AI 中文总结
本文证明在亏格一曲线中，两种Gromov-Witten理论（穿孔与根叠）的负接触环在推出后相等，通过普适靶比较和Crumplin主分量实现。
AI 中文摘要
设$D$是光滑射影复簇$X$中的光滑除子。对于具有指定带符号接触阶的算术亏格一连通曲线，我们证明Battistella--Nabijou--Ranganathan的细化穿孔环与Fan--Wu--You的负接触环在推出到带除子赋值稳定映射的公共模空间后相等。因此，推出的细化穿孔环是推出的根叠虚拟类的常数系数，每个负接触按根阶的一次幂归一化。关键步骤是对普适靶的比较。在由固定对$(X,D)$和数值数据$\Gamma$确定的有限型开子叠上限制后，我们证明正BNR空间有限地且具有一般次数一映射到Crumplin的主分量上。利用Crumplin的亏格一分量描述和次数公式，我们在根阶比较下将该分量的基本环与普适orbifold虚拟类的常数系数等同。细化零截面拉回恢复负接触，相容的虚拟拉回和根遗忘推出将所得恒等式转移到$(X,D)$。
英文摘要
Let $D$ be a smooth divisor in a smooth projective complex variety $X$. For connected curves of arithmetic genus one with prescribed signed contact orders, we prove that the refined punctured cycle of Battistella--Nabijou--Ranganathan and the negative-contact cycle of Fan--Wu--You agree after pushforward to the common moduli space of stable maps with divisor evaluations. Thus the pushed-forward refined punctured cycle is the constant coefficient of the pushed-forward root-stack virtual class, normalized by one power of the root order for each negative contact. The key step is a comparison for the universal target. After restricting to finite-type open substacks determined by the fixed pair $(X,D)$ and numerical data $Γ$, we prove that the positive BNR space maps finitely and with generic degree one onto Crumplin's main component. Using Crumplin's genus-one component description and degree formulas, we identify this component's fundamental cycle with the constant coefficient of the universal orbifold virtual class under comparison of root orders. Refined zero-section pullback recovers the negative contacts, and compatible virtual pullbacks and root-forgetting pushforwards transfer the resulting identity to $(X,D)$.
Comments67 pages. Comments are welcome
