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International Mathematics Research Notices (2007) Vol. 2007 : article ID rnm016, 14 pages, doi:10.1093/imrn/rnm016 published on May 24, 2007
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Copyright © The Author 2007. Published by Oxford University Press.

Stringy Hodge Numbers for a Class of Isolated Singularities and for Threefolds

Jan Schepers1, and Willem Veys2

1 Universiteit Leiden, Mathematisch Instituut, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands
2 Katholieke Universiteit Leuven, DepartementWiskunde, Celestijnenlaan 200B, 3001 Leuven, Belgium

Correspondence: Correspondence to be sent to: Jan Schepers, Universiteit Leiden, Mathematisch Instituut, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands. e-mail: jschepers{at}math.leidenuniv.nl

Batyrev has defined the stringy E-function for complex varieties with at most log terminal singularities. It is a rational function in two variables if the singularities are Gorenstein. Furthermore, if the variety is projective and its stringy E-function is a polynomial, Batyrev defined its stringy Hodge numbers essentially as the coefficients of this E-function, generalizing the usual notion of Hodge numbers of a nonsingular projective variety. He conjectured that they are nonnegative. We prove this for a class of ‘mild’ isolated singularities (the allowed singularities depend on the dimension). As a corollary we obtain a proof of Batyrev's conjecture for threefolds in full generality. In these cases, we also give an explicit description of the stringy Hodge numbers, and we suggest a possible generalized definition of stringy Hodge numbers if the E-function is not a polynomial.

Communicated by Enrico Arbarello



References

  1. Abramovich D., Karu K., Matsuki K., Wlodarczyk J. Torification and Factorization of Birational Maps. Journal of the American Mathematical Society (2002) 15:531–572.[CrossRef][Web of Science]
  2. Batyrev V. Integrable Systems and Algebraic Geometry, Kobe/Kyoto 1997 (1999) River Edge, NJ: World Scientific Publishing. 1–32. Stringy Hodge Numbers of Varieties with Gorenstein Canonical Singularities.
  3. Batyrev V. Non-Archimedian Integrals and Stringy Euler Numbers of Log Terminal Pairs. Journal of the European Mathematical Society (1999) 1:5–33.[CrossRef]
  4. Dais D. On the String-Theoretic Euler Number of a Class of Absolutely Isolated Singularities. Manuscripta Mathematica (2001) 105:143–174.[CrossRef][Web of Science]
  5. Dais D., Roczen M. On the String-Theoretic Euler Number of 3-dimensional A-D-E Singularities. Advances in Geometry (2001) 1:373–426.[CrossRef]
  6. de Cataldo M. A., Migliorini L. The Hodge Theory of Algebraic Maps. Annales Scientifiques de l’École Normale Supérieure 4e série (2005) t.38:693–750.
  7. de Cataldo M. A., Migliorini L. Intersection forms, Topology of Maps and Motivic Decomposition for Resolutions of Threefolds. (2005) Preprint Arxiv. math.AG/0504554.
  8. Deligne P. Théorie de Hodge, II. Publications Mathématiques de l'IHES (1971) 40:5–57.
  9. Deligne P. Théorie de Hodge, III. Publications Mathématiques de l'IHES (1974) 44:5–77.
  10. Kaup L. Eine topologische Eigenschaft Steinscher Räume. Nachrichten der Akademie der Wissenschaften Göttingen (1966) 8:213–224.
  11. Kulikov V. S., Kurchanov P. F. Complex algebraic varieties: Periods of integrals and Hodge structures. Algebraic Geometry III in Encyclopaedia of Mathematical Sciences (1998) 36:1–217.
  12. Matsuki K. Introduction to the Mori Program (2002) New York: Universitext: Springer.
  13. Reid M. Algebraic varieties and analytic varieties—Iitaka S., ed. (1983) Vol. 1. Tokyo: Kinokuniya. 131–180. Advanced Studies in Pure Mathematics. Minimal Models of Canonical Threefolds.
  14. Reid M. Young Person's Guide to Canonical Singularities. Algebraic Geometry Bowdoin, Proceedings of Symposia in Pure Mathematics (1987) 46:345–414.
  15. Rotman J. J. An Introduction to Homological Algebra (1979) New York: Academic Press.
  16. Schepers J. Stringy E-Functions of Varieties with A-D-E Singularities. Manuscripta Mathematica (2006) 119:129–157.[CrossRef][Web of Science]
  17. Srinivas V. The Hodge Characteristic. (2002) Preprint.
  18. Steenbrink J. H. M. Mixed Hodge Structures Associated with Isolated Singularities. Proceedings of Symposia in Pure Mathematics (1983) 40:513–536. Part 2.[Web of Science]
  19. Veys W. Arc Spaces, Motivic Integration and Stringy Invariants. In: Proceedings of Singularity Theory and Its applications. September 16–25, 2003: Sapporo, Japan. Advanced Studies in Pure Mathematics 43 (2006): 529–572.
  20. Veys W. Stringy Invariants of Normal Surfaces. Journal of Algebraic Geometry (2004) 13:115–141.[Web of Science]

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This Article
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