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On some aspects of the geometry of the space of arcs on an analytic space. (Sur quelques aspects de la géométrie de l’espace des arcs tracés sur un espace analytique.) (French) Zbl 1076.32001

Let \((X, x)\) be a germ of real or complex analytic space \(\mathbb K\) and \(\mathcal A(X,x)\) the space of germs of arcs on \((X,x)\). Let \(F_x:(X,x) \to (Y, y)\) be a germ of a morphism and denote by \(\mathcal F_x :\mathcal A_{(X,x)} \to \mathcal A_{(Y,y)}\) the induced morphism at the level of arcs.
The author emphasizes the analogies between the metric or local topological properties of \(F_x\) and those of \(\mathcal F_x\). The author then defines the notions of Nash sequence of multiplicities along an arc \(\varphi\in\mathcal A_{\mathbb K,0}\), Nash sequence of Hilbert-Samuel functions of \((X,x)\) along \(\varphi\), Nash sequence of diagrams of initial exponents of \((X,x)\) along an arc \(\varphi\) for all germs \((X,x)\), and studies some of their basic properties. Some elementary connections between these notions and motivic integration theory are also provided.
Recently, S. Ishii and J. Kollár considered the hypersurface of \(\mathbb C^5\) defined by \(X_1^3 + X_2^3 + X_3^5 + X_4^5 + X_5^6=0\) and showed that it provides in dimension 4 the counterexample to the Nash problem [J. F. Nash jun., Duke Math. J. 81, 31–38 (1995; Zbl 0880.14010)]. As application of the obtained results and illustration of the role played by the Nash sequences, the author calculates the motivic volume of such hypersurfaces. Connections between the spaces of arcs and rank condition of Gabrielov are also considered.

MSC:

32B99 Local analytic geometry
32S45 Modifications; resolution of singularities (complex-analytic aspects)

Citations:

Zbl 0880.14010
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References:

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