发表机构
School of Mathematics and Statistics, Shaanxi Normal University; Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS); Research Institute of Intelligent Complex Systems, Fudan University（陕西师范大学数学与统计学院; 上海数学与交叉学科研究院（SIMIS）; 复旦大学智能复杂体系研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究带停止点的分次曲面由映射类群和导出Picard群重构的问题，证明了在亏格至少2时这些代数不变量可决定曲面结构，并构造了具有相同不变量但不等价的范畴族。
AI 中文摘要
我们研究了带停止点的分次曲面$(S,M,\eta)$如何由其映射类群及其部分缠绕Fukaya范畴的导出Picard群重构。在亏格至少为2时，我们证明了未分次的映射类群决定了带停止点的曲面$(S,M)$，而带有平移区分的导出Picard群决定了底层的分次曲面$(S,\eta)$。然而，同时区分平移和Serre函子的导出Picard群具有可数无穷的纤维。我们还构造了任意大的有限族，这些族中的范畴两两不等价，但具有相同的不变量和Grothendieck秩。对于由无处消失向量场诱导的线场，我们证明了有理Serre表示在任意亏格下决定了带停止点的曲面$(S,M)$。对于亏格至少为2的任意线场，带有平移区分的完整积分导出Picard表示的纤维基数至多为4。当$S$具有一个或至少五个边界分支时，它完全决定了带停止点的分次曲面$(S,M,\eta)$，而对于两个、三个和四个边界分支的情况则会出现非单元素纤维。
英文摘要
We study reconstruction of graded surfaces with stops $(S,M,η)$ from their mapping class groups and the derived Picard groups of their partially wrapped Fukaya categories. In genus at least two, we prove that the ungraded mapping class group determines the surface with stops $(S,M)$, while the derived Picard group with the shift distinguished determines the underlying graded surface $(S,η)$. Nevertheless, the derived Picard group with both the shift and the Serre functor distinguished has countably infinite fibres. We also construct arbitrarily large finite families of pairwise non-equivalent categories with the same invariant and Grothendieck rank. For line fields induced by nowhere-vanishing vector fields, we show that the rational Serre representation determines the surface with stops $(S,M)$ in every genus. For arbitrary line fields in genus at least two, the full integral derived Picard representation, with the shift distinguished, has fibres of cardinality at most four. It completely determines the graded surface with stops $(S,M,η)$ when $S$ has one or at least five boundary components, while non-singleton fibres occur for two, three, and four boundary components.
Comments64 pages, comments welcome!
