Strictly nef divisors on K-trivial fourfolds

Abstract

In this note, we prove that for a strictly nef divisor L on a K-trivial fourfold X, L is ample if and only if κ(L) ≥ 0, that is, L is Q-effective. In the case ν(L) = 2, we show that L is Q-effective, thus ample. In the case ν(L) = 2, we show that for any metric h on O X (L) with semipositive curvature current, we must have dim V m = 2 for m ≫ 0, where V m is the closed subscheme defined by the multiplier ideal I(h ⊗m ).

On a numerical criterion for Fano fourfolds

“…In Serrano's work [24], Campana-Peternell's conjecture was put into a more general framework by viewing as a special case of the so-called Serrano's conjecture, and then could be treated by induction on the dimension. In this paper, we follow Serrano's idea (as [1,18] did) and give an affirmative answer to Campana-Peternell's conjecture in dimension 4 under the assumption that c 2 1 (X) • c 2 (X) = 0. Theorem 1.3 (= Corollary 3.2 + Theorem 4.1).…”

“…The organization of this paper follows very closely to [18], by the same reason that the unknown case of Serrano's conjecture for Calabi-Yau threefolds is far from reaching. Hence, we have to get around this difficulty by carefully using the strict nefness; however, we can not solve the problem completely as in [18]. In this paper, the case c 2 1 (X)•c 2 (X) = 0 is still remaining.…”

“…As stated at the beginning of this paper, Campana-Peternell's conjecture can be viewed as a special case of Serrano's conjecture, which is a weak analogue of Fujita's conjecture for strictly nef divisors. Then, as in Fujita's conjecture, it is natural to generalize Serrano's conjecture to log pairs (see [8,Question 3.5] and [18,Conjecture 1.5]). For manifolds with strictly nef anti-canonical divisors, the log version of Serrano's conjecture can be stated as follows, which is called the weak Campana-Peternell's conjecture in this paper: Conjecture 2.4 (Weak Campana-Peternell's conjecture).…”

“…Let µ : T → V be the composition of the normalization of V and the relative canonical model (see [18,Subsection 2.3] for the definition) of its normalization. As in [18,Theorem 3.3], there exists an effective divisor B such that…”

“…Prime Calabi-Yau divisors. As in [18], we also have to consider the so-called prime Calabi-Yau divisors in this paper, since the remaining case of Serrano's conjecture for Calabi-Yau threefolds is far from reaching. Precisely, a prime divisor D on a normal variety X is said to be a prime Calabi-Yau divisor if D is a normal variety with at worst canonical singularities satisfying that K D ∼ 0 and H 1 (D, O D ) = 0.…”