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

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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.

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[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