2.1 Mixed Hodge–Tate structures and their period matrices

We call  $H_{\text{B}}$ and  $H_{\text{dR}}$ the Betti realization and the de Rham realization of the mixed Hodge–Tate structure, and  $\unicode[STIX]{x1D6FC}$ the comparison isomorphism. The filtration  $W$ on  $H_{\text{B}}$ is called the weight filtration. The grading on  $H_{\text{dR}}$ is called the weight grading, and the corresponding filtration  $W_{2n}H_{\text{dR}}:=\bigoplus _{k\leqslant n}(H_{\text{dR}})_{2k}$ the weight filtration.

Remark 2.2. More classically, a mixed Hodge–Tate structure is defined to be a mixed Hodge structure [Reference DeligneDel71, Reference DeligneDel74] whose weight-graded quotients are of Tate type, i.e., of type  $(p,p)$ for some integer  $p$ . One passes from that classical definition to Definition 2.1 by setting $H_{\text{B}}:=H$ and  $H_{\text{dR}}:=\bigoplus _{n}W_{2n}H/W_{2(n-1)}H$ . The isomorphism  $\unicode[STIX]{x1D6FC}$ is induced by the inverses of the isomorphisms

(7) $$\begin{eqnarray}(W_{2n}H/W_{2(n-1)}H)\otimes _{\mathbb{Q}}\mathbb{C}\stackrel{\cong }{\longleftarrow }(W_{2n}H\otimes _{\mathbb{Q}}\mathbb{C})\cap F^{n}(H\otimes _{\mathbb{ Q}}\mathbb{C})\end{eqnarray}$$

(multiplied by  $(2\unicode[STIX]{x1D70B}i)^{n}$ ) which express the fact that the weight-graded quotients are of Tate type.

It is convenient to view the comparison isomorphism  $\unicode[STIX]{x1D6FC}:H_{\text{dR}}\otimes _{\mathbb{Q}}\mathbb{C}\stackrel{\simeq }{\longrightarrow }H_{\text{B}}\otimes _{\mathbb{Q}}\mathbb{C}$ as a pairing

(8) $$\begin{eqnarray}H_{\text{B}}^{\vee }\otimes _{\mathbb{ Q}}H_{\text{dR}}\longrightarrow \mathbb{C},\quad \unicode[STIX]{x1D711}\otimes v\mapsto \langle \unicode[STIX]{x1D711},v\rangle ,\end{eqnarray}$$

where  $(\cdot )^{\vee }$ denotes the linear dual. The weight filtration on  $H_{\text{B}}^{\vee }$ is defined by

$$\begin{eqnarray}W_{-2n}H_{\text{B}}^{\vee }:=(H_{\text{B}}/W_{2(n-1)}H_{\text{B}})^{\vee },\end{eqnarray}$$

so that we have

$$\begin{eqnarray}W_{-2n}H_{\text{B}}^{\vee }/W_{-2(n+1)}H_{\text{B}}^{\vee }\cong (W_{2n}H_{\text{B}}/W_{2(n-1)}H_{\text{B}})^{\vee }.\end{eqnarray}$$

The pairing (8) is compatible with the weight filtrations in that we have  $\langle \unicode[STIX]{x1D711},v\rangle =0$ for  $\unicode[STIX]{x1D711}\in W_{-2m}H_{\text{B}}^{\vee }$ ,  $v\in W_{2n}H_{\text{dR}}$ and  $m<n$ .

If we choose bases for the  $\mathbb{Q}$ -vector spaces  $H_{\text{dR}}$ and  $H_{\text{B}}$ , then the matrix of  $\unicode[STIX]{x1D6FC}$ in these bases, or equivalently the matrix of the pairing (8), is called a period matrix of the mixed Hodge–Tate structure. We will always make the following assumptions on the choice of bases.

1. – The basis of  $H_{\text{B}}$ is compatible with the weight filtration.

2. – The basis of  $H_{\text{dR}}$ is compatible with the weight grading.

3. – For every  $n$ , the matrix of the comparison isomorphism  $\unicode[STIX]{x1D6FC}_{n}$ in the corresponding basis is  $(2\unicode[STIX]{x1D70B}i)^{n}$ times the identity.

This implies that any period matrix is block upper-triangular with successive blocks of  $(2\unicode[STIX]{x1D70B}i)^{n}\,\text{Id}$ on the diagonal. Conversely, any block upper-triangular matrix with successive blocks of  $(2\unicode[STIX]{x1D70B}i)^{n}\,\text{Id}$ on the diagonal is a period matrix of a mixed Hodge–Tate structure.

Example 2.3. Any matrix of the form

$$\begin{eqnarray}\left(\begin{array}{@{}ccccc@{}}1 & \ast & \ast & \ast & \ast \\ 0 & 2\unicode[STIX]{x1D70B}i & 0 & \ast & \ast \\ 0 & 0 & 2\unicode[STIX]{x1D70B}i & \ast & \ast \\ 0 & 0 & 0 & (2\unicode[STIX]{x1D70B}i)^{2} & 0\\ 0 & 0 & 0 & 0 & (2\unicode[STIX]{x1D70B}i)^{2}\end{array}\right)\end{eqnarray}$$

defines a mixed Hodge–Tate structure  $H$ such that  $H_{\text{dR}}=(H_{\text{dR}})_{0}\oplus (H_{\text{dR}})_{2}\oplus (H_{\text{dR}})_{4}$ has graded dimension  $(1,2,2)$ .

2.2 The category of mixed Hodge–Tate structures

We denote by  $\mathsf{MHTS}$ the category of mixed Hodge–Tate structures. It is a neutral tannakian category over  $\mathbb{Q}$ , which means in particular that it is an abelian  $\mathbb{Q}$ -linear category equipped with a  $\mathbb{Q}$ -linear tensor product  $\otimes$ . We note that an object  $H\in \mathsf{MHTS}$ is endowed with a canonical weight filtration  $W$ by subobjects such that the morphisms in  $\mathsf{MHTS}$ are strictly compatible with  $W$ . We have two natural fiber functors

(9) $$\begin{eqnarray}\unicode[STIX]{x1D714}_{\text{B}}:\mathsf{MHTS}\rightarrow \mathsf{Vect}_{\mathbb{Q}}\quad \text{and}\quad \unicode[STIX]{x1D714}_{\text{dR}}:\mathsf{MHTS}\rightarrow \mathsf{Vect}_{\mathbb{Q}}\end{eqnarray}$$

from  $\mathsf{MHTS}$ to the category of finite-dimensional vector spaces over  $\mathbb{Q}$ , which only remember the Betti realization  $H_{\text{B}}$ and the de Rham realization  $H_{\text{dR}}$ respectively. We note that the de Rham realization functor  $\unicode[STIX]{x1D714}_{\text{dR}}$ factors through the category of finite-dimensional graded vector spaces. The comparison isomorphisms  $\unicode[STIX]{x1D6FC}$ gives an isomorphism between the complexifications of the two fiber functors:

(10) $$\begin{eqnarray}\unicode[STIX]{x1D714}_{\text{dR}}\otimes _{\mathbb{Q}}\mathbb{C}\stackrel{\simeq }{\longrightarrow }\unicode[STIX]{x1D714}_{\text{B}}\otimes _{\mathbb{Q}}\mathbb{C}.\end{eqnarray}$$

For an integer  $n$ , we denote by  $\mathbb{Q}(-n)$ the mixed Hodge–Tate structure whose period matrix is the  $1\times 1$ matrix  $((2\unicode[STIX]{x1D70B}i)^{n})$ . Its weight grading and filtration are concentrated in weight  $2n$ , hence we call it the pure Tate structure of weight  $2n$ . For  $H$ a mixed Hodge–Tate structure, the tensor product  $H\otimes \mathbb{Q}(-n)$ is simply denoted by  $H(-n)$ and called the  $n$ th Tate twist of  $H$ . A period matrix of  $H(-n)$ is obtained by multiplying a period matrix of  $H$ by  $(2\unicode[STIX]{x1D70B}i)^{n}$ . The weight grading and filtration of  $H(-n)$ are those of  $H$ , shifted by  $2n$ .

2.3 Extensions between pure Tate structures

The pure Tate structures  $\mathbb{Q}(-n)$ are the only simple objects of the category  $\mathsf{MHTS}$ . The extensions between them are easily described. Up to a Tate twist, it is enough to describe the extensions of  $\mathbb{Q}(-n)$ by  $\mathbb{Q}(0)$ for some integer  $n$ . The corresponding extension group is given by

$$\begin{eqnarray}\text{Ext}_{\mathsf{MHTS}}^{1}(\mathbb{Q}(-n),\mathbb{Q}(0))=\left\{\begin{array}{@{}ll@{}}\mathbb{C}/(2\unicode[STIX]{x1D70B}i)^{n}\mathbb{Q}\quad & \text{if}\;n>0,\\ 0\quad & \text{otherwise.}\end{array}\right.\end{eqnarray}$$

More concretely, the extension corresponding to a number  $z\in \mathbb{C}/(2\unicode[STIX]{x1D70B}i)^{n}\mathbb{Q}\,$ has a period matrix

$$\begin{eqnarray}\left(\begin{array}{@{}cc@{}}1 & z\\ 0 & (2\unicode[STIX]{x1D70B}i)^{n}\end{array}\right).\end{eqnarray}$$

We note that the higher extension groups vanish:  $\text{Ext}_{\mathsf{MHTS}}^{r}(H,H^{\prime })=0$ for  $r\geqslant 2$ and  $H$ ,  $H^{\prime }$ two mixed Hodge–Tate structures.

2.4 Mixed Tate motives over  $\mathbb{Q}$

Let  $\mathsf{DM}(\mathbb{Q})$ denote Voevodsky’s triangulated category of motives over  $\mathbb{Q}$  [Reference VoevodskyVoe00]. It is a  $\mathbb{Q}$ -linear triangulated tensor category whose objects can be described in terms of complexes of varieties and whose morphisms can be described in terms of algebraic cycles (in particular, in terms of Bloch’s higher Chow groups). There are invertible objects  $\mathbb{Q}(-n)\in \mathsf{DM}(\mathbb{Q})$ , where  $\mathbb{Q}(-1)$ is the reduced motive of the multiplicative group  $\mathbb{G}_{m}$ , shifted by  $-1$ (we work with cohomological conventions). The triangulated subcategory of  $\mathsf{DM}(\mathbb{Q})$ generated by these objects is denoted by  $\mathsf{DMT}(\mathbb{Q})$ . By using the relation between higher Chow groups and rational  $K$ -theory [Reference BlochBlo86] and Borel’s computation of the rational  $K$ -theory of number fields [Reference BorelBor77], Levine defined a natural  $t$ -structure on  $\mathsf{DMT}(\mathbb{Q})$  [Reference LevineLev93]. The heart of this  $t$ -structure is denoted by  $\mathsf{MT}(\mathbb{Q})$ and called the category of mixed Tate motives over  $\mathbb{Q}$ . It is a (neutral) tannakian  $\mathbb{Q}$ -linear category which contains the objects  $\mathbb{Q}(-n)$ .

There is a faithful and exact functor

(11) $$\begin{eqnarray}\mathsf{MT}(\mathbb{Q})\rightarrow \mathsf{MHTS}\end{eqnarray}$$

from  $\mathsf{MT}(\mathbb{Q})$ to the category  $\mathsf{MHTS}$ of mixed Hodge–Tate structures, which is called the Hodge realization functor ([Reference Deligne and GoncharovDG05, § 2.13], see also [Reference HuberHub00, Reference HuberHub04]). It sends the object  $\mathbb{Q}(-n)\in \mathsf{MT}(\mathbb{Q})$ to the object  $\mathbb{Q}(-n)\in \mathsf{MHTS}$ . Composing (11) with the fiber functors (9) gives the Betti and de Rham realization functors, still denoted by

(12) $$\begin{eqnarray}\unicode[STIX]{x1D714}_{\text{B}}:\mathsf{MT}(\mathbb{Q})\rightarrow \mathsf{Vect}_{\mathbb{Q}}\quad \text{and}\quad \unicode[STIX]{x1D714}_{\text{dR}}:\mathsf{MT}(\mathbb{Q})\rightarrow \mathsf{Vect}_{\mathbb{Q}},\end{eqnarray}$$

and we still have a comparison isomorphism (10). We note that any object in  $\mathsf{MT}(\mathbb{Q})$ is endowed with a canonical weight filtration  $W$ by subobjects such that the morphisms in  $\mathsf{MT}(\mathbb{Q})$ are strictly compatible with  $W$ . The realization morphisms are compatible with the weight filtrations.

The extension groups between the objects  $\mathbb{Q}(-n)$ are computed by the rational  $K$ -theory of  $\mathbb{Q}$  [Reference LevineLev93, § 4] and hence given, after Borel [Reference BorelBor77], by

(13) $$\begin{eqnarray}\text{Ext}_{\mathsf{MT}(\mathbb{Q})}^{1}(\mathbb{Q}(-n),\mathbb{Q}(0))=\left\{\begin{array}{@{}ll@{}}\bigoplus _{p\;\text{prime}}\mathbb{Q}\quad & \text{if}\;n=1,\\ \mathbb{Q}\quad & \text{if}\;n\;\text{is odd}\geqslant 3,\\ 0\quad & \text{otherwise.}\end{array}\right.\end{eqnarray}$$

As in the category  $\mathsf{MHTS}$ , the higher extension groups vanish in  $\mathsf{MT}(\mathbb{Q})$ . The morphisms

(14) $$\begin{eqnarray}\text{Ext}_{\mathsf{MT}(\mathbb{Q})}^{1}(\mathbb{Q}(-n),\mathbb{Q}(0))\longrightarrow \text{Ext}_{\mathsf{MHTS}}^{1}(\mathbb{Q}(-n),\mathbb{Q}(0))\cong \mathbb{C}/(2\unicode[STIX]{x1D70B}i)^{n}\mathbb{Q}\end{eqnarray}$$

induced by (11) are easy to describe. For  $n=1$ , the image of the direct summand indexed by a prime  $p$ is the line spanned by the class of the Kummer extension of parameter  $p$ , i.e., by  $\log (p)\in \mathbb{C}/(2\unicode[STIX]{x1D70B}i)\mathbb{Q}$ . For  $n\geqslant 3$ odd, the image is the line spanned by  $\unicode[STIX]{x1D701}(n)\in \mathbb{C}/(2\unicode[STIX]{x1D70B}i)^{n}\mathbb{Q}$ . Thus, the morphism (14) is injective for every  $n$ . This implies the following theorem [Reference Deligne and GoncharovDG05, Proposition 2.14].

This theorem is very helpful, since it allows one to compute in the category  $\mathsf{MT}(\mathbb{Q})$ with period matrices; in other words, a mixed Tate motive over  $\mathbb{Q}$ is uniquely determined by its period matrix.

