Computing motivic zeta functions on log smooth models
Abstract
We give an explicit formula for the motivic zeta function in terms of a log smooth model. It generalizes the classical formulas for sncmodels, 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 secondnamed author and his collaborators. As a further illustration, we explain how the formula for Newton nondegenerate polynomials can be viewed as a special case of our results.
Keywords
Motivic zeta functions Logarithmic geometry Monodromy conjectureMathematics Subject Classification
14E18 14M251 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 padic 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 socalled 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 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 completely. A special case of our formula has appeared in the literature under a different guise, namely, the calculation of motivic zeta functions of hypersurfaces that are nondegenerate with respect to their Newton polyhedron [11]. We will explain the precise connection (and correct some small errors in [11]) in Sect. 8.3.
Our results apply not only to Denef and Loeser’s motivic zeta function, but also to the motivic zeta functions of degenerations of Calabi–Yau varieties that were introduced in [12]. Here the formula in terms of log smooth models is particularly relevant in the context of the Gross–Siebert program on toric degenerations and Mirror Symmetry, where log smooth models appear naturally in the constructions. We have already applied our formula to compute the motivic zeta function of the degeneration of the quartic surface in [27]. Our formula is also used in an essential way in [13] to prove an analog of the monodromy conjecture for a large and interesting class of degenerations of Calabi–Yau varieties (namely, the degenerations with monodromyequivariant Kulikov models).
The main results in this paper form a part of the first author’s Ph.D. thesis [5]. They were announced in [6].
1.1 Notations and conventions
The general theory of logarithmic schemes that we will use is explained in [10, 16, 17]. For the reader’s convenience, we have included a userfriendly introduction to regular log schemes and their fans in Sect. 3. These results are not new, but they are scattered in the literature. This is not meant to be a selfcontained introduction to logarithmic geometry: we assume that the reader is familiar with the basic definitions of the theory, as explained for instance in Sects. 1–4 of [16]. In the remainder of the paper, we will frequently refer back to this section for auxiliary results on logarithmic geometry.
Unless explicitly stated otherwise, all logarithmic structures in this paper are defined with respect to the Zariski topology, like in [17], and all the log schemes are Noetherian and fs (fine and saturated). This means that they satisfy condition (S) in [17, 1.5], and are, in addition, quasicompact. We will discuss a generalisation of our results to the Nisnevich topology in Sect. 6.2.
Log schemes will be denoted by symbols of the form \(\mathcal {X}^\dagger \), and the underlying scheme will be denoted by \(\mathcal {X}\). We write \(M_{\mathcal {X}^\dagger }\) for the sheaf of monoids on \(\mathcal {X}^\dagger \). We will follow the convention in [10] and speak of regular log schemes and smooth morphisms between log schemes instead of log regular log schemes and log smooth morphisms. When we refer to geometric properties of the underlying schemes instead, this will always be clearly indicated.
2 Monoids
2.1 Generalities
For general background on the algebraic theory of monoids, we refer to [10, §4]. All monoids are assumed to be commutative, and we will most often use the additive notation \((M,+)\), with neutral element 0. The embedding of the category of abelian groups into the category of monoids has a left adjoint \((\cdot )^{\mathrm {gp}}\), which is called the groupification functor. For every monoid M, we have a canonical morphism of monoids \(M\rightarrow M^{\mathrm {gp}}\) by adjunction. The monoid M is called integral if this morphism is injective. This is equivalent to saying that the addition on M satisfies the cancellation property, that is, \(x+z=y+z\) implies \(x=y\) for all x, y and z in M. An integral monoid M is called saturated if an element m in \(M^{\mathrm {gp}}\) belongs to M if and only if there exists an integer \(d>0\) such that dm belongs to M. We say that a monoid M is fine if it is finitely generated and integral.
The embedding of the category of integral monoids into the category of monoids has a left adjoint, which is denoted by \((\cdot )^{\mathrm {int}}\). For every monoid M, we have a canonical morphism \(M\rightarrow M^{\mathrm {int}}\) by adjunction, and this morphism is surjective. Since every group is integral, the morphism \(M\rightarrow M^{\mathrm {gp}}\) factors through \(M^{\mathrm {int}}\). The induced morphism \(M^{\mathrm {int}}\rightarrow M^{\mathrm {gp}}\) is injective, so that we can identify \(M^{\mathrm {int}}\) with the image of the groupification morphism \(M\rightarrow M^{\mathrm {gp}}\). Likewise, the embedding of the category of saturated monoids into the category of monoids has a left adjoint \((\cdot )^{\mathrm {sat}}\), and we can identify \(M^{\mathrm {sat}}\) with the submonoid of \(M^{\mathrm {gp}}\) consisting of the elements m such that dm belongs to \(M^{\mathrm {int}}\) for some integer \(d>0\). A monoid M is integral, resp. saturated, if and only if the natural morphisms \(M\rightarrow M^{\mathrm {int}}\), resp. \(M\rightarrow M^{\mathrm {sat}}\), are isomorphisms.
For every monoid M, we denote by \(M^\times \) the submonoid of invertible elements of M. The monoid M is called sharp when \(M^{\times }=\{0\}\). Note that sharp monoids are, in particular, torsion free, because all torsion elements are invertible. We denote by \(M^\sharp \) the sharp monoid \(M/M^\times \), called the sharpification of M. We set \(M^+ = M\backslash M^\times \), the unique maximal ideal of M. For every monoid M, we denote by \(M^\vee \) its dual monoid: \(M^{\vee }=\mathrm {Hom}(M,\mathbb {N})\). We will also consider the submonoid \(M^{\vee ,\mathrm {loc}}\) of \(M^{\vee }\) consisting of local homomorphisms \(M\rightarrow \mathbb {N}\), that is, morphisms \(\varphi :M\rightarrow \mathbb {N}\) such that \(\varphi (m)\ne 0\) for every \(m\in M^+\).
Remark 2.1.1
When working with monoids, it is useful to keep in mind the following more concrete description: if M is a fine, saturated and torsion free monoid, then we can identify M with the monoid of integral points of the convex rational polyhedral cone \(\sigma \) generated by M in the vector space \(M^{\mathrm {gp}}\otimes _{\mathbb {Z}}\mathbb {R}\). Conversely, for every convex rational polyhedral cone \(\sigma \) in \(\mathbb {R}^n\), the intersection \(M=\sigma \cap \mathbb {Z}^n\) is a fine, saturated and torsion free monoid, by Gordan’s Lemma (see [10, 4.3.22]). The monoid M is sharp if and only if \(\sigma \) is strictly convex. This correspondence between fine, saturated and torsion free monoids and convex rational polyhedral cones preserves the faces, by [10, 4.4.7]: the faces of M are precisely the intersections of the faces of \(\sigma \) with the lattice \(M^{\mathrm {gp}}\).
2.2 The root index
Definition 2.2.1
Let M be a fine and saturated monoid endowed with a morphism of monoids \(\varphi :\mathbb {N}\rightarrow M\). The root index of \(\varphi \) is defined to be 0 if \(\varphi (1)\) is invertible. Otherwise, it is the largest positive integer \(\rho \) such that the residue class of \(\varphi (1)\) in \(M^{\sharp }\) is divisible by \(\rho \).
