On Nonvanishing for uniruled log canonical pairs

Abstract

We prove the Nonvanishing conjecture for uniruled projective log canonical pairs of dimension n, assuming the Nonvanishing conjecture for smooth projective varieties in dimension n−1. We also show that the existence of good minimal models for non-uniruled projective klt pairs in dimension n implies the existence of good minimal models for projective log canonical pairs in dimension n.

The Nonvanishing Problem for varieties with nef anticanonical bundle

“…By passing to a minimal model of X we may assume that K X is nef, and then the conclusion follows from the cases treated above. If the variety is uniruled, then the conclusion follows from [54,Theorem 1.1]. As above, the argument is inductive, hence it works in principle also in higher dimensions.…”

“…We first assume that X admits a nontrivial morphism to an abelian variety. Then K X C is semiample: this is the content of [36,Corollary 1.2] and [53,Corollary 3.2], see also [54,Lemma 4.1], and the argument is inductive, hence it works in principle also in higher dimensions. One of the main ingredients in the proof of [53,Corollary 3.2] is the subadditivity of the Iitaka dimension from [49].…”

“…If L is a nef divisor as in Theorem A (b), then we may write L D K X C C M for a nef divisor M on X. The method, which is also present in a similar form in [31,58,59], is to modify L by making and M smaller until the new divisor hits the boundary of the pseudoeffective cone, in which case a slight modification of one of the main results of [61] yields that this new divisor is num-effective; a similar strategy in the context of usual pairs was employed in [54]. Some additional technical considerations then allow us to deduce that the divisor L is also num-effective.…”

The Nonvanishing problem for varieties with nef anticanonical bundle

“…We first assume that X has a nontrivial morphism to an abelian variety. Then K X + ∆ is semiample: this is the content of [Hu16, Corollary 1.2] and [Laz19, Corollary 3.2], see also [LM21,Lemma 4.1], and the argument is inductive, hence works in principle also in higher dimensions. One of the main ingredients in the proof of [Laz19, Corollary 3.2] is the subadditivity of the Iitaka dimension from [KP17].…”

“…If L is a nef divisor as in Theorem A(b), then we may write L = K X + ∆ + M for a nef divisor M on X. The method -also present in a similar form in [LP20a, LP20b, HL20] -is to modify L by making ∆ and M smaller until the new divisor hits the boundary of the pseudoeffective cone, in which case a slight modification of one of the main results of [LT21] yields that this new divisor is num-effective; a similar strategy in the context of usual pairs was employed in [LM21]. Some additional technical considerations then allow to deduce that the divisor L is also num-effective.…”