Zariski decomposition of b-divisors. (English) Zbl 1271.14019
The Zariski decomposition on surfaces is one of the fundamental tools for studying their geometry. There have been several attempts to generalise this notion to higher dimensions. The paper under review proves a result about a decomposition of an effective b-divisor \(\mathbf{D}\) (introduced by Shokurov in the context of the Minimal Model Program) into “positive” and “negative” parts which satisfy some favourable properties that translate from the surface case; the main result of the paper, Theorem D, is slightly too technical to be presented here. In particular, the “positive” part carries information about the sections of multiples of \(\mathbf{D}\).
The result is strongly influenced by the work of Nakayama on the \(\sigma\)-decomposition [N. Nakayama, Zariski-decomposition and abundance. Tokyo: Mathematical Society of Japan (2004; Zbl 1061.14018)], and by a recent proof of the Zariski decomposition on surfaces in [T. Bauer, J. Algebr. Geom. 18, No. 4, 789–793 (2009; Zbl 1184.14007)].
The result is strongly influenced by the work of Nakayama on the \(\sigma\)-decomposition [N. Nakayama, Zariski-decomposition and abundance. Tokyo: Mathematical Society of Japan (2004; Zbl 1061.14018)], and by a recent proof of the Zariski decomposition on surfaces in [T. Bauer, J. Algebr. Geom. 18, No. 4, 789–793 (2009; Zbl 1184.14007)].
Reviewer: Vladimir Lazic (Bayreuth)
MSC:
| 14E30 | Minimal model program (Mori theory, extremal rays) |
References:
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