A great number of theoretical results are known about log Gromov–Witten invariants (Abramovich and Chen in Asian J Math 18:465–488, 2014; Chen in Ann Math (2) 180:455–521, 2014; Gross and Siebert J Am Math Soc 26: 451–510, 2013), but few calculations are worked out. In this paper we restrict to surfaces and to genus 0 stable log maps of maximal tangency. We ask how various natural components of the moduli space contribute to the log Gromov–Witten invariants. The first such calculation (Gross et al. in Duke Math J 153:297–362, 2010, Proposition 6.1) by Gross–Pandharipande–Siebert deals with multiple covers over rigid curves in the log Calabi–Yau setting. As a natural continuation, in this paper we compute the contributions of non-rigid irreducible curves in the log Calabi–Yau setting and that of the union of two rigid curves in general position. For the former, we construct and study a moduli space of “logarithmic” 1-dimensional sheaves and compare the resulting multiplicity with tropical multiplicity. For the latter, we explicitly describe the components of the moduli space and work out the logarithmic deformation theory in full, which we then compare with the deformation theory of the analogous relative stable maps.

关于 log Gromov–Witten 不变量的大量理论结果是已知的（Abramovich and Chen in Asian J Math 18:465–488, 2014; Chen in Ann Math (2) 180:455–521, 2014; Gross and Siebert J Am Math Soc 26: 451–510, 2013)，但很少计算。在本文中，我们限制为曲面和最大相切的 0 属稳定对数图。我们询问模空间的各种自然分量如何对 log Gromov-Witten 不变量做出贡献。Gross-Pandharipande-Siebert 的第一个这样的计算（Gross 等人在 Duke Math J 153:297-362, 2010, Proposition 6.1）处理 log Calabi-Yau 设置中刚性曲线上的多个覆盖。作为自然的延续，在本文中，我们计算了 log Calabi-Yau 设置中非刚性不可约曲线的贡献以及一般位置的两条刚性曲线的并集。对于前者，我们构建并研究了“对数”一维滑轮的模空间，并将由此产生的多重性与热带多重性进行了比较。对于后者，我们明确地描述了模空间的分量并完整地计算出对数变形理论，然后我们将其与类似的相对稳定映射的变形理论进行比较。