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...
