Seshadri constants for vector bundles

Abstract

Abstract We introduce Seshadri constants for line bundles in a relative setting. They generalize the classical Seshadri constants of line bundles on projective varieties and their extension to vector bundles studied by Beltrametti–Schneider–Sommese and Hacon. There are similarities to the classical theory. In particular, we give a Seshadri-type ampleness criterion, and we relate Seshadri constants to jet separation and to asymptotic base loci. Smoothness is generally not part of our assumptions. Thus we improve on some of the known results already for line bundles. We give two applications of our new version of Seshadri constants. First, a celebrated result of Mori can be restated as saying that any Fano manifold whose tangent bundle has positive Seshadri constant at a point is isomorphic to a projective space. We conjecture that the Fano condition can be removed. Among other results in this direction, we prove the conjecture for surfaces. Second, we prove that our Seshadri constants can be used to control separation of jets for direct images of pluricanonical bundles, in the spirit of a relative Fujita-type conjecture of Popa and Schnell.