发表机构
Research Institute of Mathematics, Seoul National University（首尔大学数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文构造了单奇异纤维的周期单值群莱夫谢茨纤维化，借鉴松本思路将其分裂为莱夫谢茨纤维，确定消失圈并给出相关分裂与构造步骤。
AI 中文摘要
莱夫谢茨纤维化是一种光滑4维流形，它除了有限多个奇异纤维外，都具有曲面在另一个曲面上的纤维丛结构，且这些奇异纤维的奇点仅为结点型。从奇异纤维的结构来看，每个奇异纤维的单值群由沿纤维中某条曲线（称为消失圈）的右旋德恩扭转给出。收集所有奇异纤维的单值群数据，可得到一个由右旋德恩扭转构成的单值群分解，该分解完全决定了莱夫谢茨纤维化。本文中，我们构造了一个具有单个奇异纤维的纤维化，其单值群是周期的（即单值同胚同痕于一个周期映射）。遵循松本的思路，我们将该奇异纤维分裂为莱夫谢茨纤维，并确定了一组周期单值群对应的消失圈。我们描述了分裂奇异纤维的构造过程，以及利用纤维的两个分支覆盖结构读取消失圈的步骤。我们还给出了与一族周期作用相关的奇异纤维到莱夫谢茨纤维的分裂，并确定了它们的消失圈。
英文摘要
A Lefschetz fibration is a smooth 4-manifold admitting a surface bundle structure over a surface except at finitely many singular fibers, whose singularities are only of nodal type. From the structure of the singular fibers, the monodromy of each singular fiber is given by a right-handed Dehn twist along a curve, called a vanishing cycle, in the fiber. Collecting all monodromy data from the singular fibers, we obtain a monodromy factorization into right-handed Dehn twists, which completely determines the Lefschetz fibration. In this paper, we construct a fibration with one singular fiber whose monodromy is periodic (that is, the monodromy homeomorphism is isotopic to a periodic map). Following the idea of Matsumoto, we give a splitting of the singular fiber into Lefschetz fibers and determine their vanishing cycles for a collection of periodic monodromies. We describe the construction of the splitting singular fibers and the procedure for reading vanishing cycles using two branched covering structures of the fibers. We also give splittings of singular fibers into Lefschetz fibers related to a family of periodic actions and determine their vanishing cycles.
Comments48 pages, 17 figures
