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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.

14M20 Rational and unirational varieties
14C20 Divisors, linear systems, invertible sheaves
37F75 Dynamical aspects of holomorphic foliations and vector fields
Full Text: DOI
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