Seshadri constants for vector 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.Contents 28 8. Direct images of pluricanonical sheaves 31 References 36

Note on compact Kähler manifold with strongly pseudo-effective tangent bundle

“…Now, we give another proof of Corollary 1.3 of [Iwa18] using Segre current. This result was also proved by Fulger and Murayama [FM19] by using Seshadri constants of vector bundles.…”

“…the definition below), the Albanese torus reduces to a point from which we infer that the manifold is rational connected. In fact, the only possibility is the projective space as shown by [Iwa18], [FM19]. In Section 3, using Serge currents, we give new proof of this fact.…”