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Characterizations of projective spaces and hyperquadrics. (English) Zbl 1296.14038
Let $$X$$ be a complex projective manifold of dimension $$n$$. Since S. Mori’s famous characterisation [Ann. Math. (2) 110, 593–606 (1979; Zbl 0423.14006)] of the projective space as the only projective manifold such that the tangent bundle $$T_X$$ is ample, many authors have studied the relation between the positivity of $$T_X$$ and the biregular geometry of $$X$$. Recently C. Araujo et al. [Invent. Math. 174, No. 2, 233–253 (2008; Zbl 1162.14037)] showed that if $$X$$ admits an ample line bundle $$L$$ such that $$H^0(X, \bigwedge^k T_X \otimes L^{-k}) \neq 0$$ for some $$k \in \{ 1, \dots, n \}$$, then $$X$$ is a projective space or a hyperquadric. If the Picard number is equal to one they also gave an analogous statement involving the tensor powers $$T_X^{\otimes k} \otimes L^{-k}$$.
In the paper under review the authors generalise these statements to projective manifolds polarised by an ample vector bundle of sufficiently high rank. The main theorem can be stated as follows: let $$X$$ be a projective manifold, and let $$E$$ be ample vector bundle on $$X$$ of rank $$r$$. If $$H^0(X, T_X^{\otimes k} \otimes \det E^{-1}) \neq 0$$ for some $$k \leq r$$, then $$X$$ is a projective space or hyperquadric.

##### MSC:
 14M20 Rational and unirational varieties
##### Keywords:
rational varieties; projective space; quadric hypersurface
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