The first main theorem on complements: from global to local

Prokhorov, Yuri; Shokurov, V. V.

doi:10.1070/im2001v065n06abeh000366

Cited by 20 publications

(15 citation statements)

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“…This proves our claim. Now the same arguments as in [PS01,§3] show that K X + (1 − δ)D is n-complemented near f −1 (o). Since D ∈ P n , there is an n-complement K X + D + of K X + D near f −1 (o) and moreover, a(E, X, D + ) = −1.…”

Section: Two Important Particular Cases Of Effective Adjunctionsupporting

confidence: 53%

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Towards the second main theorem on complements

2008

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We prove the boundedness of complements modulo two conjectures: Borisov-Alexeev conjecture and effective adjunction for fibre spaces. We discuss the last conjecture and prove it in two particular cases.We give a sketch of the proof of our main results in Section 4. One can see that our proof essentially uses reduction to lower-dimensional global pairs. However it is expected that an improvement of our method can use reduction to local questions in the same dimension. In fact we hope that the hypothesis in our main theorem 1.4 should be the existence of local complements and Conjecture 1.1 for ε-lt Fano varieties (without a boundary), where ε ≥ ε o > 0, ε o is a constant depending only on the dimension (cf. Addendum 1.6 and Corollary 1.7). If dim X = 2, we can take ε o = 1/7. Note also that our main theorem 1.4 is weaker than one can expect. We think that the pair (X, B) can be taken arbitrary log-semi-Fano (possibly not klt and not FT) and possible boundary multiplicities can be taken arbitrary real numbers in [0, 1] (not only in Φ(R)). The only hypothesis we have to assume is the existence of an Rcomplement B + ≥ B (cf. [Sho00]). However the general case actually needs a finite set of natural numbers for complements, and there are no such universal number for all complements (cf. [Sho93, Example 5.2.1]). Preliminaries2.1. Notation. All varieties are assumed to be algebraic and defined over an algebraically closed field k of characteristic zero. Actually, the main results hold for any k of characteristic zero not necessarily algebraically closed since they are related to singularities of general members of linear systems (see [Sho93, 5.1]). We use standard terminology and notation of the Log Minimal Model Program (LMMP) [KMM87], [Kol92], and [Sho93]. For the definition of complements and their properties we refer to [Sho93], [Sho00], [Pro01] and [PS01]. Recall that a log pair (or a log variety) is a pair (X, D) consisting of a normal variety X and a boundary D, i.e., an R-divisor D = d i D i with multiplicities 0 ≤ d i ≤ 1. As usual K X denotes the canonical (Weil) divisor of a variety X. Sometimes we will write K instead of K X if no confusion is likely. Everywhere below a(E, X, D) denotes the discrepancy of E with respect to K X + D. Recall the standard notation:In this paper we use the following strong version of ε-log terminal and ε-log canonical property. Definition 2.2. A log pair (X, B) is said to be ε-log terminal (ε-log canonical) if totaldiscr(X, B) > −1 + ε (resp., totaldiscr(X, B) ≥ −1 + ε). Licensed to New York Univ, Courant Inst. Prepared on Sat Jul 4 02:39:10 EDT 2015 for download from IP 128.122.253.228. License or copyright restrictions may apply to redistribution; see http://www.ams.org/license/jour-dist-license.pdf TOWARDS THE SECOND MAIN THEOREM ON COMPLEMENTS 155 Usually we work with R-divisors. An R-divisor is an R-linear combination of prime Weil divisors. AnTwo R-divisors D and D are said to be Q-(resp., R-)linearly equivalent if D − D is a Q-(resp., R-)linear combination of principal divi...

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Section: Two Important Particular Cases Of Effective Adjunctionsupporting

confidence: 53%

“…Therefore D ≡ D h is nef and big over Z. Now apply construction of [PS01,§3] to (X, D) over Z. There are two cases:…”

Section: Two Important Particular Cases Of Effective Adjunctionmentioning

confidence: 99%

Towards the second main theorem on complements

2008

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“…It is worth mentioning that Corollary 2 follows almost directly from the proof of [31, Theorem 4.4] and [6, Theorem 1.1]. These two results together with [18] and [12] are the main motivation of Theorem 1.…”

Section: Introductionmentioning

confidence: 80%

“…A common technique to study singularities using birational geometry is to apply certain monoidal transformation to the singularity to extract an exceptional projective divisor over it and then try to deduce some local information of the singularity from the global information of the exceptional divisor. This approach has been successful in many cases: the study of dual complexes of singularities [10, 26], the finiteness of the algebraic fundamental group of a klt singularity [37], the ascending chain condition for log canonical thresholds [9, 16], the study of the normalized volume function on klt singularities [27–29], and the theory of log canonical complements [7, 31, 32], among others.…”

Section: Introductionmentioning

confidence: 99%

On minimal log discrepancies and kollár components

Moraga¹

2021

Proceedings of the Edinburgh Mathematical Society

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In this article, we prove a local implication of boundedness of Fano varieties. More precisely, we prove that $d$ -dimensional $a$ -log canonical singularities with standard coefficients, which admit an $\epsilon$ -plt blow-up, have minimal log discrepancies belonging to a finite set which only depends on $d,\,a$ and $\epsilon$ . This result gives a natural geometric stratification of the possible mld's in a fixed dimension by finite sets. As an application, we prove the ascending chain condition for minimal log discrepancies of exceptional singularities. We also introduce an invariant for klt singularities related to the total discrepancy of Kollár components.

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“…Observe that the coefficients of the log pair ( , ) belong to the hyperstandard set H N 1 ∩ [0, 1] . Note that this set depends only on M. Hence, we may apply the effective canonical bundle formula (see, e.g., [PS01] ). We obtain a log pair , with…”

Section: Proposition 24 the Quotient Of A Fano-type Variety By A Finite Automorphism Group Is Of Fano Typementioning

confidence: 99%