On a question of Demailly-Peternell-Schneider

TL;DR

This paper proves that for a surjective morphism from a log canonical pair with nef anti-canonical divisor, the target variety's anti-canonical divisor is pseudo-effective, answering a longstanding open question in algebraic geometry.

Contribution

It provides an affirmative answer to an open question by Demailly-Peternell-Schneider and Peternell regarding the pseudo-effectiveness of the anti-canonical divisor under certain morphisms.

Findings

If $-(K_X+D)$ is nef, then $-K_Y$ is pseudo-effective.

The result applies to surjective morphisms from log canonical pairs to ${f Q}$-Gorenstein varieties.

The paper resolves a question posed in 2001 about the behavior of anti-canonical divisors under morphisms.

Abstract

This note aims to give an affirmative answer to an open question posed by Demailly-Peternell-Schneider [DPS] in 2001 and recently by Peternell [P] again. Let $f:X\mapsto Y$ be a surjective morphism from a log canonical pair (X,D) onto a ${\mathbb Q}$-Gorenstein variety $Y$. If $-(K_X+D)$ is nef, we show that $-K_Y$ is pseudo-effective.