Motivic zeta functions of hyperplane arrangements

Motivic zeta functions of hyperplane arrangements

Max Kutler and Jeremy Usatine Max Kutler, Department of Mathematics, University of Kentucky max.kutler@uky.edu Jeremy Usatine, Department of Mathematics, Yale University jeremy.usatine@yale.edu
Abstract.

For each central essential hyperplane arrangement over an algebraically closed field, let denote the Denef-Loeser motivic zeta function of . We prove a formula expressing in terms of the Milnor fibers of related hyperplane arrangements. We use this formula to show that the map taking each complex arrangement to the Hodge-Deligne specialization of is locally constant on the realization space of any loop-free matroid. We also prove a combinatorial formula expressing the motivic Igusa zeta function of in terms of the characteristic polynomials of related arrangements.

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1. Introduction

We study hyperplane arrangements and the motivic zeta functions of Denef and Loeser. Let be an algebraically closed field, and let be a central essential arrangement of hyperplanes in . If are linear forms defining , respectively, then we can consider the Denef-Loeser motivic zeta function of and the motivic Igusa zeta function of .

Inspired by Kontsevich’s theory of motivic integration [Kon95], Denef and Loeser defined zeta functions [DL98, DL01, DL02] that are power series with coefficients in a Grothendieck ring of varieties. These zeta functions are related to multiple well-known invariants in singularity theory and birational geometry, and they have implications for Igusa’s monodromy conjecture, a longstanding conjecture concerning the poles of Igusa’s local zeta function. There has been interest in understanding these motivic zeta functions, and the closely related topological zeta function, in the case of polynomials defining hyperplane arrangements [BSY11, BMT11, vdV18].

In this paper, we prove a formula for in terms of the classes of Milnor fibers of certain related hyperplane arrangements. We use this formula and a result in [KU18] to show that certain specializations of , including the Hodge-Deligne specialization, remain constant as we vary the arrangement within the same connected component of a matroid’s realization space. We also prove a combinatorial formula for in terms of the characteristic polynomials of certain related matroids.

1.1. Statements of main results

Throughout this paper, will be an algebraically closed field. Before we state our results, we need to set some notation.

For each , let be the group of -th roots of unity, let be the -equivariant Grothendieck ring of -varieties, let be the class of with the trivial -action, and let . Let , and let be the image of for any .

Let , and let be the Grassmannian of -dimensional linear subspaces in . For each , let denote the corresponding linear subspace, let be the scheme theoretic intersection of with the closed subscheme of defined by , and endow with the restriction of the -action on where each acts by scalar multiplication. Let be the Denef-Loeser motivic zeta function of , and let be the Denef-Loeser motivic zeta function of at the origin of .

If is not contained in a coordinate hyperplane of , then the restrictions of the coordinates define a central essential hyperplane arrangement in , the Milnor fiber of that hyperplane arrangement is , the -action on is the monodromy action, and and are the Denef-Loeser motivic zeta functions associated to that arrangement. Note that we are using a definition of the Milnor fiber that takes advantage of the fact that a hyperplane arrangement is defined by a homogeneous polynomial. This definition is common in the hyperplane arrangement literature, and it allows us to consider the Milnor fiber as a variety.

Remark 1.1.

If is a central essential hyperplane arrangement in , then any choice of linear forms defining gives a linear embedding of into , and is the arrangement associated to the resulting subspace of . Therefore, we lose no generality by considering the arrangements associated to -dimensional linear subspaces in .

Let be a rank loop-free matroid on , let be the Bergman fan of , and let be the locus parametrizing linear subspaces whose associated hyperplane arrangements have combinatorial type . For any , there exists a rank loop-free matroid on such that for all , the initial degeneration is equal to for some unique . We refer to Section 2.4 for the definition of . Let be the set of bases in , and set

In this paper, we will prove the following formulas that express the motivic zeta functions and in terms of classes of the Milnor fibers .

Theorem 1.1.

Let . Then

and

In the course of proving LABEL:*hyperplanearrangementDLzetaformula, we prove LABEL:*zetafunctionschon and LABEL:*tropicalzetaformulahomogeneous, which give formulas for motivic zeta functions when certain tropical hypotheses are satisfied. We think of LABEL:*zetafunctionschon and LABEL:*tropicalzetaformulahomogeneous as being in the spirit of the formulas for zeta functions of so-called Newton non-degenerate hypersurfaces [DH01, Gui02, BV16, BN16]. To prove LABEL:*zetafunctionschon and LABEL:*tropicalzetaformulahomogeneous, we use certain -schemes whose special fibers are the initial degenerations that arise in tropical geometry. These -schemes have played an essential role in much of tropical geometry. See for example [Gub13]. We also use Sebag’s [Seb04] theory of motivic integration for Greenberg schemes, which are non-constant coefficient versions of arc schemes. For our proofs to account for the -action, we use Hartmann’s [Har15] equivariant version of Sebag’s motivic integration.

