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Nearby motives and motivic nearby cycles

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Abstract

We prove that the construction of motivic nearby cycles, introduced by Jan Denef and François Loeser, is compatible with the formalism of nearby motives, developed by Joseph Ayoub. Let \(k\) be an arbitrary field of characteristic zero, and let \(X\) be a smooth quasi-projective \(k\)-scheme. Precisely, we show that, in the Grothendieck group of constructible étale motives, the image of the nearby motive associated with a flat morphism of \(k\)-schemes \(f:X\rightarrow \mathbb A ^1_k\), in the sense of Ayoub’s theory, can be identified with the image of Denef and Loeser’s motivic nearby cycles associated with \(f\). In particular, it provides a realization of the motivic Milnor fiber of \(f\) in the “non-virtual” motivic world.

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Notes

  1. Note that in that definition, and more generally in the present work, we do not take the monodromy action into account. Besides, the given expression of \(Z^1_n\) requires \(X\) to be of pure dimension \(d\). Otherwise, one has to work connected components by connected components.

  2. Note that the statement of Theorem 1.2 holds true even without the assumption of flatness. Indeed, if \(f\) is not flat and \(X\) is connected, then by [10, Corollary 4.3.10] the morphism \(f\) is constant. In that case, both terms in the formulas of Theorem 1.2 are zero. If \(X\) is not connected, we conclude by working componentwise.

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Acknowledgments

The authors are very grateful to the referee for his careful reading of the text and his comments.

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  1. Institut de recherche mathématique de Rennes, UMR 6625 du CNRS, Université de Rennes 1, Campus de Beaulieu, 35042 , Rennes Cedex, France

    Florian Ivorra & Julien Sebag

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  1. Florian Ivorra
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  2. Julien Sebag
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Correspondence to Julien Sebag.

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Ivorra, F., Sebag, J. Nearby motives and motivic nearby cycles. Sel. Math. New Ser. 19, 879–902 (2013). https://doi.org/10.1007/s00029-012-0111-5

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