On ampleness of vector bundles (Q1979883)

The paper under review gives a necessary and sufficient condition for the ampleness of a semistable holomorphic bundle with zero discriminant on a complex projective variety. Recall that a holomorphic bundle \(E\) on a projective complex variety \(X\) is \textit{ample} if the associated tautological bundle \(\mathcal{O}_{\mathbb{P}(E)}(1)\) on the projective bundle \(\mathbb{P}(E)\) is ample, namely if \(\mathcal{O}_{\mathbb{P}(E)}(1)\) supports a smooth hermitian metric with positive curvature. On the other hand, \(E\) is called \textit{Griffiths positive} whenever \(E\) supports on itself a smooth hermitian metric with positive curvature. It is known that if \(E\) is Griffiths positive then \(E\) is ample and a famous conjecture states the converse. Moreover, it is also well-known that ampleness of \(E\) implies ampleness of \(det(E)\) but the converse is not true in general. The main contribution of this paper is to close this circle of statements for semistable bundle with zero discriminant (Theorem 1): in this particular case, let \((E,h)\) be a vector bundle on a projective complex variety with hermetian metric \(h\) such that \(det(E)\) is ample. Then \((E,h)\) is Griffiths positive.