2.5 Mixed Tate motives over  $\mathbb{Z}$

Let  $\mathsf{MT}(\mathbb{Z})$ denote the category of mixed Tate motives over  $\mathbb{Z}$ , as defined in [Reference Deligne and GoncharovDG05]. By definition, it is a full tannakian subcategory

$$\begin{eqnarray}\mathsf{MT}(\mathbb{Z}){\hookrightarrow}\mathsf{MT}(\mathbb{Q})\end{eqnarray}$$

of the category of mixed Tate motives over  $\mathbb{Q}$ , which contains the pure Tate motives  $\mathbb{Q}(-n)$ for every integer  $n$ . An object of $\mathsf{MT}(\mathbb{Q})$ is in $\mathsf{MT}(\mathbb{Z})$ if and only if it has no subquotient isomorphic to a non-split extension of $\mathbb{Q}(-n)$ by $\mathbb{Q}(-n+1)$ .

The extension groups in the category $\mathsf{MT}(\mathbb{Z})$ satisfy the following properties:

1. (1) $\text{Ext}_{\mathsf{MT}(\mathbb{Z})}^{1}(\mathbb{Q}(-1),\mathbb{Q}(0))=0$ ;

2. (2) the natural morphism $\text{Ext}_{\mathsf{MT}(\mathbb{Z})}^{1}(\mathbb{Q}(-n),\mathbb{Q}(0))\rightarrow \text{Ext}_{\mathsf{MT}(\mathbb{Q})}^{1}(\mathbb{Q}(-n),\mathbb{Q}(0))$ is an isomorphism for  $n\neq 1$ .

As in the categories  $\mathsf{MHTS}$ and  $\mathsf{MT}(\mathbb{Q})$ , the higher extension groups vanish in  $\mathsf{MT}(\mathbb{Z})$ .

For  $n\geqslant 3$ odd, there is an essentially unique non-trivial extension of  $\mathbb{Q}(-n)$ by  $\mathbb{Q}(0)$ in the category  $\mathsf{MT}(\mathbb{Q})$ , which actually lives in  $\mathsf{MT}(\mathbb{Z})$ . A period matrix for such an extension is

$$\begin{eqnarray}\left(\begin{array}{@{}cc@{}}1 & \unicode[STIX]{x1D701}(n)\\ 0 & (2\unicode[STIX]{x1D70B}i)^{n}\end{array}\right).\end{eqnarray}$$

Apart from the case  $n=3$ (see [Reference BrownBro16, Corollary 11.3] or Proposition 4.11 below), we do not know of any geometric construction of these extensions.

3.1 The definition

Let  $n\geqslant 1$ be an integer. In the affine  $n$ -space  $X_{n}=\mathbb{A}_{\mathbb{Q}}^{n}$ we consider the hypersurfaces

$$\begin{eqnarray}\displaystyle & A_{n}=\{x_{1}\cdots x_{n}=1\}\quad \text{and} & \displaystyle \nonumber\\ \displaystyle & B_{n}=\mathop{\bigcup }_{1\leqslant i\leqslant n}\{x_{i}=0\}\cup \mathop{\bigcup }_{1\leqslant i\leqslant n}\{x_{i}=1\}. & \displaystyle \nonumber\end{eqnarray}$$

The union  $A_{n}\cup B_{n}$ is almost a normal crossing divisor inside  $X_{n}$ : around the point  $P_{n}=$ $(1,\ldots ,1)$ , it looks like  $z_{1}\cdots z_{n}(z_{1}+\cdots +z_{n})=0$ (set  $x_{i}=\exp (z_{i})$ ). Let

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{n}:\widetilde{X}_{n}\rightarrow X_{n}\end{eqnarray}$$

be the blow-up at  $P_{n}$ , and  $E_{n}=\unicode[STIX]{x1D70B}_{n}^{-1}(P_{n})$ be the exceptional divisor. We denote respectively by  $\widetilde{A}_{n}$ and  $\widetilde{B}_{n}$ the strict transforms of  $A_{n}$ and  $B_{n}$ along  $\unicode[STIX]{x1D70B}_{n}$ . The union  $\widetilde{A}_{n}\cup \widetilde{B}_{n}\cup E_{n}$ is a simple normal crossing divisor inside  $\widetilde{X}_{n}$ .

There is an object  ${\mathcal{Z}}^{(n)}\in \mathsf{MT}(\mathbb{Q})$ , which we may abusively denote by

$$\begin{eqnarray}{\mathcal{Z}}^{(n)}=H^{n}(\widetilde{X}_{n}-\widetilde{A}_{n},(\widetilde{B}_{n}\cup E_{n})-(\widetilde{B}_{n}\cup E_{n})\cap \widetilde{A}_{n}),\end{eqnarray}$$

such that its Betti and de Rham realizations (12) are ( $?\in \{\text{B},\text{dR}\}$ )

$$\begin{eqnarray}{\mathcal{Z}}_{\,?}^{(n)}=H_{?}^{n}(\widetilde{X}_{n}-\widetilde{A}_{n},(\widetilde{B}_{n}\cup E_{n})-(\widetilde{B}_{n}\cup E_{n})\cap \widetilde{A}_{n}).\end{eqnarray}$$

We now give the precise definition of  ${\mathcal{Z}}^{(n)}$ , along the lines of [Reference GoncharovGon02, Proposition 3.6]. Let us write  $Y=\widetilde{X}_{n}-\widetilde{A}_{n}$ and  $\unicode[STIX]{x2202}Y=(\widetilde{B}_{n}\cup E_{n})-(\widetilde{B}_{n}\cup E_{n})\cap \widetilde{A}_{n}$ , viewed as schemes defined over  $\mathbb{Q}$ . We have a decomposition into smooth irreducible components  $\unicode[STIX]{x2202}Y=\bigcup _{i}\unicode[STIX]{x2202}_{i}Y$ , where  $i$ runs in a set of cardinality  $2n+1$ . For a set  $I=\{i_{1},\ldots ,i_{r}\}$ of indices, we denote by  $\unicode[STIX]{x2202}_{I}Y=\unicode[STIX]{x2202}_{i_{1}}Y\cap \cdots \cap \unicode[STIX]{x2202}_{i_{r}}Y$ the corresponding intersection; it is either empty or a smooth subvariety of  $X$ of codimension  $r$ .

We thus get an object

(15) $$\begin{eqnarray}\cdots \rightarrow \bigsqcup _{|I|=3}\unicode[STIX]{x2202}_{I}Y\rightarrow \bigsqcup _{|I|=2}\unicode[STIX]{x2202}_{I}Y\rightarrow \bigsqcup _{|I|=1}\unicode[STIX]{x2202}_{I}Y\rightarrow Y\rightarrow 0\end{eqnarray}$$

in Voevodsky’s triangulated category  $\mathsf{DM}(\mathbb{Q})$ , see § 2.4. The differentials are the alternating sums of the natural closed immersions. One readily checks that the complex (15) lives in the triangulated subcategory  $\mathsf{DMT}(\mathbb{Q})$ . By definition, the object  ${\mathcal{Z}}^{(n)}$ in  $\mathsf{MT}(\mathbb{Q})$ is the  $n$ th cohomology group of the complex (15) with respect to the  $t$ -structure.

Remark 3.2. For  $n=1$ , the blow-up map  $\unicode[STIX]{x1D70B}_{1}:\widetilde{X}_{1}\rightarrow X_{1}$ is an isomorphism and  $\widetilde{A}_{1}=\varnothing$ , so that we get  ${\mathcal{Z}}^{(1)}=H^{1}(\mathbb{A}_{\mathbb{Q}}^{1},\{0,1\})$ . We have a long exact sequence in relative cohomology

$$\begin{eqnarray}0\rightarrow H^{0}(\mathbb{A}_{\mathbb{ Q}}^{1},\{0,1\})\rightarrow H^{0}(\mathbb{A}_{\mathbb{ Q}}^{1})\rightarrow H^{0}(\{0\})\oplus H^{0}(\{1\})\rightarrow {\mathcal{Z}}^{(1)}\rightarrow 0,\end{eqnarray}$$

which shows that  $H^{0}(\mathbb{A}_{\mathbb{Q}}^{1},\{0,1\})=0$ and that we have an isomorphism  ${\mathcal{Z}}^{(1)}\simeq \mathbb{Q}(0)$ .

3.2 Betti and de Rham realizations, 1

We now give a first description of the Betti and de Rham realizations of the zeta motive  ${\mathcal{Z}}^{(n)}$ .

We let  $C_{\bullet }$ denote the functor which assigns to a topological space the complex of singular chains with rational coefficients. By definition, the dual of the Betti realization  ${\mathcal{Z}}_{\text{B}}^{(n),\vee }$ is the  $n$ th homology group of the total complex of the double complex

(16)

obtained by applying the functor  $C_{\bullet }$ to the complex (15). One readily verifies that this complex is quasi-isomorphic to the quotient complex  $C_{\bullet }(Y(\mathbb{C}))/C_{\bullet }(\unicode[STIX]{x2202}Y(\mathbb{C}))$ , classically used to define the relative homology groups  $H_{\bullet }^{\text{B}}(Y,\unicode[STIX]{x2202}Y)=H_{\bullet }^{\text{sing}}(Y(\mathbb{C}),\unicode[STIX]{x2202}Y(\mathbb{C}))$ .

We let  $\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x2202}_{I}Y}^{\bullet }$ denote the complex of sheaves of algebraic differential forms on the smooth variety  $\unicode[STIX]{x2202}_{I}Y$ , extended by zero to  $Y$ . By definition, the de Rham realization  ${\mathcal{Z}}_{\text{dR}}^{(n)}$ is the hypercohomology of the total complex of the double complex of sheaves

(17)

where the vertical arrows are the exterior derivatives and the horizontal arrows are the alternating sums of the natural restriction maps as in the complex (15).

The comparison morphism between the Betti and de Rham realizations of  ${\mathcal{Z}}^{(n)}$ is induced, after complexification, by the morphism from the double complex (17) to the double complex (16) given by integration. Note that one first has to replace (16) by the double complex of sheaves of singular cochains.

3.3 Betti and de Rham realizations, 2

We now give descriptions of the Betti and de Rham realizations of  ${\mathcal{Z}}^{(n)}$ that allow one to work directly in the affine space  $X_{n}$ and do not require to work in the blow-up  $\widetilde{X}_{n}$ . The justification of the blow-up process goes as follows. Suppose that one wants to find a motive whose periods include all absolutely convergent integrals of the form

(18) $$\begin{eqnarray}\int _{[0,1]^{n}}\frac{P(x_{1},\ldots ,x_{n})}{(1-x_{1}\cdots x_{n})^{N}}\,dx_{1}\cdots dx_{n},\end{eqnarray}$$

where  $P(x_{1},\ldots ,x_{n})$ is a polynomial with rational coefficients, and  $N\geqslant 0$ is an integer. On the Betti side, we note that the boundary of  $[0,1]^{n}$ intersects the divisor  $A_{n}(\mathbb{C})$ of poles of the differential forms at the point  $P_{n}(\mathbb{C})$ . The blow-up process is thus required in order to have a class that represents the integration domain. On the de Rham side, the blow-up process is required in order to only consider absolutely convergent integrals of the form (18). This is made precise by Propositions 3.4 and 3.6 below.

We start with the Betti realization. Let us write  ${A\unicode[STIX]{x0030A}}_{n}=A_{n}-P_{n}$ and note that this is not a closed subset, but only a locally closed subset, of  $X_{n}$ .

Proposition 3.4. The blow-up morphism  $\unicode[STIX]{x1D70B}_{n}:\widetilde{X}_{n}\rightarrow X_{n}$ induces an isomorphism

$$\begin{eqnarray}{\mathcal{Z}}_{\text{B}}^{(n),\vee }\stackrel{\cong }{\longrightarrow }H_{n}^{\text{sing}}(X_{n}(\mathbb{C})-{A\unicode[STIX]{x0030A}}_{n}(\mathbb{C}),B_{n}(\mathbb{C})-B_{n}(\mathbb{C})\cap {A\unicode[STIX]{x0030A}}_{n}(\mathbb{C})).\end{eqnarray}$$

As a consequence of Proposition 3.4, we see that the unit  $n$ -square  $\Box ^{n}=[0,1]^{n}\subset X_{n}(\mathbb{C})-{A\unicode[STIX]{x0030A}}_{n}(\mathbb{C})$ defines a class

$$\begin{eqnarray}[\Box ^{n}]\in {\mathcal{Z}}_{\text{B}}^{(n),\vee }.\end{eqnarray}$$

When viewed in  $\widetilde{X}_{n}(\mathbb{C})-\widetilde{A}_{n}(\mathbb{C})$ , it is the class of the strict transform  $\widetilde{\Box }^{n}$ , which has the combinatorial structure of an  $n$ -cube truncated at one of its vertices.

We now turn to a description of the de Rham realization of  ${\mathcal{Z}}^{(n)}$ . Instead of giving a general description in terms of algebraic differential forms on  $X_{n}-A_{n}$ , we will only give a way of defining many classes in  ${\mathcal{Z}}_{\text{dR}}^{(n)}$ , which will turn out to be enough for our purposes.

Definition 3.5. An algebraic differential  $n$ -form on  $X_{n}-A_{n}$ is said to be integrable if it can be written as a linear combination of forms of the type

(19) $$\begin{eqnarray}\unicode[STIX]{x1D714}={\displaystyle \frac{(1-x_{1})^{v_{1}-1}\cdots (1-x_{n})^{v_{n}-1}f(x_{1},\ldots ,x_{n})}{(1-x_{1}\cdots x_{n})^{N}}}\,dx_{1}\cdots dx_{n}\end{eqnarray}$$

with  $v_{1},\ldots ,v_{n}\geqslant 1$ and  $N\geqslant 0$ integers such that  $v_{1}+\cdots +v_{n}\geqslant N+1$ , and  $f(x_{1},\ldots ,x_{n})$ a polynomial with rational coefficients.

The terminology is justified by the following proposition.

Proposition 3.6. Let  $\unicode[STIX]{x1D714}$ be an algebraic differential  $n$ -form on  $X_{n}-A_{n}$ . If  $\unicode[STIX]{x1D714}$ is integrable, then  $\unicode[STIX]{x1D70B}_{n}^{\ast }(\unicode[STIX]{x1D714})$ does not have a pole along  $E_{n}$ , and thus defines a class in  ${\mathcal{Z}}_{\text{dR}}^{(n)}$ . In particular, the integral

$$\begin{eqnarray}\int _{\widetilde{\Box }^{n}}\unicode[STIX]{x1D70B}_{n}^{\ast }(\unicode[STIX]{x1D714})=\int _{\Box ^{n}}\unicode[STIX]{x1D714}\end{eqnarray}$$

is absolutely convergent and is a period of  ${\mathcal{Z}}^{(n)}$ .

