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IMRN International Mathematics Research Notices 2005, No. 31 Nearby Cycles and Composition with a Nondegenerate Polynomial Gil Guibert, Franc¸ois Loeser, andMichel Merle 1 Introduction Let Xj be smooth varieties over a field k of characteristic zero, for 1 ≤ j ≤ p. Consider a family f of p functions fj : Xj ? A1k. We will denote also by fj the function on the product X = ∏ j Xj obtained by composition with the projection. We denote by X0(f) the set of common zeroes in X of the functions fj. Let P ? k[y1, . . . , yp] be a polynomial, which we assume to be nondegenerate with respect to its Newton polyhedron. In the present paper, we will compute the motivic nearby cycles SP(f) on X0(f) of the composed function P(f) on X as a sum over the set of compact faces ? of the Newton polyhedron of P. For every such ?, we denote by P? the corresponding quasihomogeneous polynomial. We associate to such a quasihomogeneous polynomial a convolution operator ?P ? , which in the special case where P? is the polynomial ? = y1 + y2 is nothing but the operator ?? considered in [9]. For such a compact face ?, one may also define generalized nearby cycles S?(?) f , constructed as the limit, as T ?∞, of certain truncated motivic zeta functions.

  • gm-action

  • nondegenerate polynomial

  • grothendieck ring

  • gsm ?

  • ?? k0

  • diagonal morphism

  • morphism ?

  • var ?

  • sr ??


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IMRNInternationalMathematicsResearchNotices2005,No.31NearbyCyclesandCompositionwithaNondegeneratePolynomialGilGuibert,Fran¸coisLoeser,andMichelMerle1IntroductionLetXjbesmoothvarietiesoverafieldkofcharacteristiczero,for1jp.Considerafamilyfofpfunctionsfj:XjAk1.WewilldenotealsobyfjthefunctionontheproductX=jXjobtainedbycompositionwiththeprojection.WedenotebyX0(f)thesetofcommonzeroesinXofthefunctionsfj.LetPk[y1,...,yp]beapolynomial,whichweassumetobenondegeneratewithrespecttoitsNewtonpolyhedron.Inthepresentpaper,wewillcomputethemotivicnearbycyclesSP(f)onX0(f)ofthecomposedfunctionP(f)onXasasumoverthesetofcompactfacesδoftheNewtonpolyhedronofP.Foreverysuchδ,wedenotebyPδthecorrespondingquasihomogeneouspolynomial.WeassociatetosuchaquasihomogeneouspolynomialaconvolutionoperatorΨPδ,whichinthespecialcasewherePδisthepolynomialΣ=y1+y2isnothingbuttheoperatorΨΣconsideredin[9].Forsuchacompactfaceδ,onemayalsodefinegeneralizednearbycyclesSfσ(δ),constructedasthelimit,asT→∞,ofcertaintruncatedmotiviczetafunctions.Ourmainresult,Theorem3.2,followsfromadditivityfromthefollowingstate-ment,Theorem3.3:σ(δ)iSP(f),U=ΨPδSf.1.1)ΓδHereUdenotesthecomplementofthelocuswhereatleastonefunctionfjvanishes,ΓdenotesthesetofcompactfacesoftheNewtonpolyhedronofPnotcontainedinanyReceived8February2005.Revisionreceived3June2005.(