Seshadri constants for vector bundles

TL;DR

This paper introduces a new concept of Seshadri constants for vector bundles in a relative setting, extending classical ideas and providing applications in Fano manifolds, nef cone conjectures, and jet separation.

Contribution

It generalizes Seshadri constants to vector bundles in a relative context and applies this to problems in algebraic geometry such as Fano manifolds and nef cones.

Findings

Fano manifolds with positive Seshadri tangent bundle are projective spaces

New nef classes identified on self-products of curves

Seshadri constants control jet separation for pluricanonical bundles

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. We give three 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 restate a classical conjecture on the nef cone of self-products of curves in terms of semistability of higher conormal sheaves, which we use to identify new nef classes on self-products of curves. Third, 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.