Proof. We write  $\unicode[STIX]{x1D714}$ as in (19). We note that the only problem for absolute convergence is around the point  $(1,\ldots ,1)$ . Let us thus make the change of variables  $y_{i}=1-x_{i}$ for  $i=1,\ldots ,n$ , and  $g(y_{1},\ldots ,y_{n})=(-1)^{n}\,f(x_{1},\ldots ,x_{n})$ . We write  $h(y_{1},\ldots ,y_{n})=1-(1-y_{1})\cdots (1-y_{n})$ so that we have

$$\begin{eqnarray}\unicode[STIX]{x1D714}={\displaystyle \frac{y_{1}^{v_{1}-1}\cdots y_{n}^{v_{n}-1}g(y_{1},\ldots ,y_{n})}{h(y_{1},\ldots ,y_{n})^{N}}}\,dy_{1}\cdots dy_{n}.\end{eqnarray}$$

There are  $n$ natural affine charts for the blow-up  $\unicode[STIX]{x1D70B}_{n}:\widetilde{X}_{n}\rightarrow X_{n}$ of the point  $(0,\ldots ,0)$ , and by symmetry it is enough to work in the first one. We then have local coordinates  $(z_{1},\ldots ,z_{n})$ on  $\widetilde{X}_{n}$ , which are linked to the coordinates  $(y_{1},\ldots ,y_{n})=\unicode[STIX]{x1D70B}_{n}(z_{1},\ldots ,z_{n})$ by the formula

$$\begin{eqnarray}(y_{1},\ldots ,y_{n})=(z_{1},z_{1}z_{2},\ldots ,z_{1}z_{n}).\end{eqnarray}$$

The problem of convergence occurs in the neighborhood of the exceptional divisor  $E_{n}$ , which is defined by the equation  $z_{1}=0$ . Since  $h(0,\ldots ,0)=0$ , we may write

$$\begin{eqnarray}h(z_{1},z_{1}z_{2},\ldots ,z_{1}z_{n})=z_{1}\,\widetilde{h}(z_{1},\ldots ,z_{n})\end{eqnarray}$$

with  $\widetilde{h}(z_{1},\ldots ,z_{n})$ a polynomial such that  $\widetilde{h}(0,\ldots ,0)=1$ . The strict transform  $\widetilde{A}_{n}$ of  $A_{n}$ is thus defined by the equation  $\widetilde{h}(z_{1},\ldots ,z_{n})=0$ . We note that we have  $dy_{1}\cdots dy_{n}=z_{1}^{n-1}dz_{1}\cdots dz_{n}$ , so that we can write

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{n}^{\ast }(\unicode[STIX]{x1D714})={\displaystyle \frac{z_{1}^{v_{1}-1}(z_{1}z_{2})^{v_{2}-1}\cdots (z_{1}z_{n})^{v_{n}-1}g(z_{1},z_{1}z_{2},\ldots ,z_{1}z_{n})}{z_{1}^{N}\widetilde{h}(z_{1},\ldots ,z_{n})^{N}}}\,z_{1}^{n-1}\,dz_{1}\cdots dz_{n}=z_{1}^{v_{1}+\cdots +v_{n}-N-1}\unicode[STIX]{x1D6FA},\end{eqnarray}$$

where  $\unicode[STIX]{x1D6FA}$ has a pole along  $\widetilde{A}_{n}$ but not along  $E_{n}$ . The claim follows.◻

We make an abuse of notation and denote by

$$\begin{eqnarray}[\unicode[STIX]{x1D714}]\in {\mathcal{Z}}_{\text{dR}}^{(n)}\end{eqnarray}$$

the class of the pullback  $\unicode[STIX]{x1D70B}_{n}^{\ast }(\unicode[STIX]{x1D714})$ for  $\unicode[STIX]{x1D714}$ integrable, so that the comparison isomorphism reads

$$\begin{eqnarray}\langle [\Box ^{n}],[\unicode[STIX]{x1D714}]\rangle =\int _{\Box ^{n}}\unicode[STIX]{x1D714}.\end{eqnarray}$$

We note the converse of Proposition 3.6, which we will not use.

Proof. In the coordinates  $y_{i}=1-x_{i}$ , we write

$$\begin{eqnarray}\unicode[STIX]{x1D714}={\displaystyle \frac{P(y_{1},\ldots ,y_{n})}{h(y_{1},\ldots ,y_{n})^{N}}}\,dy_{1}\cdots dy_{n}\end{eqnarray}$$

with  $P(y_{1},\ldots ,y_{n})$ a polynomial with rational coefficients. If the integral  $\int _{\Box ^{n}}\unicode[STIX]{x1D714}$ is absolutely convergent in the neighborhood of the point  $(0,\ldots ,0)$ , then after the change of variables

$$\begin{eqnarray}\unicode[STIX]{x1D719}(z_{1},\ldots ,z_{n})=(z_{1},z_{1}z_{2},\ldots ,z_{1}z_{n})\end{eqnarray}$$

we get an absolutely convergent integral in the neighborhood of  $z_{1}=0$ . We write, as in the proof of Proposition 3.6,

$$\begin{eqnarray}\unicode[STIX]{x1D719}^{\ast }(\unicode[STIX]{x1D714})={\displaystyle \frac{P(z_{1},z_{1}z_{2},\ldots ,z_{1}z_{n})}{z_{1}^{N-n+1}\widetilde{h}(z_{1},\ldots ,z_{n})^{N}}}\,dz_{1}\cdots dz_{n}.\end{eqnarray}$$

Let us write

$$\begin{eqnarray}P(y_{1},\ldots ,y_{n})=\mathop{\sum }_{\text{}\underline{a}}\unicode[STIX]{x1D706}_{\text{}\underline{a}}\,y_{1}^{a_{1}-1}\cdots y_{n}^{a_{n}-1}\end{eqnarray}$$

with  $\unicode[STIX]{x1D706}_{\text{}\underline{a}}\in \mathbb{Q}$ for every multi-index  $\text{}\underline{a}=(a_{1},\ldots ,a_{n})$ . We then have

$$\begin{eqnarray}P(z_{1},z_{1}z_{2},\ldots ,z_{1}z_{n})=\mathop{\sum }_{\text{}\underline{a}}\unicode[STIX]{x1D706}_{\text{}\underline{a}}\,z_{1}^{a_{1}+\cdots +a_{n}-n}z_{2}^{a_{2}-1}\cdots z_{n}^{a_{n}-1}.\end{eqnarray}$$

Let  $v$ denote the smallest integer such that there exists a multi-index  $\text{}\underline{a}$ with  $|\text{}\underline{a}|:=a_{1}+\cdots +a_{n}=v$ . We then have an equivalence

$$\begin{eqnarray}P(z_{1},z_{1}z_{2},\ldots ,z_{1}z_{n}){\sim}_{z_{1}\rightarrow 0}z_{1}^{v-n}Q(z_{2},\ldots ,z_{n}),\end{eqnarray}$$

where  $Q(z_{2},\ldots ,z_{n})=\sum _{|\text{}\underline{a}|=v}\unicode[STIX]{x1D706}_{\text{}\underline{a}}\,z_{2}^{a_{2}-1}\cdots z_{n}^{a_{n}-1}$ . We also have the equivalence

$$\begin{eqnarray}\widetilde{h}(z_{1},\ldots ,z_{n}){\sim}_{z_{1}\rightarrow 0}1+z_{2}+\cdots +z_{n},\end{eqnarray}$$

from which we deduce

$$\begin{eqnarray}\unicode[STIX]{x1D719}^{\ast }(\unicode[STIX]{x1D714}){\sim}_{z_{1}\rightarrow 0}z_{1}^{v-N-1}\,dz_{1}{\displaystyle \frac{Q(z_{2},\ldots ,z_{n})}{(1+z_{2}+\cdots +z_{n})^{N}}}\,dz_{2}\cdots dz_{n}.\end{eqnarray}$$

This gives an absolutely convergent integral in the neighborhood of  $z_{1}=0$ if and only if  $v\geqslant N+1$ , which is exactly the integrability condition.◻

3.4 The Eulerian differential forms

Recall that the family of Eulerian polynomials  $E_{r}(x)$ ,  $r\geqslant 0$ , is defined by the equation

(20) $$\begin{eqnarray}\frac{E_{r}(x)}{(1-x)^{r+1}}=\mathop{\sum }_{j\geqslant 0}(j+1)^{r}x^{j}.\end{eqnarray}$$

We refer to [Reference FoataFoa10] for a survey on Eulerian polynomials. If  $r\geqslant 1$ , then (20) is equivalent to

$$\begin{eqnarray}\frac{E_{r}(x)}{(1-x)^{r+1}}=\frac{1}{x}\biggl(x\frac{d}{dx}\biggr)^{r}\frac{1}{1-x}.\end{eqnarray}$$

For instance, we have  $E_{0}(x)=E_{1}(x)=1$ ,  $E_{2}(x)=1+x$ ,  $E_{3}(x)=1+4x+x^{2}$ . The Eulerian polynomials satisfy the recurrence relation

(21) $$\begin{eqnarray}E_{r+1}(x)=x(1-x)E_{r}^{\prime }(x)+(1+rx)E_{r}(x).\end{eqnarray}$$

For integers  $n\geqslant 2$ and  $k=2,\ldots ,n$ , we define a differential form

$$\begin{eqnarray}\unicode[STIX]{x1D714}_{k}^{(n)}={\displaystyle \frac{E_{n-k}(x_{1}\cdots x_{n})}{(1-x_{1}\cdots x_{n})^{n-k+1}}}\,dx_{1}\cdots dx_{n}.\end{eqnarray}$$

Note that we have  $\unicode[STIX]{x1D714}_{n}^{(n)}=dx_{1}\cdots dx_{n}/(1-x_{1}\cdots x_{n})$ .

Lemma 3.8. For  $k=2,\ldots ,n$ , the form  $\unicode[STIX]{x1D714}_{k}^{(n)}$ defines a class  $[\unicode[STIX]{x1D714}_{k}^{(n)}]\in {\mathcal{Z}}_{\text{dR}}^{(n)}$ and we have

(22) $$\begin{eqnarray}\langle [\Box ^{n}],[\unicode[STIX]{x1D714}_{k}^{(n)}]\rangle =\int _{\Box ^{n}}\unicode[STIX]{x1D714}_{k}^{(n)}=\unicode[STIX]{x1D701}(k).\end{eqnarray}$$

Proof. The first statement follows from Proposition 3.6. The computation of the period is then straightforward using the definition (20) of the Eulerian polynomials:

$$\begin{eqnarray}\int _{\Box ^{n}}\unicode[STIX]{x1D714}_{k}^{(n)}=\mathop{\sum }_{j\geqslant 0}(j+1)^{n-k}\int _{[0,1]^{n}}(x_{1}\cdots x_{n})^{j}\,dx_{1}\cdots dx_{n}=\mathop{\sum }_{j\geqslant 0}(j+1)^{-k}=\unicode[STIX]{x1D701}(k).\Box\end{eqnarray}$$

For every  $n\geqslant 0$ , we define  $\unicode[STIX]{x1D714}_{0}^{(n)}=dx_{1}\cdots dx_{n}$ ; we also have the class  $[\unicode[STIX]{x1D714}_{0}^{(n)}]\in {\mathcal{Z}}_{n,\text{dR}}$ , whose pairing with the class  $[\Box ^{n}]$ is

$$\begin{eqnarray}\langle [\Box ^{n}],[\unicode[STIX]{x1D714}_{0}^{(n)}]\rangle =\int _{\Box ^{n}}\unicode[STIX]{x1D714}_{0}^{(n)}=1.\end{eqnarray}$$

We call the differential forms  $\unicode[STIX]{x1D714}_{k}^{(n)}$ , for  $k=0,2,\ldots ,n$ , the Eulerian differential forms.

3.5 An inductive system

For  $n\geqslant 2$ there are natural morphisms

(23) $$\begin{eqnarray}i^{(n)}:{\mathcal{Z}}^{(n-1)}\rightarrow {\mathcal{Z}}^{(n)}\end{eqnarray}$$

in the category  $\mathsf{MT}(\mathbb{Q})$ , that we now define. We fix the identification  $X_{n-1}=\{x_{n}=1\}\subset X_{n}$ , which implies the equality  $A_{n-1}=A_{n}\cap X_{n-1}$ . Let us set

$$\begin{eqnarray}B_{n}^{\prime }=\mathop{\bigcup }_{1\leqslant i\leqslant n}\{x_{i}=0\}\cup \mathop{\bigcup }_{1\leqslant i\leqslant n-1}\{x_{i}=1\},\end{eqnarray}$$

so that we have  $B_{n}=B_{n}^{\prime }\cup X_{n-1}$ , and  $B_{n-1}=B_{n}^{\prime }\cap X_{n-1}$ .

In the blow-up  $\widetilde{X}_{n}$ , we thus get an embedding  $\widetilde{X}_{n-1}\subset \widetilde{X}_{n}$ and identifications  $\widetilde{A}_{n-1}=\widetilde{A}_{n}\cap \widetilde{X}_{n-1}$ ,  $\widetilde{B}_{n-1}=\widetilde{B}_{n}^{\prime }\cap \widetilde{X}_{n-1}$ and  $E_{n-1}=E_{n}\cap \widetilde{X}_{n-1}$ . Thus, the complex in  $\mathsf{DM}(\mathbb{Q})$ that we have used to define  ${\mathcal{Z}}^{(n-1)}$ is the subcomplex

(24) $$\begin{eqnarray}\cdots \rightarrow \bigsqcup _{\substack{ |I|=3 \\ \unicode[STIX]{x2202}_{I}Y\subset \widetilde{X}_{n-1}}}\unicode[STIX]{x2202}_{I}Y\rightarrow \bigsqcup _{\substack{ |I|=2 \\ \unicode[STIX]{x2202}_{I}Y\subset \widetilde{X}_{n-1}}}\unicode[STIX]{x2202}_{I}Y\rightarrow \widetilde{X}_{n-1}\rightarrow 0\rightarrow 0\end{eqnarray}$$

of the complex (15) that we have used to define  ${\mathcal{Z}}^{(n)}$ , shifted by  $1$ . Taking the  $n$ th cohomology groups with respect to the  $t$ -structure gives the morphism (23).

In the Betti and the de Rham realizations, the morphism (23) is also induced by the inclusion of double subcomplexes of (16) and (17).

We define the ind-motive

$$\begin{eqnarray}{\mathcal{Z}}=\lim _{\stackrel{\longrightarrow }{n}}{\mathcal{Z}}^{(n)},\end{eqnarray}$$

viewed as an ind-object in the category  $\mathsf{MT}(\mathbb{Q})$ , and simply call it the zeta motive.

The map  $i_{\text{B}}^{(n),\vee }:{\mathcal{Z}}_{\text{B}}^{(n),\vee }\rightarrow {\mathcal{Z}}_{\text{B}}^{(n-1),\vee }$ given by the transpose of the Betti realization of  $i^{(n)}$ satisfies

(25) $$\begin{eqnarray}i_{\text{B}}^{(n),\vee }([\Box ^{n-1}])=[\Box ^{n}].\end{eqnarray}$$

More generally and loosely speaking, if  $\unicode[STIX]{x1D70E}$ is a chain on  $\widetilde{X}_{n}(\mathbb{C})-\widetilde{A}_{n}(\mathbb{C})$ whose boundary is on  $\widetilde{B}_{n}(\mathbb{C})\cup E_{n}(\mathbb{C})$ , then  $i_{\text{B}}^{(n),\vee }([\unicode[STIX]{x1D70E}])$ is the class of ‘the component of the boundary of  $\unicode[STIX]{x1D70E}$ that lives on  $\widetilde{X}_{n-1}(\mathbb{C})$ ’. According to Proposition 3.4, one can also work with chains on  $X_{n}(\mathbb{C})-{A\unicode[STIX]{x0030A}}_{n}(\mathbb{C})$ . We note that (25) allows us to define a class

$$\begin{eqnarray}[\Box ]\in {\mathcal{Z}}_{\text{B}}^{\vee }:=\lim _{\stackrel{\longleftarrow }{n}}{\mathcal{Z}}_{\text{B}}^{(n),\vee }.\end{eqnarray}$$

The next proposition shows that the Eulerian differential forms  $\unicode[STIX]{x1D714}_{k}^{(n)}$ are compatible with the inductive structure on the zeta motives.

