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Motivic-type invariants of blow-analytic equivalence. (English) Zbl 1062.14006
In this difficult and interesting work the authors develop techniques that allow to study and distinguish different blow-analytic classes of analytic function germs $$f:(\mathbb R^n,0)\to (\mathbb R,0)$$. They adapt and apply to the real analytic set-up the ideas coming from motivic integrations, in particular the concept of motivic zeta functions due to J. Denef and F. Loeser [J. Algebr. Geom. 7, No.3, 505–537 (1998; Zbl 0943.14010); Invent. Math. 135, 201–232 (1999; Zbl 0928.14004); Duke Math. J. 99, No. 2, 285–309 (1999; Zbl 0966.14015)]. The blow-analytic equivalence is interesting because it does not allow continuous moduli for families of isolated singularities and it preserves a deep information on the algebraic structure of the singularity.
On the other hand for real singularities, unlike for the complex ones, the topological classification is very weak. The blow-analytic equivalence was proposed by T. C. Kuo [Invent. Math. 82, 257–262 (1985; Zbl 0587.32018)] in order to overcome this problem. Moreover the authors suggest by some examples that the blow-up analytic equivalence of real analytic function germs behaves in a similar way to the topological equivalence in the complex case. Until now there were very few results allowing to distinguish different blow-analytic types and hence to attempt a classification even for simplest analytic singularities. The main result of the paper is to introduce new invariants and to start such a classification.

##### MSC:
 14B05 Singularities in algebraic geometry 32S15 Equisingularity (topological and analytic)
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##### References:
 [1] Polyèdre de Newton et trivialité en famille, J. Math. Soc. Japan, 54, 513-550, (2002) · Zbl 1031.58024 [2] Arc-analytic functions, Invent. Math., 101, 411-424, (1990) · Zbl 0723.32005 [3] Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math., 128, 207-302, (1997) · Zbl 0896.14006 [4] Topological triviality of deformations of functions and Newton filtrations, Invent. Math., 72, 335-358, (1983) · Zbl 0519.58021 [5] Motivic igusa zeta functions, J. Alg. Geom., 7, 505-537, (1998) · Zbl 0943.14010 [6] Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math., 135, 201-232, (1999) · Zbl 0928.14004 [7] Motivic exponential integrals and a motivic thom-sebastiani theorem, Duke Math. J., 99, 289-309, (1999) · Zbl 0966.14015 [8] Geometry of arc spaces of algebraic varieties, European Congress of Math. (Barcelona, July 10-14, 2000), Vol. 1, 327-348, (2001) · Zbl 1079.14003 [9] Lefschetz numbers of iterates of the monodromy and truncated arcs, Topology, 41, 1031-1040, (2002) · Zbl 1054.14003 [10] The modified analytic trivialization of family of real analytic functions, Invent. Math., 82, 467-477, (1985) · Zbl 0559.58005 [11] Seeking invariants for blow-analytic equivalence, Comp. Math., 105, 95-107, (1997) · Zbl 0873.32008 [12] Blow-analytic equisingularities, properties, problems and progress, real analytic and algebraic singularities, 381, 8-29, (1998) · Zbl 0954.26012 [13] Modified analytic trivialization for weighted homogeneous function-germs, J. Math. Soc. Japan, 52, 433-446, (2000) · Zbl 0964.32023 [14] Existence of moduli for bi-Lipschitz equivalence of analytic functions, Comp. Math., 136, 217-235, (2003) · Zbl 1026.32055 [15] Invariants of bi-Lipschitz equivalence of real analytic functions · Zbl 1059.32006 [16] Resolution of singularities of an algebraic variety over a field of characteristic zero: I, II, Ann. of Math., 79, 109-302, (1964) · Zbl 0122.38603 [17] Computations and stability of the fukui invariant, Comp. Math., 130, 49-73, (2002) · Zbl 1007.58023 [18] (1995) [19] Examples in the theory of sufficiency of jets, Proc. Amer. Math. Soc., 96, 163-166, (1986) · Zbl 0594.58008 [20] $$C^1$$-equivalence of functions near isolated critical points, Symp. Infinite Dimensional Topology, Baton Rouge, 1967, 69, 199-218, (1972), Princeton Univ. Press · Zbl 0236.58001 [21] On $$C^0$$-sufficiency of jets of potential functions, Topology, 8, 167-171, (1969) · Zbl 0183.04601 [22] The modified analytic trivialization of singularities, J. Math. Soc. Japan, 32, 605-614, (1980) · Zbl 0509.58007 [23] On classification of real singularities, Invent. Math., 82, 257-262, (1985) · Zbl 0587.32018 [24] Ensembles semi-algébriques symétriques par arcs, Math. Ann., 282, 445-462, (1988) · Zbl 0686.14027 [25] Injective endomorphisms of real algebraic sets are surjective, Math. Ann., 282, 1-14, (1998) · Zbl 0933.14036 [26] Topologie des singularités des hypersurfaces complexes, Singularités à Cargèse, 7 & 8, 171-182, (1973) · Zbl 0331.32009 [27] Ensembles semi-analytiques, (1965), I.H.E.S. [28] Motivic Measures, exposé 874, (2000) · Zbl 0996.14011 [29] Complex monodromy and the topology of real algebraic sets, Comp. Math., 106, 211-233, (1997) · Zbl 0949.14037 [30] Isolated singularities defined by weighted homogeneous polynomials, Topology, 9, 385-393, (1970) · Zbl 0204.56503 [31] Topological invariance of weights for weighted homogeneous singularities, Kodai Math. J., 9, 188-190, (1986) · Zbl 0612.32001 [32] Espace des germes d’arcs réels et série de Poincaré d’un ensemble semi-algébrique, Ann. Inst. Fourier, 51, 1, 43-67, (2001) · Zbl 0967.14037 [33] Topological invariance of weights for weighted homogeneous isolated singularities in $$\bb C^3,$$ Proc. Amer. Math. Soc., 103, 995-999, (1988) · Zbl 0656.32009 [34] Cycles évanescents, sections planes, et conditions de Whitney, Singularités à Cargèse, 7 & 8, 285-362, (1973) · Zbl 0295.14003 [35] The topological zeta function associated to a function on a normal surface germ, Topology, 38, 439-456, (1999) · Zbl 0947.32020 [36] Topological types and multiplicity of isolated quasihomogeneous surface singularities, Bull. Amer. Math. Soc., 19, 447-454, (1988) · Zbl 0659.32013 [37] Topological types of quasihomogeneous singularities in $$\bb C^2,$$ Topology, 18, 113-116, (1979) · Zbl 0428.32004
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