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.

On asymptotic base loci of relative anti-canonical divisors of algebraic fiber spaces

“…Algebraic proof of Campana-Cao-Matsumura's equality. The following theorem is a slight extension of [CCM19, Theorem 1.2], which generalizes Hacon-M c Kernan's question proved in [EG19]. In this subsection, we give an algebraic proof for this equality.…”

“…All the fibers X y are isomorphic to F in our situation, but we will use the notations X y and G y (which looks like depending on y) to carefully treat the automorphisms of F . In this step, we will compare the above semi-ample fibrations and prove that the Stein factorization of the restriction of Φ to X y actually coincides with ϕ by using the solution of Hacon-M c Kernan's question proved by Ejiri-Gongyo (see [HM07,EG19] and see [CCM19] for its generalization).…”

“…By applying the result of [EG19] to our new morphism φ : X → Y again (see Step 2), we can see that the restriction map of sections of −mK X/Y to X y is injective. Hence it is sufficient to prove that any section in H 0 (X y , −mK Xy ) can be extended to a section in H 0 (X, −mK X/Y ).…”

Iitaka conjecture for anticanonical divisors in positive characteristics

“…Nonetheless, if we work in low dimensions, the same proofs go through with little modifications. In particular, the results in [EG19] permit to prove C − n,1 . Theorem 0.2 (C − n,1 , see theorem 2.1).…”

“…We may wonder then, does the same inequality also hold in positive characteristics? In the paper [EG19], S. Ejiri and Y. Gongyo studied positivity properties of the relative anticanonical divisor in a fibration f : X → Z, also in positive characteristics. The paper [Cha22], that proves C − n,m in characteristic 0, combines the same tools with techniques involving canonical bundle formula results as studied in [Amb05].…”