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Arc spaces, motivic measure and Lipschitz geometry of real algebraic sets

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Abstract

We investigate connections between Lipschitz geometry of real algebraic varieties and properties of their arc spaces. For this purpose we develop motivic integration in the real algebraic set-up. We construct a motivic measure on the space of real analytic arcs. We use this measure to define a real motivic integral which admits a change of variables formula not only for the birational but also for generically one-to-one Nash maps. As a consequence we obtain an inverse mapping theorem which holds for continuous rational maps and, more generally, for generically arc-analytic maps. These maps appeared recently in the classification of singularities of real analytic function germs. Finally, as an application, we characterize in terms of the motivic measure, germs of arc-analytic homeomorphism between real algebraic varieties which are bi-Lipschitz for the inner metric.

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Notes

  1. For ease of reading, in the introduction we avoid varieties admitting points which have a structure of smooth submanifold of smaller dimension as in the handle of the Whitney umbrella \(\{x^2=zy^2\}\subset \mathbb R^3\).

  2. Actually, noticing that \(f_1=\cdots =f_s=0\Leftrightarrow f_1^2+\cdots +f_s^2=0\), we may always describe a real algebraic set as the zero-set of only one polynomial.

  3. A subset of \(\mathbb P_\mathbb R^N\) is semialgebraic if it is for \(\mathbb P_\mathbb R^N\) seen as an algebraic subset of some \(\mathbb R^M\), or, equivalently, if the intersection of the set with each canonical affine chart is semialgebraic.

  4. i.e. for every \(x\in B\) there is \(U\subset B\) an \(\mathcal {AS}\)-open subset containing x such that \(p^{-1}(U)\simeq U\times F\).

  5. i.e. \(\mathcal F^{m+1}\mathcal M\subset \mathcal F^m\mathcal M\) and \(\mathcal F^m\mathcal M\cdot \mathcal F^n\mathcal M\subset \mathcal F^{m+n}\mathcal M\). The last condition induces a ring structure on the group \(\widehat{\mathcal M}\).

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Authors and Affiliations

  1. Aix Marseille University, CNRS, Centrale Marseille, I2M, Marseille, France

    Jean-Baptiste Campesato

  2. Department of Mathematics, Faculty of Science, Saitama University, 255 Shimo-Okubo, Sakura-ku, Saitama, 338-8570, Japan

    Toshizumi Fukui

  3. University Grenoble Alpes, University Savoie Mont Blanc, CNRS, LAMA, 73000, Chambéry, France

    Krzysztof Kurdyka

  4. Université Côte d’Azur, Université Nice Sophia Antipolis, CNRS, LJAD, Nice, France

    Adam Parusiński

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  1. Jean-Baptiste Campesato
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  2. Toshizumi Fukui
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  3. Krzysztof Kurdyka
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  4. Adam Parusiński
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Correspondence to Adam Parusiński.

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Communicated by Jean-Yves Welschinger.

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The second author was supported by JSPS KAKENHI Grant Number JP26287011.

The two last authors were partially supported by ANR project LISA (ANR-17-CE40-0023-03)

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Campesato, JB., Fukui, T., Kurdyka, K. et al. Arc spaces, motivic measure and Lipschitz geometry of real algebraic sets. Math. Ann. 374, 211–251 (2019). https://doi.org/10.1007/s00208-019-01805-8

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