Proof. Since all the differential forms that we are manipulating have no poles along the exceptional divisors  $E_{n-1}$ and  $E_{n}$ , it is safe to do the computations in the affine spaces  $X_{n-1}$ and  $X_{n}$ ; we leave it to the reader to turn them into computations in  $\widetilde{X}_{n-1}$ and  $\widetilde{X}_{n}$ by working in local charts as in the proof of Proposition 3.6. Let us first assume that  $k\in \{2,\ldots ,n-1\}$ . We put

$$\begin{eqnarray}\unicode[STIX]{x1D702}_{k}^{(n-1)}={\displaystyle \frac{x_{n}E_{n-1-k}(x_{1}\cdots x_{n})}{(1-x_{1}\cdots x_{n})^{n-k}}}\,dx_{1}\cdots dx_{n-1},\end{eqnarray}$$

viewed as a form on  $X_{n}$ . Then we have  $(\unicode[STIX]{x1D702}_{k}^{(n-1)})_{|X_{n-1}}=\unicode[STIX]{x1D714}_{k}^{(n-1)}$ and  $(\unicode[STIX]{x1D702}_{k}^{(n-1)})_{|B_{n-1}^{\prime }}=0$ . A diagram chase in the double complex (17) shows that  $i_{\text{dR}}^{(n)}([\unicode[STIX]{x1D714}_{k}^{(n-1)}])$ is the class of 

$$\begin{eqnarray}(-1)^{n-1}\,(d(\unicode[STIX]{x1D702}_{k}^{(n-1)}))\end{eqnarray}$$

(the sign is here to be consistent with the Betti version, see Remark 3.9). We have

$$\begin{eqnarray}(-1)^{n-1}\,d(\unicode[STIX]{x1D702}_{k}^{(n-1)})=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{n}}\biggl({\displaystyle \frac{x_{n}E_{n-1-k}(x_{1}\cdots x_{n})}{(1-x_{1}\cdots x_{n})^{n-k}}}\biggr)\,dx_{1}\cdots dx_{n}\end{eqnarray}$$

and one easily sees that setting  $x=x_{1}\cdots x_{n}$ we have

$$\begin{eqnarray}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{n}}\biggl({\displaystyle \frac{x_{n}E_{n-1-k}(x_{1}\cdots x_{n})}{(1-x_{1}\cdots x_{n})^{n-k}}}\biggr)={\displaystyle \frac{x(1-x)E_{n-1-k}^{\prime }(x)+(1+(n-1-k)x)E_{n-1-k}(x)}{(1-x)^{n-k+1}}}.\end{eqnarray}$$

Using the recurrence relation (21), one then concludes that

$$\begin{eqnarray}(-1)^{n-1}d(\unicode[STIX]{x1D702}_{k}^{(n-1)})={\displaystyle \frac{E_{n-k}(x_{1}\cdots x_{n})}{(1-x_{1}\cdots x_{n})^{n-k+1}}}\,dx_{1}\cdots dx_{n}=\unicode[STIX]{x1D714}_{k}^{(n)}.\end{eqnarray}$$

For  $k=0$ , this is the same computation with  $\unicode[STIX]{x1D702}_{0}^{(n)}=x_{n}\,dx_{1}\cdots dx_{n-1}$ and

$$\begin{eqnarray}(-1)^{n-1}\,d(\unicode[STIX]{x1D702}_{0}^{(n-1)})=dx_{1}\cdots dx_{n}=\unicode[STIX]{x1D714}_{0}^{(n)}.\Box\end{eqnarray}$$

Proposition 3.10 allows us to unambiguously define classes

$$\begin{eqnarray}[\unicode[STIX]{x1D714}_{k}]\in {\mathcal{Z}}_{\text{dR}}\end{eqnarray}$$

for  $k=0,2,3,\ldots ,$ whose pairing with the class  $[\Box ]\in {\mathcal{Z}}_{\text{B}}^{\vee }$ is

$$\begin{eqnarray}\langle [\Box ],[\unicode[STIX]{x1D714}_{0}]\rangle =1\quad \text{and}\quad \langle [\Box ],[\unicode[STIX]{x1D714}_{k}]\rangle =\unicode[STIX]{x1D701}(k)\;\;(k\geqslant 2).\end{eqnarray}$$

Remark 3.11. The proof of Proposition 3.10 can be thought of as a cohomological version of the relation

$$\begin{eqnarray}\int _{\Box ^{n}}\unicode[STIX]{x1D714}_{k}^{(n)}=\int _{\Box ^{n-1}}\unicode[STIX]{x1D714}_{k}^{(n-1)},\end{eqnarray}$$

which may be proved using Stokes’s theorem and the recurrence relation (21).

Proof. For  $k=0$ , Proposition 3.10 and the fact that the maps  $i_{\text{dR}}^{(n)}$ are compatible with the weight gradings implies that it is enough to do the proof for  $n=1$ ; this case is easy since  ${\mathcal{Z}}^{(1)}\cong \mathbb{Q}(0)$ only has weight  $0$ . We now turn to the case  $k=2,\ldots ,n$ . Thanks to Proposition 3.10 and the fact that the maps  $i_{\text{dR}}^{(n)}$ are compatible with the weight gradings, it is enough to check it for  $k=n$ .

By (7), we need to prove that the class of  $\unicode[STIX]{x1D714}_{n}^{(n)}$ is in  $F^{n}{\mathcal{Z}}_{\text{dR}}^{(n)}$ . Let  $Y$ be a smooth projective variety of dimension  $n$ ,  $D$ be a normal crossing divisor inside  $Y$ , and  $Z$ be a closed subvariety of  $Y$ of dimension ${\leqslant}n-1$ . Then we have

$$\begin{eqnarray}F^{n}H_{\text{dR}}^{n}(Y-D,Z-Z\cap D)=\text{Im}(H^{0}(\unicode[STIX]{x1D6FA}_{Y}^{n}(\log D))\longrightarrow H_{\text{dR}}^{n}(Y-D,Z-Z\cap D)).\end{eqnarray}$$

Thus, it is enough to prove that there is a compactification  $Y$ of  $\widetilde{X}_{n}-\widetilde{A}_{n}$ such that  $Y-(\widetilde{X}_{n}-\widetilde{A}_{n})$ is a normal crossing divisor  $D$ , and such that  $\unicode[STIX]{x1D714}_{n}^{(n)}$ has logarithmic singularities along  $D$ . Since  $\unicode[STIX]{x1D714}_{n}^{(n)}$ does not have poles along  $E_{n}$ , we can work on  $X_{n}-A_{n}$ instead. Let us start with  $Y_{1}=(\mathbb{P}^{1})^{n}$ with coordinates  $((x_{1}:y_{1}),\ldots ,(x_{n}:y_{n}))$ , and  $D_{1}$ the divisor given by the union of the subvarieties  $\{y_{i}=0\}$ for  $i=1,\ldots ,n$ , and the subvariety  $\{x_{1}\cdots x_{n}=y_{1}\cdots y_{n}\}$ (this is the closure of  $A_{n}$ ). This is not enough since $D_{1}$ is not a normal crossing divisor. We then finish thanks to the following lemma.◻

Proof. This is checked locally on the standard affine cover of  $Y$ , consisting of  $2^{n}$ affine spaces. By symmetry of the variables, it is enough to look at the charts  $U_{r}$ with affine coordinates

$$\begin{eqnarray}(y_{1},\ldots ,y_{r},x_{r+1},\ldots ,x_{n}),\end{eqnarray}$$

for  $r=0,\ldots ,n$ . We note that in the chart  $U_{r}$ the divisor  $D_{1}$ is the union of the subvarieties  $\{y_{i}=0\}$ for  $i=1,\ldots ,r$ , and the subvariety  $\{y_{1}\cdots y_{r}=x_{r+1}\cdots x_{n}\}$ . In that chart the differential form that we are looking at is (up to a sign)

$$\begin{eqnarray}\unicode[STIX]{x1D714}={\displaystyle \frac{dy_{1}\cdots dy_{r}\,dx_{r+1}\cdots dx_{n}}{y_{1}\cdots y_{r}(y_{1}\cdots y_{r}-x_{r+1}\cdots x_{n})}}.\end{eqnarray}$$

We proceed by induction on  $r$ . For  $r=0$ ,  $D_{1}$ only consists of  $\{x_{1}\cdots x_{n}=1\}$ , which is a normal crossing divisor, and  $\unicode[STIX]{x1D714}$ has logarithmic singularities along  $D_{1}$ . The subvarieties  $Z_{i,j}$ do not intersect  $U_{0}$ , so the blow-ups do not change anything. For a given  $r=1,\ldots ,n$ , let us look at the blow-up of  $Z_{1,n}=\{y_{1}=x_{n}=0\}$ (this is enough for reasons of symmetry) in the chart  $U_{r}$ . There are two natural affine charts  $\mathbb{A}^{n}\rightarrow U_{r}$ for the blow-up.

1. (i) On the first chart, the blow-up map is given by

   $$\begin{eqnarray}(y_{1},\ldots ,y_{r},x_{r+1},\ldots ,x_{n})=(v_{1},\ldots ,v_{r},u_{r+1},\ldots ,u_{n-1},v_{1}u_{n}).\end{eqnarray}$$
   The preimage of  $D_{1}$ consists of the subvarieties  $\{v_{i}=0\}$ for  $i=1,\ldots ,r$ , and the subvariety  $\{v_{2}\cdots v_{r}=u_{r+1}\cdots u_{n}\}$ . The pullback of  $\unicode[STIX]{x1D714}$ is
   $$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D714}}=\frac{dv_{1}\cdots dv_{r}\,du_{r+1}\cdots du_{n}}{v_{1}\cdots v_{r}(v_{2}\cdots v_{r}-u_{r+1}\cdots u_{n})}=\frac{dv_{1}}{v_{1}}\wedge \unicode[STIX]{x1D714}^{\prime }\quad \text{with }\unicode[STIX]{x1D714}^{\prime }=\frac{dv_{2}\cdots dv_{r}\,du_{r+1}\cdots du_{n}}{v_{2}\cdots v_{r}(v_{2}\cdots v_{r}-u_{r+1}\cdots u_{n})}.\end{eqnarray}$$
   We note that  $\{v_{1}=0\}$ is the exceptional divisor and that the total transforms of the subvarieties  $Z_{1,j}$ are empty in this chart. By the induction hypothesis (with $n$ replaced by $n-1$ ), the pullback of $\unicode[STIX]{x1D714}^{\prime }$ by the successive blow-up of the subvarieties $Z_{i,j}$ with $i\neq 1$ has logarithmic singularities. Since $dv_{1}/v_{1}$ has logarithmic singularities along $\{v_{1}=0\}$ , we are done.

2. (ii) On the second chart, the blow-up map is given by

   $$\begin{eqnarray}(y_{1},\ldots ,y_{r},x_{r+1},\ldots ,x_{n})=(v_{1}u_{n},v_{2},\ldots ,v_{r},u_{r+1},\ldots ,u_{n}).\end{eqnarray}$$
   The same argument as in the first chart applies. ◻

3.6 A long exact sequence

We now show that the morphism  $i^{(n)}:{\mathcal{Z}}^{(n-1)}\rightarrow {\mathcal{Z}}^{(n)}$ fits into a long exact sequence. We first define objects of  $\mathsf{MT}(\mathbb{Q})$ :

$$\begin{eqnarray}{\mathcal{Z}}^{(n),r}=H^{r}(\widetilde{X}_{n}-\widetilde{A}_{n},(\widetilde{B}_{n}\cup E_{n})-(\widetilde{B}_{n}\cup E_{n})\cap \widetilde{A}_{n})\end{eqnarray}$$

and

$$\begin{eqnarray}^{\prime }{\mathcal{Z}}^{(n),r}=H^{r}(\widetilde{X}_{n}-\widetilde{A}_{n},(\widetilde{B}_{n}^{\prime }\cup E_{n})-(\widetilde{B}_{n}^{\prime }\cup E_{n})\cap \widetilde{A}_{n}),\end{eqnarray}$$

so that  ${\mathcal{Z}}^{(n)}={\mathcal{Z}}^{(n),n}$ . We leave it to the reader to fill in the technical definitions of these objects by mimicking that of  ${\mathcal{Z}}^{(n)}$ from § 3.1.

Proposition 3.14. For  $n\geqslant 2$ , we have a long exact sequence in  $\mathsf{MT}(\mathbb{Q})$ :

(26) $$\begin{eqnarray}\cdots \rightarrow {\mathcal{Z}}^{(n-1),r-1}\rightarrow {{\mathcal{Z}}^{(n),r}\rightarrow }^{\prime }{\mathcal{Z}}^{(n),r}\rightarrow {\mathcal{Z}}^{(n-1),r}\rightarrow {\mathcal{Z}}^{(n),r+1}\rightarrow \cdots \,.\end{eqnarray}$$

Proof. The objects  ${\mathcal{Z}}^{(n-1),\bullet }$ ,  ${\mathcal{Z}}^{(n),\bullet }$ and  $^{\prime }{\mathcal{Z}}^{(n),\bullet }$ are defined via objects in  $\mathsf{DMT}(\mathbb{Q})$ that we denote by  $C^{(n-1)}$ ,  $C^{(n)}$ and  $^{\prime }C^{(n)}$ respectively,  $C^{(n)}$ being the complex (15) and  $C^{(n-1)}$ the subcomplex (24). Now there is an obvious exact triangle

$$\begin{eqnarray}C^{(n-1)}[-1]\longrightarrow {C^{(n)}\longrightarrow }^{\prime }C^{(n)}\stackrel{+1}{\longrightarrow },\end{eqnarray}$$

in  $\mathsf{DMT}(\mathbb{Q})$ , which gives the desired long exact sequence after taking the cohomology with respect to the  $t$ -structure.◻

We note that the map  ${\mathcal{Z}}^{(n-1),n-1}\rightarrow {\mathcal{Z}}^{(n),n}$ in the long exact sequence (26) is exactly  $i^{(n)}$ .

4.1 The Gysin long exact sequence

Since the divisor  $A_{n}$ is smooth, it is natural to decompose the motives  ${\mathcal{Z}}^{(n),r}$ thanks to a Gysin long exact sequence. In the next proposition, the definition of the objects  $H^{\bullet }(X_{n},B_{n})$ and  $H^{\bullet }(A_{n},B_{n}\cap A_{n})$ of  $\mathsf{MT}(\mathbb{Q})$ is similar to that of  ${\mathcal{Z}}^{(n)}$ from § 3.1.