Note that such a largest \(\rho \) exists because \(M^{\sharp }\) is a submonoid of the free abelian group of finite rank \((M^{\sharp })^{\mathrm {gp}}\). The importance of the root index lies in the following properties.
Proposition 2.2.2
 (1)
The monoid M(d) is fine for every \(d>0\).
 (2)If d divides \(\rho \), then \(M^{\sharp }\rightarrow (M(d)^{\mathrm {sat}})^{\sharp }\) is an isomorphism, and the morphismhas root index \(\rho /d\).$$\begin{aligned} \varphi _d:\frac{1}{d}\mathbb {N}\rightarrow M(d)^{\mathrm {sat}} \end{aligned}$$
 (3)If d is prime to \(\rho \), then the morphismsare isomorphisms. In particular, if M is sharp, then so are M(d) and \(M(d)^{\mathrm {sat}}\).$$\begin{aligned} M^{\times }\rightarrow M(d)^{\times },\qquad M(d)^{\times }\rightarrow (M(d)^{\mathrm {sat}})^{\times } \end{aligned}$$
Proof
 (1)
It is obvious that M(d) is finitely generated. It is also integral because we can apply the criteria of [16, 4.1] to the morphism \(\mathbb {N}\rightarrow (1/d)\mathbb {N}\).
 (2)Assume that d divides \(\rho \). We may suppose that M is sharp, because the morphismis an isomorphism by the universal properties of sharpification, coproduct and saturation. Let m be an element of M such that \(\rho m=\varphi (1)\). Then$$\begin{aligned} (M(d)^{\mathrm {sat}})^{\sharp }\rightarrow (M^{\sharp }(d)^{\mathrm {sat}})^{\sharp } \end{aligned}$$so that \((\rho /d)m\varphi _d(1/d)\) is a unit in M(d) and \((\rho /d)m=\varphi _d(1/d)\) in \((M(d)^{\mathrm {sat}})^{\sharp }\). Using the universal properties of the coproduct, saturation and sharpification, together with the fact that M is sharp and saturated, we obtain a morphism of monoids \((M(d)^{\mathrm {sat}})^{\sharp }\rightarrow M\) that sends the residue class of \(\varphi _d(1/d)\) to \((\rho /d)m\) and that is inverse to \(M\rightarrow (M(d)^{\mathrm {sat}})^{\sharp }\). It follows at once that$$\begin{aligned} d((\rho /d)m\varphi _d(1/d))=0, \end{aligned}$$has root index \(\rho /d\).$$\begin{aligned} \varphi _d:\frac{1}{d}\mathbb {N}\rightarrow M(d)^{\mathrm {sat}} \end{aligned}$$
 (3)
Assume that d is prime to \(\rho \). It suffices to prove that the composed morphism \(M^{\times }\rightarrow (M(d)^{\mathrm {sat}})^{\times }\) is an isomorphism, because \(M(d)^{\times }\rightarrow (M(d)^{\mathrm {sat}})^{\times }\) is injective since M(d) is integral. Let x be an invertible element in \(M(d)^{\mathrm {sat}}\). We must show that x lies in M. Since \(M\rightarrow M(d)^{\mathrm {sat}}\) is exact by [10, 4.4.42(vi)], it is enough to prove that \(x\in M^{\mathrm {gp}}\). We can write x as (m, i / d) with \(m\in M^{\mathrm {gp}}\) and \(i\in \{0,\ldots ,d1\}\). Since M is saturated, the element dx lies in M, and hence in \(M^{\times }\) because we can apply the same argument to the inverse of x. This means that \(dm=i\varphi (1)\) in \(M^{\sharp }\). But d is prime to \(\rho \), so that d divides i. Hence, \(i=0\) and \(x\in M^{\mathrm {gp}}\). \(\square \)
3 Regular log schemes
3.1 Kato’s definition of regularity
The notion of regularity for logarithmic schemes was introduced by Kato [17]. It can be viewed as a generalization of the theory of toroidal embeddings in [18]. An important advantage of the logarithmic approach is that it works equally well in mixed characteristic; moreover, the monoidal structure on logarithmic schemes keeps track in an efficient way of the cones that describe the local toroidal structure.
The idea behind the definition of regularity for log schemes is that, when \(\mathcal {X}^{\dagger }\) is regular at x, the lack of regularity of the scheme \(\mathcal {X}\) at x is encoded in the characteristic monoid \(M^{\sharp }_{\mathcal {X}^{\dagger },x}\). By Remark, 2.1.1, we can think of \(M^{\sharp }_{\mathcal {X}^{\dagger },x}\) as the monoid of integral points in a strictly convex rational polyhedral cone, and this cone describes the toroidal structure of \(\mathcal {X}\) at x. An explicit description of the completed local ring of \(\mathcal {X}\) at x in terms of the characteristic monoid can be found in [17, 3.2].
Example 3.1.1
The log scheme \(\mathcal {X}^{\dagger }=(\mathcal {X},M)\) is not fine and saturated, in general. However, if \(\mathcal {X}\) is regular and the reduced divisor \(D_{\mathrm {red}}\) has strict normal crossings, then \(\mathcal {X}^{\dagger }\) is regular. If x is a point of \(\mathcal {X}\) and \((z_1,\ldots ,z_n)\) is a regular system of local parameters in \({\mathcal {O}}_{\mathcal {X},x}\) such that \(D_{\mathrm {red}}\) is defined by \(z_1\cdot \ldots \cdot z_r=0\) locally at x, for some \(0\le r\le n\), then \(I_{\mathcal {X}^\dagger ,x}\) is the ideal generated by \((z_1,\ldots ,z_r)\), and \(M_x/M^{\times }_x\) is a free monoid of rank r. A basis for this monoid is given by the residue classes of \(z_1,\ldots ,z_r\).
Note, however, that \(\mathcal {X}\) is not necessarily regular. For instance, if \(\mathcal {X}\) is a toric variety over a field, and D is its toric boundary, then the divisorial log structure induced by D makes \(\mathcal {X}\) into a regular log scheme (see Example 3.2.3).
3.2 Fans and log stratifications
Let \(\mathcal {X}^\dagger \) be a regular log scheme, and consider its associated fan \(F(\mathcal {X}^\dagger )\) in the sense of [17, §10]. This is a sharp monoidal space whose underlying topological space is the subspace of \(\mathcal {X}\) consisting of the points x such that \(M_{\mathcal {X}^\dagger ,x}^+\) generates the maximal ideal of \({\mathcal {O}}_{\mathcal {X},x}\). The sheaf of monoids on \(F(\mathcal {X}^\dagger )\) is the pullback of the sheaf \(M^{\sharp }_{\mathcal {X}^{\dagger }}=M_{\mathcal {X}^\dagger }/{\mathcal {O}}_{\mathcal {X}}^{\times }\) on \(\mathcal {X}\). A dictionary between this notion of fan and the usual notion in toric geometry is provided in Example 3.2.3 below.
