Varieties fibered by good minimal models

By Ching-Jui Lai. 2009

121 citations recorded by the source corpus.

Let f : X → Y be an algebraic fiber space such that the general fiber has a good minimal model. We show that if f is the Iitaka fibration or if f is the Albanese map with relative dimension no more than three, then X has a good minimal model.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1eb9c9bceb2ff3be2202c0bae7895f0ac20d449f

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Boundedness of log Calabi–Yau pairs of Fano type (2014, background): “By [Lai11, 4.4], we may replace (X,B) by its good minimal model and hence we may assume that KX +B is semiample.”
Reduction maps and minimal model theory (2011, background): “Definition 2.14 ([Nak04, III, § 5.a], [Lai11, Definition 2.9] and [Tak08, Definition 2.4]).”
Existence of log canonical closures (2011, background): “We follow the strategy of [Lai11].”
The global moduli theory of symplectic varieties (2018, background): “As the general fiber has a good minimal model—unique by the assumption on the Picard rank—the latter terminates by [Lai11, Prop.”
Algebraic fiber spaces over abelian varieties: around a recent theorem by Cao and Paun (2016, methodology): “The following proposition is the same as [Hac04, Corollary 3.2(4)], since it can be seen that the assumption on F imposed there is equivalent to that of being a GV-sheaf.”
Existence of log canonical flips and a special LMMP (2011, background): “18] (see also [25], [13] for more recent adaptations).”
Special test configurations and $K$-stability of Fano varieties (2011, background): “For Ev, by our assumption, it is of the insufficient fiber type (cf. [Lai11, 2.9]), so by [Lai11, 2.10] it is contained in B−(KY + ∆Y + L/C) ⊂ B(KY + ∆Y + L/C).”
On Generalised Abundance, I (2018, unknown): “If additionally f∗D is semiample, the map f is a good model for D. Note that if (X,∆) is a klt pair, then it has a good model if and only if there exists a Minimal Model Program with scaling of an ample divisor which terminates with a good model of (X,∆), cf. [Lai11, Propositions 2.4 and 2.5].”
Remarks on degenerations of hyper-K\"ahler manifolds (2017, methodology): “More precisely, a higher dimensional analogue of the KPP theorem is given by Fujino [21] and Lai [48] (building on [7]).”