We show that, after the change of variables \(q=e^{iu}\), refined floor diagrams for \({\mathbb {P}}^2\) and Hirzebruch surfaces compute generating series of higher genus relative Gromov–Witten invariants with insertion of a lambda class. The proof uses an inductive application of the degeneration formula in relative Gromov–Witten theory and an explicit result in relative Gromov–Witten theory of \({\mathbb {P}}^1\). Combining this result with the similar looking refined tropical correspondence theorem for log Gromov–Witten invariants, we obtain a non-trivial relation between relative and log Gromov–Witten invariants for \({\mathbb {P}}^2\) and Hirzebruch surfaces. We also prove that the Block–Göttsche invariants of \({\mathbb {F}}_0\) and \({\mathbb {F}}_2\) are related by the Abramovich–Bertram formula.

我们表明，在变量\(q=e^{iu}\)发生变化后，\({\mathbb {P}}^2\)和 Hirzebruch 曲面的精制平面图计算了更高属相对 Gromov– 的生成系列插入 lambda 类的 Witten 不变量。该证明使用了相对 Gromov-Witten 理论中退化公式的归纳应用和相对 Gromov-Witten 理论中\({\mathbb {P}}^1\)的明确结果。将此结果与对数 Gromov–Witten 不变量的类似的精炼热带对应定理相结合，我们获得了\({\mathbb {P}}^2\)和 Hirzebruch 曲面的相对和对数 Gromov–Witten 不变量之间的非平凡关系. 我们还证明了 Block-Göttsche 不变量\({\mathbb {F}}_0\)和\({\mathbb {F}}_2\)由 Abramovich-Bertram 公式相关。