Log abundance of the moduli b-divisors of lc-trivial fibrations

Abstract

We prove that the moduli b-divisor of an lc-trivial fibration from a log canonical pair is log abundant. This result follows from a theorem on the restriction of the moduli b-divisor, which is also obtained by E. Floris and V. Lazić [FL19]. We give an alternative proof in Section 4.3, based on a theory of lc-trivial morphisms. We also prove a theorem on extending a finite cover over a closed subvariety of the same degree, on a variety over an arbitrary field.

Existence of flips for generalized lc pairs

“…1) f, ϕ are birational morphisms, π ′ is an equidimensional contraction, Y only has Qfactorial toroidal singularities, and V is smooth, and (2) there exist two R-divisors B Y and E on Y , such that (a)K Y + B Y = f * (K X + B) + E, (b) B Y ≥ 0, E ≥ 0, and B Y ∧ E = 0, (c) (Y, B Y) is lc quasi-smooth, and any lc center of (Y, B Y ) on X is an lc center of (X, B).Proof. This result follows from[AK00], see also[Hu20, Theorem B.6], [Kaw15, Theorem 2] and [Has19, Step 2 of Proof of Lemma 3.2].…”

On generalized lc pairs with $\mathrm{\textbf b}$-log abundant nef part

“…Moreover, f : (X, B, M) → Z is called good if γ * M Z = M Z , and f : (X, B, M) → Z is called a gdlt-trivial morphism/U if (Z, B Z , M Z ) and (X, B, M) are both gdlt. We remark that if M = 0, then a gdlt-trivial morphism/U is called a good dlt model in [Hu20].…”

“…for any prime divisor D over Z. By applying the weak-semistable reduction [AK00] (see [Hu20,Theorem B.6] for details), we only need to consider finitely many prime divisors D over Z. The theorem follows from Lemma 2.22, [Hu20, Lemma 3.3], and [HL19, Lemma 4.1] after replacing (X, B i , M i ) and a i for each i so that ||B i − B|| is sufficiently small and B i is contained in the minimal affine subspace of Weil Q (X) which contains B.…”

An abundance theorem for generalised pairs

“…In this subsection we will introduce the notion of nef and abundant divisor and elementary properties in the setting of R-divisors. Most contents are taken from [Hu20a].…”

“…The prototype of generalised pairs is derived from a canonical bundle formula for a certain class of algebraic fibre spaces, namely lc-trivial fibrations. For a quick and recent survey of these objects we refer to [Hu20a,3.1]. The approach of [BZh16] allows us to apply most philosophy of minimal model program to log pairs whose boundaries birationally contain nef parts.…”