Proposition 3.2.1
Proof
This is stated without proof in [17, 10.2]; as observed in [10, 10.6.9(i)], it follows from the construction of the retraction \(\pi \) in [10, 10.6.9(ii)]. \(\square \)
It follows that, for every point x of \(E(\tau )^o\), the monoid \(M_{\mathcal {X}^{\dagger },x}^{\sharp }\) has dimension \(r(\tau )\); we say that \(E(\tau )^o\) is a stratum of rank \(r(\tau )\).
Example 3.2.2
Example 3.2.3
Let k be a field, and let \(Y=Y(\Sigma )\) be a toric variety over k, associated with a fan \(\Sigma \) in \(\mathbb {R}^n\). We endow Y with the divisorial log structure induced by the toric boundary divisor D; the resulting log scheme will be denoted by \(Y^{\dagger }\).
3.3 Boundary divisor and divisorial valuations
Example 3.3.1
Let \(\mathcal {X}\) be a quasicompact regular scheme and let D be a strict normal crossings divisor on \(\mathcal {X}\). Let \(\mathcal {X}^{\dagger }\) be the log scheme that we obtain by endowing \(\mathcal {X}\) with the divisorial log structure induced by D (see Example 3.1.1). Then \(\mathcal {X}^{\dagger }\) is log regular and its boundary coincides with D. Conversely, if \(\mathcal {X}^{\dagger }\) is a regular log scheme, then the underlying scheme \(\mathcal {X}\) is regular if and only if \(M_{F(\mathcal {X}^{\dagger }),\tau }\) is isomorphic to \(\mathbb {N}^{r(\tau )}\) for every \(\tau \) in \(F(\mathcal {X}^{\dagger })\) [10, 10.5.35]. In that case, the boundary divisor D of \(\mathcal {X}^{\dagger }\) has strict normal crossings [33, 2.7].
Let \(\mathcal {X}^{\dagger }\) be any regular log scheme, and fix a point \(\tau \) of the fan \(F(\mathcal {X}^{\dagger })\). Let \(F(\tau )\) be the subspace of \(F(\mathcal {X}^{\dagger })\) consisting of the points \(\sigma \) such that \(\tau \) is contained in the closure of \(\{\sigma \}\). We denote by \(M_{F(\tau )}\) the restriction of the sheaf of monoids \(M_{F(\mathcal {X}^{\dagger })}\) to \(F(\tau )\). Then the monoidal space \((F(\tau ),M_{F(\tau )})\) is canonically isomorphic to the spectrum of the characteristic monoid \(M_{F(\mathcal {X}^{\dagger }),\tau }\), by [17, 10.1] and its proof. This implies, in particular, that the prime ideals of height one in \(M_{F(\mathcal {X}^{\dagger }),\tau }\) are in bijective correspondence with the strata \(E(\sigma )\) such that \(\sigma \) is a codimension one point in \(F(\mathcal {X}^{\dagger })\) whose closure contains \(\tau \); these are precisely the irreducible components of D that pass through \(\tau \).
3.4 Subdivisions
3.5 Charts
Let \(\mathcal {X}^{\dagger }\) be a fine and saturated log scheme. A chart for \(\mathcal {X}^{\dagger }\) is a strict morphism of log schemes \(\mathcal {X}^{\dagger }\rightarrow {{\,\mathrm{Spec}\,}}^\dagger \mathbb {Z}[N]\), where N is a monoid and we denote by \({{\,\mathrm{Spec}\,}}^{\dagger }\mathbb {Z}[N]\) the scheme \({{\,\mathrm{Spec}\,}}\mathbb {Z}[N]\) endowed with the log structure induced by \(N\rightarrow \mathbb {Z}[N]\). Here strict means that the log structure on \(\mathcal {X}^{\dagger }\) is the pullback of the log structure on \({{\,\mathrm{Spec}\,}}^{\dagger } \mathbb {Z}[N]\). If \(\mathcal {X}^{\dagger }\rightarrow {{\,\mathrm{Spec}\,}}^{\dagger } \mathbb {Z}[N]\) is a chart for \(\mathcal {X}^{\dagger }\), then for every point x of \(\mathcal {X}\), the morphism of monoids \(N\rightarrow M_{\mathcal {X}^{\dagger },x}\) induces a surjection \(N\rightarrow M^{\sharp }_{\mathcal {X}^{\dagger },x}\). Thus, up to a multiplicative factor, every element in \(M^+_{\mathcal {X}^{\dagger },x}\) lifts to an element of \(N^+\). Therefore, if \(\mathcal {X}^{\dagger }\) is regular, then locally around x, the logarithmic stratum that contains x is the zero locus of the image of \(N^+\) in \({\mathcal {O}}_{\mathcal {X},x}\). We will use this description in the proof of Lemma 4.1.2.
For every fine and saturated log scheme \(\mathcal {X}^{\dagger }\) and every point x of \(\mathcal {X}\), we can find locally around x a chart of the form \(\mathcal {X}^{\dagger }\rightarrow {{\,\mathrm{Spec}\,}}^{\dagger } \mathbb {Z}[M^{\sharp }_{\mathcal {X}^{\dagger },x}]\), by the proof of [10, 10.1.36(i)]. Then the induced morphism of monoids \(M^{\sharp }_{\mathcal {X}^{\dagger },x}\rightarrow M_{\mathcal {X}^{\dagger },x}\) is a section of the projection morphism \(M_{\mathcal {X}^{\dagger },x}\rightarrow M^{\sharp }_{\mathcal {X}^{\dagger },x}\).
3.6 Smoothness versus regularity
For the applications in Sect. 8 we will also need the following result, which relates log regularity and log smoothness. For the definition of regularity for log schemes with respect to the étale topology, we refer to [30, 2.2]. It is the direct analog of the Zariski case, replacing points by geometric points and local rings by their strict henselizations. The basic theory of logarithmic smoothness can be found in [16, §3.3].
Proposition 3.6.1
 (1)
If \(\mathcal {X}^{\dagger }\) is regular, then \(\mathcal {X}\) is flat over S.
 (2)
If \(\mathcal {X}^{\dagger }\) is smooth over \(S^{\dagger }\), then it is regular.
 (3)
If k has characteristic zero and \(\mathcal {X}^{\dagger }\) is regular, then \(\mathcal {X}^{\dagger }\) is smooth over \(S^{\dagger }\) .
Proof
(2) Every smooth fs log scheme over \(S^{\dagger }\) is regular by [17, 8.2] (we can reduce to the Zariski case by passing to an étale cover of \(\mathcal {X}\) where the log structure on \(\mathcal {X}^{\dagger }\) is Zariski in the sense of [30, 2.1.1]).
