New outlook on the minimal model program. I. (English) Zbl 1261.14007

Cascini, Paolo; Lazić, Vladimir

Duke Math. J. 161, No. 12, 2415-2467 (2012).

Let \(X\) be a smooth complex projective variety and \(\Delta\) a \(\mathbb Q\)-divisor with simple normal crossings such that \(\lfloor \Delta \rfloor =0\). It is proven in [C. Birkar et al., J. Am. Math. Soc. 23, No. 2, 405–468 (2010; Zbl 1210.14019)] that the log canonical ring \[ R(K_X+\Delta )=\bigoplus _{m\geq 0} H^0(X,\mathcal O _X(m(K_X+\Delta ))) \] is finitely generated. This is one of the fundamental results of the minimal model program (MMP) and the original proof heavily relies and builds on techniques developed for the MMP. The main result of the paper under review is a new and self contained proof of the above result which avoids the techniques of the MMP.

Reviewer: Christopher Hacon (Salt Lake City)

MSC:

14E30

Minimal model program (Mori theory, extremal rays)

Keywords:

minimal models

Citations:

Zbl 1210.14019

References:

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