In this paper, we study projective klt pairs (X, ∆) with nef anti-log canonical divisor −(KX + ∆) and their maximal rationally connected fibration ψ: X −−• Y. We prove that the numerical dimension of −(KX +∆) on X coincides with that of −(KXy +∆Xy ) on a general fiber Xy of ψ: 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–McKernan’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–Höring’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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