(3) Assume that k has characteristic zero, and that \(\mathcal {X}^{\dagger }\) is regular. Let \({\overline{x}}\) be a geometric point on \(\mathcal {X}_k\), and set \(M=M^{\sharp }_{\mathcal {X}^{\dagger },{\overline{x}}}\). We choose a chart \(\mathcal {X}^{\dagger }\rightarrow {{\,\mathrm{Spec}\,}}^{\dagger }\mathbb {Z}[M]\) étalelocally around \({\overline{x}}\). We also choose a uniformizer t in R; this choice determines a chart \(S^{\dagger }\rightarrow {{\,\mathrm{Spec}\,}}^{\dagger }\mathbb {Z}[\mathbb {N}]\) such that the induced morphism \(\mathbb {N}\rightarrow M_{S^{\dagger },s}\) maps 1 to t. Then we can find an element m in M and a unit u in \({\mathcal {O}}_{\mathcal {X},{\overline{x}}}\) such that \(t=um\) in \({\mathcal {O}}_{\mathcal {X},{\overline{x}}}\). Since k has characteristic zero, we can take arbitrary roots of invertible functions on \(\mathcal {X}\) locally in the étale topology on \(\mathcal {X}\). Thus there exists a morphism \(\psi \) from the free abelian group \(M^{\mathrm {gp}}\) to \({\mathcal {O}}^{\times }_{\mathcal {X},{\overline{x}}}\) that maps m to u. Multiplying the morphism of monoids \(M\rightarrow {\mathcal {O}}_{\mathcal {X},{\overline{x}}}\) with the restriction of \(\psi \) to M, we obtain a new chart \(\mathcal {X}^{\dagger }\rightarrow {{\,\mathrm{Spec}\,}}^{\dagger }\mathbb {Z}[M]\) étalelocally around \({\overline{x}}\) such that the pullback of m is equal to t.
Beware that Proposition 3.6.1(3) does not extend to the case where k has characteristic \(p>0\). The problem is that we cannot take pth roots of all invertible functions locally in the étale topology, and that the order of the torsion part of the cokernel of the morphism \(\mathbb {Z}\rightarrow M^{\mathrm {gp}}\) may not be invertible in k. A sufficient condition for log smoothness is given by the following statement. Let \(\mathcal {X}^{\dagger }\) be a regular log scheme of finite type over \(S^{\dagger }\) (with respect to the étale topology). Suppose moreover that k is perfect, the log structure on \(\mathcal {X}^{\dagger }\) is vertical (that is, it is trivial on \(\mathcal {X}_K\)), the generic fiber \(\mathcal {X}_K\) is smooth over K, and the multiplicities of the components in the special fiber are prime to p. Then \(\mathcal {X}^{\dagger }\) is smooth over \(S^{\dagger }\). This follows from the same argument as in the proof of Proposition 3.6.1.
3.7 Fine and saturated fibered products
The most important class of fs fibered products for our purposes is described in the following proposition.
Proposition 3.7.1
Let R be a complete discrete valuation ring with quotient field K. Let \(K'\) be a finite extension of K and denote by \(R'\) the integral closure of R in \(K'\). We denote by \(S^{\dagger }\) the scheme \(S={{\,\mathrm{Spec}\,}}R\) endowed with its standard log structure (see Example 3.2.2). The log scheme \((S')^{\dagger }\) is defined analogously, replacing R by \(R'\).
Let \(\mathcal {X}^{\dagger }\) be a fine and saturated log scheme, and let \(\mathcal {X}^{\dagger }\rightarrow S^+\) be a smooth morphism of log schemes. Then the underlying scheme \(\mathcal {Y}\) of \(\mathcal {Y}^{\dagger }=\mathcal {X}^{\dagger }\times ^{\mathrm {fs}}_{S^{\dagger }}(S')^{\dagger }\) is the normalization of \(\mathcal {X}\times _S S'\).
Proof
Smoothness is preserved by fine and saturated base change, so that the log scheme \(\mathcal {Y}^{\dagger }\) is smooth over \((S')^{\dagger }\). Since \(S^{\dagger }\) and \((S')^{\dagger }\) are regular, the log schemes \(\mathcal {X}^{\dagger }\) and \(\mathcal {Y}^{\dagger }\) are regular, as well, by Proposition 3.6.1. Thus \(\mathcal {X}\) and \(\mathcal {Y}\) are normal. It also follows from Proposition 3.6.1 that \(\mathcal {X}\) is flat over S and \(\mathcal {Y}\) is flat over \(S'\). The morphism \(\mathcal {Y}\rightarrow \mathcal {X}\times _S S'\) is an isomorphism on the generic fibers, because the log structures on S and \(S'\) are trivial at their generic points, so that \(\mathcal {X}^{\dagger }\times _{S^{\dagger }}{{\,\mathrm{Spec}\,}}(K')\) is already saturated. Thus \(\mathcal {Y}\rightarrow \mathcal {X}\times _S S'\) is birational; since it is also finite and \(\mathcal {Y}\) is normal, it is a normalization morphism. \(\square \)
4 Smooth log schemes over discrete valuation rings
4.1 Log modifications and ramified base change
Example 4.1.1
If \(\tau \) is a point of \(F(\mathcal {X}^{\dagger })\cap \mathcal {X}_k\) and \(\rho (\tau )\) is not divisible by the characteristic of k, then \({\widetilde{E}}(\tau )^o\rightarrow E(\tau )^o\) has a canonical structure of a \(\mu _{\rho (\tau )}\)torsor, which is described explicitly in [26, §2.3].
The following results constitute a key step in the calculation of the motivic zeta function.
Lemma 4.1.2
Proof
By Sect. 3.5, all the elements in the image of the morphism \(M^+\times \mathbb {Z}\rightarrow {\mathcal {O}}_{\mathcal {Y},y}\) vanish in \({\mathcal {O}}_{{\widetilde{E}}(\tau )^o,y}\). By construction, \((m/\rho )N^+\) is contained in \(M^+\times \mathbb {Z}\). Thus for every element h in the image of \(N^+\rightarrow {\mathcal {O}}_{\mathcal {W},y}\), we have that \(h^{m/\rho }\) vanishes in \({\mathcal {O}}_{{\widetilde{E}}(\tau )^o,y}\). Since \({\mathcal {O}}_{{\widetilde{E}}(\tau )^o,y}\) is reduced, we conclude that the morphism \(\mathbb {Z}[N]\rightarrow {\mathcal {O}}_{{\widetilde{E}}(\tau )^o,y}\) factors through \(\mathbb {Z}[N]/\langle N^+\rangle \), where \(\langle N^+\rangle \) denotes the ideal generated by \(N^+\).
Proposition 4.1.3
Proof
We will prove that the morphism h induced by the subdivision \(\psi \) is compatible with fs base change, in the following sense.
5 Motivic zeta functions
We denote by R a complete discrete valuation ring with residue field k and quotient field K. We assume that k is perfect and we fix a uniformizer t in R. For every positive integer n, we write \(R(n)=R[u]/(u^nt)\) and \(K(n)=K[u]/(u^nt)\). We write \(S^\dagger \) and \(S(n)^\dagger \) for the schemes \(S={{\,\mathrm{Spec}\,}}R\) and \(S(n)={{\,\mathrm{Spec}\,}}R(n)\) endowed with their standard log structures.
