Counting elliptic curves in $\mathrm {K3}$ surfaces
Authors:
Junho Lee and Naichung Conan Leung
Journal:
J. Algebraic Geom. 15 (2006), 591-601
DOI:
https://doi.org/10.1090/S1056-3911-06-00439-5
Published electronically:
May 2, 2006
MathSciNet review:
2237262
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Abstract |
References |
Additional Information
Abstract: We compute the genus $g=1$ family GW-invariants of K3 surfaces for non-primitive classes. These calculations verify the Göttsche-Yau-Zaslow formula for non-primitive classes with index two. Our approach is to use the genus two topological recursion formula and the symplectic sum formula to establish relationships among various generating functions.
- Jim Bryan and Naichung Conan Leung, The enumerative geometry of $K3$ surfaces and modular forms, J. Amer. Math. Soc. 13 (2000), no. 2, 371â410. MR 1750955, DOI https://doi.org/10.1090/S0894-0347-00-00326-X
- Jim Bryan and Naichung Conan Leung, Counting curves on irrational surfaces, Surveys in differential geometry: differential geometry inspired by string theory, Surv. Differ. Geom., vol. 5, Int. Press, Boston, MA, 1999, pp. 313â339. MR 1772273, DOI https://doi.org/10.4310/SDG.1999.v5.n1.a3
- E. Getzler, Topological recursion relations in genus $2$, Integrable systems and algebraic geometry (Kobe/Kyoto, 1997) World Sci. Publ., River Edge, NJ, 1998, pp. 73â106. MR 1672112
- Lothar Göttsche, A conjectural generating function for numbers of curves on surfaces, Comm. Math. Phys. 196 (1998), no. 3, 523â533. MR 1645204, DOI https://doi.org/10.1007/s002200050434
- Eleny-Nicoleta Ionel and Thomas H. Parker, The symplectic sum formula for Gromov-Witten invariants, Ann. of Math. (2) 159 (2004), no. 3, 935â1025. MR 2113018, DOI https://doi.org/10.4007/annals.2004.159.935
- M. Kontsevich and Yu. Manin, Relations between the correlators of the topological sigma-model coupled to gravity, Comm. Math. Phys. 196 (1998), no. 2, 385â398. MR 1645019, DOI https://doi.org/10.1007/s002200050426
- Junho Lee, Family Gromov-Witten invariants for KĂ€hler surfaces, Duke Math. J. 123 (2004), no. 1, 209â233. MR 2060027, DOI https://doi.org/10.1215/S0012-7094-04-12317-6
l2 ---, Counting Curves in Elliptic Surfaces by Symplectic Methods, preprint, math. SG/0307358, to appear in Comm. Anal. and Geom.
- Xiaobo Liu, Quantum product on the big phase space and the Virasoro conjecture, Adv. Math. 169 (2002), no. 2, 313â375. MR 1926225, DOI https://doi.org/10.1006/aima.2001.2062
- Junho Lee and Naichung Conan Leung, Yau-Zaslow formula on $K3$ surfaces for non-primitive classes, Geom. Topol. 9 (2005), 1977â2012. MR 2175162, DOI https://doi.org/10.2140/gt.2005.9.1977
- 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, Int. Press, Cambridge, MA, 1998, pp. 47â83. MR 1635695
m D. Mumford, Towards an enumerating geometry of the moduli space of curves, in Arithmetic and Geometry, M. Artin and J. Tate, eds., BirkhĂ€user, 1995, 401â417.
- Yongbin Ruan and Gang Tian, Higher genus symplectic invariants and sigma models coupled with gravity, Invent. Math. 130 (1997), no. 3, 455â516. MR 1483992, DOI https://doi.org/10.1007/s002220050192
- Shing-Tung Yau and Eric Zaslow, BPS states, string duality, and nodal curves on $K3$, Nuclear Phys. B 471 (1996), no. 3, 503â512. MR 1398633, DOI https://doi.org/10.1016/0550-3213%2896%2900176-9
bl1 J. Bryan and N.C. Leung, The enumerative geometry of K3 surfaces and modular forms, J. Amer. Math. Soc. 13 (2000), 371â410.
bl2 ---, Counting curves on irrational surfaces, Survey of Differential Geometry. 5 (1999), 313â339 .
g E. Getzler, Topological recursion relations in genus 2, In âIntegrable systems and algebraic geometry (Kobe/Kyoto, 1997).â World Sci. Publishing, River Edge, NJ, 198, pp. 73â106.
go L. Göttsche, A conjectural generating function for numbers of curves on surfaces, Comm. Math. Phys. 196 (1998), no. 3, 523â533.
ip E. Ionel and T. Parker, The Symplectic Sum Formula for Gromov-Witten Invariants, Ann. Math. 159 (2004), 935â1025.
km1 M. Kontsevich and Y.I. Manin, Relations between the correlators of the topological sigma model coupled to gravity, Commun. Math. Phys. 196 (1998), 385â398.
l1 J. Lee, Family Gromov-Witten Invariants for KĂ€hler Surfaces, Duke Math. J. 123 (2004), no. 1, 209â233.
l2 ---, Counting Curves in Elliptic Surfaces by Symplectic Methods, preprint, math. SG/0307358, to appear in Comm. Anal. and Geom.
li Liu Xiaobo, Quantum product on the big phase space and the Virasoro conjecture, Adv. math. 169 (2002), 313â375.
ll J. Lee and N.C. Leung, Yau-Zaslow formula on K3 surfaces for non-primitive classes, Geom. Topol. 9 (2005), 1977â2012.
lt J. Li and G. Tian, Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds, Topics in symplectic $4$-manifolds (Irvine, CA, 1996), 47â83, First Int. Press Lect. Ser., I, International Press, Cambridge, MA, 1998.
m D. Mumford, Towards an enumerating geometry of the moduli space of curves, in Arithmetic and Geometry, M. Artin and J. Tate, eds., BirkhĂ€user, 1995, 401â417.
rt Y. Ruan and G. Tian, Higher genus symplectic invariants and sigma models coupled with gravity, Invent. Math. 130 (1997), 455â516.
yz S.T. Yau and E. Zaslow, BPS States, String Duality, and Nodal Curves on K3, Nuclear Phys. B 471 (1996), 503â512.
Additional Information
Junho Lee
Affiliation:
921 D Cherry Lane, East Lansing, Michigan 48823
Email:
leejunho@msu.edu
Naichung Conan Leung
Affiliation:
Institute of Mathematical Sciences, The Chinese University of Hong Kong, Shatin, NT, Hong Kong
MR Author ID:
610317
Email:
leung@ims.cuhk.edu.hk
Received by editor(s):
April 29, 2004
Received by editor(s) in revised form:
October 9, 2005
Published electronically:
May 2, 2006
Additional Notes:
The second author is partially supported by NSF/DMS-0103355, CUHK/2060275, and CUHK/2160256.
