Big $q$-ample Line Bundles

Abstract

A recent paper of Totaro develops a theory of $q$-ample bundles in characteristic 0. Specifically, a line bundle $L$ on $X$ is $q$-ample if for every coherent sheaf $\mathcal{F}$ on $X$, there exists an integer $m_0$ such that $m\ge m_0$ implies $H^i(X,\mathcal{F}\otimes \mathcal{O}(mL))=0$ for $i>q$. We show that a line bundle $L$ on a complex projective scheme $X$ is $q$-ample if and only if the restriction of $L$ to its augmented base locus is $q$-ample. In particular, when $X$ is a variety and $L$ is big but fails to be $q$-ample, then there exists a codimension 1 subscheme $D$ of $X$ such that the restriction of $L$ to $D$ is not $q$-ample.

arXiv comment9 pages, 2 pdf figures

References in corpus (1)

• Ample subvarieties and q-ample divisors

Cited by in corpus (3)

• A partial converse to the Andreotti-Grauert theorem
• RC-positive metrics on rationally connected manifolds
• Partially ample line bundles and base loci