# Rational connectedness modulo the non-nef locus

### Amaël Broustet

Université Lille I, Villeneuve d'Ascq, France### Gianluca Pacienza

Université de Strasbourg, France

## Abstract

It is well known that a smooth projective Fano variety is rationally connected. Recently Zhang [Z2] (and later Hacon and McKernan [HM] as a special case of their work on the Shokurov RC-conjecture) proved that the same conclusion holds for a klt pair $(X,\Delta)$ such that $-(K_X+\Delta)$ is big and nef. We prove here a natural generalization of the above result by dropping the nefness assumption. Namely we show that a klt pair $(X,\Delta)$ such that $-(K_X+\Delta)$ is big is rationally connected modulo the non-nef locus of $-(K_X+\Delta)$. This result is a consequence of a more general structure theorem for arbitrary pairs $(X,\Delta)$ with $-(K_X+\Delta)$ pseff.

## Cite this article

Amaël Broustet, Gianluca Pacienza, Rational connectedness modulo the non-nef locus. Comment. Math. Helv. 86 (2011), no. 3, pp. 593–607

DOI 10.4171/CMH/235