Proposition 4.1. For  $n\geqslant 1$ , we have a long exact sequence in  $\mathsf{MT}(\mathbb{Q})$ :

(27) $$\begin{eqnarray}\cdots \rightarrow H^{r}(X_{n},B_{n})\rightarrow {\mathcal{Z}}^{(n),r}\rightarrow H^{r-1}(A_{n},B_{n}\cap A_{n})(-1)\rightarrow H^{r+1}(X_{n},B_{n})\rightarrow {\mathcal{Z}}^{(n),r+1}\rightarrow \cdots \,.\end{eqnarray}$$

Proof. Recall from [Reference VoevodskyVoe00, (3.5.4)] the existence of a Gysin exact triangle in the category  $\mathsf{DM}(\mathbb{Q})$ . For the pair  $(\widetilde{X}_{n},\widetilde{A}_{n})$ , it reads (with cohomological conventions)

$$\begin{eqnarray}\widetilde{X}_{n}\longrightarrow \widetilde{X}_{n}-\widetilde{A}_{n}\longrightarrow \widetilde{A}_{n}(-1)[-1]\stackrel{+1}{\longrightarrow }\end{eqnarray}$$

and is an exact triangle in the category  $\mathsf{DMT}(\mathbb{Q})$ . Applying this triangle to every pair  $(\unicode[STIX]{x2202}_{I}Y,\unicode[STIX]{x2202}_{I}Y\cap \widetilde{A}_{n})$ in the complex (15) and taking the cohomology with respect to the  $t$ -structure leads to a long exact sequence

$$\begin{eqnarray}\cdots \rightarrow H^{r}(\widetilde{X}_{n},\widetilde{B}_{n}\cup E_{n})\rightarrow H^{r}(\widetilde{X}_{n}-\widetilde{A}_{n},(\widetilde{B}_{n}\cup E_{n})-\widetilde{A}_{n})\rightarrow H^{r-1}(\widetilde{A}_{n},(\widetilde{B}_{n}\cup E_{n})\cap \widetilde{A}_{n})(-1)\rightarrow \cdots\end{eqnarray}$$

in  $\mathsf{MT}(\mathbb{Q})$ . One finishes with the fact that the natural morphisms

$$\begin{eqnarray}H^{r}(\widetilde{X}_{n},\widetilde{B}_{n}\cup E_{n})\rightarrow H^{r}(X_{n},B_{n})\quad \text{and}\quad H^{r-1}(\widetilde{A}_{n},(\widetilde{B}_{n}\cup E_{n})\cap \widetilde{A}_{n})\rightarrow H^{r-1}(A_{n},B_{n}\cap A_{n})\end{eqnarray}$$

are isomorphisms. This can be checked in the Betti realization (see Remark 2.5), where it is a consequence of the excision theorem as in the proof of Proposition 3.4. ◻

4.2 The motives  $H^{\bullet }(X_{n},B_{n})$

The computation of the motives  $H^{\bullet }(X_{n},B_{n})$ appearing in the long exact sequence (27) is relatively easy.

4.3 The motives  $H^{\bullet }(A_{n},B_{n}\cap A_{n})$

For  $n\geqslant 1$ , we realize the  $n$ -torus as  $T^{n}=\{x_{1}\cdots x_{n+1}=1\}$ , and we have subtori  $T_{i}^{n-1}=\{x_{i}=1\}\subset T^{n}$ for  $i=1,\ldots ,n+1$ . We define

$$\begin{eqnarray}{\mathcal{T}}^{(n),r}=H^{r}\biggl(T^{n},\mathop{\bigcup }_{1\leqslant i\leqslant n+1}T_{i}^{n-1}\biggr)\quad \text{and}\quad ^{\prime }{\mathcal{T}}^{(n),r}=H^{r}\biggl(T^{n},\mathop{\bigcup }_{1\leqslant i\leqslant n}T_{i}^{n-1}\biggr),\end{eqnarray}$$

which are objects in  $\mathsf{MT}(\mathbb{Q})$ (whose definition is similar to that of  ${\mathcal{Z}}^{(n)}$ from § 3.1) and write

$$\begin{eqnarray}{\mathcal{T}}^{(n)}={\mathcal{T}}^{(n),n}\quad \text{and}\quad ^{\prime }{\mathcal{T}}^{(n)}=^{\prime }{\mathcal{T}}^{(n),n}.\end{eqnarray}$$

We then have

$$\begin{eqnarray}H^{r-1}(A_{n},B_{n}\cap A_{n})\cong {\mathcal{T}}^{(n-1),r-1}.\end{eqnarray}$$

By mimicking the proof of Proposition 3.14, one produces a long exact sequence in  $\mathsf{MT}(\mathbb{Q})$ :

(28) $$\begin{eqnarray}\cdots \rightarrow {\mathcal{T}}^{(n-1),r-1}\rightarrow {{\mathcal{T}}^{(n),r}\rightarrow }^{\prime }{\mathcal{T}}^{(n),r}\rightarrow {\mathcal{T}}^{(n-1),r}\rightarrow {\mathcal{T}}^{(n),r+1}\rightarrow \cdots \,.\end{eqnarray}$$

Proposition 4.3. (i) We have  $^{\prime }{\mathcal{T}}^{(n),r}=0$ for  $r\neq n$ , and an isomorphism  $^{\prime }{\mathcal{T}}^{(n)}\cong H^{n}(T^{n})\cong \mathbb{Q}(-n)$ .

(ii) We have  ${\mathcal{T}}^{(n),r}=0$ for  $r\neq n$ , and short exact sequences in  $\mathsf{MT}(\mathbb{Q})$ :

(29) $$\begin{eqnarray}0\rightarrow {\mathcal{T}}^{(n-1)}\stackrel{j^{(n)}}{\longrightarrow }{\mathcal{T}}^{(n)}\rightarrow H^{n}(T^{n})\rightarrow 0.\end{eqnarray}$$

Proof. If (i) is proved, then (ii) follows from the long exact sequence (28). By choosing coordinates  $(x_{1},\ldots ,x_{n})$ on  $T^{n}$ we see that we have

$$\begin{eqnarray}^{\prime }{\mathcal{T}}^{(n),\bullet }=H^{\bullet }\biggl((\mathbb{A}_{\mathbb{ Q}}^{1}-\{0\})^{n},\mathop{\bigcup }_{1\leqslant i\leqslant n}\{x_{i}=1\}\biggr)\cong H^{\bullet }(\mathbb{A}_{\mathbb{Q}}^{1}-\{0\},\{1\})^{\otimes n}= (^{\prime }{\mathcal{T}}^{(1),\bullet } )^{\otimes n},\end{eqnarray}$$

where we have used the relative Künneth formula. Thus, it is enough to prove (i) for  $n=1$ , which is easy since  $^{\prime }{\mathcal{T}}^{(1),\bullet }$ is nothing but the reduced cohomology of  $\mathbb{A}_{\mathbb{Q}}^{1}-\{0\}$ .◻

We note that we have  ${\mathcal{T}}^{(0)}=H^{0}(\text{pt},\text{pt})=0$ , so that Proposition 4.3 implies that we have

$$\begin{eqnarray}\text{gr}_{2k}^{W}{\mathcal{T}}^{(n)}=\left\{\begin{array}{@{}ll@{}}\mathbb{Q}(-k)\quad & \text{if}\;k\in \{1,\ldots ,n\},\\ 0\quad & \text{otherwise.}\end{array}\right.\end{eqnarray}$$

In the next proposition, we will prove that the weight filtration of  ${\mathcal{T}}^{(n)}$ actually splits in  $\mathsf{MT}(\mathbb{Q})$ . For that we introduce the involution  $\unicode[STIX]{x1D70F}$ which acts on the tori  $T^{n}$ by

$$\begin{eqnarray}\unicode[STIX]{x1D70F}:(x_{1},\ldots ,x_{n+1})\mapsto (x_{1}^{-1},\ldots ,x_{n+1}^{-1}).\end{eqnarray}$$

This induces an involution, still denoted by  $\unicode[STIX]{x1D70F}$ , on the objects  ${\mathcal{T}}^{(n),r}$ and  $^{\prime }{\mathcal{T}}^{(n),r}$ of  $\mathsf{MT}(\mathbb{Q})$ , such that all the maps in the long exact sequence (28) commute with  $\unicode[STIX]{x1D70F}$ .

Proposition 4.5. (i) The short exact sequences (29) split in  $\mathsf{MT}(\mathbb{Q})$ , hence we have isomorphisms:

$$\begin{eqnarray}{\mathcal{T}}^{(n)}\cong \mathbb{Q}(-1)\oplus \mathbb{Q}(-2)\oplus \cdots \oplus \mathbb{Q}(-n).\end{eqnarray}$$

Thus, a period matrix for  ${\mathcal{T}}^{(n)}$ is the diagonal matrix  $\,\text{Diag}(2\unicode[STIX]{x1D70B}i,(2\unicode[STIX]{x1D70B}i)^{2},\ldots ,(2\unicode[STIX]{x1D70B}i)^{n})$ .

(ii) The involution  $\unicode[STIX]{x1D70F}$ acts on the direct summand  $\mathbb{Q}(-k)$ of  ${\mathcal{T}}^{(n)}$ by multiplication by  $(-1)^{k}$ .

Proof. We first note that  $\unicode[STIX]{x1D70F}$ acts on  $H^{1}(T^{1})$ by multiplication by  $-1$ . It is enough to prove it in the de Rham realization, where it follows from  $\unicode[STIX]{x1D70F}.\,\text{dlog}(x_{1})=-\text{dlog}(x_{1})$ . Thus,  $\unicode[STIX]{x1D70F}$ acts on  $\text{gr}_{2n}^{W}{\mathcal{T}}^{(n)}\cong H^{n}(T^{n})\cong H^{1}(T^{1})^{\otimes n}$ by multiplication by  $(-1)^{n}$ , and we are left with proving (i). We denote by  ${\mathcal{T}}^{(n)}={\mathcal{T}}_{+}^{(n)}\oplus {\mathcal{T}}_{-}^{(n)}$ the direct sum decomposition of  ${\mathcal{T}}^{(n)}$ into its invariant and anti-invariant parts with respect to  $\unicode[STIX]{x1D70F}$ . We have to prove that we have isomorphisms

$$\begin{eqnarray}{\mathcal{T}}_{+}^{(2n)}\cong {\mathcal{T}}_{+}^{(2n+1)}\cong \mathbb{Q}(-2)\oplus \mathbb{Q}(-4)\oplus \cdots \oplus \mathbb{Q}(-2n)\end{eqnarray}$$

and

$$\begin{eqnarray}{\mathcal{T}}_{-}^{(2n+1)}\cong {\mathcal{T}}_{-}^{(2n+2)}\cong \mathbb{Q}(-1)\oplus \mathbb{Q}(-3)\oplus \cdots \oplus \mathbb{Q}(-(2n+1)).\end{eqnarray}$$

We only prove the statements corresponding to the invariant parts, the statements corresponding to the anti-invariant parts being proved similarly. We use induction on  $n$ , the case  $n=0$ being trivial:  ${\mathcal{T}}_{+}^{(0)}={\mathcal{T}}_{+}^{(1)}=0$ . The short exact sequences (29) imply that we have short exact sequences

$$\begin{eqnarray}0\rightarrow {\mathcal{T}}_{+}^{(2n+1)}\rightarrow {\mathcal{T}}_{+}^{(2n+2)}\rightarrow \mathbb{Q}(-(2n+2))\rightarrow 0\quad \text{and}\quad 0\rightarrow {\mathcal{T}}_{+}^{(2n+2)}\rightarrow {\mathcal{T}}_{+}^{(2n+3)}\rightarrow 0\rightarrow 0.\end{eqnarray}$$

Using the induction hypothesis we see that we have

$$\begin{eqnarray}\displaystyle \text{Ext}_{\mathsf{MT}(\mathbb{Q})}^{1}(\mathbb{Q}(-(2n+2)),{\mathcal{T}}_{+}^{(2n+1)}) & \cong & \displaystyle \text{Ext}_{\mathsf{MT}(\mathbb{Q})}^{1}(\mathbb{Q}(-(2n+2)),\mathbb{Q}(-2)\oplus \mathbb{Q}(-4)\oplus \cdots \oplus \mathbb{Q}(-2n))\nonumber\\ \displaystyle & \cong & \displaystyle \bigoplus _{1\leqslant k\leqslant n}\text{Ext}_{\mathsf{MT}(\mathbb{Q})}^{1}(\mathbb{Q}(-2k),\mathbb{Q}(0))\nonumber\\ \displaystyle & = & \displaystyle 0,\nonumber\end{eqnarray}$$

where we have used (13). Thus, the first short exact sequence splits. The second short exact sequence then completes the induction. ◻

4.4 The structure of the zeta motives

We can now determine the structure of the zeta motives  ${\mathcal{Z}}^{(n)}$ , for  $n\geqslant 1$ .

Theorem 4.7. (i) We have a short exact sequence in  $\mathsf{MT}(\mathbb{Q})$ :

(30) $$\begin{eqnarray}0\rightarrow \mathbb{Q}(0)\rightarrow {\mathcal{Z}}^{(n)}\stackrel{p^{(n)}}{\longrightarrow }{\mathcal{T}}^{(n-1)}(-1)\rightarrow 0,\end{eqnarray}$$

with  ${\mathcal{T}}^{(n-1)}(-1)\cong \mathbb{Q}(-2)\oplus \cdots \oplus \mathbb{Q}(-n)$ .

(ii) We have a short exact sequence in  $\mathsf{MT}(\mathbb{Q})$ :

(31) $$\begin{eqnarray}0\rightarrow {\mathcal{Z}}^{(n-1)}\stackrel{i^{(n)}}{\longrightarrow }{\mathcal{Z}}^{(n)}\rightarrow \mathbb{Q}(-n)\rightarrow 0.\end{eqnarray}$$

(iii) These short exact sequences fit into a commutative diagram

(32)

where all rows and columns are exact.

Theorem 4.9. (i) The classes

$$\begin{eqnarray}v_{k}^{(n)}:=[\unicode[STIX]{x1D714}_{k}^{(n)}]\quad (k=0,2,\ldots ,n)\end{eqnarray}$$

of the Eulerian differential forms provide a basis  $(v_{0}^{(n)},v_{2}^{(n)},\ldots ,v_{n}^{(n)})$ of the de Rham realization  ${\mathcal{Z}}_{\text{dR}}^{(n)}$ which is compatible with the weight grading.

