Characterizations of projective spaces and hyperquadrics.

*(English)*Zbl 1296.14038Let \(X\) 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.

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.

Reviewer: Andreas Höring (Nice)

##### MSC:

14M20 | Rational and unirational varieties |