Nef anti-canonical divisors and rationally connected fibrations

Abstract

We study the Iitaka-Kodaira dimension of nef relative anti-canonical divisors. As a consequence, we prove that given a complex projective variety with klt singularities, if the anti-canonical divisor is nef, then the dimension of a general fibre of the maximal rationally connected fibration is at least the Iitaka-Kodaira dimension of the anti-canonical divisor.

13 pages, v2: an error in Theorem 3.1 corrected

References in corpus (6)

• Globally $F$-regular and log Fano varieties
• Bertini theorems for F-singularities
• F-singularities in families
• Generalized test ideals, sharp F-purity, and sharp test elements
• Shokurov's Rational Connectedness Conjecture
• On positivity and base loci of vector bundles

Cited by in corpus (6)

• Projective klt pairs with nef anti-canonical divisor
• On asymptotic base loci of relative anti-canonical divisors of algebraic fiber spaces
• Two Non-Vanishing results concerning the Anti-Canonical Bundle
• On the superadditivity of anticanonical Iitaka dimension
• On the Iitaka conjecture for anticanonical divisors in positive characteristic
• Positivity of anticanonical divisors in algebraic fibre spaces