论文标题
Neumann和Wahl的Milnor纤维猜想，以及其证明的概述
The Milnor fiber conjecture of Neumann and Wahl, and an overview of its proof
论文作者
论文摘要
Neumann和Wahl于2002年推出的剪接类型表面奇异性提供了迄今已知的所有示例，这些示例是迄今为止的积分同源球，这些示例似乎是二维尺寸的复杂分离的完整相交的链接。通过称为剪接图的装饰树，它们被确定为一种平等性的形式。 2005年，诺伊曼（Neumann）和瓦尔（Wahl）提出了他们的米尔诺（Milnor）纤维猜想，并指出，剪接图的内部边缘的任何选择都将特殊的分解分解为相关奇点的米尔诺纤维部分。这些碎片是由由剪接类型奇异性的Milnor纤维构成的，由所选边缘两侧的细分布确定。在本文中，我们概述了这种猜想的概述，并根据Fontaine和Illusie意义上的热带几何形状和日志几何形状的技术进行了详细的证明概述。至关重要的对数几何成分是Kato和Nakayama于1999年引入的复杂对数空间的圆形操作。这是对实际取向爆炸的操作的功能概括。 1975年，A'Campo率先使用了后者来研究Milnor纤维化。
Splice type surface singularities, introduced in 2002 by Neumann and Wahl, provide all examples known so far of integral homology spheres which appear as links of complex isolated complete intersections of dimension two. They are determined, up to a form of equisingularity, by decorated trees called splice diagrams. In 2005, Neumann and Wahl formulated their Milnor fiber conjecture, stating that any choice of an internal edge of a splice diagram determines a special kind of decomposition into pieces of the Milnor fibers of the associated singularities. These pieces are constructed from the Milnor fibers of the splice type singularities determined by the subdiagrams on both sides of the chosen edge. In this paper we give an overview of this conjecture and a detailed outline of its proof, based on techniques from tropical geometry and log geometry in the sense of Fontaine and Illusie. The crucial log geometric ingredient is the operation of rounding of a complex logarithmic space introduced in 1999 by Kato and Nakayama. It is a functorial generalization of the operation of real oriented blowup. The use of the latter to study Milnor fibrations was pioneered by A'Campo in 1975.