(ii) There exists a unique basis  $(\unicode[STIX]{x1D711}_{0}^{(n)},\unicode[STIX]{x1D711}_{2}^{(n)},\ldots ,\unicode[STIX]{x1D711}_{n}^{(n)})$ for the dual of the Betti realization  ${\mathcal{Z}}_{\text{B}}^{(n),\vee }$ which is compatible with the weight filtration and such that the period matrix for  ${\mathcal{Z}}^{(n)}$ in the  $v$ -basis and the  $\unicode[STIX]{x1D711}$ -basis is

(33) $$\begin{eqnarray}\left(\begin{array}{@{}ccccccc@{}}1 & \unicode[STIX]{x1D701}(2) & \unicode[STIX]{x1D701}(3) & \cdots \, & \cdots \, & \unicode[STIX]{x1D701}(n-1) & \unicode[STIX]{x1D701}(n)\\ & (2\unicode[STIX]{x1D70B}i)^{2} & & & & & \\ & & (2\unicode[STIX]{x1D70B}i)^{3} & & & 0 & \\ & & & \ddots & & & \\ & & & & \ddots & & \\ & 0 & & & & (2\unicode[STIX]{x1D70B}i)^{n-1} & \\ & & & & & & (2\unicode[STIX]{x1D70B}i)^{n}\end{array}\right).\end{eqnarray}$$

We have already noted that the classes  $v_{k}^{(n)}$ are compatible with the inductive system of the zeta motives. By the uniqueness statement in Theorem 4.9, this is also the case for the classes  $\unicode[STIX]{x1D711}_{k}^{(n)}$ , and the zeta motive  ${\mathcal{Z}}$ has an infinite period matrix

$$\begin{eqnarray}\left(\begin{array}{@{}ccccccc@{}}1 & \unicode[STIX]{x1D701}(2) & \unicode[STIX]{x1D701}(3) & \unicode[STIX]{x1D701}(4) & \cdots \, & \cdots \, & \cdots \\ & (2\unicode[STIX]{x1D70B}i)^{2} & & & & & \\ & & (2\unicode[STIX]{x1D70B}i)^{3} & & & 0 & \\ & & & (2\unicode[STIX]{x1D70B}i)^{4} & & & \\ & & & & \ddots & & \\ & 0 & & & & \ddots & \\ & & & & & & \ddots \end{array}\right).\end{eqnarray}$$

4.5 The odd zeta motive

Let us write  ${\mathcal{T}}^{(n-1)}={\mathcal{T}}_{+}^{(n-1)}\oplus {\mathcal{T}}_{-}^{(n-1)}$ for the direct sum decomposition into its invariant and anti-invariant parts with respect to  $\unicode[STIX]{x1D70F}$ , and let us write  $p^{(n)}:{\mathcal{Z}}^{(n)}\rightarrow {\mathcal{T}}^{(n-1)}(-1)$ for the surjection appearing in the short exact sequence (30).

Definition 4.10. The  $n$ th odd zeta motive  ${\mathcal{Z}}^{(n),\text{odd}}$ is the object of  $\mathsf{MT}(\mathbb{Q})$ defined by

$$\begin{eqnarray}{\mathcal{Z}}^{(n),\text{odd}}:=(p^{(n)})^{-1}({\mathcal{T}}_{+}^{(n-1)}(-1)).\end{eqnarray}$$

We obviously have a short exact sequence

(34) $$\begin{eqnarray}0\rightarrow \mathbb{Q}(0)\rightarrow {\mathcal{Z}}^{(n),\text{odd}}\rightarrow {\mathcal{T}}_{+}^{(n-1)}(-1)\rightarrow 0\end{eqnarray}$$

with

$$\begin{eqnarray}{\mathcal{T}}_{+}^{(n-1)}(-1)\cong \bigoplus _{3\leqslant 2k+1\leqslant n}\mathbb{Q}(-(2k+1)).\end{eqnarray}$$

We note that there are morphisms

$$\begin{eqnarray}i^{(n),\text{odd}}:{\mathcal{Z}}^{(n-1),\text{odd}}\rightarrow {\mathcal{Z}}^{(n),\text{odd}}\end{eqnarray}$$

such that  $i^{(2n),\text{odd}}$ is an isomorphism for every integer  $n$ . The limit

$$\begin{eqnarray}{\mathcal{Z}}^{\text{odd}}:=\lim _{\stackrel{\longrightarrow }{n}}{\mathcal{Z}}^{(n),\text{odd}}\end{eqnarray}$$

is an ind-object in  $\mathsf{MT}(\mathbb{Q})$ that we simply call the odd zeta motive.

Proposition 4.11. (i) We have a direct sum decomposition

(35) $$\begin{eqnarray}{\mathcal{Z}}^{(n)}\cong {\mathcal{Z}}^{(n),\text{odd}}\oplus \bigoplus _{2\leqslant 2k\leqslant n}\mathbb{Q}(-2k).\end{eqnarray}$$

(ii) A period matrix for  ${\mathcal{Z}}^{(2n+1),\text{odd}}\cong {\mathcal{Z}}^{(2n+2),\text{odd}}$ is

(36) $$\begin{eqnarray}\left(\begin{array}{@{}ccccccc@{}}1 & \unicode[STIX]{x1D701}(3) & \unicode[STIX]{x1D701}(5) & \cdots \, & \cdots \, & \unicode[STIX]{x1D701}(2n-1) & \unicode[STIX]{x1D701}(2n+1)\\ & (2\unicode[STIX]{x1D70B}i)^{3} & & & & & \\ & & (2\unicode[STIX]{x1D70B}i)^{5} & & & 0 & \\ & & & \ddots & & & \\ & & & & \ddots & & \\ & 0 & & & & (2\unicode[STIX]{x1D70B}i)^{2n-1} & \\ & & & & & & (2\unicode[STIX]{x1D70B}i)^{2n+1}\end{array}\right).\end{eqnarray}$$

Proposition 4.11 implies that the odd zeta motive  ${\mathcal{Z}}^{\text{odd}}$ has an infinite period matrix (6). In particular, ${\mathcal{Z}}^{(3),\text{odd}}$ is the essentially unique non-trivial extension of $\mathbb{Q}(-3)$ by $\mathbb{Q}(0)$ in $\mathsf{MT}(\mathbb{Q})$ .

Proof. This is a consequence of the short exact sequence (30) and the vanishing of the extension groups $\text{Ext}_{\mathsf{MT}(\mathbb{Q})}^{1}(\mathbb{Q}(-2k),\mathbb{Q}(0))$ , see (13). An alternative proof which does not use the computation of extension groups goes as follows. A basis for  ${\mathcal{Z}}_{\text{dR}}^{(n),\text{odd}}$ is given by  $v_{0}^{(n)}$ and the  $v_{2k+1}^{(n)}$ , for  $3\leqslant 2k+1\leqslant n$ , and a basis for  ${\mathcal{Z}}_{\text{B}}^{(n),\text{odd},\vee }$ is given by  $\unicode[STIX]{x1D711}_{0}^{(n)}$ and the  $\unicode[STIX]{x1D711}_{2k+1}^{(n)}$ , for  $3\leqslant 2k+1\leqslant n$ . This gives the desired shape for the period matrix (36). Now, Euler’s solution to the Basel problem implies that we have  $\unicode[STIX]{x1D701}(2k)=\unicode[STIX]{x1D706}_{2k}(2\unicode[STIX]{x1D70B}i)^{2k}$ for every integer  $k\geqslant 1$ , with  $\unicode[STIX]{x1D706}_{2k}=-B_{2k}/2(2k)!\in \mathbb{Q}$ . Thus, we may replace the basis  $(\unicode[STIX]{x1D711}_{0}^{(n)},\unicode[STIX]{x1D711}_{2}^{(n)},\ldots ,\unicode[STIX]{x1D711}_{n}^{(n)})$ of Theorem 4.9 by the basis  $(^{\prime }\unicode[STIX]{x1D711}_{0}^{(n)},\unicode[STIX]{x1D711}_{2}^{(n)},\ldots ,\unicode[STIX]{x1D711}_{n}^{(n)} )$ with

$$\begin{eqnarray}^{\prime }\unicode[STIX]{x1D711}_{0}^{(n)}=\unicode[STIX]{x1D711}_{0}^{(n)}-\mathop{\sum }_{2\leqslant 2k\leqslant n}\unicode[STIX]{x1D706}_{2k}\,\unicode[STIX]{x1D711}_{2k}^{(n)}\end{eqnarray}$$

to get a period matrix similar to (33) where the even zeta values  $\unicode[STIX]{x1D701}(2k)$ in the first row are replaced by  $0$ . This implies the direct sum decomposition (35).◻

We finish by proving that all the objects in  $\mathsf{MT}(\mathbb{Q})$ considered earlier actually live in the full subcategory  $\mathsf{MT}(\mathbb{Z})$ .

Proof. Thanks to the direct sum decomposition (35), it is enough to prove it for the odd zeta motives. Let us recall the definition [Reference Deligne and GoncharovDG05, Définition 1.4] of the category  $\mathsf{MT}(\mathbb{Z})$ . According to the tannakian formalism, the de Rham realization functor  $\mathsf{MT}(\mathbb{Q})\rightarrow \mathsf{grVect}_{\mathbb{Q}}$ induces an equivalence of categories

$$\begin{eqnarray}\mathsf{MT}(\mathbb{Q})\cong \mathsf{grRep}(\mathfrak{g}_{\text{dR}}^{\mathbb{Q}})\end{eqnarray}$$

between  $\mathsf{MT}(\mathbb{Q})$ and the category of graded finite-dimensional representations of a graded Lie algebra  $\mathfrak{g}_{\text{dR}}^{\mathbb{Q}}$ . The degree in  $\mathfrak{g}_{\text{dR}}^{\mathbb{Q}}$ is half the weight. This Lie algebra is non-positively graded. The category  $\mathsf{MT}(\mathbb{Z})$ is defined as the full subcategory of  $\mathsf{MT}(\mathbb{Q})$ consisting of objects  $H$ such that the degree  $-1$ component of  $\mathfrak{g}_{\text{dR}}^{\mathbb{Q}}$ acts trivially on  $H_{\text{dR}}$ . This is obviously the case for  ${\mathcal{Z}}^{(n),\text{odd}}$ , which is concentrated in weights  $0$ and  $2(2k+1)$ with  $2k+1\geqslant 3$ by the short exact sequence (34).◻

Remark 4.13. A tannakian interpretation of the odd zeta motive goes as follows. Let  $\mathfrak{g}^{\mathbb{Z},\vee }$ be the graded dual of the fundamental Lie algebra  $\mathfrak{g}^{\mathbb{Z}}$ of the tannakian category  $\mathsf{MT}(\mathbb{Z})$ . It is an ind-object in  $\mathsf{MT}(\mathbb{Z})$ , independent of the choice of a fiber functor [Reference DeligneDel89, Définition 6.1]. Then one has a short exact sequence

$$\begin{eqnarray}0\rightarrow \mathbb{Q}(0)\rightarrow \mathfrak{g}^{\mathbb{Z},\vee }\rightarrow \mathfrak{u}^{\mathbb{Z},\vee }\rightarrow 0,\end{eqnarray}$$

where  $\mathfrak{u}^{\mathbb{Z}}$ is the pro-unipotent radical of  $\mathfrak{g}^{\mathbb{Z}}$ . One views  ${\mathcal{Z}}^{\text{odd}}$ inside the exact subsequence

$$\begin{eqnarray}0\rightarrow \mathbb{Q}(0)\rightarrow {\mathcal{Z}}^{\text{odd}}\rightarrow \mathfrak{u}^{\mathbb{Z},\text{ab},\vee }\rightarrow 0,\end{eqnarray}$$

where  $\mathfrak{u}^{\mathbb{Z},\text{ab},\vee }\cong \bigoplus _{k\geqslant 1}\mathbb{Q}(-(2k+1))$ is the graded dual of the abelianization of  $\mathfrak{u}^{\mathbb{Z}}$ .

4.6 An elementary computation of the motives  ${\mathcal{T}}^{(n)}$

We give an elementary proof of Proposition 4.5, which only uses basic algebraic topology. The proof is Hodge-theoretic, and the only drawback is that we have to use the full faithfulness of the Hodge realization (Theorem 2.6). Let us consider the relative homology group

$$\begin{eqnarray}{\mathcal{T}}_{\text{B}}^{(n),\vee }=H_{n}^{\text{sing}}\biggl((\mathbb{C}^{\ast })^{n},\mathop{\bigcup }_{1\leqslant i\leqslant n}\{x_{i}=1\}\cup \{x_{1}\cdots x_{n}=1\}\biggr).\end{eqnarray}$$

By homotopy invariance, one may replace every  $\mathbb{C}^{\ast }$ by the unit circle  $S^{1}=\{|x|=1\}{\hookrightarrow}\mathbb{C}^{\ast }$ and the divisor  $\{x_{1}\cdots x_{n}=1\}$ by its intersection with  $(S^{1})^{n}$ , and we get

$$\begin{eqnarray}{\mathcal{T}}_{\text{B}}^{(n),\vee }\cong H_{n}^{\text{sing}}\biggl((S^{1})^{n},\mathop{\bigcup }_{1\leqslant i\leqslant n}\{x_{i}=1\}\cup \{x_{1}\cdots x_{n}=1\}\biggr).\end{eqnarray}$$

Let us look at the projection  $[0,1]^{n}\rightarrow (S^{1})^{n},(t_{1},\ldots ,t_{n})\mapsto (e^{2\unicode[STIX]{x1D70B}it_{1}},\ldots ,e^{2\unicode[STIX]{x1D70B}it_{n}})$ . Then by excision we can write

$$\begin{eqnarray}{\mathcal{T}}_{\text{B}}^{(n),\vee }\cong H_{n}^{\text{sing}}\biggl([0,1]^{n},\mathop{\bigcup }_{1\leqslant i\leqslant n}\{t_{i}\in \mathbb{Z}\}\cup \{t_{1}+\cdots +t_{n}\in \mathbb{Z}\}\biggr).\end{eqnarray}$$

This is simply the singular homology of the unit hypercube  $[0,1]^{n}$ relative to the union of its faces  $\{t_{i}=0\}$ and  $\{t_{i}=1\}$ , for  $1\leqslant i\leqslant n$ , and the hyperplanes  $\{t_{1}\,+\cdots +\,t_{n}=k\}$ for  $k=0,1,\ldots ,n$ . We note that these hyperplanes cut the unit hypercube into polytopes

$$\begin{eqnarray}\unicode[STIX]{x1D6E5}(n,k)=\{(t_{1},\ldots ,t_{n})\in [0,1]^{n}\,\,|\,\,k\leqslant t_{1}+\cdots +t_{n}\leqslant k+1\},\end{eqnarray}$$

for  $k=0,\ldots ,n-1$ . We note that  $\unicode[STIX]{x1D6E5}(n,0)$ is the usual  $n$ -simplex; the polytopes  $\unicode[STIX]{x1D6E5}(n,k)$ are usually called hypersimplices.

The Eulerian numbers are the coefficients of the Eulerian polynomials and are denoted by symbols  $\langle \!\begin{smallmatrix}n\\ k\end{smallmatrix}\!\rangle$ :

$$\begin{eqnarray}E_{n}(x)=\mathop{\sum }_{k=0}^{n-1}\langle \begin{array}{@{}c@{}}n\\ k\end{array}\rangle \,x^{k}.\end{eqnarray}$$

They satisfy many beautiful identities, including the following recursion, which can be deduced from (21):

$$\begin{eqnarray}\langle \begin{array}{@{}c@{}}n\\ k\end{array}\rangle =(n-k)\langle \begin{array}{@{}c@{}}n-1\\ k-1\end{array}\rangle +(k+1)\langle \begin{array}{@{}c@{}}n-1\\ k\end{array}\rangle .\end{eqnarray}$$

The following lemma is a classical result due to Laplace [Reference FoataFoa77, § 2].

