标题： 实定向爆破在环面、环面化和对数几何中的介绍 显示英文标题

标题： An introduction to real oriented blowups in toric, toroidal and logarithmic geometries

摘要： 本文是关于复数对数空间（也称为Kato-Nakayama或Betti实现）的四舍五入在奇点理论中的应用的介绍。 按照Fontaine和Illusie意义上的对数空间，首先由Kato在1988年的论文中发表。 复数对数空间的四舍五入是由Kato和Nakayama在1999年的论文中引入的，它是A'Campo于1975年提出的实定向爆破概念的函子性推广。 它允许沿着其圆锥边界$\partial X$，以规范的方式切割任何复数圆锥流形$X$，从而产生一个带边界的拓扑流形，其边界是$\partial X$在$X$中的任何管状邻域边界的规范代表。 在奇点理论中，一旦选择了奇点或其平滑的圆锥解析，可以使用四舍五入来获得孤立复解析奇点的链接和复奇点平滑的Milnor纤维的规范代表。 本文首先介绍了不一定正常的仿射扇形流形，然后转向圆锥流形及其实定向爆破。 接着介绍了对数空间和复数对数空间的四舍五入。 最后介绍了Nakayama和Ogus关于特定类型对数态射的四舍五入局部平凡性的一个重要定理。 仿射扇形流形、实定向爆破、对数结构和四舍五入的概念是通过经典的极坐标转换引入的。 显示英文摘要

摘要： This text is an introduction to the applications of rounding of complex log spaces (also known as Kato-Nakayama or Betti realization) to singularity theory. Log spaces in the sense of Fontaine and Illusie were first described in print by Kato, in a 1988 paper. Rounding of complex log spaces was introduced in a 1999 paper by Kato and Nakayama and is a functorial generalization of A'Campo's 1975 notion of a real oriented blowup. It allows to cut canonically any complex toroidal variety $X$ along its toroidal boundary $\partial X$, producing a topological manifold-with-boundary, whose boundary is a canonical representative of the boundary of any tubular neighborhood of $\partial X$ in $X$. In singularity theory, roundings may be used to get canonical representatives of links of isolated complex analytic singularities and of Milnor fibers of smoothings of complex singularities, once toroidal resolutions of the singularity or of the smoothing are chosen. The text starts with introductions to not necessarily normal toric varieties, it passes then to toroidal varieties and to their real oriented blowups. It continues with introductions to log spaces and to rounding of complex log spaces. It concludes with an important theorem of Nakayama and Ogus about the local triviality of the rounding of special types of log morphisms. The notions of affine toric variety, real oriented blowup, log structure and rounding are introduced by means of the classical passage to polar coordinates.