Ann. Sci. Éc. Norm. Supér. (4), 25, 5, 539-545, (1992) · Zbl 0783.14022
[11] Campana, F., Orbifolds, special varieties and classification theory, Ann. Inst. Fourier (Grenoble), 54, 3, 499-630, (2004) · Zbl 1062.14014
[12] Campana, F.; Peternell, T., Rational curves and ampleness properties of the tangent bundle of algebraic varieties, Manuscripta Math., 97, 1, 59-74, (1998) · Zbl 0932.14024
[13] Fania, M. L.; Sato, E.-I.; Sommese, A. J., On the structure of 4-folds with a hyperplane section which is a P1 bundle over a surface that fibres over a curve, Nagoya Math. J., 108, 1-14, (1987) · Zbl 0602.14014
[14] Grothendieck, A., Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes, Publ. Math. Inst. Hautes Études Sci., 8, 222, (1961)
[15] Grothendieck, A., Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Publ. Math. Inst. Hautes Études Sci., 24, 231, (1965) · Zbl 0135.39701
[16] Grothendieck, A., Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Publ. Math. Inst. Hautes Études Sci., 28, 255, (1966) · Zbl 0144.19904
[17] Hartshorne, R., Ample vector bundles, Publ. Math. Inst. Hautes Études Sci., 29, 63-94, (1966) · Zbl 0173.49003
[18] Hartshorne, R., Algebraic Geometry, (1977), Springer: Springer, New York
[19] Kollár, J., Rational Curves on Algebraic Varieties, (1996), Springer: Springer, Berlin
[20] Kollár, J., Singularities of the Minimal Model Program, (2013), Cambridge University Press: Cambridge University Press, Cambridge
[21] Kollár, J.; Miyaoka, Y.; Mori., S., Rational connectedness and boundedness of Fano manifolds, J. Differential Geom., 36, 3, 765-779, (1992) · Zbl 0759.14032
[22] Kubota, K., Ample sheaves, J. Fac. Sci. Univ. Tokyo Sect. I A Math., 17, 421-430, (1970) · Zbl 0212.26102
[23] Litt, D., Manifolds containing an ample P1 -bundle, Manuscripta Math., 152, 3, 533-537, (2017) · Zbl 1359.14048
[24] Miyaoka, Y., Deformations of a morphism along a foliation and applications, Algebraic Geometry, Bowdoin 1985 (Brunswick, Maine, 1985), 245-268, (1987), American Mathematical Society: American Mathematical Society, Providence, RI
[25] Mori, S., Projective manifolds with ample tangent bundles, Ann. of Math. (2), 110, 3, 593-606, (1979) · Zbl 0423.14006
[26] Sommese, A. J., On manifolds that cannot be ample divisors, Math. Ann., 221, 1, 55-72, (1976) · Zbl 0306.14006
[27] Wahl, J. M., A cohomological characterization of Pn, Invent. Math., 72, 2, 315-322, (1983) · Zbl 0544.14013
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.