Counting curves in elliptic surfaces by symplectic methods
type: Item
instance of: scholarly article
title: Counting curves in elliptic surfaces by symplectic methods
zbMATH Open document ID: 1117.53063
author: Junho Lee
published in: Communications in Analysis and Geometry
publication date: 2007-03-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0307358
review text: 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…
reviewed by: Akira Asada
Mathematics Subject Classification ID: 53D45
Mathematics Subject Classification ID: 14J27
zbMATH DE Number: 5135051
zbMATH Keywords: family of Gromov-Witten invariants
zbMATH Keywords: elliptic surface
zbMATH Keywords: topological recursion relation
MaRDI profile type: Publication
arXiv ID: math/0307358
OpenAlex ID: W2052556682
DOI: 10.4310/CAG.2006.V14.N1.A5
description: scientific article