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Arc spaces and additive invariants in real algebraic and analytic geometry. (English) Zbl 1155.14313

Coste, Michel et al., Arc spaces and additive invariants in real algebraic and analytic geometry. Paris: Société Mathématique de France (ISBN 978-2-85629-236-5/pbk). Panoramas et Synthèses 24, iii-vi (2007).
Summary: 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.
For the entire collection see [Zbl 1135.14001].

MSC:

14Pxx Real algebraic and real-analytic geometry
14P05 Real algebraic sets
14P10 Semialgebraic sets and related spaces
14P25 Topology of real algebraic varieties
32B20 Semi-analytic sets, subanalytic sets, and generalizations
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