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Varieties fibered by good minimal models. (English) Zbl 1221.14018
Given a normal complex projective variety $$X$$ with sufficiently mild singularities and pseudoeffective canonical divisor, the theory of the minimal model program predicts:
- Existence of minimal models: There exists a a variety $$Y$$ birational to $$X$$ such that the canonical divisor $$K_Y$$ is nef;
- Abundance conjecture: $$K_Y$$ is semiample, i.e. $$mK_Y$$ is base point free for some integer $$m>0$$.
If a variety has a minimal model with semiample canonical divisor we say that it has a good minimal model.
In the paper under review the author proves that every $$\mathbb{Q}$$-factorial variety $$X$$ with at most terminal singularities admits a good minimal model if $$X$$ has Kodaira dimension $$\kappa(X)=0$$ and the general fiber of the Albanese morphism has a good minimal model or if $$\kappa(X)\geq 0$$ and the general fiber of the Iitaka fibration has a good minimal model. As a corollary the author proves Iitaka’s conjecture C for algebraic fiber spaces $$f$$ with general fiber $$F$$ such that the general fiber of the Iitaka fibration of $$F$$ has a good minimal model, giving in such a way a positive answer to a question posed by S. Mori [in: Algebraic geometry, Proc. Summer Res. Inst., Brunswick/Maine 1985, part 1, Proc. Symp. Pure Math. 46, 269–331 (1987; Zbl 0656.14022)].
The proofs of the main results use many of the techniques introduced in famous work C. Birkar, P. Cascini, C. D. Hacon and J. McKernan [J. Am. Math. Soc. 23, No. 2, 405–468 (2010; Zbl 1210.14019)].

##### MSC:
 1.4e+31 Minimal model program (Mori theory, extremal rays)
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##### References:
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