Effective base point free theorem for log canonical pairs. II: Angehrn-Siu type theorems. (English) Zbl 1201.14010
In the paper under review, the author gives a simple new proof of the effective base point free theorem for log canonical pairs. In the case of Kawamata log terminal pairs this result was first proven by J. Kollár [in: Algebraic geometry. Proceedings of the Summer Research Institute, Santa Cruz, CA, USA, July 9–29, 1995. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 62(pt.1), 221–287 (1997; Zbl 0905.14002)] and for log canonical pairs in the first part [Tohoku Math. J. (2) 61, No. 4, 475–481 (2009; Zbl 1189.14025)]. The method of proof is based on the inversion of log canonicity [cf. M. Kawakita, Invent. Math. 167, No. 1, 129–133 (2007; Zbl 1114.14009)] and new cohomological techniques developed by F. Ambro [Proc. Steklov Inst. Math. 240, 214–233 (2003; Zbl 1081.14021)].
Reviewer: Christopher Hacon (Salt Lake City)
MSC:
| 14E30 | Minimal model program (Mori theory, extremal rays) |
References:
| [1] | F. Ambro, Quasi-log varieties, Tr. Mat. Inst. Steklova 240 (2003), 220–239 (Russian); English translation in Proc. Steklov Inst. Math. 240 (2003), 214–233. · Zbl 1081.14021 |
| [2] | U. Angehrn and Y.-T. Siu, Effective freeness and point separation for adjoint bundles, Invent. Math. 122 (1995), 291–308. · Zbl 0847.32035 · doi:10.1007/BF01231446 |
| [3] | O. Fujino, Introduction to the theory of quasi-log varieties, Classification of algebraic varieties (Schiermonnikoog, May 2009) (to appear). |
| [4] | ——, On injectivity, vanishing and torsion-free theorems for algebraic varieties, Proc. Japan Acad. Ser. A Math. Sci. 85 (2009), 95–100. · Zbl 1189.14024 · doi:10.3792/pjaa.85.95 |
| [5] | ——, Introduction to the log minimal model program for log canonical pairs, preprint, 2009. |
| [6] | ——, Non-vanishing theorem for log canonical pairs, J. Algebraic Geom. (to appear). · Zbl 1258.14018 |
| [7] | ——, Effective base point free theorem for log canonical pairs—Kollár type theorem, Tôhoku Math. J. (2) 61 (2009), 475–481. · Zbl 1189.14025 · doi:10.2748/tmj/1264084495 |
| [8] | ——, Fundamental theorems for the log minimal model program, preprint, 2009. |
| [9] | M. Kawakita, Inversion of adjunction on log canonicity, Invent. Math. 167 (2007), 129–133. · Zbl 1114.14009 · doi:10.1007/s00222-006-0008-z |
| [10] | J. Kollár, Singularities of pairs, Algebraic geometry (Santa Cruz, 1995), Proc. Sympos. Pure Math., 62, pp. 221–287, Amer. Math. Soc., Providence, RI, 1997. |
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