Ample and spanned vector bundles of minimal curve genus
Let \({\mathcal E}\) be an ample and spanned vector bundle of rank \(n - 1\) on a compact complex manifold \(X\) of dimension \(n \geq 3\) and let \(g(X, {\mathcal E})\) be the curve genus, defined by \[ 2g (X, {\mathcal E}) - 2 = (K_X + c_1 ({\mathcal E})) c_{n - 1} ({\mathcal E}). \] The fact that \({\mathcal E}\) is ample and spanned implies the inequality \(g(X, {\mathcal E}) \geq h^{1,0} (X)\). The main result of the paper is the characterization of pairs \((X, {\mathcal E})\) as above satisfying the condition \[ g(X, {\mathcal E}) = h^{1,0} (X)\tag{*} \] A consequence of this is that (*) implies the vanishing of \(h^0 (m(K_X + \text{det} {\mathcal E}))\) for all integers \(m \geq 1\). The conjecture related to the equivalence between (*) and the vanishing of \(h^0 (K_X + \text{det} {\mathcal E})\), suggested by well known adjunction theoretic facts, is also discussed and proved in some special instances.
scientific article; zbMATH DE number 937059
scientific article