Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Stringy invariants of normal surfaces

Author: Willem Veys
Journal: J. Algebraic Geom. 13 (2004), 115-141
Published electronically: September 3, 2003
MathSciNet review: 2008717
Full-text PDF

Abstract | References | Additional Information

Abstract: The stringy Euler number and $E$-function of Batyrev for log terminal singularities in dimension 2 can also be considered for a normal surface singularity with all log discrepancies nonzero in its minimal log resolution. Here we obtain a structure theorem for resolution graphs with respect to log discrepancies, implying that these stringy invariants can be defined in a natural way, even when some log discrepancies are zero, and more precisely for all normal surface singularities which are not log canonical. We also show that the stringy $E$-functions of log terminal surface singularities are polynomials (with rational powers) with nonnegative coefficients, yielding well defined (rationally graded) stringy Hodge numbers.

References [Enhancements On Off] (What's this?)

  • [A] V. Alexeev, Log canonical surface singularities: arithmetical approach, Seminar, Salt Lake City 1991, In Flips and abundance for algebraic threefolds, J. Kollár ed., Astérisque 211 (1992), 47-58.
  • [ACLM] E. Artal, P. Cassou-Noguès, I. Luengo and A. Melle, Monodromy conjecture for some surface singularities.
  • [B1] V. Batyrev, Stringy Hodge numbers of varieties with Gorenstein canonical singularities, Proc. Taniguchi Symposium 1997, In Integrable Systems and Algebraic Geometry, Kobe/Kyoto 1997, World Sci. Publ. (1999), 1-32.
  • [B2] -, Non-Archimedian integrals and stringy Euler numbers of log terminal pairs, J. Europ. Math. Soc. 1 (1999), 5-33.
  • [BPV] W. Barth, C. Peters and A. Van de Ven, Compact complex surfaces, Springer Verlag, Berlin, 1984.
  • [D] A. Dimca, Singularities and topology of hypersurfaces, Springer Verlag, New York, 1992.
  • [DL1] J. Denef and F. Loeser, Motivic Igusa zeta functions, J. Alg. Geom. 7 (1998), 505-537.
  • [DL2] -, Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math. 135 (1999), 201-232.
  • [DM] J. Denef and D. Meuser, A functional equation of Igusa's local zeta function, Amer. J. Math. 113 (1991), 1135-1152.
  • [K] M. Kontsevich, Lecture at Orsay (December 7, 1995).
  • [KM] J. Kollár and S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics 134, Cambridge Univ. Press, 1998.
  • [KMM] Y. Kawamata, K. Matsuda and K. Matsuki, Introduction to the Minimal Model Program, Algebraic Geometry, Sendai, T. Oda ed., Kinokuniya, Adv. Stud. Pure Math. 10 (1987), 283-360.
  • [L] A. Langer, Logarithmic orbifold Euler numbers of surfaces with applications, math.AG/ 0012180 (2000).
  • [M] D. Mumford, The topology of normal singularities of an algebraic surface and a criterion for simplicity, Publ. Math. I.H.E.S. 9 (1961), 5-22.
  • [OW] P. Orlik and Ph. Wagreich, Equivariant resolution of singularities with $\mathbb{C} ^{*}$-action, Proceedings of the Second Conference on Compact Transformation Groups II, Lecture Notes in Mathematics, vol. 299, Springer Verlag, Berlin, 1972, pp. 270-290.
  • [R] Y. Ruan, Stringy geometry and topology of orbifolds, Symposium in Honor of C.H. Clemens (Salt Lake City, 2000), Contemp. Math. 312, Amer. Math. Soc., Providence, RI, 2002, pp. 187-233.
  • [V1] W. Veys, Zeta functions for curves and log canonical models, Proc. London Math. Soc. 74 (1997), 360-378.
  • [V2] -, The topological zeta function associated to a function on a normal surface germ, Topology 38 (1999), 439-456.
  • [V3] -, Zeta functions and Kontsevich invariants on singular varieties, Canadian J. Math. 53 (2001), 834-865.
  • [Wag] Ph. Wagreich, The structure of quasihomogeneous singularities, Proc. Symp. Pure Math. (Arcata Singularities Conference), vol. 40 (2), A.M.S., 1983, pp. 593-611.
  • [Wah] J. Wahl, A characteristic number for links of surface singularities, J. Amer. Math. Soc. 3 (1990), 625-637.

Additional Information

Willem Veys
Affiliation: K.U.Leuven, Departement Wiskunde, Celestijnenlaan 200B, B–3001 Leuven, Belgium

Received by editor(s): June 8, 2001
Published electronically: September 3, 2003

Journal of Algebraic Geometry
The Journal of Algebraic Geometry
is sponsored by the Department of Mathematical Sciences
of Tsinghua University
and is distributed by the American Mathematical Society
for University Press, Inc.
Online ISSN 1534-7486; Print ISSN 1056-3911
© 2017 University Press, Inc.
AMS Website