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Flops connect minimal models. (English) Zbl 1145.14014
Let $$(X,B)$$ and $$(X',B')$$ be two Kawamata log terminal pairs and $$p:X\to Z$$ and $$q:X'\to Z$$ be two birational morphisms that contract no divisors. The rational map $$(q^{-1})\circ p:X\dasharrow X'$$ is a flop if $$\rho (X/Z)=\rho (X'/Z)=1$$, $$B'=(q^{-1}\circ p)_*B$$ and both $$K_X+B$$ and $$K_{X'}+B'$$ are numerically equivalent to the pull back of a divisor from $$Z$$. The minimal model program predicts that if $$(X,B)$$ and $$(X',B')$$ are two minimal models, then $$q^{-1}\circ p$$ is given by a finite sequence of flops. In the paper under review, the author shows that this is indeed the case. More precisely, he shows that if $$(X,B)$$ and $$(X',B')$$ are two projective $$\mathbb Q$$-factorial terminal pairs where $$K_X+B$$ and $$K_{X'}+B'$$ are nef and if $$\alpha : X \dasharrow X'$$ is a birational map such that $$\alpha _* B=B'$$, then $$\alpha$$ may be decomposed in to a sequence of flops $$\alpha = \alpha _t\circ \ldots \circ \alpha _1$$. The proof is based on a result of C. Birkar, P. Cascini, J. McKernan and the reviewer [Existence of minimal models for varieties of log general type, Preprint, arXiv:math/0610203] and on a result concerning the boundedness of the length of extremal rays.

##### MSC:
 14E30 Minimal model program (Mori theory, extremal rays) 14E05 Rational and birational maps 14J32 Calabi-Yau manifolds (algebro-geometric aspects)
##### Keywords:
minimal models; flops
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##### References:
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