Formal deformations of algebraic spaces and generalizations of the motivic Igusa-zeta function.

Progress has been made recently in understanding certain motivic generating series associated to the germ of a variety. Originally introduced by Schoutens in [Scha] and [Schb], the auto-Igusa zeta series is a motivic generating series whose coefficients are determined by the algebra automorphisms of the local ring corresponding to a point on a variety. First calculations of this series were completed by Schoutens and secondary calculations were carried out by the author of this paper with the use of computer algebra software in [Sto17]. This led to several conjectures concerning the structure of the auto-Igusa zeta function. The author later confirmed that these conjectures are indeed true in [Sto18] for plane curve singularities, yet the veracity of these conjectures for broader classes of singularities remains unknown.

In this paper, the author generalizes the auto-Igusa zeta function to stacks. However, we choose to frame almost all of the work in the language of algebraic spaces, as there is a bottleneck for Artin stacks related to the occasional lack of representability of the hom functor for general Artin stacks. The representability results are discussed in §2.

One notable development in this paper concerns the introduction of a new type of moduli stack, which takes into account the canonical group action on the space of endormophism by conjugation of all automorphisms. We call this the moduli stack of Jordan-Normal forms. This space is in many ways more natural than the space of all endomrophism (from which the auto-Igusa zeta series is formed), yet it is unclear to the author under what conditions this moduli stack is itself an algebraic space or even when it is an Artin stack. Regardless, one may form a generating series from this stack, which we then term the canonical auto-Igusa zeta function.

Another important development in this paper is that it is shown in §6 that both the canonical zeta function and the auto-Igusa zeta function may both be regarded as generalizations of the motivic Igusa zeta function, which has been studied by Denef, Loeser, Cluckers, Nicaise, Mustata, and many others. It is hoped that the motivic Igusa zeta series would offer a key to proving the monodromy conjecture first introduced by Igusa. However, as of yet, the general conjecture remains unproven.

In §7, the author rephrases his work in [Sto18] in this new context and offers possible directions for future research. As mentioned at the beginning of this section, the possible rationality of the auto-Igusa zeta function for varieties (and other conjectures) other than plane curves remains mostly elusive to direct proof. It is the hope of the author that the work carried out in this paper will offer progress toward such a proof as well as possible keys to further understanding of local zeta functions in general.

Let $\mathbf{C}$ be a site. It follows from the Yoneda lemma that the Grothendieck topos $\tau=\mbox{Sh}(\mathbf{C})$ is a locally cartesian closed monoidal category. In particular, given two objects $\mathcal{X}$ and $\mathcal{Y}$ of $\tau$, we have a presheaf which sends an object $U$ of $\mathbf{C}$ to the set $\mbox{Hom}_{\tau}(\mathcal{X}\otimes U,\mathcal{Y}\otimes U)$, and, moreover, this presheaf is in fact a sheaf – i.e., it is an object of $\tau$, which we denote by $\underline{\mbox{Hom}}_{\tau}(\mathcal{X},\mathcal{Y})$. In other words, there is a natural bijection

In fact, a similar result holds in higher category theory–i.e., if $\tau$ is an $(\infty,1)$-topos, then $\tau$ is a locally cartesian closed monoidal $(\infty,1)$-category. This follows from the $(\infty,1)$-Yoneda lemma (cf. Proposition 5.1.3.1 of [Lur09]). In particular, if $\mathcal{X}$ and $\mathcal{Y}$ are two stacks over an algebraic space $S$, then one defines $\underline{\mbox{Hom}}_{S}(\mathcal{X},\mathcal{Y})$ to be the fibered category of groupoids over the site of $S$-schemes $(\mathbf{Sch}/S)_{J}$ with a given Grothendieck topology $J$, which to any obect $U\to S$ of $(\mathbf{Sch}/S)_{J}$ associates the groupoid of functors from $\mathcal{X}\times_{S}U$ to $\mathcal{Y}\times_{S}U$ over $U$, and, moreover, this fibered category $\underline{\mbox{Hom}}_{S}(\mathcal{X},\mathcal{Y})$ is in fact a stack. It is possible to show that $\underline{\mbox{Hom}}_{S}(\mathcal{X},\mathcal{Y})$ is an Artin stack (or, Deligne-Mumford stack, or algebraic space) depending on certain conditions on $\mathcal{X}$ and $\mathcal{Y}$ (cf. Theorem 1.1 of [Ols06]). For example, we have the following result due to Artin.

