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Singularities on normal varieties

Published online by Cambridge University Press:  01 March 2009

Tommaso de Fernex
Affiliation:
Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, UT 48112-0090, USA (email: defernex@math.utah.edu)
Christopher D. Hacon
Affiliation:
Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, UT 48112-0090, USA (email: hacon@math.utah.edu)
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Abstract

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In this paper we generalize the definitions of singularities of pairs and multiplier ideal sheaves to pairs on arbitrary normal varieties, without any assumption on the variety being ℚ-Gorenstein or the pair being log ℚ-Gorenstein. The main features of the theory extend to this setting in a natural way.

Type
Research Article
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Problems raised during the Workshop on Numerical invariants of singularities and higher-dimensional algebraic varieties (American Institute of Mathematics, Palo Alto, 2006), www.aimath.org/WWN/singularvariety/problemsV1.pdf.Google Scholar
[2]Blickle, M., Multiplier ideals and modules on toric varieties, Math. Z. 248 (2004), 113121.CrossRefGoogle Scholar
[3]Ein, L. and Mustaţă, M., Invariants of singularities of pairs, International Congress of Mathematicians, vol. II (European Mathematical Society, Zürich, 2006), 583602.Google Scholar
[4]Elkik, R., Rationalité des singularités canoniques, Invent. Math. 64 (1981), 16.CrossRefGoogle Scholar
[5]Esnault, H. and Viehweg, E., Dyson’s lemma for polynomials in several variables (and the theorem of Roth), Invent. Math. 78 (1984), 445490.CrossRefGoogle Scholar
[6]Francia, P., Some remarks on minimal models. I, Composito Math. 40 (1980), 301313.Google Scholar
[7]Grothendieck, A., Éleménts de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Publ. Math. Inst. Hautes Études Sci. 24 (1965), 231.Google Scholar
[8]Hara, N., Geometric interpretation of tight closure and test ideals, Trans. Amer. Math. Soc. 353 (2001), 18851906.Google Scholar
[9]Hara, N. and Takagi, S., On a generalization of test ideals, Nagoya Math. J. 175 (2004), 5974.CrossRefGoogle Scholar
[10]Hara, N. and Yoshida, K.-I., A generalization of tight closure and multiplier ideals, Trans. Amer. Math. Soc. 355 (2003), 31433174.Google Scholar
[11]Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109203.CrossRefGoogle Scholar
[12]Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 205326.Google Scholar
[13]Kawamata, Y., On Fujita’s freeness conjecture for 3-folds and 4-folds, Math. Ann. 308 (1997), 491505.CrossRefGoogle Scholar
[14]Kawamata, Y., Subadjunction of log canonical divisors. II, Amer. J. Math. 120 (1998), 893899.CrossRefGoogle Scholar
[15]Kawamata, Y., Deformations of canonical singularities, J. Amer. Math. Soc. 12 (1999), 8592.CrossRefGoogle Scholar
[16]Kawamata, Y., On the extension problem of pluricanonical forms, in Algebraic geometry: Hirzebruch 70. Proceedings of the algebraic geometry conference in honor of F. Hirzebruch’s 70th birthday, Stefan Banach International Mathematical Center, Warszawa, Poland, May 11–16, 1998, Contemporary Mathematics, vol. 241, eds P. Pragacz et al. (American Mathematical Society, Providence, RI, 1999), 193207.Google Scholar
[17]Kawamata, Y., Matsuda, K. and Matsuki, K., Introduction to the minimal model problem, in Algebraic Geometry, Sendai, 1985, Advanced Studies in Pure Mathematics, vol. 10 (North-Holland, Amsterdam, 1987), 283360.CrossRefGoogle Scholar
[18]Kollár, J., Flips and abundance for algebraic threefolds. A summer seminar at the University of Utah (Salk Lake City, 1991), Astérisque 211 (1992).Google Scholar
[19]Kollár, J., Singularities of pairs, in Algebraic geometry—Santa Cruz 1995, Proceedings of Symposia in Pure Mathematics, vol. 62, part 1 (American Mathematical Society, Providence, RI, 1997), 221287.Google Scholar
[20]Kollár, J. and Mori, S., Birational Geometry of Algebraic Varieties, Cambridge Tracts in Mathematics, vol. 134 (Cambridge University Press, Cambridge, 1998).Google Scholar
[21]Kovács, S., Schwede, K. and Smith, K.E., Cohen-Macaulay semi-log canonical singularities are Du Bois, Preprint, arXiv:0801.1541.Google Scholar
[22]Lazarsfeld, R., Positivity in Algebraic Geometry. I, in Classical Setting: Line Bundles and Linear Series. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, A Series of Modern Surveys in Mathematics, vol. 48 (Springer, Berlin, 2004).Google Scholar
[23]Lazarsfeld, R., Positivity in Algebraic Geometry. II. Positivity for Vector Bundles, and Multiplier Ideals. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, A Series of Modern Surveys in Mathematics, vol. 49 (Springer, Berlin, 2004).Google Scholar
[24]Nadel, A., Multiplier ideal sheaves and existence of Kähler-Einstein metrics of positive scalar curvature, Proc. Natl. Acad. Sci. U.S.A. 86 (1989), 72997300.CrossRefGoogle ScholarPubMed
[25]Nadel, A., Multiplier ideal sheaves and Kähler-Einstein metrics of positive scalar curvature, Ann. of Math. (2) 132 (1990), 549596.CrossRefGoogle Scholar
[26]Nakayama, N., Zariski-Decomposition and Abundance, MSJ Memoirs, vol. 14 (Mathematical Society of Japan, Tokyo, 2004).CrossRefGoogle ScholarPubMed
[27]Shokurov, V. V., Three-dimensional log perestroikas. In Russian, with an appendix in English by Yujiro Kawamata, Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), 105203 (Engl. transl. Izv. Math. 40 (1993), 95–202).Google Scholar
[28]Schwede, K., A simple characterization of Du Bois singularities, Composito. Math. 143 (2007), 813828.Google Scholar
[29]Siu, Y.-T., Invariance of plurigenera, Invent. Math. 134 (1998), 661673.CrossRefGoogle Scholar
[30]Smith, K. E., The multiplier ideal is a universal test ideal, Comm. Algebra 28 (2000), 59155929.Google Scholar
[31]Takagi, S., An interpretation of multiplier ideals via tight closure, J. Algebraic Geom. 13 (2004), 393415.Google Scholar
[32]Takayama, S., Pluricanonical systems on algebraic varieties of general type, Invent. Math. 165 (2006), 551587.CrossRefGoogle Scholar