Recall from Remark 4.6 that, for every integer  $n\geqslant 1$ ,  ${\mathcal{T}}_{\text{dR}}^{(n)}$ has a basis  $(w_{1}^{(n)},\ldots ,w_{n}^{(n)})$ which is compatible with the weight grading and with the morphisms  $j_{\text{dR}}^{(n)}:{\mathcal{T}}_{\text{dR}}^{(n-1)}\rightarrow {\mathcal{T}}_{\text{dR}}^{(n)}$ . We let  $P_{n}$ be the period matrix of  ${\mathcal{T}}^{(n)}$ with respect to the  $w$ -basis and the  $\unicode[STIX]{x1D6E5}$ -basis from Lemma 4.14. The first period matrix  $P_{1}$ is simply the  $1\times 1$ matrix  $(2\unicode[STIX]{x1D70B}i)$ . Let us introduce the following  $n\times n$ integer matrix encoding the family of Eulerian numbers:

$$\begin{eqnarray}A_{n}=\left(\begin{array}{@{}ccccccc@{}}1 & & & & & & \left\langle \begin{array}{@{}c@{}}n\\ 0\end{array}\right\rangle \\ -1 & 1 & & & 0 & & \left\langle \begin{array}{@{}c@{}}n\\ 1\end{array}\right\rangle \\ & -1 & 1 & & & & \left\langle \begin{array}{@{}c@{}}n\\ 2\end{array}\right\rangle \\ & & \ddots & \ddots & & & \\ & & & \ddots & \ddots & & \\ & 0 & & & -1 & 1 & \left\langle \begin{array}{@{}c@{}}n\\ n-2\end{array}\right\rangle \\ & & & & & -1 & \left\langle \begin{array}{@{}c@{}}n\\ n-1\end{array}\right\rangle \end{array}\right).\end{eqnarray}$$

Proposition 4.17. The period matrices  $P_{n}$ satisfy the recurrence relation

Proof. Recall the short exact sequence (29)

$$\begin{eqnarray}0\rightarrow {\mathcal{T}}^{(n-1)}\stackrel{j^{(n)}}{\longrightarrow }{\mathcal{T}}^{(n)}\rightarrow H^{n}(T^{n})\rightarrow 0\end{eqnarray}$$

and the fact (see Remark 4.6) that the morphism  $j^{(n)}$ is compatible with the  $w$ -bases. Then Lemma 4.14 shows that the first  $(n-1)$ columns of  $P_{n}$ are as stated. It only remains to compute the entries in the last column, i.e., compute the integral of the  $n$ -form  $dx_{1}/x_{1}\wedge \cdots \wedge dx_{n}/x_{n}$ on a hypersimplex  $\unicode[STIX]{x1D6E5}(n,k)$ . After the change of variables  $(x_{1},\ldots ,x_{n})=(e^{2\unicode[STIX]{x1D70B}it_{1}},\ldots ,e^{2\unicode[STIX]{x1D70B}it_{n}})$ , one sees that this integral is simply  $(2\unicode[STIX]{x1D70B}i)^{n}$ times the volume of  $\unicode[STIX]{x1D6E5}(n,k)$ , and completes the proof thanks to Lemma 4.16.◻

We note that the period matrices  $P_{n}$ are not block upper-triangular. This is because the  $\unicode[STIX]{x1D6E5}$ -basis is not compatible with the weight filtration. We thus have to introduce a change of basis. Let  $(Q_{n})_{n\geqslant 1}$ be the family of matrices (with rational entries) defined by  $Q_{1}=(1)$ and the recurrence relation

The first terms are

$$\begin{eqnarray}Q_{1}=\left(\begin{array}{@{}c@{}}1\end{array}\right),\quad Q_{2}=\left(\begin{array}{@{}cc@{}}{\textstyle \frac{1}{2}} & -{\textstyle \frac{1}{2}}\\ 1 & 1\end{array}\right),\quad Q_{3}=\left(\begin{array}{@{}ccc@{}}{\textstyle \frac{1}{3}} & -{\textstyle \frac{1}{6}} & {\textstyle \frac{1}{3}}\\ 1 & 0 & -1\\ 1 & 1 & 1\end{array}\right),\quad Q_{4}=\left(\begin{array}{@{}cccc@{}}{\textstyle \frac{1}{4}} & -{\textstyle \frac{1}{12}} & {\textstyle \frac{1}{12}} & -{\textstyle \frac{1}{4}}\\ {\textstyle \frac{11}{12}} & -{\textstyle \frac{1}{12}} & -{\textstyle \frac{1}{12}} & {\textstyle \frac{11}{12}}\\ {\textstyle \frac{3}{2}} & {\textstyle \frac{1}{2}} & -{\textstyle \frac{1}{2}} & -{\textstyle \frac{3}{2}}\\ 1 & 1 & 1 & 1\end{array}\right).\end{eqnarray}$$

Let us put

$$\begin{eqnarray}\left(\begin{array}{@{}c@{}}\unicode[STIX]{x1D6F4}_{1}^{(n)}\\ \unicode[STIX]{x1D6F4}_{2}^{(n)}\\ \vdots \\ \unicode[STIX]{x1D6F4}_{n}^{(n)}\end{array}\right)=Q_{n}\left(\begin{array}{@{}c@{}}\unicode[STIX]{x1D6E5}(n,0)\\ \unicode[STIX]{x1D6E5}(n,1)\\ \vdots \\ \unicode[STIX]{x1D6E5}(n,n-1)\end{array}\right).\end{eqnarray}$$

We view  $\unicode[STIX]{x1D6F4}_{k}^{(n)}$ as a relative cycle with rational coefficients. The change of indexing is here to remind the reader that  $\unicode[STIX]{x1D6F4}_{k}^{(n)}$ lives in weight ${\leqslant}2k$ . We have thus proved the following result.

By using Theorem 2.6, we thus get an alternate (Hodge-theoretic) proof of Proposition 4.5.

5.1 Integral formulas for the coefficients

Theorem 5.1. For  $\unicode[STIX]{x1D714}$ an integrable algebraic differential form on  $X_{n}-A_{n}$ , we have

(37) $$\begin{eqnarray}\int _{[0,1]^{n}}\unicode[STIX]{x1D714}=a_{0}(\unicode[STIX]{x1D714})+a_{2}(\unicode[STIX]{x1D714})\unicode[STIX]{x1D701}(2)+\cdots +a_{n}(\unicode[STIX]{x1D714})\unicode[STIX]{x1D701}(n)\end{eqnarray}$$

with  $a_{k}(\unicode[STIX]{x1D714})$ a rational number for every  $k$ , given for  $k=2,\ldots ,n$ by the formula

(38) $$\begin{eqnarray}a_{k}(\unicode[STIX]{x1D714})=(2\unicode[STIX]{x1D70B}i)^{-k}\,\langle \unicode[STIX]{x1D711}_{k}^{(n)},[\unicode[STIX]{x1D714}]\rangle .\end{eqnarray}$$

Proof. According to Proposition 3.6, the class  $[\unicode[STIX]{x1D714}]$ defines an element in  ${\mathcal{Z}}_{n,\text{dR}}$ , hence we may write

$$\begin{eqnarray}[\unicode[STIX]{x1D714}]=a_{0}(\unicode[STIX]{x1D714})v_{0}+a_{2}(\unicode[STIX]{x1D714})v_{2}+\cdots +a_{n}(\unicode[STIX]{x1D714})v_{n}\end{eqnarray}$$

with  $a_{k}(\unicode[STIX]{x1D714})\in \mathbb{Q}$ for every  $k$ . Pairing with the class  $\unicode[STIX]{x1D711}_{0}^{(n)}=[\Box ^{n}]$ gives the equality (37), and pairing with the class  $\unicode[STIX]{x1D711}_{k}^{(n)}$ ,  $k=2,\ldots ,n$ , gives the equality (38).◻

Remark 5.2. If we represent the class  $\unicode[STIX]{x1D711}_{k}^{(n)}$ by a relative cycle  $\unicode[STIX]{x1D70E}_{k}^{(n)}$ , then (38) becomes

$$\begin{eqnarray}a_{k}(\unicode[STIX]{x1D714})=(2\unicode[STIX]{x1D70B}i)^{-k}\int _{\unicode[STIX]{x1D70E}_{k}^{(n)}}\unicode[STIX]{x1D714}.\end{eqnarray}$$

Here we will not give explicit representatives for the classes  $\unicode[STIX]{x1D711}_{k}^{(n)}$ . Recall from the proof of Theorem 4.9 that the class  $\unicode[STIX]{x1D711}_{k}^{(n)}$ is the image by the map  $p_{\text{B}}^{(n),\vee }:{\mathcal{T}}_{\text{B}}^{(n-1),\vee }\rightarrow {\mathcal{Z}}_{\text{B}}^{(n),\vee }$ of an element  $\unicode[STIX]{x1D713}_{k-1}^{(n-1)}$ , which by Remark 4.19 can be represented by the cycle  $\unicode[STIX]{x1D6F4}_{k-1}^{(n-1)}$ . The question is then how to compute the map  $p_{B}^{(n),\vee }$ at the level of cycles. Such a task would involve the following ingredients. Let  $T\subset \mathbb{C}^{n}$ be a tubular neighborhood of  $A_{n}(\mathbb{C})$ in  $\mathbb{C}^{n}$ . Let us denote by  $\unicode[STIX]{x1D70C}:T\rightarrow A_{n}(\mathbb{C})$ the corresponding projection, and by  $\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}:\unicode[STIX]{x2202}T\rightarrow A_{n}(\mathbb{C})$ the projection corresponding to the boundary of the tubular neighborhood; it is an  $S^{1}$ -bundle. The natural map  $H_{r}^{\text{sing}}(A_{n}(\mathbb{C}))\rightarrow H_{r+1}^{\text{sing}}(\mathbb{C}^{n}-A_{n}(\mathbb{C}))$ can be computed at the level of singular chains by mapping an  $r$ -cycle  $\unicode[STIX]{x1D70E}$ to the  $(r+1)$ -cycle  $(\unicode[STIX]{x2202}\unicode[STIX]{x1D70C})^{-1}(\unicode[STIX]{x1D70E})$ . We note that since  $A_{n}(\mathbb{C})$ does not intersect the hyperplanes  $\{x_{i}=0\}$ , we can do the computation with a tubular neighborhood inside  $(\mathbb{C}^{\ast })^{n}$ and get representatives in  $(\mathbb{C}^{\ast })^{n}$ . Now if we want to play this game for the relative homology groups  ${\mathcal{Z}}_{\text{B}}^{(n),\vee }$ , we need the tubular neighborhood to be ‘compatible’ with the subvariety  $B_{n}(\mathbb{C})$ , in the sense that  $\unicode[STIX]{x1D70C}$ should pull back  $A_{n}(\mathbb{C})\cap B_{n}(\mathbb{C})$ to  $B_{n}(\mathbb{C})$ . At this point, it is probably easier to ask for something weaker than a tubular neighborhood, i.e., something that is a tubular neighborhood on a dense open subset of  $A_{n}(\mathbb{C})$ (this does not change anything for the integral formulas). We will not try to give formulas here and postpone this discussion to a future article. Nevertheless, we can give more explicit formulas than (38) in two situations.

5.1.1 The highest weight coefficient

Let us fix real numbers  $\unicode[STIX]{x1D70C}_{1},\ldots ,\unicode[STIX]{x1D70C}_{n-1},\unicode[STIX]{x1D70C}_{n}>0$ and let us introduce the cycle  $S^{(n)}\subset \mathbb{C}^{n}-A_{n}(\mathbb{C})$ defined by the conditions

$$\begin{eqnarray}|x_{1}|=\unicode[STIX]{x1D70C}_{1},\ldots ,|x_{n-1}|=\unicode[STIX]{x1D70C}_{n-1},\quad \biggl|x_{n}-\frac{1}{x_{1}\cdots x_{n-1}}\biggr|=\unicode[STIX]{x1D70C}_{n}.\end{eqnarray}$$

Proposition 5.3. Let  $\unicode[STIX]{x1D714}$ be an integrable differential form on  $X_{n}-A_{n}$ . Then the highest weight coefficient  $a_{n}(\unicode[STIX]{x1D714})$ from Theorem 5.1 is given by the integral formula

$$\begin{eqnarray}a_{n}(\unicode[STIX]{x1D714})=(2\unicode[STIX]{x1D70B}i)^{-n}\int _{S^{(n)}}\unicode[STIX]{x1D714}.\end{eqnarray}$$

Proof. The integral formula is obviously independent of the choice of  $\unicode[STIX]{x1D70C}_{1},\ldots ,\unicode[STIX]{x1D70C}_{n-1},\unicode[STIX]{x1D70C}_{n}$ and we can assume that we have  $\unicode[STIX]{x1D70C}_{1}=\cdots =\unicode[STIX]{x1D70C}_{n-1}=\unicode[STIX]{x1D70C}_{n}=1$ . We have noted in Remark 4.20 that the highest weight basis vector  $\unicode[STIX]{x1D713}_{n-1}^{(n-1)}$ of  ${\mathcal{T}}_{\text{B}}^{(n-1),\vee }$ can be represented by the  $(n-1)$ -torus  $\{|x_{1}|=\cdots =|x_{n-1}|=1\}$ . Since this has an empty boundary we can make the computation explained in Remark 5.2 with the choice of any tubular neighborhood of  $A_{n}(\mathbb{C})$ in  $\mathbb{C}^{n}$ , for instance the one defined by $|x_{n}-1/(x_{1}\cdots x_{n-1})|\leqslant 1$ , with projection map $\unicode[STIX]{x1D70C}(x_{1},\ldots ,x_{n})=(x_{1},\ldots ,x_{n-1},1/(x_{1}\cdots x_{n-1}))$ . The pullback of the  $(n-1)$ -torus by the projection  $\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}$ is exactly  $S^{(n)}$ .◻

The case  $n=2$ is Rhin and Viola’s contour integral for  $\unicode[STIX]{x1D701}(2)$  [Reference Rhin and ViolaRV96, Lemma 2.6].

5.1.2 The case of forms with simple poles

We say that a differential form on  $X_{n}-A_{n}$ has a simple pole along  $A_{n}$ if it can be written as

$$\begin{eqnarray}\unicode[STIX]{x1D714}=\unicode[STIX]{x1D6FC}+\text{dlog}(1-x_{1}\cdots x_{n})\wedge \unicode[STIX]{x1D6FD},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ do not have poles along  $A_{n}$ . The residue of such a form along  $A_{n}$ is the restriction

$$\begin{eqnarray}\text{Res}(\unicode[STIX]{x1D714})=\unicode[STIX]{x1D6FD}_{|A_{n}}.\end{eqnarray}$$

Recall that the relative cycles  $\unicode[STIX]{x1D6F4}_{k-1}^{(n-1)}$ were defined in § 4.6.

