## Geometry & Topology

### Yau–Zaslow formula on K3 surfaces for non-primitive classes

#### Abstract

We compute the genus zero family Gromov–Witten invariants for K3 surfaces using the topological recursion formula and the symplectic sum formula for a degeneration of elliptic K3 surfaces. In particular we verify the Yau–Zaslow formula for non-primitive classes of index two.

#### Article information

Source
Geom. Topol., Volume 9, Number 4 (2005), 1977-2012.

Dates
Revised: 16 October 2005
Accepted: 24 April 2005
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513799673

Digital Object Identifier
doi:10.2140/gt.2005.9.1977

Mathematical Reviews number (MathSciNet)
MR2175162

Zentralblatt MATH identifier
1088.53059

#### Citation

Lee, Junho; Leung, Naichung Conan. Yau–Zaslow formula on K3 surfaces for non-primitive classes. Geom. Topol. 9 (2005), no. 4, 1977--2012. doi:10.2140/gt.2005.9.1977. https://projecteuclid.org/euclid.gt/1513799673

#### References

• A Beauville, Counting rational curves on $K3$ surfaces, Duke Math. J. 97 (1999) 99–108
• J Bryan, N C Leung, The enumerative geometry of $K3$ surfaces and modular forms, J. Amer. Math. Soc. 13 (2000) 371–410
• J Bryan, N C Leung, Counting curves on irrational surfaces, from: “Surveys in differential geometry: differential geometry inspired by string theory”, Surv. Differ. Geom. 5, Int. Press, Boston, MA (1999) 313–339
• X Chen, Rational curves on $K3$ surfaces, J. Algebraic Geom. 8 (1999) 245–278
• A Gathmann, The number of plane conics 5-fold tangent to a given curve.
• E Getzler, Topological recursion relations in genus $2$, from: “Integrable systems and algebraic geometry (Kobe/Kyoto, 1997)”, World Sci. Publishing, River Edge, NJ (1998) 73–106
• E-N Ionel, T H Parker, Relative Gromov-Witten invariants, Ann. of Math. (2) 157 (2003) 45–96
• E-N Ionel, T H Parker, The symplectic sum formula for Gromov-Witten invariants, Ann. of Math. (2) 159 (2004) 935–1025
• S Ivashkovich, V Shevchishin, Gromov compactness theorem for $J$–complex curves with boundary, Internat. Math. Res. Notices (2000) no. 22 1167–1206
• A W Knapp, Elliptic curves, Mathematical Notes 40, Princeton University Press, Princeton, NJ (1992)
• M Kontsevich, Y Manin, Relations between the correlators of the topological sigma-model coupled to gravity, Comm. Math. Phys. 196 (1998) 385–398
• M Kaneko, D Zagier, A generalized Jacobi theta function and quasimodular forms, from: “The moduli space of curves (Texel Island, 1994)”, Progr. Math. 129, Birkhäuser Boston, Boston, MA (1995) 165–172
• J Lee, Family Gromov-Witten invariants for Kähler surfaces, Duke Math. J. 123 (2004) 209–233
• J Lee, Counting Curves in Elliptic Surfaces by Symplectic Methods, to appear in Comm. Anal. and Geom. \arxivmath.SG/0307358
• J Lee, N C Leung, Counting Elliptic Curves in K3 Surfaces.
• J Li, A note on enumerating rational curves in a $K3$ surface, from: “Geometry and nonlinear partial differential equations (Hangzhou, 2001)”, AMS/IP Stud. Adv. Math. 29, Amer. Math. Soc., Providence, RI (2002) 53–62
• J Li, G Tian, Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds, from: “Topics in symplectic $4$–manifolds (Irvine, CA, 1996)”, First Int. Press Lect. Ser., I, Internat. Press, Cambridge, MA (1998) 47–83
• T-J Li, A-K Liu, Counting curves on elliptic ruled surface, Topology Appl. 124 (2002) 347–353
• A-K Liu, Family blowup formula, admissible graphs and the enumeration of singular curves. I, J. Differential Geom. 56 (2000) 381–579
• T Parker, Compactified moduli spaces of pseudo-holomorphic curves, from: “Mirror symmetry III (Montreal PQ 1995)”, AMS/IP Stud. Adv. Math. 10, Amer. Math. Soc. Providence RI (1999) 77–113
• T Parker, J Wolfson, Pseudo-holomorphic maps and bubble trees, Jour. Geometric Analysis 3 (1993) 63–98
• Y Ruan, G Tian, Higher genus symplectic invariants and sigma models coupled with gravity, Invent. Math. 130 (1997) 455–516
• S-T Yau, E Zaslow, BPS states, string duality, and nodal curves on $K3$, Nuclear Phys. B 471 (1996) 503–512