5.1 Grothendieck rings and geometric series
If R has equal characteristic, then for every noetherian kscheme X, we denote by \({\mathcal {M}}_{X}\) the Grothendieck ring of varieties over X, localized with respect to the class \(\mathbb {L}\) of the affine line \(\mathbb {A}^1_{X}\). If R has mixed characteristic, we use the same notation, but we replace the Grothendieck ring of varieties by its modified version, which means that we identify the classes of universally homeomorphic Xschemes of finite type—see [29, §3.8]. In the calculation of the motivic zeta function, we will need to consider some specific geometric series in \(\mathbb {L}^{1}\). The standard technique is to pass to the completion \(\widehat{{\mathcal {M}}}_X\) of \({\mathcal {M}}_X\) with respect to the dimensional filtration. However, since it is not known whether the completion morphism \({\mathcal {M}}_X\rightarrow \widehat{{\mathcal {M}}}_X\) is injective, we will use a different method to avoid any loss of information. We start with an elementary lemma.
Lemma 5.1.1
Proof
5.2 Definition of the motivic zeta function
Let \(\mathcal {X}\) be an Rscheme of finite type with smooth generic fiber \(\mathcal {X}_K\), and let \(\omega \) be a volume form on \(\mathcal {X}_K\) (that is, a nowhere vanishing differential form of maximal degree on each connected component of \(\mathcal {X}_K\)). A Néron smoothening of \(\mathcal {X}\) is a morphism of finite type \(h:\mathcal {Y}\rightarrow \mathcal {X}\) such that \(\mathcal {Y}\) is smooth over R, \(h_K:\mathcal {Y}_K\rightarrow \mathcal {X}_K\) is an isomorphism, and the natural map \(\mathcal {Y}(R')\rightarrow \mathcal {X}(R')\) is a bijection for every finite unramified extension \(R'\) of R. Such a Néron smoothening always exists, by [3, 3.1.3]. For every connected component C of \(\mathcal {Y}_k\), we denote by \({{\,\mathrm{ord}\,}}_C\omega \) the unique integer a such that \(t^{a}\omega \) extends to a relative volume form on \(\mathcal {Y}\) locally around the generic point of C.
Definition 5.2.1
It is a deep fact that this definition does not depend on the choice of a Néron smoothening; the proof relies on the theory of motivic integration [23]. Definition 5.2.1 can be interpreted as a motivic upgrade of the integral of a volume form on a compact padic manifold [23, §4.6].
The motivic zeta function of the pair \((\mathcal {X},\omega )\) is a generating series that measures how the motivic integral in Definition 5.2.1 changes under ramified extensions of R. For every positive integer n, we set \(\mathcal {X}(n)=\mathcal {X}\times _R R(n)\), and we denote by \(\omega (n)\) the pullback of \(\omega \) to the generic fiber of \(\mathcal {X}(n)\).
Definition 5.2.2
Beware that this definition depends on the choice of the uniformizer t, except when k has characteristic zero and contains all the roots of unity: in that case, K(n) is the unique degree n extension of K, up to Kisomorphism.
If \(h:\mathcal {X}'\rightarrow \mathcal {X}\) is a proper morphism of Rschemes such that \(h_K:\mathcal {X}'_K\rightarrow \mathcal {X}_K\) is an isomorphism, then it follows immediately from the definition that we can recover \(Z_{\mathcal {X},\omega }(T)\) from \(Z_{\mathcal {X}',\omega }(T)\) by specializing the coefficients with respect to the forgetful group homomorphism \({\mathcal {M}}_{\mathcal {X}'_k}\rightarrow {\mathcal {M}}_{\mathcal {X}_k}\). Thus we can compute \(Z_{\mathcal {X},\omega }(T)\) after a suitable proper modification of \(\mathcal {X}\). The principal aim of this paper is to establish an explicit formula for \(Z_{\mathcal {X},\omega }(T)\) in the case where \(\mathcal {X}\) is smooth over \(S^\dagger \) with respect to a suitable choice of log structure on \(\mathcal {X}\).
5.3 Explicit formula on a log smooth model
Let \(\mathcal {X}^{\dagger }\) be a smooth fs log scheme of finite type over \(S^\dagger \), and denote by D its reduced boundary divisor, which was defined in Sect. 3.2. We write \(F=F(\mathcal {X}^{\dagger })\) for the fan associated with \(\mathcal {X}^{\dagger }\), and we denote by \(e_t\) the image of the uniformizer t in the monoid of global sections of \(M_{\mathcal {X}^{\dagger }}\). We write \(F_k\) for the set \(F\cap \mathcal {X}_k\). This is a finite set, consisting of the points in the special fiber \(\mathcal {X}_k\) whose Zariski closure is a connected component of an intersection of irreducible components of D (this follows from the description of the logarithmic stratification in Section 3.2).
Let \(\omega \) be a differential form of maximal degree on \(\mathcal {X}_K\) that is nowhere vanishing on \(\mathcal {X}_K{\setminus } D\). Then we can view \(\omega \) as a rational section of the relative canonical bundle \(\omega _{\mathcal {X}^{\dagger }/S^\dagger }\). As such, it defines a Cartier divisor on \(\mathcal {X}\), which we denote by \(\mathrm {div}_{\mathcal {X}^{\dagger }}(\omega )\). This divisor is supported on D. Let \(\tau \) be a point of F. For every element u of \(M_{F,\tau }^{\vee ,\mathrm {loc}}\), we set \(u(\omega )=u^{\mathrm {gp}}({\overline{f}})\in \mathbb {Z}\), where \({\overline{f}}\) is the residue class in \(M^{\mathrm {gp}}_{F,\tau }\) of any element \(f\in M^{\mathrm {gp}}_{\mathcal {X}^{\dagger },\tau }\) such that \(\mathrm {div}(f)=\mathrm {div}_{\mathcal {X}^{\dagger }}(\omega )\) locally at \(\tau \). This definition does not depend on the choice of f. Note that \(u(\omega )>0\) if \(\tau \) is not contained in \(F_k\), because \(\mathrm {div}_{\mathcal {X}^{\dagger }}(\omega )\ge D\) on \(\mathcal {X}_K\).
Theorem 5.3.1
Proof
We break up the proof into four steps.
Step 1: the expression (2) is welldefined. Since \(\omega \) is a volume form on \(\mathcal {X}_K\), the horizontal part of the divisor \(\mathrm {div}_{\mathcal {X}^{\dagger }}(\omega )\) coincides with the horizontal part of the reduced boundary divisor D of \(\mathcal {X}^{\dagger }\). This means that \(u(\omega )=1\) for every \(\tau \in F_k\) and every generator u of a onedimensional face of \(M_{F,\tau }\) such that \(u(e_t)=0\). Hence, Lemma 5.1.1 guarantees that (2) is a welldefined element of \({\mathcal {M}}_{\mathcal {X}_k}\llbracket T \rrbracket \).
