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On birational boundedness of Fano fibrations. (English) Zbl 1404.14021
Fano varieties form a fundamental class in birational geometry according to minimal model program. Kollár, Miyaoka and Mori showed in [J. Kollár et al., J. Differ. Geom. 36, No. 3, 765–779 (1992; Zbl 0759.14032)] that Fano manifolds with fixed dimension form a bounded family. In birational geometry, especially in the minimal model program, it is more natural to consider log pairs with wild singularities. However, it is well-known that the boundedness does not hold without assumption on the singularities.
Recall that a normal projective variety $$X$$ is of $$\epsilon$$-Fano type if there exists an effective $$\mathbb{Q}$$-divisor $$B$$ such that $$(X,B)$$ is an $$\epsilon$$-klt log Fano pair. The following conjecture due to A. Borisov, L.Borisov and V. Alexeev (also called BAB conjecture) is one of the most important conjectures about Fano varieties.
Conjecture. Fix an integer $$n>0$$, $$0<\epsilon<1$$. Then the set of all $$n$$-dimensional varieties of $$\epsilon$$-Fano type is bounded.
In dimension $$2$$, this conjecture was proved by V. Alexeev in [Int. J. Math. 5, No. 6, 779–810 (1994; Zbl 0838.14028)] with a simplied argument by V. Alexeev and S. Mori [in: Algebra, arithmetic and geometry with applications. Papers from Shreeram S. Ahhyankar’s 70th birthday conference, Purdue University, West Lafayette, IN, USA, July 19–26, 2000. Berlin: Springer. 143–174 (2003; Zbl 1103.14021)].
The paper under review studies the birational boundedness of varieties of $$\epsilon$$-Fano type. To be precise, the author proved the following conjecture in dimension $$3$$.
Conjecture. Fix an integer $$n>0$$, $$0<\epsilon<1$$. The set of all $$n$$-dimensional varieties of $$\epsilon$$-Fano type is birationally bounded.
The key observation is that to prove birational boundedness of varieties of $$\epsilon$$-Fano type, it is enough to consider those with a Mori fibration structure. Then one can apply some highly non-trivial inductive arguments to prove the theorem in higher dimension from lower dimension.
Finally, we remark that the BAB conjecture has already been solved by C. Birkar in his remarkable paper [“Singularities of linear systems and boundedness of Fano varieties”, arXiv:1609.05543].
Reviewer: Jie Liu (Nice)

MSC:
 14E30 Minimal model program (Mori theory, extremal rays) 14J45 Fano varieties 14D06 Fibrations, degenerations in algebraic geometry 14E05 Rational and birational maps
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