LABEL:*hyperplanearrangementDLzetaformula allows us to use results about additive invariants of the Milnor fibers to obtain results about specializations of the Denef-Loeser motivic zeta functions. To state such an application, we first define some terminology that can apply to additive invariants. Let be the polynomial ring over the symbol , and endow with the -algebra structure given by .

Definition 1.1.

Let be a -module, and let be a -module morphism. We say that is constant on smooth projective families with -action if the following always holds.

  • If is a connected separated finite type -scheme with trivial -action and is a -equivariant smooth projective morphism from a scheme with -action, then the map is constant, where denotes the fiber of over .

Remark 1.2.

If and is the morphism that sends the class of each variety to its Hodge-Deligne polynomial, then is constant on smooth projective families with -action.

Note that if and are in the same connected component of , then are in the same connected component of . See for example [KU18, Fact 2.4]. Therefore the following theorem is a direct consequence of LABEL:*hyperplanearrangementDLzetaformula and [KU18, Theorem 1.4].

Theorem 1.2.

Let be a torsion-free -module, let be a -module morphism that is constant on smooth projective families with -action, and assume that the characteristic of does not divide .

If are in the same connected component of , then

and

Remark 1.3.

In the statement of LABEL:*specializationinvarianceinconnectedcomponent, by applied to a power series, we mean the power series obtained by applying to each coefficient.

In particular, LABEL:*specializationinvarianceinconnectedcomponent implies that the Hodge-Deligne specialization of the Denef-Loeser motivic zeta function remains constant as we vary the linear subspace within the same connected component of . There has been much interest in understanding how invariants of hyperplane arrangements, particularly those invariants arising in singularity theory, vary as the arrangements vary with fixed combinatorial type. For example, a major open conjecture predicts that when , the Betti numbers of a hyperplane arrangement’s Milnor fiber depend only on combinatorial type, i.e., they depend only on the matroid. Budur and Saito proved that a related invariant, the Hodge spectrum, depends only on the combinatorial type [BS10]. Randell proved that the diffeomorphism type, and thus Betti numbers, of the Milnor fiber is constant in smooth families of hyperplane arrangements with fixed combinatorial type [Ran97]. See [Suc17] for a survey on such questions. Our perspective on LABEL:*specializationinvarianceinconnectedcomponent is in the context of that literature, and we hope it illustrates the use of LABEL:*hyperplanearrangementDLzetaformula in answering related questions.

Our final main result consists of combinatorial formulas for the motivic Igusa zeta functions of a hyperplane arrangement. It is well known that the motivic Igusa zeta functions are combinatorial invariants. For example, one can see this by using De Concini and Procesi’s wonderful models [DCP95] and Denef and Loeser’s formula for the motivic Igusa zeta function in terms of a log resolution [DL01, Corollary 3.3.2]. Regardless, we believe it is worth stating LABEL:*hyperplanearrangementIgusazetaformula below, as it follows from the methods of this paper with little extra effort, and because we are not aware of these particular formulas having appeared in the literature.

Let be the Grothendieck ring of -varieties, let be the class of , and let . For each , let be the motivic Igusa zeta function of , and let be the motivic Igusa zeta function of at the origin of .

Theorem 1.3.

Let . Then

and

where is the characteristic polynomial of evaluated at .

Acknowledgements.

We would like to acknowledge useful discussions with Dori Bejleri, Daniel Corey, Netanel Friedenberg, Dave Jensen, Kalina Mincheva, Sam Payne, and Dhruv Ranganathan. The second named author was supported by NSF Grant DMS-1702428 and a Graduate Research Fellowship from the NSF.

2. Preliminaries

In this section, we will set some notation and recall facts about the equivariant Grothendieck ring of varieties, the motivic zeta functions of Denef and Loeser, Hartmann’s equivariant motivic integration, and linear subspaces and matroids.

2.1. The equivariant Grothendieck ring of varieties

Suppose is a separated finite type scheme over . We will let denote the Grothendieck ring of varieties over , we will let denote the class of , and for each separated finite type -scheme , we will let denote the class of . We will let denote the ring obtained by inverting in , and by slight abuse of notation, we will let denote the images of , respectively, in .

We will let and denote and , respectively, and for each separated finite type -scheme , we will let in both and .

