Ample subvarieties and q-ample divisors

TL;DR

This paper introduces a new concept of ampleness for higher-codimension subvarieties using q-ample line bundles, explores their geometric properties, and constructs a counterexample to a classical vanishing theorem.

Contribution

It defines a novel notion of ampleness for subvarieties of higher codimension and studies their geometric and cohomological properties.

Findings

Ample subvarieties satisfy Lefschetz hyperplane theorems.

They exhibit numerical positivity properties.

Counterexample to the converse of the Andreotti-Grauert vanishing theorem was constructed.

Abstract

We introduce a notion of ampleness for subschemes of higher codimension using the theory of q-ample line bundles. We also investigate certain geometric properties satisfied by ample subvarieties, e.g. the Lefschetz hyperplane theorems and numerical positivity. Using these properties, we also construct a counterexample to the converse of the Andreotti-Grauert vanishing theorem.