Counting curves in elliptic surfaces by symplectic methods
The author introduced a family of Gromov-Witten (GW) invariants for Kähler surfaces with \(p_g> 0\) [\textit{J. Lee}, Family Gromov-Witten invariants for Kähler surfaces, Duke Math. J. 123, No. 1, 209--233 (2004; Zbl 1059.53070)]. In this paper, the generating function of the family of GW invariants of a standard elliptic surface \(E(n)\) with a section of self-intersection \(-n\), for \(S+ dF\), \(S\) and \(F\), the homology classes of the section and fiber, respectively, is computed to be \[ \sum_{d\geq 0} GW^{{\mathcal H}}_{S+ dF,g}(E(n))(pt^g)\,t^d= (tG'(t))^g \prod_{d\geq 1} \Biggl({1\over 1- t^d}\Biggr)^{12n}, \] where \(G(t)= \sum_{d\geq 1}\sigma(t)\,t^d\), \(\sigma(d)= \sum_{k|d} k\) (Theorem 0.1). For \(E(1)\) (rational elliptic surfaces), and \(E(2)\) (\(K3\) surfaces), this formula was already proved by \textit{J. Bryan} and \textit{N. C. Leung} [J. Am. Math. Soc. 13, No. 2, 371--410 (2000; Zbl 0963.14031)]. Bryan and Leung also showed that this formula counts holomorphic curves in the primitive classes for the generic complex structure on those surfaces. But the author says it is not clear whether that is true for \(E(n)\) with \(n\geq 3\). The definition of the family GW invariants is reviewed in \S1. They are defined relative to \(S+ dF\) and unchanged under deformation of Kähler structure (Proposition 1.4). In \S2, an outline of the proof of Theorem 0.1 is described as follows: Let \[ F_g(t)= \sum_{d\geq 0} \text{GW}^{{\mathcal H}}_{S+ dF,g}(pt^g)\,t^d,\quad H(t)= \sum_{d\geq 0} \text{GW}^{{\mathcal H}}_{S+ dF,1}(\tau(F))\,t^d, \] where \(\tau(F)= \psi_1\cup ev^*(F^*)\), \(\psi_1\) is the first Chern class of a line bundle of a certain moduli space, then \(H(t)\) satisfies the topological recursion relation \[ H(t)= {1\over 12} tF_0'(t)- {1\over 12} F_0(t)+ (2- n)F_0(t) G(t). \] This is proved in \S3. Then the equations \[ H(t)= -{1\over 12} F_0(t)+ 2F_0(t) G(t),\quad F_g(t)= F_{g-1}(t) tG'(T), \] are established in \S4 to \S8. Hence we get \[ tF_0'(t)= 12nG(t) F_0(t). \] Since \(\text{GW}^{{\mathcal H}}_{S,0}(E(n))= 1\) (Proposition 4.6), the initial condition of this differential equation is \(F_0(0)= 1\). Therefore we get \[ F_0(t)= \prod_{d\geq 1} \Biggl({1\over 1- t^d}\Biggr)^{12n}. \] This proves Theorem 0.1.
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