Suppose is a finite abelian group. An action of on a scheme is said to be good if each orbit is contained in an affine open subscheme. For example, any -action on any quasiprojective -scheme is good. Suppose is a separated finite type -scheme with a good -action. We will let denote the -equivariant Grothendieck ring of varieties over . For the precise definition of , we refer to [Har15, Definition 4.1]. We will let denote the class of with the action induced by the trivial -action on and the given -action on , and for each separated finite type -scheme with good -action making the structure morphism -equivariant, we will let denote the class of with its given -action. We will let denote the ring obtained by inverting in , and by slight abuse of notation, we will let denote the images of , respectively, in .

If is a separated finite type -scheme with no specified -action and we refer to or , then we are considering with the trivial -action. We will let and denote and , respectively, and for each separated finite type -scheme with good -action making the structure morphism -equivariant, we will let in both and .

For each , we will let denote the group of -th roots of unity.

Remark 2.1.

We will only consider as a finite group, so when the characteristic of divides , we will not consider the non-reduced scheme structure of .

For each , there is a morphism . Suppose that is a separated finite type scheme over . Then for each , the morphism induces ring morphisms and . We will let and . We will let denote the image of for any , and similarly we will let denote the image of for any . For each and each separated finite type -scheme with good -action making the structure morphism -equivariant, we will let denote the image of , and we will similarly let denote the image of .

We will let and denote and , respectively, and for each and each separated finite type -scheme with good -action making the structure morphism -equivaraint, we will let in both and .

2.2. The motivic zeta functions of Denef and Loeser

Let be a smooth, pure dimensional, separated, finite type -scheme. For each , we will let denote -th jet scheme of , and for each , we will let denote the truncation morphism. We will let denote the arc scheme of , and for each , we will let denote the canonical morphism. The following is a special case of a theorem of Bhatt’s [Bha16, Theorem 1.1].

Theorem 2.1 (Bhatt).

The -scheme represents the functor taking each -algebra to , and under this identification, each morphism is the truncation morphism.

A subset of is called a cylinder if it is the preimage, under , of a constructible subset of for some . We will let denote the motivic measure on , which assigns a motivic volume in to each cylinder.

Suppose is a regular function on . If has residue field , then it corresponds to a -morphism , and we will let denote . For each , the order of at will refer to the order of in the power series , and the angular component of at will refer to the leading coefficient of the power series . We will let denote the function taking each to the order of at . We will let denote the motivic Igusa zeta function of . Then

Remark 2.2.

In the literature, the motivic Igusa zeta function is sometimes referred to as the naive zeta function of Denef and Loeser.

We will let denote the Denef-Loeser motivic zeta function of . We briefly recall the definition of . The constant term of is equal to . Let , and let be the closed subscheme of where is equal to . For each -algebra , there is a -action on given by for each , and these actions induce a -action on making invariant. Note also that the truncation morphism restricts to a -equivariant morphism . Then the coefficient of in is defined to be equal to .

Remark 2.3.

Denef and Loeser defined versions of these zeta functions with coefficients in and [DL98, DL02], and Looijenga introduced versions with coefficients in the relative Grothendieck rings and [Loo02]. See [DL01] for the definitions we are using for and , but note that compared to those definitions, ours differ by a normalization factor of .

2.3. Hartmann’s equivariant motivic integration

For the remainder of this paper, let , the ring of power series over . We will set up some notation and recall facts for Greenberg schemes and Hartmann’s equivariant motivic integration [Har15], which is an equivariant version of Sebag’s motivic integration for formal schemes [Seb04]. For the non-equivariant version of this theory, we also recommend the book [CNS18].

Remark 2.4.

In [Har15], Hartmann uses formal -schemes. The analogous theory for algebraic -schemes, as stated here, directly follows by taking -adic completion.

Let be a smooth, pure relative dimensional, separated, finite type -scheme. We will let denote the special fiber of . For each , we will let denote the -th Greenberg scheme of . Thus represents the functor taking each -algebra to . For each , we will let denote the truncation morphism. We will let denote the Greenberg scheme of , and for each , we will let denote the canonical morphism. As for arc schemes, the following is a special case of [Bha16, Theorem 1.1]. See for example [CNS18, Chapter 4, Proposition 3.1.7].

Theorem 2.2 (Bhatt).

The -scheme represents the functor taking each -algebra to , and under this identification, each morphism is the truncation morphism.

A subset of is called a cylinder if it is the preimage, under , of a constructible subset of for some . We will let denote the motivic measure on , which assigns a motivic volume in to each cylinder.

Suppose is a regular function on . If has residue field , then it corresponds to an -morphism , and we will let denote . As for arc schemes, this is used to define the order and angular component of at and the order function .

Now suppose is a finite abelian group acting on , and suppose that each element of acts on by a -adically continuous -algebra morphism. Endow with a good -action making the structure morphism -equivariant, and endow with the restriction of the -action on . The -action on induces good -actions on and each . We refer to [Har15, Section 3.2] for the construction and properties of these -actions on the Greenberg schemes. We will let denote the -equivariant motivic measure on , which assigns a motivic volume in to each -invariant cylinder in . We refer to [Har15, Section 4.2] for the definition of .

