Arc spaces and additive invariants in real algebraic and analytic geometry
Arc spaces and additive invariants in real algebraic and analytic geometry
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实代数和解析几何中的弧空间和加性不变量
DOI：
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发表时间：
2007
期刊：
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通讯作者：
M. Coste
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作者：
M. Coste
- 在这卷中，我们提出了一些新的趋势，在真实的代数几何的基础上研究弧空间和添加剂不变量的真实的代数集。一般来说，真实的代数几何使用它自己的方法，通常与更广为人知的复代数几何方法有很大的不同。这一特点在研究真实的代数集的基本拓扑和几何性质时尤为明显;丰富的代数结构通常是隐藏的，无法从拓扑中恢复。弧空间和加法不变量的使用部分地消除了这个缺点。此外，这些方法往往是平行的复代数几何的基本方法。我们的介绍包含建设的局部拓扑不变量的真实的代数集的代数构造函数。这种技术被扩展到更广泛的家庭的弧对称半代数集。此外，后者定义了一个自然拓扑，填补了Zurski拓扑和欧几里得拓扑之间的空白。在真实的等奇异性理论中，Kuo的真实的解析函数芽的吹解析等价提供了一个等价关系，它对应于复解析结构中的拓扑等价。在其他应用中，弧对称几何，通过motivic集成方法，给出了新的不变量，这种等价性，允许一些初始的分类结果。该卷载有两门课程和两篇调查文章，是为广大读者，特别是学生和青年研究人员设计的。c © Panoramas et Synthèses 24，SMF 2007
— In this volume we present some new trends in real algebraic geometry based on the study of arc spaces and additive invariants of real algebraic sets. Generally, real algebraic geometry uses methods of its own that usually differ sharply from the more widely known methods of complex algebraic geometry. This feature is particularly apparent when studying the basic topological and geometric properties of real algebraic sets; the rich algebraic structures are usually hidden and cannot be recovered from the topology. The use of arc spaces and additive invariants partially obviates this disadvantage. Moreover, these methods are often parallel to the basic approaches of complex algebraic geometry. Our presentation contains the construction of local topological invariants of real algebraic sets by means of algebraically constructible functions. This technique is extended to the wider family of arc-symmetric semialgebraic sets. Moreover, the latter family defines a natural topology that fills a gap between the Zariski topology and the euclidean topology. In real equisingularity theory, Kuo’s blow-analytic equivalence of real analytic function germs provides an equivalence relation that corresponds to topological equivalence in the complex analytic set-up. Among other applications, arc-symmetric geometry, via the motivic integration approach, gives new invariants of this equivalence, allowing some initial classification results. The volume contains two courses and two survey articles that are designed for a wide audience, in particular students and young researchers. c © Panoramas et Synthèses 24, SMF 2007