This result can be found in [Art69], and generalizes¹¹1Essentially, Artin relaxed the condition that $X\to S$ is projective to merely proper, but this came at the cost of moving from the category of schemes to the larger category of algebraic spaces. the following result of Grothendieck. If $S$ is a locally Noetherian scheme and if $X$ and $Y$ are in $\mathbf{Sch}/S$ with $X$ projective and flat over $S$ and $Y$ quasi-projective over $S$, then the functor from $\mathbf{Sch}/S$ to $\mathbf{Sets}$ defined by sending $U$ to $\mbox{Hom}_{S}(X\times_{S}U,Y\times_{S}U)$ is represented by a scheme $\underline{\mbox{Hom}}_{S}(X,Y)$ which is a separated and locally of finite type over $S$. This result can be found at the end of page 221-19 of [Gro62].

In particular, if $X$ is finite and flat over a locally Noetherian scheme $S$, then for any quasi-projective object $Y$ in $\mathbf{Sch}/S$, $\underline{\mbox{Hom}}_{S}(X,Y)$ is a scheme over $S$. This is found in Proposition 5.7 on pages 221-27 of [Gro62], and it usually goes by the name Weil restriction or restriction of scalars. If $Y\to S$ is separated (respectively, affine, finite type), then $\underline{\mbox{Hom}}_{S}(X,Y)\to S$ is separated (respectively, affine, finite type) over $S$ (cf. Section 7 on pages 221-27 and 221-28 of [Gro62]).

Let $X\to S$ be a proper and flat morphism of algebraic spaces with $S$ locally Noetherian. There is a functor $\mbox{Hilb}_{X/S}:\mathbf{Sch}/S\to\mathbf{Sets}$ defined by

where $[j:Z\hookrightarrow X\times_{S}T$ denotes the isomorphism class of $j$. This is in fact represented by a separated algebraic space locally of finite type over $S$ (cf. Section 6 of [Art69]) which we denote by $\underline{\mbox{Hilb}}_{X/S}$. The natural transformation $\iota:\mbox{Hom}_{S}(X,Y)\hookrightarrow\mbox{Hilb}_{(X\times_{S}Y)/S}$ defined by

where $\Gamma_{f}$ denotes the graph of $f$ induces an open immersion of algebraic spaces $\underline{\mbox{Hom}}_{S}(X,Y)\hookrightarrow\underline{\mbox{Hilb}}_{(X\times_{S}Y)/S}$ provided $X\to S$ is proper and flat and $Y\to S$ is separated and of finite type.

Now, if $f:S^{\prime\prime}\rightarrow S^{\prime}$ is an $S$-morphism of algebraic spaces, then there is a natural pullback $f^{*}$ given by the natural transformation $f^{*}:\mbox{Hilb}_{S^{\prime}/S}\to\mbox{Hilb}_{S^{\prime\prime}/S}$ defined by

Given two $S$-morphisms $f:X^{\prime}\to X$ and $g:Y^{\prime}\to Y$ of algebraic spaces with $X^{\prime}\to S$ and $X\to S$ proper and flat and with $Y^{\prime}\to S$ and $Y\to S$ separated and finite type, we have the canonical morphism $(f,g):X^{\prime}\times_{S}Y^{\prime}\to X\times_{S}Y$ and the corresponding pullback morphism of Hilbert spaces

If, in addition, $g:Y^{\prime}\hookrightarrow Y$ is a closed immersion, then the restriction of $(f,g)^{*}$ to $\underline{\mbox{Hom}}(X,Y)$ induces a well-defined morphism of algebraic spaces

Note that $\underline{\mbox{Hom}}_{S}(X,Y)$ is contravariant in the first argument and covariant in the second; however, the morphism $(f,g)^{*}$ sends a morphism $h:X\to Y$ to $h\circ f:X^{\prime}\to Y$ and then restricts the range of $h\circ f$ to the closed subset $Y^{\prime}$ of $Y$.

