We prove a conjecture of Maulik, Pandharipande and Thomas expressing the Gromov–Witten invariants of K3 surfaces for divisibility 2 curve classes in all genera in terms of weakly holomorphic quasi-modular forms of level 2. Then we establish the holomorphic anomaly equation in divisibility 2 in all genera. Our approach involves a refined boundary induction, relying on the top tautological group of the moduli space of smooth curves, together with a degeneration formula for the reduced virtual fundamental class with imprimitive curve classes. We use double ramification relations with target variety as a new tool to prove the initial condition. The relationship between the holomorphic anomaly equation for higher divisibility and the conjectural multiple cover formula of Oberdieck and Pandharipande is discussed in detail and illustrated with several examples.

中文翻译：

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可整除的 K3 曲面上的曲线 2

我们证明了 Maulik、Pandharipande 和 Thomas 的猜想，用 2 级的弱全纯拟模形式表达了 K3 曲面的 Gromov-Witten 不变量，用于所有属中的可除性 2 曲线类。然后我们建立了可除性 2 的全纯异常方程在所有属中。我们的方法涉及精细的边界归纳，依赖于平滑曲线模空间的顶部重言式群，以及具有原始曲线类的简化虚拟基本类的退化公式。我们使用与目标品种的双重派生关系作为证明初始条件的新工具。