Subvarieties with q-ample normal bundle and q-ample subvarieties

TL;DR

This paper investigates subvarieties with q-ample normal bundles in projective varieties, establishing their G2 and G3 properties, and introduces the concept of partially ample subvarieties with applications to connectedness problems.

Contribution

It generalizes previous results on ample subvarieties by defining and analyzing partially ample subvarieties and their properties, including G2 and G3 conditions.

Findings

Subvarieties with q-ample normal bundle are G2 in the ambient space.

Partially ample subvarieties satisfy the G3 property.

Applications to connectedness problems in algebraic geometry.

Abstract

The goal of this article is twofold. On one hand, we study the subvarieties of projective varieties which possess partially ample normal bundle; we prove that they are G2 in the ambient space. This generalizes results of Hartshorne and B\u{a}descu-Schneider. We work with the cohomological partial ampleness introduced by Totaro. On the other hand, we define the concept of a partially ample subvariety, which generalizes the notion of an ample subvariety introduced by Ottem. We prove that partially ample subvarieties enjoy the stronger G3 property. Moreover, we present an application to a connectedness problem posed by Fulton-Hansen and Hartshorne. The results are illustrated with examples.