On positivity and base loci of vector bundles

Abstract

The aim of this note is to shed some light on the relationships among some notions of positivity for vector bundles that arose in recent decades. Our purpose is to study several of the positivity notions studied for vector bundles with some notions of asymptotic base loci that can be defined on the variety itself, rather than on the projectivization of the given vector bundle. We relate some of the different notions conjectured to be equivalent with the help of these base loci, and we show that these can help simplify the various relationships between the positivity properties present in the literature. In particular, we define augmented and restricted base loci B + (E) and B − (E) of a vector bundle E on the variety X , as generalizations of the corresponding TB was partially supported by DFG Grant BA 1559/6-1. SJK notions studied extensively for line bundles. As it turns out, the asymptotic base loci defined here behave well with respect to the natural map induced by the projectivization of the vector bundle E.

Nef anti-canonical divisors and rationally connected fibrations

“…We first consider the case when E = 0 and G is locally free. If f is generically finite, then the assertion is proved in the same way as [32,Lemma 2.15,2]. Otherwise, we need to recall some definition in [2].…”

“…If f is generically finite, then the assertion is proved in the same way as [32,Lemma 2.15,2]. Otherwise, we need to recall some definition in [2]. For a vector bundle E on a quasi-projective variety X, we define Bs(E) to be the set of points x ∈ X such that the stalk E x is not generated by global sections of E.…”

“…) ≥ 2 by the assumption. Next, we show (2). We first consider the case when E = 0 and G is locally free.…”

“…The weak positivity of in the sense of the above definition is actually equivalent to pseudo-effectivity in the sense of [BKKM + 15, Definition 5.1] when is a vector bundle, and it is equivalent to the usual pseudo-effectivity when is a line bundle.…”

Holomorphic symmetric differentials and a birational characterization of abelian varieties

“…The aim of this work is to answer positively a question raised in the framework of the investigation on stable base loci for vector bundles started in [1].…”