Numerical triviality and pullbacks

TL;DR

This paper proves that a pseudo-effective divisor on a smooth projective variety, which is numerically trivial along the fibers of a surjective morphism, is pulled back from a divisor on the base variety.

Contribution

It establishes that f-numerically trivial pseudo-effective divisors are pullbacks of divisors on the base, extending understanding of divisor behavior under morphisms.

Findings

f-numerically trivial divisors are pullbacks of divisors on the base variety

The result applies to smooth complex projective varieties with connected fibers

Provides a criterion for divisors to be pullbacks in the context of algebraic geometry

Abstract

Let f: X \to Z be a surjective morphism of smooth complex projective varieties with connected fibers. Suppose that L is a pseudo-effective divisor on X that is f-numerically trivial. We show that there is a divisor D on Z such that L is numerically equivalent to f^*D.