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Reduction maps and minimal model theory. (English) Zbl 1264.14025
The Minimal Model Program is a way to chose in the birational class of a (log) variety a good representative using the numerical class of the canonical sheaf. The MMP conjecture predict that if the canonical class is pseudo effective then it is semiample. This is known when the Kodaira dimension is either maximal [C. Birkar, J. Am. Math. Soc. 23, No. 2, 405–468 (2010; Zbl 1210.14019)] or zero [S. Druel, Math. Z. 267, No. 1–2, 413–423 (2011; Zbl 1216.14007)]. The paper under review approaches the existence of good models for log terminal pairs using a $$(K_X+\Delta)$$-trivial reduction map. The authors are able to prove that the existence of a good model for a log terminal pair $$(X,\Delta)$$ can be detected by the existence of a good model of the base of a $$(K_X+\Delta)$$-trivial reduction map. The latter map can be interpreted as a way to contracts curves that have zero intersection with $$K_X+\Delta$$. Unfortunately due to technical problems this is not enough to prove the conjecture via an induction argument but it is an interesting property.

##### MSC:
 14E30 Minimal model program (Mori theory, extremal rays) 14J40 $$n$$-folds ($$n>4$$)
##### Keywords:
minimal model; reduction map; abundance
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##### References:
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