Step 2: invariance under log modifications. We will show that the right hand side of (3) is invariant under the log modification \(h:\mathcal {(X')}^{\dagger }\rightarrow \mathcal {X}^{\dagger }\) induced by any fs proper subdivision \(\psi :F'\rightarrow F\) that is an isomorphism over \(F\cap \mathcal {X}_K\), or equivalently, such that \(h_K:\mathcal {X}'_K\rightarrow \mathcal {X}_K\) is an isomorphism.
First, we consider the case where the log structure at \(\tau \) is vertical (this means that every irreducible component of the boundary divisor D that passes through \(\tau \) is contained in the special fiber \(\mathcal {X}_k\)). Then \(\tau \) is the generic point of \(C(\tau )\), the monoid \(M_{F,\tau }\) is isomorphic to \(\mathbb {N}\), and \(e_t\) is its unique generator. Thus the only morphism u contributing to the sum in the left hand side of (5) is the identity morphism \(u:\mathbb {N}\rightarrow \mathbb {N}\). Now the equality follows from the fact that locally around \(\tau \), we have a canonical isomorphism \(\omega _{\mathcal {X}/S}\cong \omega _{\mathcal {X}^{\dagger }/S^{\dagger }}\) because the morphism \(\mathcal {X}^{\dagger }\rightarrow S^{\dagger }\) is strict at \(\tau \).
As a special case of Theorem 5.3.1, we recover a generalization to arbitrary characteristic of the formula for strict normal crossings models from [28, 7.7]. Beware that in [28], the motivic integrals were renormalized by multiplying them with \(\mathbb {L}^{d}\), where d is the relative dimension of \(\mathcal {X}\) over R.
Corollary 5.3.2
Proof
5.4 Poles of the motivic zeta function
Theorem 5.3.1 yields interesting information on the poles of the motivic zeta function. Since the localized Grothendieck ring of varieties is not a domain, the notion of a pole requires some care; see [32]. To circumvent this issue, we introduce the following definition.
Definition 5.4.1
For any reasonable definition of a pole (in particular, the one in [32]), the set of rational poles is included in every set of candidate poles.
Proposition 5.4.2
Proof
Proposition 5.4.2 tells us that, in order to find a set of candidate poles of \(Z_{\mathcal {X},\omega }(T)\), it is not necessary to take a log resolution of the pair \((\mathcal {X},\mathcal {X}_k)\), which would introduce many redundant candiate poles. This observation is particularly useful in the context of the monodromy conjecture for motivic zeta functions; see Sect. 8.
6 Generalizations
6.1 Formal schemes
The definition of the motivic zeta function (Definition 5.2.2) can be generalized to the case where \(\mathcal {X}\) is a formal scheme satisfying a suitable finiteness condition (a socalled special formal scheme in the sense of [1], which is also called a formal scheme formally of finite type in the literature). This generalization is carried out in [25], and it is not difficult to extend our formula from Theorem 5.3.1 to this setting. The main reason why we have chosen to work in the category of schemes in this article is the lack of suitable references for the basic properties of logarithmic formal schemes on which the proof of our formula relies. However, the proofs for log schemes carry over easily to the formal case, so that the reader who would want to apply Theorem 5.3.1 to formal schemes should have no difficulties in making the necessary verifications.
6.2 Nisnevich log structures
We will now show how Theorem 5.3.1 can be adapted to log schemes in the Nisnevich topology. This allows us to compute motivic zeta functions on a larger class of models with components with “mild” selfintersections in the special fiber. This generality is needed, for instance, for the applications to motivic zeta functions of Calabi–Yau varieties in [13]. We will explain in Example 6.2.3 what is the advantage of the Nisnevich topology over the étale topology when computing motivic zeta functions.
These covering families generate a Grothendieck topology, which is called the Nisnevich topology. Let \(M_Y\rightarrow {\mathcal {O}}_Y\) be an fs log structure on Y with respect to the étale topology (as in [16]). We say that the log structure \(M_Y\) is Nisnevich if we can find charts for the log structure \(M_Y\) locally in the Nisnevich topology on Y.
Let \(\mathcal {X}^{\dagger }\) be a smooth fs Nisnevich log scheme of finite type over \(S^\dagger \). Then the sheaf of monoids \(M^{\sharp }_{\mathcal {X}^{\dagger }}\) is constructible on the Nisnevich site of \(\mathcal {X}\), by the same proof as in [10, 10.2.21]. We choose a partition \({\mathscr {P}}\) of \(\mathcal {X}_k\) into irreducible locally closed subsets U such that the restriction of \(M^{\sharp }_{\mathcal {X}^{\dagger }}\) to the Nisnevich site on U is constant. We denote by P the set consisting of the generic points of all the strata U in \({\mathscr {P}}\). For every point \(\tau \) in P we will write \(E(\tau )^o\) for the unique stratum in \({\mathscr {P}}\) containing \(\tau \), and we denote by \(r(\tau )\) the dimension of the monoid \(M^{\sharp }_{\mathcal {X}^{\dagger },\tau }\). We define the root index \(\rho (\tau )\) and the scheme \({\widetilde{E}}(\tau )^o\) in exactly the same way as before, and we write \(e_t\) for the image of t in the monoid of global sections of \(M^{\sharp }_{\mathcal {X}^{\dagger }}\). If \(\mathcal {X}_K\) is smooth over K and \(\omega \) is a volume form on \(\mathcal {X}_K\), then we can also simply copy the definition of the value \(u(\omega )\) for every local morphism \(u:M^{\sharp }_{\mathcal {X}^{\dagger },\tau }\rightarrow \mathbb {N}\).
Theorem 6.2.1
Proof
Corollary 6.2.2
Proof
The proof is almost identical to the proof of the Zariski case (Proposition 5.4.2), using the formula in Theorem 6.2.1 instead of Theorem 5.3.1. We no longer have a bijective correspondence between the generators u of onedimensional faces of \((M^{\sharp }_{\mathcal {X}^{\dagger },\tau })^{\vee }\) and the irreducible components of the boundary D containing \(\tau \), in general, because one component may have multiple formal branches at \(\tau \) and each of these will give rise to a onedimensional face of \((M^{\sharp }_{\mathcal {X}^{\dagger },\tau })^{\vee }\). However, it remains true that for every generator u of a onedimensional face of \((M^{\sharp }_{\mathcal {X}^{\dagger },\tau })^{\vee }\), there exists an irreducible component E of D such that \(u(e_t)\) equals the multiplicity of D in \(\mathcal {X}_k\) and \(u(\omega )\) equals the multiplicity of E in \(\mathrm {div}_{\mathcal {X}^{\dagger }}(\omega )\). This is sufficient to prove the result. \(\square \)
The following example shows that the formula in Theorem 6.2.1 may fail if we replace the Nisnevich topology by the étale topology.
Example 6.2.3
7 The monodromy action
7.1 Equivariant Grothendieck rings

Generators: Isomorphism classes [Y] of Xschemes Y of finite type endowed with a good action of G such that the morphism \(Y\rightarrow X\) is Gequivariant; here the isomorphism class is taken with respect to Gequivariant isomorphisms.
