Projective klt pairs with nef anti-canonical divisor

Abstract

In this paper, we study projective klt pairs (X, ) with nef anti-log canonical divisor -(K X + ) and their maximal rationally connected fibration : X Y . We prove that the numerical dimension of -(K X +) on X coincides with that of -(K Xy + Xy ) on a general fiber X y of : X Y , which is an analogue of Ejiri-Gongyo's result formulated for the Kodaira dimension. As a corollary, we obtain a relation between the positivity of the anti-canonical divisor and the rational connectedness, which provides a sharper estimate than that in Hacon-M c Kernan's question. Moreover, in the case of X being smooth, we show that X admits a "holomorphic" maximal rationally connected fibration to a smooth projective variety Y with numerically trivial canonical divisor, and also that this is locally trivial with respect to the pair (X, ), which generalizes Cao-Hring's structure theorem to the case of klt pairs. Finally, we consider slope rationally connected quotients of (X, ) and obtain a structure theorem for projective orbifold surfaces.

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