Computing motivic zeta functions on log smooth models
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Mathematische Zeitschrift
Computing motivic zeta functions on log smooth models Emmanuel Bultot1 · Johannes Nicaise1,2 Received: 23 October 2017 / Accepted: 11 June 2019 © The Author(s) 2019
Abstract We give an explicit formula for the motivic zeta function in terms of a log smooth model. It generalizes the classical formulas for snc-models, but it gives rise to much fewer candidate poles, in general. This formula plays an essential role in recent work on motivic zeta functions of degenerating Calabi–Yau varieties by the second-named author and his collaborators. As a further illustration, we explain how the formula for Newton non-degenerate polynomials can be viewed as a special case of our results. Keywords Motivic zeta functions · Logarithmic geometry · Monodromy conjecture Mathematics Subject Classification 14E18 · 14M25
1 Introduction Denef and Loeser’s motivic zeta function is a subtle invariant associated with hypersurface singularities over a field k of characteristic zero. It can be viewed as a motivic upgrade of Igusa’s local zeta function for polynomials over p-adic fields. The motivic zeta function is a power series over a suitable Grothendieck ring of varieties, and it can be specialized to more classical invariants of the singularity, such as the Hodge spectrum. The main open problem in this context is the so-called monodromy conjecture, which predicts that each pole of the motivic zeta function is a root of the Bernstein polynomial of the hypersurface. One of the principal tools in the study of the motivic zeta function is its explicit computation on a log resolution [8, 3.3.1]. While this formula gives a complete list of candidate poles of the zeta function, in practice most of these candidates tend to cancel out for reasons that are not well understood. Understanding this cancellation phenomenon is the key to the monodromy conjecture. The aim of this paper is to establish a formula for the motivic
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Johannes Nicaise [email protected] Emmanuel Bultot [email protected]
1
KU Leuven, Department of Mathematics, Celestijnenlaan 200B, 3001 Heverlee, Belgium
2
Department of Mathematics, Imperial College, South Kensington Campus, London SW7 2AZ, UK
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E. Bultot, J. Nicaise
zeta function in terms of log smooth models instead of log resolutions (Theorem 5.3.1). These log smooth models can be viewed as partial resolutions with toroidal singularities. Our formula generalizes the computation on log resolutions, but typically gives substantially fewer candidate poles (Proposition 5.4.2). A nice bonus is that, even for log resolutions, the language of log geometry allows for a cleaner and more conceptual proof of the formula for the motivic zeta function in [28], and to extend the results to arbitrary characteristic (Corollary 5.3.2). We will also indicate in Sect. 8.2 how our formula gives a conceptual explanation for the determination of the set of poles of the motivic zeta function of a curve singularity; this is the only dimension in which the monodromy conjecture has been proven complete
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