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Log abundance theorem for threefolds. (English) Zbl 0818.14007
The abundance theorem of Y. Kawamata and Y. Miyaoka states that the canonical divisor, of a minimal model of a threefold – an endproduct of the minimal model program of Mori – is semiample. In this paper, the abundance theorem is generalized for the log-extension of the minimal model program due to Shokurov. The log-abundance theorem, proved by the authors of this paper, states:
Let the pair \((X,D)\) consist of a threefold \(X\) and a boundary \(D\) (i.e., \(D\) is a \(\mathbb{Q}\)-Weil divisor, such that any irreducible component \(D_ i\) of \(D\) has multiplicity \(d_ i \in [0,1])\), and let \(K_ X + D\) be nef and log-canonical. Then some integer multiple of \(K_ X + D\) is base-point free.
Reviewer: A.Iliev (Sofia)

MSC:
14E30 Minimal model program (Mori theory, extremal rays)
14J30 \(3\)-folds
14C20 Divisors, linear systems, invertible sheaves
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