If is a -invariant cylinder and is a function whose fibers are -invariant cylinders, then the integral of is defined to be

Remark 2.5.

By the quasi-compactness of the construcible topology, takes finitely many values, so the above sum is well defined. See [CNS18, Chaper 6, Section 1.2].

We now state the equivariant version of the motivic change of variables formula [Har15, Theorem 4.18]. If is a morphism of -schemes, then we let denote the order function of the jacobian ideal of .

Theorem 2.3 (Hartmann).

Suppose is not divisible by the characteristic of . Let be smooth, pure relative dimensional, separated, finite type -schemes with good -action making the structure morphisms equivariant, and let be a -equivariant morphism that induces an open immersion on generic fibers. Let be -invariant cylinders in , respectively, such that induces a bijection for all extensions of .

If is a function whose fibers are -invariant cylinders, then is a function whose fibers are -invariant cylinders, and

Remark 2.6.

Hartmann stated the formula when and , but the same proof works when replacing and with -invariant cylinders. See for example the proof of the non-equivariant version in [CNS18].

We note that for all , the characteristic of never divides .

2.4. Linear subspaces and matroids

Let . We will let denote the Grassmannian of -dimensional linear subspaces in . We will let denote the complement of the coordinate hyperplanes, and we will let denote the closed subscheme of defined by . For each , we will let denote the corresponding linear subspace. If is not contained in a coordinate hyperplane of , then the restrictions to of the coordinates define a central essential hyperplane arrangement in . We let and denote this arrangement’s complement and Milnor fiber, respectively, and we endow with the restriction of the -action on where each acts by scalar multiplication. In the context of tropical geometry, we will consider both and as closed subschemes of the algebraic torus . We will let and denote the Denef-Loeser motivic zeta function and the motivic Igusa zeta function, respectively, of the restriction of to . We will let (resp. ) denote the power series obtained by pushing forward each coefficient of (resp. ) along the structure morphism of . We will let (resp. ) denote the power series obtained by pulling back each coefficient of (resp. ) along the inclusion of the origin into .

Remark 2.7.

The zeta functions , and are as denoted in the introduction of this paper.

Let be a rank loop-free matroid on . We will let denote the characteristic polynomial of evaluated at , so

where is the rank function of applied to . We will let denote the set of bases of , and we will let denote the function . For each , we will set

Then is the set of bases for a rank matroid on , and we will let denote that matroid. We let denote the Bergman fan of , so

We will let denote the locus parametrizing linear subspaces whose associated hyperplane arrangements have combinatorial type . For all , the fact that is loop-free implies that is not contained in a coordinate hyperplane. Note that if , then and

For each and each , we will let denote the unique point such that .

Before concluding the preliminaries, we recall two propositions proved in [KU18] that will be used in Section 6. If and , then we will let denote the fundamental circuit in of with respect to , so is the unique circuit in contained in . For each circuit in and each , we will let denote a linear form in the ideal defining in such that the coefficient of in is nonzero if and only if . Such an exists and is unique up to scaling by a unit in . Once and for all, we fix such an for all and .

Proposition 2.1 (Proposition 3.6 in [Ku18]).

Let , let , and let . Then

generates the ideal of in , and

generates the ideal of in .

Proposition 2.2 (Proposition 3.2 in [Ku18]).

Let , let , and let . Then

For additional information on matroids and the tropical geometry of linear subspaces, we refer to [MS15, Chapter 4].

3. Equivariant motivic integration and the motivic zeta function

Let , and throughout this section, endow with the -action where each acts on by the -adically continuous -morphism .

Let be a smooth, pure dimensional, finite type, separated scheme over . We will endow and each with -actions that make the truncation morphisms -equivariant as follows. Let , let be a -algebra, let be the morphism whose pullback is the -adically continuous -algebra morphism , and let be the morphism whose pullback is the -algebra morphism .

If corresponds to a -morphism

then let correspond to the -morphism

This action is clearly functorial in , so it defines a -action on . Similarly, if corresponds to a -morphism

then let correspond to the -morphism

This action is also functorial in , so it defines a -action on . We also see that these -actions make the truncation morphisms -equivariant.

Proposition 3.1.

Let be a regular function on . Then has constant order on any -orbit of . Furthermore, has constant angular component on any -orbit of on which has order .

Proof.

Let , let be its action on , let for some extension of , let , and let be the morphism whose pullback is the -adically continuous -algebra morphism . Then corresponds to a -morphism

and corresponds to the -morphism