Kyoto Journal of Mathematics

Du Bois pairs and vanishing theorems

Sándor J Kovács
Source: Kyoto J. Math. Volume 51, Number 1 (2011), 47-69.

Abstract

The main purpose of this article is to define the notion of Du Bois singularities for pairs and prove a vanishing theorem by using this new notion. The main vanishing theorem specializes to a new vanishing theorem for resolutions of log canonial singularities.

First Page: Show Hide
Primary Subjects: 14J17, 14F17
Full-text: Access denied (no subscription detected)
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.kjm/1298669425
Digital Object Identifier: doi:10.1215/0023608X-2010-020
Mathematical Reviews number (MathSciNet): MR2784747
Zentralblatt MATH identifier: 1218.14021

References

[AKMW] D. Abramovich, K. Karu, K. Matsuki, and J. Włodarczyk, Torification and factorization of birational maps, J. Amer. Math. Soc. 15 (2002), 531–572.
Mathematical Reviews (MathSciNet): MR1896232
Zentralblatt MATH: 1032.14003
Digital Object Identifier: doi:10.1090/S0894-0347-02-00396-X
[BL] L. Borisov and A. Libgober, McKay correspondence for elliptic genera, Ann. of Math. (2) 161 (2005), 1521–1569.
Mathematical Reviews (MathSciNet): MR2180406
Zentralblatt MATH: 1153.58301
Digital Object Identifier: doi:10.4007/annals.2005.161.1521
[Car] J. A. Carlson, Polyhedral resolutions of algebraic varieties, Trans. Amer. Math. Soc. 292 (1985), 595–612.
Mathematical Reviews (MathSciNet): MR808740
Zentralblatt MATH: 0602.14012
Digital Object Identifier: doi:10.1090/S0002-9947-1985-0808740-3
[Del] P. Deligne, Théorie de Hodge, III, Inst. Hautes Études Sci. Publ. Math. 44 (1974), 5–77.
Mathematical Reviews (MathSciNet): MR498552
Digital Object Identifier: doi:10.1007/BF02685881
[DB] P. Du Bois, Complexe de de Rham filtré d’une variété singulière, Bull. Soc. Math. France 109 (1981), 41–81.
Mathematical Reviews (MathSciNet): MR613848
[DJ] P. Du Bois and P. Jarraud, Une propriété de commutation au changement de base des images directes supérieures du faisceau structural, C. R. Acad. Sci. Paris Sér. A 279 (1974), 745–747.
Mathematical Reviews (MathSciNet): MR376678
[GKKP] D. Greb, S. Kebekus, S. J. Kovács, and T. Peternell, Differential forms on log canonical spaces, preprint, arXiv:1003.2913v3 [math.AG]
[GNPP] F. Guillén, V. Navarro Aznar, P. Pascual-Gainza, and F. Puerta, Hyperrésolutions cubiques et descente cohomologique, Lecture Notes in Math. 1335, Springer, Berlin, 1988.
Mathematical Reviews (MathSciNet): MR972984
[HK] C. D. Hacon and S. J. Kovács, Classification of Higher Dimensional Algebraic Varieties, Oberwolfach Seminars 41, Birkhäuser, Basel, 2010.
[Kol] J. Kollár, Shafarevich maps and automorphic forms, M. B. Porter Lectures, Princeton Univ. Press, Princeton, 1995.
[KK1] J. Kollár and S. J. Kovács, Log canonical singularities are Du Bois, J. Amer. Math. Soc. 23 (2010), 791–813.
[KK2] J. Kollár and S. J. Kovács, Rational pairs, preprint, 2009.
[KM] J. Kollár and S. Mori, Birational geometry of algebraic varieties, with the collaboration of C. H. Clemens and A. Corti, trans. from the 1998 Japanese original, Cambridge Tracts in Math. 134, Cambridge Univ. Press, Cambridge, 1998.
[Kov1] S. J. Kovács, Rational, log canonical, Du Bois singularities: On the conjectures of Kollár and Steenbrink, Compositio Math. 118 (1999), 123–133.
[Kov2] S. J. Kovács, A characterization of rational singularities, Duke Math. J. 102 (2000), 187–191.
[Kov3] S. J. Kovács, Rational, log canonical, Du Bois singularities, II: Kodaira vanishing and small deformations, Compositio Math. 121 (2000), 297–304.
[KS] S. J. Kovács and K. Schwede, Hodge theory meets the minimal model program: A survey of log canonical and Du Bois singularities, to appear in Topology of Stratified Spaces, Math. Sci. Res. Inst., Cambridge Univ. Press, Cambridge, 2011, preprint, arXiv:0909.0993v1 [math.AG]
[KSS] S. J. Kovács, K. Schwede, and K. E. Smith, The canonical sheaf of Du Bois singularities, Adv. Math. 224 (2010), 1618–1640.
[PS] C. A. M. Peters and J. H. M. Steenbrink, Mixed Hodge structures, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 52, Springer, Berlin, 2008.
Mathematical Reviews (MathSciNet): MR2393625
Zentralblatt MATH: 1138.14002
[Sai] M. Saito, Mixed Hodge complexes on algebraic varieties, Math. Ann. 316 (2000), 283–331.
Mathematical Reviews (MathSciNet): MR1741272
Zentralblatt MATH: 0976.14011
Digital Object Identifier: doi:10.1007/s002080050014
[Sch1] K. Schwede, A simple characterization of Du Bois singularities, Compos. Math. 143 (2007), 813–828.
Mathematical Reviews (MathSciNet): MR2339829
Zentralblatt MATH: 1125.14002
[Sch2] K. Schwede, F-injective singularities are Du Bois, Amer. J. Math. 131 (2009), 445–473.
Mathematical Reviews (MathSciNet): MR2503989
Zentralblatt MATH: 1164.14001
Digital Object Identifier: doi:10.1353/ajm.0.0049
[Sch3] K. Schwede, On Du Bois and F-injective singularities, Ph.D. thesis, University of Washington, Seattle, 2006.
[ST] K. Schwede and S. Takagi, “Rational singularities associated to pairs” in Special Volume in Honor of Melvin Hochster, Michigan Math. J. 57, Univ. Michigan Press, Ann Arbor, 2008, 625–658.
Mathematical Reviews (MathSciNet): MR2492473
Zentralblatt MATH: 1177.14028
Digital Object Identifier: doi:10.1307/mmj/1220879429
Project Euclid: euclid.mmj/1220879429
[Ste1] J. H. M. Steenbrink, “Mixed Hodge structures associated with isolated singularities” in Singularities, Part 2 (Arcata, Calif., 1981), Proc. Sympos. Pure Math. 40, Amer. Math. Soc., Providence, 1983, 513–536.
Mathematical Reviews (MathSciNet): MR713277
Zentralblatt MATH: 0515.14003
[Ste2] J. H. M. Steenbrink, “Vanishing theorems on singular spaces” in Differential Systems and Singularities (Luminy, France, 1983), Astérisque 130, Soc. Math. France, Montrouge, 1985, 330–341.
Mathematical Reviews (MathSciNet): MR804061
Zentralblatt MATH: 0582.32039
[Sza] E. Szabó, Divisorial log terminal singularities, J. Math. Sci. Univ. Tokyo 1 (1994), 631–639.

2013 © Kyoto University

Kyoto Journal of Mathematics

Kyoto Journal of Mathematics

Turn MathJax Off
What is MathJax?