Projective klt pairs with nef anti-canonical divisor

Abstract

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 $\psi :X⇢Y$. 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 $\psi :X⇢Y$, 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-${\mathrm{M}}^{\mathrm{c}}$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-Höring'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.

arXiv comment36 pages, comments are welcome

References in corpus (3)

• Relative adjoint transcendental classes and Albanese maps of compact Kaehler manifolds with nef Ricci curvature
• Nef anti-canonical divisors and rationally connected fibrations
• Orbifold slope-rational connectedness

Cited by in corpus (7)

• Structure theorem for projective klt pairs with nef anti-canonical divisor
• Pseudo-effective and numerically flat reflexive sheaves
• On asymptotic base loci of relative anti-canonical divisors of algebraic fiber spaces
• On the structure of a log smooth pair in the equality case of the Bogomolov-Gieseker inequality
• Structure of projective varieties with nef anticanonical divisor: the case of log terminal singularities
• On compact Kähler manifold with strongly pseudo-effective tangent bundle
• Positivity of anticanonical divisors in algebraic fibre spaces