Log canonical $3$-fold complements

Abstract

We expand the theory of log canonical 3-fold complements. More precisely, fix a set Λ ⊂ Q satisfying the descending chain condition with Λ ⊂ Q, and let (X, B + B ′ ) be a log canonical 3-fold with coeff(B) ∈ Λ and K X + B Q-Cartier. Then, there exists a natural number n, only depending on Λ, such that the following holds. Given a contraction f : X → T and t ∈ T with K X + B + B ′ ∼ Q 0 over t, there exists Γ ≥ 0 such that Γ ∼ −n(K X + B) over t ∈ T , and (X, B + Γ/n) is log canonical.

Boundedness of log pluricanonical representations of log Calabi-Yau pairs in dimension 2

“…The following lemma is a modification of [15, Lemma 2.16] in our situation. is injective, where [8,Proposition 8.4], there exists an integer N 1 depending only on m such that…”

“…induces a natural B-birational map g l! : [8,Proposition 8.4], there exists an integer N 1 depending only on m such that |ρ m (Bir…”

Boundedness of $(ε, n)$-Complements for Surfaces

“…It originates from his earlier work on anti-canonical systems on Fano threefolds [Sho79]. The theory is further developed in [Sho00,PS01,PS09,Bir19,HLS19,Sho20], see [FM18,Xu19b,Xu19c,FMX19] for recent works.…”

“…For the purpose of induction in birational geometry, we need the theory of complements for infinite sets Γ when we apply the adjunction formula. In order to prove the existence of complements for infinite sets Γ, a frequently-used strategy is to replace Γ by some finite set which is contained in the closure of Γ, see [PS09,Bir19,FM18,HLS19,FMX19]. Thus we need to study the theory of complements when Γ is a subset of real numbers.…”

On singularities of threefold weighted blowups

“…There are some recent progress for the theory of complements. We refer reads to [FMX19], [HLS19], [Sh19] and [Xu19] for more details.…”