We prove a higher genus version of the genus $0$ local-relative correspondence of van Garrel-Graber-Ruddat: for $(X,D)$ a pair with X a smooth projective variety and D a nef smooth divisor, maximal contact Gromov-Witten theory of $(X,D)$ with $\lambda _g$ -insertion is related to Gromov-Witten theory of the total space of ${\mathcal O}_X(-D)$ and local Gromov-Witten theory of D. Specializing to $(X,D)=(S,E)$ for S a del Pezzo surface or a rational elliptic surface and E a smooth anticanonical divisor, we show that maximal contact Gromov-Witten theory of $(S,E)$ is determined by the Gromov-Witten theory of the Calabi-Yau 3-fold ${\mathcal O}_S(-E)$ and the stationary Gromov-Witten theory of the elliptic curve E. Specializing further to $S={\mathbb P}^2$ , we prove that higher genus generating series of maximal contact Gromov-Witten invariants of $({\mathbb P}^2,E)$ are quasimodular and satisfy a holomorphic anomaly equation. The proof combines the quasimodularity results and the holomorphic anomaly equations previously known for local ${\mathbb P}^2$ and the elliptic curve. Furthermore, using the connection between maximal contact Gromov-Witten invariants of $({\mathbb P}^2,E)$ and Betti numbers of moduli spaces of semistable one-dimensional sheaves on ${\mathbb P}^2$ , we obtain a proof of the quasimodularity and holomorphic anomaly equation predicted in the physics literature for the refined topological string free energy of local ${\mathbb P}^2$ in the Nekrasov-Shatashvili limit.

中文翻译：

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局部的全纯异常方程和 Nekrasov-Shatashvili 极限

我们证明了属的更高属版本 $0$ van Garrel-Graber-Ruddat 的局部相对对应：对于 $(X,D)$ 一对X平滑的射影变体和D一个 nef 平滑除数，最大接触 Gromov-Witten 理论 $(X,D)$ 和 $\lambda_g$ -insertion 与 Gromov-Witten 的总空间理论有关 ${\mathcal O}_X(-D)$ 和局部 Gromov-Witten 理论D. 专攻 $(X,D)=(S,E)$ 为了小号del Pezzo 曲面或有理椭圆曲面和乙一个光滑的反规范除数，我们证明了最大接触 Gromov-Witten 理论 $(S,E)$ 由 Calabi-Yau 3 倍的 Gromov-Witten 理论确定 ${\mathcal O}_S(-E)$ 和椭圆曲线的平稳 Gromov-Witten 理论乙. 进一步专注于 $S={\mathbb P}^2$ ，我们证明了更高的属生成系列的最大接触 Gromov-Witten 不变量 $({\mathbb P}^2,E)$ 是拟模的并且满足一个全纯异常方程。该证明结合了准模结果和以前已知的局部全纯异常方程 ${\mathbb P}^2$ 和椭圆曲线。此外，使用最大接触 Gromov-Witten 不变量之间的联系 $({\mathbb P}^2,E)$ 和 Betti 数的半稳态一维滑轮模空间 ${\mathbb P}^2$ ，我们得到了物理学文献中预测的对局部的精化拓扑弦自由能的拟模性和全纯异常方程的证明 ${\mathbb P}^2$ 在 Nekrasov-Shatashvili 极限。