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Iterated vanishing cycles, convolution, and a motivic analogue of a conjecture of Steenbrink. (English) Zbl 1173.14301
Let \(f : X \to {\mathbb A}^1\) be a function on a smooth complex variety, and fix a point \(x\) in the fiber \(f^{-1}(0)\). J. H. M. Steenbrink [in: Real and compl. Singul., Proc. Nordic Summer Sch., Symp. Math., Oslo 1976, 525–563 (1977; Zbl 0373.14007) and in: Théorie de Hodge, Actes Colloq., Luminy/Fr. 1987, Astérisque 179–180, 163–184 (1989; Zbl 0725.14031)] introduced the Hodge spectrum \(\text{Sp}(f,x)\) of \(f\) at \(x\), a certain fractional Laurent polynomial, using the action of the monodromy on the mixed Hodge structure on the cohomology of the Milnor fiber at \(x\). When the singular locus of \(f\) is a curve, Steenbrink gave a conjectural formula for the difference \[ \text{Sp}(f+g^N,x) - \text{Sp}(f,x), \] where \(g\) is a general linear form vanishing at \(x\) and \(N \gg 0\). This formula was proven by M. Saito [Math. Ann. 289, No. 4, 703–716 (1991; Zbl 0712.14002)] and a second proof was later given by A. Nemethi and J. H. M. Steenbrink [New York J. Math. 1, 149–177 (1995; Zbl 0878.14017)]; both proofs use the theory of mixed Hodge modules.
More recently, J. Denef and F. Loeser [J. Algebr. Geom. 7, No. 3, 505–537 (1998; Zbl 0943.14010) and in: ECM 2000, Vol. I. Prog. Math. 201, 327–348 (2001; Zbl 1079.14003)] introduced the motivic Milnor fiber \({\mathcal S}_{f,x}\) of \(f\) at \(x\), a virtual variety endowed with an action of the group scheme of roots of unity from which one can retrieve the Hodge spectrum.
The article under review uses motivic integration to establish a motivic analogue of Steenbrink’s conjecture in terms of the motivic Milnor fiber. By construction, the virtual variety \(\mathcal S_{f,x}\) corresponds to nearby cycles; a slightly modified version of it, corresponding to vanishing cycles, is denoted by \(\mathcal S^{\phi}_{f,x}\). Then, for any two functions \(f\) and \(g\) vanishing at a point \(x\), and for \(N \gg 0\), the main theorem of the paper expresses the difference \[ {\mathcal S}^{\phi}_{f + g^N,x} - {\mathcal S}^{\phi}_{f,x} \] in terms of a generalization of the convolution product defined by E. Looijenga [Motivic measures, Astérisque 276, 267–297 (2002; Zbl 0996.14011)]. From this result one can recover Steenbrink’s conjecture, and in fact an extension of it to a more general setting, given the weaker assumptions on \(f\) and \(g\). A formula relating the convolution operator introduced in this paper with the convolution defined by Looijenga also gives the opportunity to the authors to recover the motivic Thom-Sebastiani theorem from their main result.

MSC:
14B05 Singularities in algebraic geometry
14B07 Deformations of singularities
14J17 Singularities of surfaces or higher-dimensional varieties
32S05 Local complex singularities
32S25 Complex surface and hypersurface singularities
32S30 Deformations of complex singularities; vanishing cycles
32S35 Mixed Hodge theory of singular varieties (complex-analytic aspects)
32S55 Milnor fibration; relations with knot theory
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