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Anti-pluricanonical systems on Fano varieties. (English) Zbl 1470.14078

In birational geometry, by the philosophy of the minimal model program, there are 3 building blocks of varieties: varieties of general type (varieties with \(K\) positive), Calabi-Yau varieties (varieties with \(K\) trivial), and Fano varieties (varieties with \(K\) negative). They are characterized by numerical positivity of the canonical divisors. It is expected that such varieties satisfy certain boundedness or finiteness property. Here a set of projective varieties is bounded if its elements can be realized as fibers of an algebraic family over a base of finite type. For example, for varieties of general type, Hacon, McKernan, and Xu showed that if fix a positive integer \(d\) and a positive rational number \(v\), then the set of \(d\)-dimensional log semi-canonical varieties \(X\) with \(K_X\) ample and \(K_X^d=v\) forms a bounded family [C. D. Hacon et al., J. Eur. Math. Soc. (JEMS) 20, No. 4, 865–901 (2018; Zbl 1464.14038)]. For Fano varieties, it was conjectured that, if fix a positive integer \(d\) and a positive real number \(\epsilon\), then the set of \(d\)-dimensional \(\epsilon\)-lc Fano varieties forms a bounded family, which is known as the Borisor-Alexeev-Borisov (BAB) conjecture. The author proves the BAB conjecture in this paper together with a subsequent paper [C. Birkar, Ann. Math. (2) 193, No. 2, 347–405 (2021; Zbl 1469.14085)].
In Proposition 7.13, the author provides a criterion for a set of klt Fano varieties to be bounded. To be more precise, given a set \(\mathcal{P}\) of klt Fano varieties of dimension \(d\), if one can find a positive integer \(m\) and positive real numbers \(v, t\) such that for any \(X\in \mathcal{P}\)
\(X\) has an \(m\)-complement, that is, there exists \(\Delta\in |-mK_X|\) such that \((X, \frac{1}{m}\Delta)\) is lc;
\(|-mK_X|\) defines a birational map;
\((-K_X)^d\leq v\);
\((X, B)\) is lc for any effective \(\mathbb{R}\)-divisor \(B\sim_\mathbb{R}-t K_X \),
then \(\mathcal{P}\) is bounded.
So the author proved the BAB conjecture by showing the existence of such \(m,v,t\) for \(\mathcal{P}\) the set of \(d\)-dimensional \(\epsilon\)-lc Fano varieties.
Section 4 is devoted to the existence of a uniform \(m\) such that \(|-mK_X|\) defines a birational map for any \(\epsilon\)-lc Fano varieties \(X\) of dimension \(d\). The method is to construct isolated non-klt centers and to apply Nadel vanishing to get point separation. Such kind of ideas were used by U. Angehrn and Y.-T. Siu [Invent. Math. 122, No. 2, 291–308 (1995; Zbl 0847.32035)], C. D. Hacon et al. [Ann. Math. (2) 177, No. 3, 1077–1111 (2013; Zbl 1281.14036); Ann. Math. (2) 180, No. 2, 523–571 (2014; Zbl 1320.14023)]. One of the main difficulty here is that to use the \(\epsilon\)-lc condition one need to develop a more presice adjunction theory for non-klt centers (this part is explained in Section 3).
Sections 6–8 are devoted to the existence of a uniform \(m\) such that \(X\) has an \(m\)-complement for any klt Fano varieties \(X\) of dimension \(d\). This result is the most important and technical part of this paper, which is originally conjectured by Shokurov. Note that here \(X\) is only assumed to be klt instead of \(\epsilon\)-lc. As a consequence, it implies that for any klt Fano varieties \(X\) of dimension \(d\), there exists a uniform \(m\) depending on \(d\) such that \(|-mK_X|\neq \emptyset\). For the proof the author introduces the concept of generalized pairs and extend the conjecture in the setting of generalized pairs. Then the author proves the conjecture by induction on dimensions.
Section 9 is devoted to the existence of a uniform \(v\) such that \((-K_X)^d\leq v\) for any \(\epsilon\)-lc Fano varieties \(X\) of dimension \(d\).

MSC:

14J45 Fano varieties
14E30 Minimal model program (Mori theory, extremal rays)
14C20 Divisors, linear systems, invertible sheaves
14E05 Rational and birational maps
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