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Birkar, Caucher; Cascini, Paolo; Hacon, Christopher D.; McKernan, James

Existence of minimal models for varieties of log general type. (English) Zbl 1210.14019

J. Am. Math. Soc. 23, No. 2, 405-468 (2010).
The paper under review is a milestone in the birational and biregular geometry of higher dimensional algebraic varieties. The main result is the existence of a log terminal model for log terminal pairs \((X,\Delta)\), with big boundary, \(\Delta\), and pseudo-effective adjoint divisor, \(K_X+\Delta\).
The Minimal Model Program aims to show that given any \(n\)-fold \(X\) there is a variety \(Y\), birational to \(X\) such that either the canonical class \(K_Y\) is nef or \(Y\) admits a fibration with ample anticanonical class. The two main lines to pursued this task are prove the finite generation of the canonical ring and kill the negative part of the canonical class via birational transformations. One of the main novelty of this paper is a way to combine both approach. Instead of trying to explain and write in details the main technical result, obtained via a complicate induction, I think it is much more profitable to list some of the outstanding consequences of it.
Any smooth variety of general type has a minimal model, a canonical model a model with Kähler–Einstein metric, and the canonical ring is finitely generated. Any klt pair \((X,\Delta)\) has the canonical ring finitely generated. Fano manifolds are Mori dream spaces [see Y. Hu and S. Keel, Mich. Math. J. 48, Spec. Vol., 331–348 (2000; Zbl 1077.14554)]. Let \((X,\Delta)\) be a klt pair. Then flips for \((X,\Delta)\) exist. If \(K_X+\Delta\) is not pseudo-effective then \(X\) is birational to a Mori fiber space. It is proven the Inversion of adjunction for arbitrary log pairs. This together with applications to moduli spaces, birational geometry and biregular geometry.
Reviewer: Massimiliano Mella (Ferrara)

Cited in 61 Reviews
Cited in 486 Documents

MSC:

14E30 Minimal model program (Mori theory, extremal rays)
14J45 Fano varieties
14J15 Moduli, classification: analytic theory; relations with modular forms
14B05 Singularities in algebraic geometry
14J40 \(n\)-folds (\(n>4\))

Keywords:

minimal models; finite generation; flip; birational geometry; MMP

Citations:

Zbl 1077.14554
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\textit{C. Birkar} et al., J. Am. Math. Soc. 23, No. 2, 405--468 (2010; Zbl 1210.14019)
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References:

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