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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:
14E30 Minimal model program (Mori theory, extremal rays)
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