Quasi-log varieties. (English. Russian original) Zbl 1081.14021

Ambro, F.

Proc. Steklov Inst. Math. 240, 214-233 (2003); translation from Tr. Mat. Inst. Im. V. A. Steklova 240, 220-239 (2003).

The minimal model programs for threefolds and for projective varieties with only log canonical singularities are based upon the cone and the contraction theorems of S. Mori and their extensions which were proved later on by several authors. In the paper under review the author considers so-called generalized log varieties and, more general, quasi-log varieties. The singularities of these varieties need not be log canonical but there are specific assumptions on extremal rays and their contractions. Quasi-log varieties can be characterized as the target spaces of log-\(0\)-contractions from a variety with only embedded normal crossings. The author proves a cone theorem and a contraction theorem for quasi-log varieties, herewith laying the ground for a minimal model program for those varieties. For the methods of proof and further references cf. Y. Kawamata, K.Matsuda, K. Matsuki [in: Algebraic geometry, Proc. Symp., Sendai/Jap. 1985, Adv. Stud. Pure Math. 10, 283–360 (1987; Zbl 0672.14006)] and J. Kollár, S. Mori [Birational geometry of algebraic varieties. With the collaboration of C. H. Clemens and A. Corti. Cambridge Tracts in Mathematics. 134. (Cambridge: Cambridge University Press) (1998; Zbl 0926.14003)]. Besides these theorems the author extends the vanishing and torsion freeness theorems of J. Kollár [Ann. Math. (2) 123, 11–42 (1986; Zbl 0598.14015)] to normal crossing pairs, using logarithmic de Rham complexes for quasi-log varieties [cf. H. Esnault and E. Viehweg, “Lectures on vanishing theorems”, DMV Seminar. 20. (Basel: Birkhäuser) (1992; Zbl 0779.14003) and Y. Kawamata, Invent. Math. 79, 567–588 (1985; Zbl 0593.14010)]. Moreover he proves a base-point-free theorem for projective quasi-log varieties, using results of V. V. Shokurov [Math. USSR, Izv. 26, 591–604 (1986); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 49, No. 3, 635–651 (1985; Zbl 0605.14006)]. The results are applied to quasi-log Fano contractions.
For the entire collection see [Zbl 1059.14001].

Reviewer: Eberhard Oeljeklaus (Bremen)

MSC:

14E30	Minimal model program (Mori theory, extremal rays)
32J18	Compact complex \(n\)-folds
14J10	Families, moduli, classification: algebraic theory

Keywords:

quasi-log variety; contraction theorem; cone theorem; minimal model program