zbMATH — the first resource for mathematics

On a conjecture of Shokurov: Characterization of toric varieties. (English) Zbl 1078.14515
Summary: We verify a special case of V. V. Shokurov’s conjecture [J. Math. Sci., New York 102, No. 2, 3876–3932 (2000; Zbl 1177.14078)] about characterization of toric varieties. More precisely, we consider three-dimensional log varieties with only purely log terminal singularities and numerically trivial log canonical divisor. In this situation we prove an inequality connecting the rank of the group of Weil divisors modulo algebraic equivalence and the sum of coefficients of the boundary. We describe such varieties for which the equality holds and show that all of them are toric.

MSC:
 14E30 Minimal model program (Mori theory, extremal rays) 14M25 Toric varieties, Newton polyhedra, Okounkov bodies
Zbl 1177.14078
Full Text:
References:
 [1] V. Alexeev, Two two-dimensional terminations, Duke Math. J. 69 (1992), 527–545. · Zbl 0791.14006 [2] Y. Kawamata, K. Matsuda and K. Matsuki, Introduction to the minimal model program, Adv. Stud. Pure Math. 10 (1987), 283–360. · Zbl 0672.14006 [3] J. Kollár, Singularities of pairs, Proc. Sympos. Pure Math. 62 (1995), 221–287. · Zbl 0905.14002 [4] J. Kollár et al., Flips and abundance for algebraic threefolds, A summer seminar at the University of Utah (Salt Lake City, 1991), Astérisque 211, 1992. · Zbl 0782.00075 [5] Y. Miyaoka and S. Mori, A numerical criterion for uniruledness, Ann. of Math. (2) 124 (1986), 65–69. JSTOR: · Zbl 0606.14030 [6] S. Mori, Threefolds whose canonical bundles are not numerically effective, Ann. of Math. (2) 115 (1982), 133–176. JSTOR: · Zbl 0557.14021 [7] V. A. Iskovskikh, On the rationality problem for conic bundles, Math. USSR-Sb. 72 (1992), 105–111. · Zbl 0776.14013 [8] V. A. Iskovskikh and Yu.G. Prokhorov, Fano Varieties, Encyclopaedia Math. Sci. 47, Springer-Verlag, Berlin, 1999. · Zbl 0912.14013 [9] Yu.G. Prokhorov, Lectures on complements on log surfaces, MSJ Mem. 10, Mathematical Society of Japan, Tokyo, 2001. · Zbl 1037.14003 [10] V. G. Sarkisov, On conic bundle structures, Math. USSR, Izv. 20 (1983), 355–390. · Zbl 0593.14034 [11] V. V. Shokurov, $$3$$-fold log flips, Russian Acad. Sci. Izv. Math. 40 (1993), 93–202. · Zbl 0828.14027 [12] V. V. Shokurov, Complements on surfaces, J. Math. Sci. 102 (2000), 3876–3932. · Zbl 1177.14078
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.