Rational connectedness modulo the non-nef locus
Résumé
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.