 Relations:
 (1)If Y is an Xscheme of finite type with good Gaction and Z is a closed subscheme of Y that is stable under the Gaction, then$$\begin{aligned}{}[Y]=[Z]+[Y{\setminus } Z]. \end{aligned}$$
 (2)If Y is an Xscheme of finite type with good Gaction and \(A\rightarrow Y\) is an affine bundle of rank r endowed with an affine lift of the Gaction on Y, thenwhere G acts trivially on \(\mathbb {A}^r_{\mathbb {Z}}\).$$\begin{aligned}{}[A]=[\mathbb {A}^r_{\mathbb {Z}}\times _{\mathbb {Z}} Y] \end{aligned}$$
 (1)
Proposition 7.1.1
Proof
7.2 Monodromy action on the motivic zeta function
Theorem 7.2.1
If k has characteristic zero, then Theorems 5.3.1 and 6.2.1 are valid already for the equivariant motivic zeta function \(Z^{{\widehat{\mu }}}_{\mathcal {X},\omega }(T)\) in \({\mathcal {M}}^{{\widehat{\mu }}}_{\mathcal {X}_k}\llbracket T \rrbracket \).
Proof
We can follow a similar strategy as in the proof of Theorem 5.3.1. Let \(k^a\) be an algebraic closure of k and set \((S^a)^{\dagger }={{\,\mathrm{Spec}\,}}k^a\llbracket t\rrbracket \) with its standard log structure. To compute the degree n coefficient of \(Z_{\mathcal {X},\omega }(T)\) we can choose a regular subdivision of the fan \(F^a(n)\) of \(\mathcal {X}(n)^{\dagger }\times _{S^\dagger }(S^a)^{\dagger }\) that is equivariant with respect to the actions of \(\mu _n(k^a)\) and the Galois group \(\mathrm {Gal}(k^a/k)\). This can be achieved by canonical equivariant resolution of singularities for toroidal embeddings (see for instance the remark on p. 33 of [36]).
The induced log modification of \(\mathcal {X}(n)\) is a regular scheme and its smooth locus is a \(\mu _n\)equivariant Néron smoothening of \(\mathcal {X}(n)\). Then a similar computation as in the last step of the proof of Theorem 5.3.1 yields the desired result.
The definition of a set of candidate poles (Definition 5.4.1) can be generalized to elements of \({\mathcal {M}}^{{\widehat{\mu }}}_{\mathcal {X}_k}\llbracket T \rrbracket \) in the obvious way. Then we can deduce the following result from Theorem 7.2.1.
Corollary 7.2.2
8 Applications to Denef and Loeser’s motivic zeta function
8.1 The motivic zeta function of Denef–Loeser
Thus Theorem 7.2.1 and Corollary 7.2.2 also apply to the motivic zeta function of Denef and Loeser. As an illustration, we will apply these results to two particular situations: the case where X has dimension 2, and the case where f is a polynomial that is nondegenerate with respect to its Newton polyhedron. These cases have been studied extensively in the literature; we will explain how some of the main results can be viewed as special cases of Theorem 5.3.1.
8.2 The surface case
Theorem 8.2.1
Proof
Set \(\mathcal {Y}=Y\times _{k[t]}R\). Contracting all the components \(E_i\) with \(i\in I_0\) yields a new model \(\mathcal {Z}\) of \(\mathcal {X}_K\) that is proper over \(\mathcal {X}\), namely, the log canonical model of \((\mathcal {X},\mathcal {X}_k)\) over \(\mathcal {X}\). We endow \(\mathcal {Z}\) with the divisorial log structure induced by \({\mathcal {Z}}_k\). It follows from [15, §3] that the resulting log scheme \(\mathcal {Z}^{\dagger }\) is regular with respect to the étale topology, and since k has characteristic zero, this implies that \(\mathcal {Z}^{\dagger }\) is smooth over \(S^\dagger \) with respect to the étale topology (Proposition 3.6.1).
The log structure on \(\mathcal {Z}^{\dagger }\) fails to be Zariski precisely at the selfintersection points of components in the strict transform of \(X_0\). If k is algebraically closed, then the log structure is Nisnevich at these points (because they have algebraically closed residue field) and our result follows immediately from Corollary 7.2.2. For general k, we can make the log structure Zariski by blowing up \(\mathcal {Z}^{\dagger }\) at each of the selfintersection points (see the proof of [30, 5.4]). These blowups are log blowups, so that the resulting morphism of log schemes \(\mathcal {W}^{\dagger }\rightarrow \mathcal {Z}^{\dagger }\) is étale. Therefore, blowing up at a selfintersection point of a component \(E_i\) yields an exceptional divisor with numerical data \(N=2N_i\) and \(\nu =2\nu _i\). This implies that \({\mathcal {P}}(\mathcal {W}^{\dagger })={\mathcal {P}}'\), so that the result follows from Corollary 7.2.2 (applied to the smooth Zariski log scheme \(\mathcal {W}^{\dagger }\)). \(\square \)
Corollary 8.2.2
There exists a set of candidate poles for \(Z_f(T)\) that consists entirely of roots of the Bernstein polynomial of f. Thus the monodromy conjecture for \(Z_f(T)\in {\mathcal {M}}_{\mathcal {X}_k}^{{\widehat{\mu }}}\llbracket T\rrbracket \) holds in dimension 2.
Proof
Loeser has proven in [21, III.3.1] that every element of \({\mathcal {P}}'\) is a root of the Bernstein polynomial of f. \(\square \)
Analogous results have previously appeared in the literature for the padic zeta function [21, 34], the topological zeta function [35] and the socalled “naïve” motivic zeta function [31].
8.3 Nondegenerate polynomials
We denote by \(\Sigma \) the dual fan of \(\Gamma (f)\) and by \(h:Y\rightarrow \mathbb {A}^n_k\) the toric modification associated with the subdivision \(\Sigma \) of \((\mathbb {R}_{\ge 0})^n\). We view Y as a k[t]scheme via the morphism \(f\circ h:Y\rightarrow {{\,\mathrm{Spec}\,}}k[t]\) and we set \(\mathcal {Y}=Y\times _{k[t]}R\). We denote by H the pullback to \(\mathcal {Y}\) of the union of the coordinate hyperplanes in \(\mathbb {A}^n_k\). We endow \(\mathcal {Y}\) with the divisorial Zariski log structure induced by the divisor \(\mathcal {Y}_k+H\). The result is a Zariski log scheme \(\mathcal {Y}^{\dagger }\) over \(S^{\dagger }\).
Proposition 8.3.1
The log scheme \(\mathcal {Y}^{\dagger }\) is fine and saturated, and smooth over \(S^{\dagger }\).
