Duke Mathematical Journal

Generating functions for the number of curves on abelian surfaces

Jim Bryan and Naichung Conan Leung

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Article information

Duke Math. J. Volume 99, Number 2 (1999), 311-328.

First available in Project Euclid: 19 February 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Digital Object Identifier

Primary: 14N10: Enumerative problems (combinatorial problems)
Secondary: 11F23: Relations with algebraic geometry and topology 14J25: Special surfaces {For Hilbert modular surfaces, see 14G35}


Bryan, Jim; Leung, Naichung Conan. Generating functions for the number of curves on abelian surfaces. Duke Math. J. 99 (1999), no. 2, 311--328. doi:10.1215/S0012-7094-99-09911-8. http://projecteuclid.org/euclid.dmj/1077227774.

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  • [1] Arnaud Beauville, Counting rational curves on $K3$ surfaces, Duke Math. J. 97 (1999), no. 1, 99–108.
  • [2] J. Bryan and N. C. Leung, The enumerative geometry of $K3$ surfaces and modular forms, preprint, http://xxx.lanl.gov/abs/alg-geom/9711031.
  • [3] O. Debarre, On the Euler characteristic of generalized Kummer varieties, preprint, http://xxx.lanl.gov/abs/alg-geom/9711035.
  • [4] S. K. Donaldson, Yang-Mills invariants of four-manifolds, Geometry of low-dimensional manifolds, 1 (Durham, 1989) eds. S. K. Donaldson and C. B. Thomas, London Math. Soc. Lecture Note Ser., vol. 150, Cambridge Univ. Press, Cambridge, 1990, Gauge Theory and Algebraic Surfaces, pp. 5–40.
  • [5] Lothar Göttsche, A conjectural generating function for numbers of curves on surfaces, Comm. Math. Phys. 196 (1998), no. 3, 523–533.
  • [6] Herbert Lange and Christina Birkenhake, Complex abelian varieties, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 302, Springer-Verlag, Berlin, 1992.
  • [7] Jun Li and Gang Tian, Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds, Topics in symplectic $4$-manifolds (Irvine, CA, 1996), First Int. Press Lect. Ser., I, Internat. Press, Cambridge, MA, 1998, pp. 47–83.
  • [8] Jun Li and Gang Tian, Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties, J. Amer. Math. Soc. 11 (1998), no. 1, 119–174.
  • [9] J. Li and G. Tian, Comparison of the algebraic and the symplectic Gromov-Witten invariants, preprint, http://xxx.lanl.gov/abs/alg-geom/9712035.
  • [10] Shing Tung Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Comm. Pure Appl. Math. 31 (1978), no. 3, 339–411.
  • [11] Shing-Tung Yau and Eric Zaslow, BPS states, string duality, and nodal curves on $K3$, Nuclear Phys. B 471 (1996), no. 3, 503–512.