On subvarieties with ample normal bundle

TL;DR

This paper proves that subvarieties with ample normal bundle impose strong positivity conditions on divisors and cycles, confirming conjectures about their bigness and numerical properties in algebraic geometry.

Contribution

It establishes that a pseudoeffective divisor trivial on such subvarieties has numerical dimension zero, and confirms that curves with ample normal bundle are big, answering Peternell's question.

Findings

Pseudoeffective divisors trivial on subvarieties with ample normal bundle have numerical dimension zero.

Cycle classes of curves with ample normal bundle are big.

Provides new positivity properties for subvarieties with ample normal bundle.

Abstract

We show that a pseudoeffective R-divisor has numerical dimension 0 if it is numerically trivial on a subvariety with ample normal bundle. This implies that the cycle class of a curve with ample normal bundle is big, giving an affirmative answer to a question of Peternell. We also give other positivity properties such subvarieties.