Projective klt pairs with nef anti-canonical divisor

In this paper, we study a projective klt pair $(X, \Delta)$ with the nef anti-log canonical divisor $-(K_X+\Delta)$ and its maximally rationally connected fibration $\psi: X \dashrightarrow Y$. We prove that the numerical dimension of the anti-log canonical divisor $-(K_X+\Delta)$ on $X$ coincides with that of the anti-log canonical divisor $-(K_{X_y}+\Delta_{X_y})$ on a general fiber $X_y$ of $\psi: X \dashrightarrow Y$, which is an analogue of Ejiri-Gongyo's result formulated for the Kodaira dimension. As a corollary, we reveal a relation between positivity of the anti-canonical divisor and the rational connectedness, which gives a sharper estimate than the question posed by Hacon-$\mathrm{M^{c}}$Kernan. Moreover, in the case of $X$ being smooth, we show that a maximally rationally connected fibration $\psi: X \to Y$ can be chosen to be a morphism to a smooth projective variety $Y$ with numerically trivial canonical divisor, and further that it is locally trivial with respect to the pair $(X, \Delta)$, which can be seen as a generalization of Cao-H\"oring's structure theorem to klt pair cases. Finally, we study the structure of the slope rationally connected quotient for a pair $(X, \Delta)$ with $-(K_X +\Delta)$ nef, and obtain a structure theorem for projective orbifold surfaces.