球状对象、传递性及Kodaira循环的自同构 via 轻柔代数
Spherical objects, transitivity and auto-equivalences of Kodaira cycles via gentle algebras
AI总结：
本文通过轻柔代数研究Kodaira循环上的球状对象，推导出其同构类的拓扑模型，并证明自同构群的传递性，同时给出简单向量包的唯一性定理。
摘要：
本文研究了任何射影线的Kodaira n-循环上的球状对象类，并以n- punctured torus上的闭合曲线参数化其同构类，这些曲线无自相交。利用最近关于轻柔代数的结果，我们推导出任何Kodaira循环的有界导出范畴的拓扑模型。这些范畴的三角自同构群被计算并证明其在球状对象同构类上作用传递。这回答了Polishchuk提出的问题，并扩展了Burban-Kreussler和Lekili-Polishchuk先前的结果。自同构的描述进一步用于证明Sibilla定义的映射类群作用的忠实性。最后一部分描述了对应于向量包和简单向量包的闭合曲线。这导致了Bodnarchuk-Drozd-Greuel结果的另一种证明，该结果指出，射影线的循环上的简单向量包由其多度、秩和行列式唯一确定。作为附带结果，我们得到了Burban-Drozd-Greuel引入的任何简单向量包在C_n上的循环序列的闭合公式。
英文摘要：
This paper studies the class of spherical objects over any Kodaira $n$-cycle of projective lines and provides a parametrization of their isomorphism classes in terms of closed curves on the $n$-punctured torus without self-intersections. Employing recent results on gentle algebras, we derive a topological model for the bounded derived category of any Kodaira cycle. The groups of triangle auto-equivalences of these categories are computed and are shown to act transitively on isomorphism classes of spherical objects. This answers a question by Polishchuk and extends earlier results by Burban-Kreussler and Lekili-Polishchuk. The description of auto-equivalences is further used to establish faithfulness of a mapping class group action defined by Sibilla. The final part describes the closed curves which correspond to vector bundles and simple vector bundles. This leads to an alternative proof of a result by Bodnarchuk-Drozd-Greuel which states that simple vector bundles on cycles of projective lines are uniquely determined by their multi-degree, rank and determinant. As a by-product we obtain a closed formula for the cyclic sequence of any simple vector bundle on $C_n$ as introduced by Burban-Drozd-Greuel.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
