Stringy invariants of normal surfaces
Author:
Willem Veys
Journal:
J. Algebraic Geom. 13 (2004), 115-141
DOI:
https://doi.org/10.1090/S1056-3911-03-00340-0
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.
[A]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]ACLM E. Artal, P. Cassou–Noguès, I. Luengo and A. Melle, Monodromy conjecture for some surface singularities.
[B1]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]B2 ---, Non–Archimedian integrals and stringy Euler numbers of log terminal pairs, J. Europ. Math. Soc. 1 (1999), 5–33.
[BPV]BPV W. Barth, C. Peters and A. Van de Ven, Compact complex surfaces, Springer Verlag, Berlin, 1984.
[D]D A. Dimca, Singularities and topology of hypersurfaces, Springer Verlag, New York, 1992.
[DL1]DL1 J. Denef and F. Loeser, Motivic Igusa zeta functions, J. Alg. Geom. 7 (1998), 505–537.
[DL2]DL2 ---, Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math. 135 (1999), 201–232.
[DM]DM J. Denef and D. Meuser, A functional equation of Igusa’s local zeta function, Amer. J. Math. 113 (1991), 1135–1152.
[K]K M. Kontsevich, Lecture at Orsay (December 7, 1995).
[KM]KM J. Kollár and S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics 134, Cambridge Univ. Press, 1998.
[KMM]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]L A. Langer, Logarithmic orbifold Euler numbers of surfaces with applications, math.AG/ 0012180 (2000).
[M]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]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]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]V1 W. Veys, Zeta functions for curves and log canonical models, Proc. London Math. Soc. 74 (1997), 360–378.
[V2]V2 ---, The topological zeta function associated to a function on a normal surface germ, Topology 38 (1999), 439–456.
[V3]V3 ---, Zeta functions and ‘Kontsevich invariants’ on singular varieties, Canadian J. Math. 53 (2001), 834–865.
[Wag]W 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]Wah J. Wahl, A characteristic number for links of surface singularities, J. Amer. Math. Soc. 3 (1990), 625–637.
[A]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]ACLM E. Artal, P. Cassou–Noguès, I. Luengo and A. Melle, Monodromy conjecture for some surface singularities.
[B1]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]B2 ---, Non–Archimedian integrals and stringy Euler numbers of log terminal pairs, J. Europ. Math. Soc. 1 (1999), 5–33.
[BPV]BPV W. Barth, C. Peters and A. Van de Ven, Compact complex surfaces, Springer Verlag, Berlin, 1984.
[D]D A. Dimca, Singularities and topology of hypersurfaces, Springer Verlag, New York, 1992.
[DL1]DL1 J. Denef and F. Loeser, Motivic Igusa zeta functions, J. Alg. Geom. 7 (1998), 505–537.
[DL2]DL2 ---, Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math. 135 (1999), 201–232.
[DM]DM J. Denef and D. Meuser, A functional equation of Igusa’s local zeta function, Amer. J. Math. 113 (1991), 1135–1152.
[K]K M. Kontsevich, Lecture at Orsay (December 7, 1995).
[KM]KM J. Kollár and S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics 134, Cambridge Univ. Press, 1998.
[KMM]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]L A. Langer, Logarithmic orbifold Euler numbers of surfaces with applications, math.AG/ 0012180 (2000).
[M]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]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]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]V1 W. Veys, Zeta functions for curves and log canonical models, Proc. London Math. Soc. 74 (1997), 360–378.
[V2]V2 ---, The topological zeta function associated to a function on a normal surface germ, Topology 38 (1999), 439–456.
[V3]V3 ---, Zeta functions and ‘Kontsevich invariants’ on singular varieties, Canadian J. Math. 53 (2001), 834–865.
[Wag]W 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]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
Email:
wim.veys@wis.kuleuven.ac.be
Received by editor(s):
June 8, 2001
Published electronically:
September 3, 2003