An injective system $\{P_{i}\}_{i\in I}$ of algebraic spaces over $S$ with affine transition maps and with $P_{i}\to S$ proper and flat gives rise to a projective system $\{\underline{\mbox{Hilb}}_{P_{i}/S}\}_{i\in I}$ of algebraic spaces with affine transition maps, and it is therefore the case that $\varprojlim\underline{\mbox{Hilb}}_{P_{i}/S}$ exists in the category of algebraic spaces over $S$. In particular, if $\{Y_{i}\}_{i\in I}$ is an injective system whose transition maps are closed immersions with $Y_{i}\to S$ separated and finite type and if $\{X_{i}\}_{i\in I}$ is any other injective system with $X_{i}\to S$ proper and flat, then $\varprojlim\underline{\mbox{Hom}}_{S}(X_{i},Y_{i})\hookrightarrow\varprojlim\underline{\mbox{Hilb}}_{X_{i}\times_{S}Y_{i}/S}$ is an open immersion of algebraic spaces over $S$.

Furthermore, if $\{X_{i}\}_{i\in I}$ is an injective system of algebraic spaces over $S$ whose transitions maps are closed immersions and where each $X_{i}\to S$ is proper and flat then there is a projective system of open immersions

where $\underline{\mbox{Aut}}(X_{i})$ is the algebraic space which represents the subfunctor of $\mbox{Hom}_{S}(X_{i},X_{i})$ which associates to $T\to S$ the set of all automorphisms of $X_{i}\times_{S}T$. Here, we write $\underline{\mbox{End}}_{S}(X_{i})$ in place of $\underline{\mbox{Hom}}_{S}(X_{i},X_{i}).$ Note that $\underline{\mbox{End}}_{S}(X_{i})$ is a monoid object and $\underline{\mbox{Aut}}_{S}(X_{i})$ is a group object in the category of algebraic spaces over $S$. Since the transition maps are closed immersions, the inclusion

is an open immersion of algebraic spaces over $S$.

Assume that $A$ is a Noetherian local ring with $B$ finite and flat over $A$, and whence free. Let $A^{n}\xrightarrow{\sim}B$ be a presentation of $B$ as a free $A$-algebra obtained by sending the standard basis elements $e_{i}$ of $A^{n}$ to the generating elements $b_{i}$ of $B$ for $i=1,\ldots,n$. Consider the polynomial ring $R=A[x_{1},\ldots,x_{n}]$ and the surjective homomorphism $\phi:R\to B$ defined by $\phi(x_{i})=b_{i}$ for $i=1,\ldots,n$. Then, by Hilbert’s Nullstellensatz, $R$ is Noetherian and therefore the kernel $\mbox{Ker}(\phi)$ is finitely generated – i.e., $\mbox{Ker}(\phi)=(f_{1},\ldots,f_{s})$. Moreover, there is a canonical isomorphism $R/\mbox{Ker}(\phi)\xrightarrow{\sim}B$ in the category of $A$-algebras. Therefore, an $A$-algebra endomorphism $\gamma$ of $B$ is given by a choice $\gamma(x_{i})=P_{i}$ with $P_{i}$ any element of $B=A[x_{1},\ldots,x_{n}]/(f_{1},\ldots,f_{s})$ for each $i=1,\ldots,n$ such that $f_{j}(P_{1},\ldots,P_{n})=0$ in $B$ for $j=1,\ldots,s$.

One simple case where Lemma 4.1 may be applied is when $S=\mbox{Spec}(A)$ with $(A,\mathfrak{m})$ an Artinian local ring. This is because if $B$ is a finitely generated $A$-module with $A$ an Artinian local ring, then $B$ will have finite length over the residue field $k=A/\mathfrak{m}$ and hence $B$ will also be an Artinian local ring. In particular, $B$ is automatically free over $A$. Thus, we may form the full subcategory $\mathbf{Fat}/A$ of $\mathbf{Sch}/A$ defined by all finite maps $X\to\mbox{Spec}(A)$. We call²²2In general, if $S$ is an algebraic space, we call the full subcategory $\mathbf{C}$ of $\mathbf{Sch}/S$ whose objects are finite, flat, and locally finitely presented over $S$ the category of fat points over $S$, and we denote this category by $\mathbf{Fat}/S$. this category the category of fat points over $A$. Usually, we assume $A=k$ is a field (and, in this case, Lemma 4.3 will apply), and often, we will further assume that it has characteristic zero and that it is algebraically closed.

In general, if $X$ is proper and flat over an algebraic space $S$, then the group action

given by conjugation is well-defined in the category of algebraic spaces over $S$. The resulting quotient stack

is termed the moduli stack of Jordan norm forms of $X$ over $S$, and it is of general interest. In the case of Lemma 4.3, assuming further that $A$ is an algebraically closed field $k$, the points of $\mathcal{M}_{\mathrm{JN}}(X)$ correspond to $d\times d$ Jordan normal forms over $k$.