Proof
By Proposition 3.6.1, we only need to show that \(\mathcal {Y}^{\dagger }\) is regular. Since f has no critical points on \({\mathbb {G}}^n_{m,k}\) by the nondegeneracy assumption, we only need to check regularity at the points y on \(H\cap \mathcal {Y}_k\). Let \(\sigma \) be the cone of \(\Sigma \) such that y lies on the associated torus orbit \(O(\sigma )\) in Y, and denote by \(\gamma \) the face of \(\Gamma (f)\) corresponding to \(\sigma \). We denote by M the characteristic monoid \(M^{\sharp }_{\mathcal {Y}^{\dagger },y}\) of \(\mathcal {Y}^{\dagger }\) at y.
If y does not lie in the closure of \(\mathrm {div}(f\circ h)\cap {\mathbb {G}}_{m,k}^n\), then locally around y, the log structure on \(\mathcal {Y}^{\dagger }\) is the pullback of the natural log structure on the toric variety Y, which was described in Example 3.2.3. Thus \(\mathcal {Y}^{\dagger }\) is fine and saturated, because the divisorial log structure on Y induced by the toric boundary has these properties. It also follows that \(M=(\sigma ^{\vee }\cap \mathbb {Z}^n)^{\sharp }\) so that \(M^{\mathrm {gp}}\) has rank \(r=\mathrm {dim}(\sigma )\). Locally at y, the maximal ideal of \(M_{\mathcal {Y}^{\dagger },y}\) defines the toric orbit \(O(\sigma )\), which is regular of codimension r in \(\mathcal {Y}\). Thus \(\mathcal {Y}^{\dagger }\) is regular at y.
Locally at y, the closed subscheme Z of \(\mathcal {Y}\) defined by the maximal ideal of \(M_{\mathcal {Y}^{\dagger },y}\) coincides with the schematic intersection of \(O(\sigma )\) with \(\mathrm {div}(f')\). Since \(O(\sigma )\) is canonically isomorphic to \({{\,\mathrm{Spec}\,}}k[(\sigma ^{\vee }\cap \mathbb {Z}^n)^{\times }]\) and Z is the closed subscheme of \(O(\sigma )\) defined by \(f_{\gamma }/x^{v}\), the assumption that \(f_{\gamma }\) has no critical points in \({\mathbb {G}}^n_{m,k}\) now implies that Z is regular at y of codimension r in \({\mathcal {Y}}\). Hence, \(\mathcal {Y}^{\dagger }\) is regular at y. \(\square \)
Theorem 8.3.2
Proof
We denote by D the logarithmic boundary divisor on \(\mathcal {Y}^{\dagger }\). By the definition of a nondegenerate polynomial, the divisor \(\mathrm {div}(f\circ h)\cap {\mathbb {G}}^n_{m,k}\) in the dense torus of Y is smooth. We denote by \(D'\) the restriction to \(\mathcal {Y}\) of the schematic closure of this divisor in Y. Then D is the sum of \(D'\) and the restriction to \(\mathcal {Y}\) of the toric boundary on Y. By the description of the logarithmic strata on a regular log scheme in Sect. 3.2, the logarithmic strata of \(\mathcal {Y}^{\dagger }\) are precisely the sets \(O(\sigma ){\setminus } D'\) and the connected components of \(O(\sigma )\cap D'\), where \(\sigma \) ranges through the cones of \(\Sigma \). Like on any regular log scheme, the points of the fan \(F(\mathcal {Y}^{\dagger })\) are the generic points of these logarithmic strata.
Let \(\gamma \) be a face of \(\Gamma (f)\), and denote by \(\sigma _{\gamma }\) the corresponding cone in the dual fan \(\Sigma \). The points in \(F(\mathcal {Y}^{\dagger })_k\) that lie on \(O(\sigma _{\gamma })\) are the generic points of \(O(\sigma _{\gamma })\cap D'\) and, provided that \(v_\gamma \ne 0\), also the generic point of \(O(\sigma _{\gamma })\). Let \(\tau \) be a point of \(F(\mathcal {Y}^{\dagger })_k\cap O(\sigma _{\gamma })\) and set \(M=M^{\sharp }_{\mathcal {Y}^{\dagger },\tau }\). As usual, we write \(e_t\) for the class of \(t=f\circ h\) in M.
Remark 8.3.3
The calculation of \(Z_f(T)\) in [11, §2.1] contains the following flaws: the \({\widehat{\mu }}\)action on the schemes \(X_\gamma (1)\) is illdefined; the term involving \([X_\gamma (1)]\) should be omitted if \(v_\gamma =0\); the factor \((\mathbb {L}1)\) after \([X_\gamma (0)]\) should be omitted; the \(X_0\)scheme structure on \(X_\gamma (0)\) and \(X_\gamma (1)\) is not specified.
Corollary 8.3.4
Proof
This follows from Theorem 8.3.2 in the same way as in the proof of Proposition 5.4.2. \(\square \)
Note that this set of candidate poles is substantially smaller than the set of candidates we would get from a toric log resolution of \((\mathbb {A}^n_k,X_0)\): the latter set would include the candidate poles associated with all the rays in a regular subdivision of the dual fan of \(\Gamma (f)\). An analogous result for Igusa’s padic zeta function was proven in [7].
The same method of proof yields similar results for the local motivic zeta function \(Z_{f,O}(T)\) of f at the origin O of \(\mathbb {A}^n_k\). This zeta function is defined as the image of \(Z_f(T)\) under the base change morphism \({\mathcal {M}}^{{\widehat{\mu }}}_{X_0}\rightarrow {\mathcal {M}}^{{\widehat{\mu }}}_{O}={\mathcal {M}}^{{\widehat{\mu }}}_{k}\). In fact, we only need to assume that f is nondegenerate with respect to the compact faces of its Newton polyhedron. This means that for every compact face \(\gamma \) of \(\Gamma (f)\), the polynomial \(f_\gamma \) has no critical points in \({\mathbb {G}}^n_{m,k}\).
Theorem 8.3.5
Proof
The nondegeneracy condition on f guarantees that \(\mathcal {Y}^{\dagger }\) is smooth over \(S^\dagger \) at every point of \(h^{1}(O)\), by the same arguments as in the proof of Proposition 8.3.1. The remainder of the argument is identical to the proof of Theorem 8.3.2: we only need to take into account that \(O(\sigma _\gamma )\) lies in \(h^{1}(O)\) if \(\gamma \) is compact, and has empty intersection with \(h^{1}(O)\) otherwise. \(\square \)
Remark 8.3.6
The monodromy conjecture for nondegenerate polynomials in at most 3 variables has been proven for the topological zeta function [20] and the padic and naïve motivic zeta functions [2] (in a weaker form, replacing roots of the Bernstein polynomial by local monodromy eigenvalues). See also [22] for partial results in arbitrary dimension in the padic setting.
Notes
Acknowledgements
We are grateful to Wim Veys for his suggestion to interpret the results in Sect. 8.2 in the context of logarithmic geometry. The first author was supported by a Ph.D. grant from the Fund of Scientific Research – Flanders (FWO). We would also like to thank the referee for their valuable and thoughtful comments. The second author was supported by the ERC Starting Grant MOTZETA (project 306610) of the European Research Council, and by long term structural funding (Methusalem grant) of the Flemish Government.
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