Numerical character of the effectivity of adjoint line bundles

Abstract

In this note we show that, for any log-canonical pair (X, ∆), K X + ∆ is Q-effective if its Chern class contains an effective Q-divisor.

Kodaira dimension of algebraic fiber spaces over abelian varieties

“…The following result due to Campana-Peternell [12, Thm 3.1] together with its generalization in [11] will allow us to conclude. 4.20.…”

“…In this case we conclude by using another crucial result due Campana-Peternell cf. [12,Thm 3.1], as well as the generalization in [11].…”

Varieties fibred over abelian varieties with fibres of log general type

“…So, by Theorem 4.1, K X + B ≡ D for some Q-divisor D ≥ 0. By Campana-Koziarz-Pǎun [8] or Kawamata [18], we may assume that K X + B ∼ Q D.…”

“…This uses the Fourier-Mukai transform of the derived category of A combined with the above LMMP and vanishing theorems (see Section 4). One can replace K X + B ≡ D with K X + B ∼ Q D [8,18]. Moreover, after an appropriate resolution, we may in addition assume that X is smooth and that Supp(B +D) has simple normal crossing singularities.…”

On generic vanishing for pluricanonical bundles

“…Proposition 3.10 ( [CKP,Corollary 3.2]). Let X be a normal projective variety, ∆ be a boundary Q-divisor on X such that K X + ∆ is Q-Cartier and (X, ∆) is log canonical, and f : X → A be a morphism to an abelian variety.…”