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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
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