Given an admissible system $\{X_{i}\}_{i\in I}$ over $S$ and a morphism $Y\to S$ of algebraic spaces, we have the trivial formal deformation given by $Y_{i}:=Y\times_{S}X_{i}$. More generally, given an ind-object $\mathcal{Y}$ in the category of algebraic spaces over $S$ – i.e., the filtered colimit $\mathcal{Y}=\varinjlim_{i\in I}Y_{i}$ where the transition maps $Y_{i}\hookrightarrow Y_{i+1}$ are closed immersions of algebraic spaces – and, an admissible system $\{X_{i}\}_{i\in I}$ with $\mathcal{X}=\varinjlim_{i\in I}X_{i}$, we let $\mathtt{Def}_{\mathcal{X}}$ denote the fibered category of formal deformations over $S$ with respect to the admissible system $\{X_{i}\}$. Thus, in particular, if $Y\to S$ is a morphism of algebraic space then the category $\mathtt{Def}_{\mathcal{X}}(Y)$ has at least one object.

An object $\mathcal{Y}=\varprojlim_{i\in I}Y_{i}$ of $\mathtt{Def}_{\mathcal{X}}$ gives rise to a projective system $\{\mathcal{A}_{i}(\mathcal{Y})\}_{i\in I}$ of algebraic spaces over $S$ defined by

We call $\mathcal{A}_{i}(\mathcal{Y})$ the truncated auto-arc space of $\mathcal{Y}$ at level $i$. Moreover, we may form the projective limit

which is termed the infinite auto-arc space of $\mathcal{Y}$. If $\mathcal{Y}$ is the trivial deformation of $Y\to S$, then

In general, the group scheme $\underline{\mbox{Aut}}_{S}(X_{i})^{\mathrm{red}}$ acts on $\mathcal{A}_{i}(\mathcal{Y})$ by conjugation for any formal deformation $\mathcal{Y}$, and this give rise to the quotient stack

In particular, when $\mathcal{Y}$ is the trivial deformation, then

where $\mathcal{M}_{JN}(X_{i})$ is the moduli stack of Jordan normal forms introduced at the end of section 4. The projective limit of quotient stacks

is of general interest.

We define $\mathbb{L}:=[\mathbb{A}_{S}^{1}]$ and call it the Leftshetz motive over $S$. We may invert this element to obtain the localized Grothendieck ring $\mathcal{G}_{S}:=\mbox{K}_{0}(\mathbf{Alg}/S)[\mathbb{L}^{-1}]$ of algebraic spaces over $S$. We say that an element of the power series ring $\mathcal{G}_{S}[[t]]$ is a motivic generating series of algebraic spaces over $S$ in one variable, or, more briefly, we will call it a motivic generating series.

Let $\mathcal{Y}$ be as in Definition 6.3 and assume further that $\mathcal{M}_{i}(\mathcal{Y})$ is an algebraic space for all $i\in I$, we may form the motivic generating function

where $m_{i}=\mbox{dim}_{S}(Y_{i})(\mbox{rank}_{S}(X_{i})-1)+e_{i}$ where $e_{i}$ is chosen to be the whole number which makes the coefficient $[\mathcal{M}_{i}(\mathcal{Y})]\mathbb{L}^{-m_{i}}$ dimensionless. We call $\eta_{\mathcal{Y}}(t)$ the canonical auto-Igusa zeta function of $\mathcal{Y}$.

We will need to refer to the following Set-up below.

The main rationality result the author has obtained is the following.

Part of the argument in [Sto18] relies on rationality results of [DF98] and [DL99] and their later generalizations in [CL08] and [CL15]. In fact, the first part of this argument holds whenever the morphisms $\varphi_{i}:Y_{i}\to X_{i}$ are smooth with $\{X_{i}\}_{i\in I}$ any admissible injective system of fat points over any algebraic space $S$ – i.e., regardless of whether or not $X_{i}$ is subscheme of a plane curve $C$, formula 7.4 holds provided $\varphi_{i}:Y_{i}\to X_{i}$ is smooth for all $i\in I$. However, as of yet, rationality is only proven for the case when $\mathcal{Y}$ is as in Set-up 7.1. Thus, there are several directions for future results: