Motivic zeta functions of degenerating Calabi–Yau varieties

Abstract

We study motivic zeta functions of degenerating families of Calabi–Yau varieties. Our main result says that they satisfy an analog of Igusa’s monodromy conjecture if the family has a so-called Galois equivariant Kulikov model; we provide several classes of examples where this condition is verified. We also establish a close relation between the zeta function and the skeleton that appeared in Kontsevich and Soibelman’s non-archimedean interpretation of the SYZ conjecture in mirror symmetry.

Appendix: motivic integration on algebraic spaces

The aim of this section is to prove that one can use models in the category of algebraic spaces to compute motivic integrals. Specifically, we will extend the computation of motivic zeta functions on log smooth models from [12] to algebraic spaces; this is required for the proof of Theorem 5.3.2. On the way, we answer a question raised by Stewart and Vologodsky in [62, A.4(b)]. In principle, one can go through the entire theory of motivic integration on schemes over discrete valuation rings and check that all the statements remain valid for algebraic spaces. Here, we will use a shortcut instead, passing through the category of formal schemes.

1.1 Weak Néron models and motivic integrals

(7.1.1) Let R be a complete discrete valuation ring with quotient field K and perfect residue field k; we do not require k to have characteristic zero. Let X be a connected smooth and proper algebraic space over K. We define a weak Néron model for X to be a separated smooth algebraic space $\mathscr {U}$ over R, endowed with an isomorphism $\mathscr {U}_K\rightarrow X$, such that for every finite unramified extension $R'$ of R with quotient field $K'$, the map $\mathscr {U}(R')\rightarrow X(K')$ is bijective. For our purposes, we will only need the case where X itself is a scheme; in general, the existence of weak Néron models can be proven in exactly the same way as for schemes, by applying the smoothening algorithm in the proof of [9, 3.4.2] to a compactification of $\mathscr {U}$ over R (see [14] for an extension of Nagata’s embedding theorem to algebraic spaces). The smoothening algorithm is functorial with respect to étale morphisms and carries over to algebraic spaces without difficulties.

(7.1.2) Let $\omega$ be a volume form on X. For every weak Néron model $\mathscr {U}$ of X and every connected component C of $\mathscr {U}_k$, we can define the order $\mathrm {ord}_C\omega$ of $\omega$ along C in the same way as for schemes: it is the unique integer m such that $\pi ^{-m}\omega$ extends to a generator of $\omega _{\mathscr {U}/R}$ at the generic point of C, where $\pi$ is a uniformizer in R.

(7.1.3) If Y is an algebraic space of finite type over k, then Y has a dense open subspace that is a scheme of finite type over k [35, II. 6.8]. Thus, by Noetherian induction, we can partition Y into finitely many subschemes of finite type over k. The sum of the classes of these subschemes in the Grothendieck ring of k-varieties $K_0(\mathrm {Var}_k)$ does not depend on the chosen partition, so that we can take this sum as the definition of the class [Y] in $K_0(\mathrm {Var}_k)$. If R has equal characteristic, we denote by $\mathcal {M}_k$ the localized Grothendieck ring of k-varieties $K_0(\mathrm {Var}_k)[\mathbb {L}^{-1}]$. If R has mixed characteristic, then $\mathcal {M}_k$ will denote the modified localized Grothendieck ring from [52, § 3.8], obtained by trivializing all universal homeomorphisms. The key result in this appendix is the following proposition.

Proposition 7.1.4

Let X be a connected smooth and proper algebraic space over K, let $\omega$ be a volume form X, and let $\mathscr {U}$ be a weak Néron model for X. Then the element

$$\begin{aligned} \sum _{C\in \pi _0(\mathscr {U}_k)}[C]\mathbb {L}^{-\mathrm {ord}_C\omega } \end{aligned}$$

(7.1.5)

of $\mathcal {M}_k$ only depends on X and $\omega$, and not on the choice of the weak Néron model of X. In particular, if X is a scheme, then

$$\begin{aligned} \int _X|\omega |=\sum _{C\in \pi _0(\mathscr {U}_k)}[C]\mathbb {L}^{-\mathrm {ord}_C\omega }. \end{aligned}$$

Proof

By [15, 4.2.1], we can consider the analytification $X^{\mathrm {rig}}$ in the category of rigid analytic K-varieties. This is a smooth and proper rigid K-variety, by [15, 2.3.1]. The volume form $\omega$ on X induces a volume form on $X^{\mathrm {rig}}$ that we will still denote by $\omega$. We claim that the expression (7.1.5) is equal to the motivic integral

$$\begin{aligned} \mathbb {L}^{\dim (X)}\int _{X^{\mathrm {rig}}}|\omega | \end{aligned}$$

(7.1.6)

defined in [43, 4.1.2]—see also [53, p. 266] for a corrigendum. This implies, in particular, that it does not depend on the choice of $\mathscr {U}$.

So let us prove our claim. In order to compute the motivic integral (7.1.6), we construct a formal weak Néron model $\mathfrak {V}$ for $X^{\mathrm {rig}}$ in the sense of [10]. This is a smooth formal R-scheme of finite type, endowed with an open immersion of rigid K-varieties $\mathfrak {V}_\eta \rightarrow X^{\mathrm {rig}}$ that is bijective on $K'$-points for every finite unramified extension of K. Then for every connected component C of $\mathfrak {V}_k$, one can define the order $\mathrm {ord}_C\omega$ in exactly the same way as before. By [43, 4.3.1], we have

$$\begin{aligned} \int _{X^{\mathrm {rig}}}|\omega |=\mathbb {L}^{-\dim (X)}\sum _{C\in \pi _0(\mathfrak {V}_k)}[C]\mathbb {L}^{-\mathrm {ord}_C\omega } \end{aligned}$$

in $\mathcal {M}_k$. The factor $\mathbb {L}^{-\dim (X)}$ comes from a different choice of normalization of the motivic measure than the one we have made in Sect. 2.2; the assumption in [43, 4.3.1] that $\mathfrak {V}$ is contained in a formal R-model of $X^{\mathrm {rig}}$ is redundant, by [53, 2.43].

By [35, II. 6.8] and the assumption that k is perfect, we can find a partition of $\mathscr {U}_k$ into finitely many connected k-smooth subschemes $U_1,\ldots ,U_r$. If we denote by $\mathfrak {U}_i$ the formal completion of the algebraic space $\mathscr {U}$ along $U_i$, then $\mathfrak {U}_i$ is a formal scheme, by [35, V. 2.5]. The reduction of $\mathfrak {U}_i$ (the closed subscheme defined by the largest ideal of definition $J_{\mathfrak {U}_i}$) is precisely $U_i$; thus it is of finite type over k, and $\mathfrak {U}_i$ is a smooth special formal R-scheme in Berkovich’s terminology used in [48, 53] (special formal R-schemes are also called formally of finite type in the literature).

For every i in $\{1,\ldots ,r\}$, we denote by $\mathfrak {V}_i\rightarrow \mathfrak {U}_i$ the dilatation centered at $U_i$. This means that $\mathfrak {V}_i$ is the maximal open formal subscheme of the blow-up of $\mathfrak {U}_i$ at $U_i$ such that the ideal $J_{\mathfrak {U}_i}\mathcal {O}_{\mathfrak {V}_i}$ is generated by a uniformizer in R. The dilatation satisfies a universal property that guarantees, in particular, that the map $\mathfrak {V}_i(R')\rightarrow \mathfrak {U}_i(R')$ is bijective for every finite unramified extension $R'$ of R [48, 2.22]. Moreover, $\mathfrak {V}_i$ is a smooth separated formal R-scheme of finite type, because $\mathfrak {U}_i$ and $U_i$ are smooth (see the proof of [48, 4.15]). It follows that the disjoint union $\mathfrak {V}$ of the formal R-schemes $\mathfrak {V}_i$ is a weak Néron model of $X^{\mathrm {rig}}$. If we denote by $c_i$ the codimension of $U_i$ in $\mathscr {U}_k$, then it is easy to check that $[(\mathfrak {V}_i)_k]=[U_i]\mathbb {L}^{c_i}$ in $K_0(\mathrm {Var}_k)$ (see again the proof of [48, 4.15]). Moreover, if we write $C_i$ for the unique connected component of $\mathscr {U}_k$ containing $U_i$, then a straightforward computation also shows that $\mathrm {ord}_{(\mathfrak {V}_i)_k}\omega =\mathrm {ord}_{C_i}\omega +c_i$. Hence,

$$\begin{aligned} \mathbb {L}^{\dim (X)}\int _{X^{\mathrm {rig}}}|\omega |=\sum _{D\in \pi _0(\mathfrak {V}_k)}[D]\mathbb {L}^{-\mathrm {ord}_D\omega }=\sum _{C\in \pi _0(\mathscr {U}_k)}[C]\mathbb {L}^{-\mathrm {ord}_C\omega } \end{aligned}$$

in $\mathcal {M}_k$, by the scissor relations in the Grothendieck ring. $\square$

Thus, when X is a scheme, we can use weak Néron models in the category of algebraic spaces to compute motivic integrals of volume forms on X. This answers the question raised in [62, A.4(b)].

1.2 Motivic zeta functions and Nisnevich covers

(7.2.1) From now on, we assume that k is an algebraically closed field of characteristic zero, and we fix an isomorphism $R\cong k[[\pi ]]$. Let X be a smooth and proper K-scheme with trivial canonical line bundle, and let $\omega$ be a volume form on X. Let $\mathscr {X}$ be a proper algebraic space over R endowed with an isomorphism of K-schemes $\mathscr {X}_K\rightarrow X$. By Proposition 5.2.3, we can find a partition of $\mathscr {X}_k$ into subschemes $U_1,\ldots ,U_r$ and, for each j in $\{1,\ldots ,r\}$, an étale morphism of finite type $\mathscr {U}_j\rightarrow \mathscr {X}$ such that $\mathscr {U}_j$ is a scheme and $\mathscr {U}_j\times _{\mathscr {X}}U_j\rightarrow U_j$ is an isomorphism. For each j, we consider the motivic zeta function $Z^{\widehat{\mu }}_{\mathscr {U}_j,\omega }(T)$ as defined in [12, § 6.2]; here we abuse notation by writing $\omega$ for the restriction of $\omega$ to the generic fiber of $\mathscr {U}_j$. This zeta function is a formal power series in T with coefficients in the localized Grothendieck ring $\mathcal {M}^{\widehat{\mu }}_{(\mathscr {U}_j)_k}$ of varieties over $(\mathscr {U}_j)_k$ with good $\widehat{\mu }$-action. We define the generating series $Z_{j}(T)$ by first applying the base change morphism

$$\begin{aligned} \mathcal {M}^{\widehat{\mu }}_{(\mathscr {U}_{j})_k}\rightarrow \mathcal {M}^{\widehat{\mu }}_{U_j} \end{aligned}$$

to the coefficients of $Z^{\widehat{\mu }}_{\mathscr {U}_j,\omega }(T)$, and then the forgetful morphism

$$\begin{aligned} \mathcal {M}^{\widehat{\mu }}_{U_j}\rightarrow \mathcal {M}^{\widehat{\mu }}_{k}. \end{aligned}$$

Proposition 7.2.2

We have

$$\begin{aligned} Z_{X,\omega }(T)=Z_1(T)+\cdots +Z_r(T) \end{aligned}$$

in $\mathcal {M}^{\widehat{\mu }}_{k}$.

Proof

The proof is similar to the one of Proposition 7.1.4. For every j in $\{1,\ldots ,r\}$, we denote by $\mathfrak {X}_j$ the formal completion of $\mathscr {X}$ along the subscheme $U_j$. It is isomorphic to the completion of $\mathscr {U}_j$ along $\mathscr {U}_j\times _{\mathscr {X}}U_j$. Thus $Z_j(T)$ is precisely the motivic zeta function of the pair $((\mathfrak {X}_j)_\eta ,\omega )$, and the result follows from the fact that the rigid varieties $(\mathfrak {X}_j)_\eta$ form a partition of $X^{\mathrm {rig}}$. $\square$