Proposition 5.4. Let  $\unicode[STIX]{x1D714}$ be an integrable differential form on  $X_{n}-A_{n}$ which has a simple pole along  $A_{n}$ . Then the coefficients  $a_{k}(\unicode[STIX]{x1D714})$ ,  $k=2,\ldots ,n$ , from Theorem 5.1 are given by the integral formulas

$$\begin{eqnarray}a_{k}(\unicode[STIX]{x1D714})=(2\unicode[STIX]{x1D70B}i)^{-k+1}\int _{\unicode[STIX]{x1D6F4}_{k-1}^{(n-1)}}\text{Res}(\unicode[STIX]{x1D714}).\end{eqnarray}$$

Proof. Recall from the proof of Theorem 4.9 that we have defined

$$\begin{eqnarray}\unicode[STIX]{x1D711}_{k}^{(n)}=p_{\text{B}}^{(n),\vee }(\unicode[STIX]{x1D713}_{k-1}^{(n-1)}),\end{eqnarray}$$

where $(\unicode[STIX]{x1D713}_{1}^{(n-1)},\ldots ,\unicode[STIX]{x1D713}_{n-1}^{(n-1)})$ is a basis of ${\mathcal{T}}_{\text{B}}^{(n-1),\vee }$ for which the period matrix is diagonal. In the light of Remark 4.19 we see that  $\unicode[STIX]{x1D713}_{k-1}^{(n-1)}$ is the class of the cycle  $\unicode[STIX]{x1D6F4}_{k-1}^{(n-1)}$ , hence we get

$$\begin{eqnarray}a_{k}(\unicode[STIX]{x1D714})=(2\unicode[STIX]{x1D70B}i)^{-k}\langle p_{\text{B}}^{(n),\vee }([\unicode[STIX]{x1D6F4}_{k-1}^{(n-1)}]),[\unicode[STIX]{x1D714}]\rangle =(2\unicode[STIX]{x1D70B}i)^{-k+1}\langle [\unicode[STIX]{x1D6F4}_{k-1}^{(n-1)}],p_{\text{dR}}^{(n)}([\unicode[STIX]{x1D714}])\,\rangle ,\end{eqnarray}$$

where the extra  $2\unicode[STIX]{x1D70B}i$ comes from the Tate twist at the target of  $p^{(n)}$ . Since  $\unicode[STIX]{x1D714}$ has a simple pole,  $p_{\text{dR}}^{(n)}([\unicode[STIX]{x1D714}])$ is simply the class of  $\text{Res}(\unicode[STIX]{x1D714})$ , hence the result.◻

5.1.3 Vanishing of coefficients

Theorem 5.5. For $\unicode[STIX]{x1D714}$ an integrable algebraic differential form on  $X_{n}-A_{n}$ , we have:

1. (i) if  $\unicode[STIX]{x1D70F}.\,\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}$ then  $a_{k}(\unicode[STIX]{x1D714})=0$ for  $k\neq 0$ even;

2. (ii) if  $\unicode[STIX]{x1D70F}.\,\unicode[STIX]{x1D714}=-\unicode[STIX]{x1D714}$ then  $a_{k}(\unicode[STIX]{x1D714})=0$ for  $k$ odd.

Proof. Let us assume that we have  $\unicode[STIX]{x1D70F}.\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}$ , and let us write  $x$ for the image of  $[\unicode[STIX]{x1D714}]$ in  ${\mathcal{T}}_{\text{dR}}^{(n-1)}$ . Then we have  $\unicode[STIX]{x1D70F}.x=x$ ; according to Proposition 4.5, this implies that  $x$ only has components of weights  $2k$ with  $k$ even. Thus,  $[\unicode[STIX]{x1D714}]\in {\mathcal{Z}}_{\text{dR}}^{(n)}$ only has components in weight  $0$ and  $2k$ with  $k$ odd, which implies that we have  $a_{k}(\unicode[STIX]{x1D714})=0$ for  $k\neq 0$ even. The second case is similar.◻

Let us write an integrable form as

(39) $$\begin{eqnarray}\unicode[STIX]{x1D714}={\displaystyle \frac{P(x_{1},\ldots ,x_{n})}{(1-x_{1}\cdots x_{n})^{N}}}\,dx_{1}\cdots dx_{n}\end{eqnarray}$$

with  $P(x_{1},\ldots ,x_{n})$ a polynomial with rational coefficients and  $N\geqslant 0$ an integer. Then we have

(40) $$\begin{eqnarray}\unicode[STIX]{x1D70F}.\unicode[STIX]{x1D714}=\pm \unicode[STIX]{x1D714}\;\;\Leftrightarrow \;\;P(x_{1},\ldots ,x_{n})=\pm (-1)^{N+n}(x_{1}\cdots x_{n})^{N-2}P(x_{1}^{-1},\ldots ,x_{n}^{-1}).\end{eqnarray}$$

5.2 The Ball–Rivoal integrals

We apply Theorems 5.1 and 5.5 to a special family of integrals.

Corollary 5.6. Let  $u_{1},\ldots ,u_{n},v_{1},\ldots ,v_{n}\geqslant 1$ and  $N\geqslant 0$ be integers such that  $v_{1}+\cdots +v_{n}\geqslant N+1$ . Then the integral

(41) $$\begin{eqnarray}\int _{[0,1]^{n}}{\displaystyle \frac{x_{1}^{u_{1}-1}\cdots x_{n}^{u_{n}-1}(1-x_{1})^{v_{1}-1}\cdots (1-x_{n})^{v_{n}-1}}{(1-x_{1}\cdots x_{n})^{N}}}\,dx_{1}\cdots dx_{n}\end{eqnarray}$$

is absolutely convergent and evaluates to a linear combination

$$\begin{eqnarray}a_{0}+a_{2}\unicode[STIX]{x1D701}(2)+a_{3}\unicode[STIX]{x1D701}(3)+\cdots +a_{n}\unicode[STIX]{x1D701}(n)\end{eqnarray}$$

with  $a_{k}$ a rational number for every  $k$ . If furthermore we have  $2u_{i}+v_{i}=N+1$ for every  $i$ , then we get:

1. (i) if  $(n+1)(N+1)$ is odd then  $a_{k}=0$ for  $k\neq 0$ even;

2. (ii) if  $(n+1)(N+1)$ is even then  $a_{k}=0$ for  $k$ odd.

Proof. This is a direct application of Theorem 5.5. The polynomial

$$\begin{eqnarray}P(x_{1},\ldots ,x_{n})=x_{1}^{u_{1}-1}\cdots x_{n}^{u_{n}-1}(1-x_{1})^{v_{1}-1}\cdots (1-x_{n})^{v_{n}-1}\end{eqnarray}$$

satisfies

$$\begin{eqnarray}P(x_{1},\ldots ,x_{n})=(-1)^{n+v_{1}+\cdots +v_{n}}x_{1}^{2u_{1}+v_{1}-3}\cdots x_{n}^{2u_{n}+v_{n}-3}P(x_{1}^{-1},\ldots ,x_{n}^{-1}).\end{eqnarray}$$

Let us assume that we have  $2u_{i}+v_{i}=N+1$ for every  $i$ , then  $v_{1}+\cdots +v_{n}\equiv n(N+1)\;(\text{mod}\;2)$ and we get

$$\begin{eqnarray}P(x_{1},\ldots ,x_{n})=-(-1)^{(n+1)(N+1)}(-1)^{N+n}(x_{1}\cdots x_{n})^{N-2}P(x_{1}^{-1},\ldots ,x_{n}^{-1}),\end{eqnarray}$$

hence the result, in view of (40). ◻

Corollary 5.6 applies in particular to the special case

$$\begin{eqnarray}N=(2r+1)m+2,\quad u_{i}=rm+1,\quad v_{i}=m+1\end{eqnarray}$$

for some integer parameters  $r,m\geqslant 0$ satisfying  $n(m+1)\geqslant (2r+1)m+3$ . We then recover the integrals considered by Ball and Rivoal [Reference Ball and RivoalBR01, Lemme 2]. The vanishing of the coefficients is [Reference Ball and RivoalBR01, Lemme 1]. The notations  $(a,n,r)$ in [Reference Ball and RivoalBR01] correspond to our notations  $(n-1,m,r)$ .

The integrals (41) can be expressed as generalized hypergeometric series

(42) $$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\biggl(\mathop{\prod }_{i=1}^{n}{\displaystyle \frac{(u_{i}-1)!(v_{i}-1)!}{(u_{i}+v_{i}-1)!}}\biggr)}\nolimits_{n+1}F_{n}\left(\begin{array}{@{}c@{}}u_{1},\ldots ,u_{n},N\\ u_{1}+v_{1},\ldots ,u_{n}+v_{n}\end{array};\displaystyle 1\right)\nonumber\\ \displaystyle & & \displaystyle \quad ={\displaystyle \frac{\mathop{\prod }_{i=1}^{n}(v_{i}-1)!}{(N-1)!}}\mathop{\sum }_{k\geqslant 0}{\displaystyle \frac{(k)_{u_{1}}\cdots (k)_{u_{n}}(k+1)_{N-1}}{(k)_{u_{1}+v_{1}}\cdots (k)_{u_{n}+v_{n}}}}.\end{eqnarray}$$

If  $2u_{i}+v_{i}=N+1$ , then the corresponding generalized hypergeometric series is said to be well-poised.

5.3 Weight drop

In the context of Theorem 5.1, we say that the integral  $\int _{[0,1]^{n}}\unicode[STIX]{x1D714}$ has weight drop if the highest weight coefficient  $a_{n}(\unicode[STIX]{x1D714})$ vanishes. This amounts to saying that the class  $[\unicode[STIX]{x1D714}]$ actually lives in the step  $W_{2(n-1)}{\mathcal{Z}}_{n,\text{dR}}$ of the weight filtration, hence the terminology. We give a sufficient condition for this phenomenon to happen.

Lemma 5.7. Let  $u,v\geqslant 1$ and  $N\geqslant 0$ be integers such that  $u+v\leqslant N$ . Then there exists a polynomial  $P(t)$ with rational coefficients such that

$$\begin{eqnarray}\int _{0}^{1}{\displaystyle \frac{x^{u-1}(1-x)^{v-1}}{(1-tx)^{N}}}\,dx={\displaystyle \frac{P(t)}{(1-t)^{N-v}}}\end{eqnarray}$$

for every  $0\leqslant t<1$ .

Proof. We can write

$$\begin{eqnarray}x^{u-1}(1-x)^{v-1}=\mathop{\sum }_{k=0}^{u+v-2}a_{k}(t)(1-tx)^{k}\end{eqnarray}$$

with  $a_{k}(t)$ a Laurent polynomial with rational coefficients for every  $k$ . We then have

$$\begin{eqnarray}{\displaystyle \frac{x^{u-1}(1-x)^{v-1}}{(1-tx)^{N}}}=\mathop{\sum }_{k=0}^{u+v-2}\frac{a_{k}(t)}{(1-tx)^{N-k}}\end{eqnarray}$$

and all the powers of  $(1-tx)$ appearing in the denominators are ${\geqslant}N-(u+v-2)\geqslant N-u-v+2\geqslant 2$ . Thus, we may integrate and get

$$\begin{eqnarray}\int _{0}^{1}{\displaystyle \frac{x^{u-1}(1-x)^{v-1}}{(1-tx)^{N}}}\,dx=\frac{Q(t)}{(1-t)^{N-1}}\end{eqnarray}$$

with  $Q(t)$ a Laurent polynomial with rational coefficients. The left-hand side has a limit when  $t$ tends to  $0$ , so  $Q(t)$ has to be a polynomial. To finish, it is enough to show that

$$\begin{eqnarray}(1-t)^{N-v}\int _{0}^{1}{\displaystyle \frac{x^{u-1}(1-x)^{v-1}}{(1-tx)^{N}}}\,dx\end{eqnarray}$$

is bounded when  $t$ approaches  $1$ . We make the change of variables  $s=1-t$ ,  $y=1-x$ , and consider integrals

$$\begin{eqnarray}s^{N-v}\int _{0}^{1}{\displaystyle \frac{(1-y)^{u-1}y^{v-1}}{(y+s-ys)^{N}}}\,dy\end{eqnarray}$$

with  $s$ approaching  $0$ . Since  $(1-y)^{u-1}\leqslant 1$ and  $y+s-ys\geqslant {\textstyle \frac{1}{2}}(y+s)$ , it is enough to prove that the quantities

$$\begin{eqnarray}s^{N-v}\int _{0}^{1}{\displaystyle \frac{y^{v-1}}{(y+s)^{N}}}\,dy\end{eqnarray}$$

are bounded when  $s$ approaches  $0$ . This equals

$$\begin{eqnarray}s^{N-v}\int _{0}^{1}\biggl({\displaystyle \frac{y}{y+s}}\biggr)^{v-1}{\displaystyle \frac{dy}{(y+s)^{N-v+1}}}\leqslant s^{N-v}\int _{0}^{1}{\displaystyle \frac{dy}{(y+s)^{N-v+1}}}={\displaystyle \frac{1}{N-v}}\biggl(1-\biggl({\displaystyle \frac{s}{1+s}}\biggr)^{N-v}\biggr)\end{eqnarray}$$

and we are done. ◻

Proposition 5.8. Let  $u_{1},\ldots ,u_{n},v_{1},\ldots ,v_{n}\geqslant 1$ and  $N\geqslant 0$ be integers such that  $v_{1}+\cdots +v_{n}\geqslant N+1$ . Let us assume that there exists an index  $i\in \{1,\ldots ,N\}$ such that

$$\begin{eqnarray}u_{i}+v_{i}\leqslant N.\end{eqnarray}$$

Then the integral

$$\begin{eqnarray}\int _{[0,1]^{n}}{\displaystyle \frac{x_{1}^{u_{1}-1}\cdots x_{n}^{u_{n}-1}(1-x_{1})^{v_{1}-1}\cdots (1-x_{n})^{v_{n}-1}}{(1-x_{1}\cdots x_{n})^{N}}}\,dx_{1}\cdots dx_{n}\end{eqnarray}$$

is absolutely convergent and evaluates to a linear combination

$$\begin{eqnarray}a_{0}+a_{2}\unicode[STIX]{x1D701}(2)+a_{3}\unicode[STIX]{x1D701}(3)+\cdots +a_{n-1}\unicode[STIX]{x1D701}(n-1)\end{eqnarray}$$

with  $a_{i}\in \mathbb{Q}$ for every  $i$ .

Proof. By symmetry, we can assume that  $u_{n}+v_{n}\leqslant N$ . Therefore, applying Lemma 5.7 to the variables  $x=x_{n}$ and  $t=x_{1}\cdots x_{n-1}$ in the integral leads to the  $(n-1)$ -dimensional integral

$$\begin{eqnarray}\int _{[0,1]^{n-1}}{\displaystyle \frac{x_{1}^{u_{1}-1}\cdots x_{n-1}^{u_{n-1}-1}(1-x_{1})^{v_{1}-1}\cdots (1-x_{n-1})^{v_{n-1}-1}P(x_{1}\cdots x_{n-1})}{(1-x_{1}\cdots x_{n-1})^{N-v_{n}}}}\,dx_{1}\cdots dx_{n-1}.\end{eqnarray}$$

Since  $v_{1}+\cdots +v_{n-1}\geqslant N-v_{n}+1$ , one can then finish thanks to Theorem 5.1.◻

Note that Proposition 5.8 applies in particular if for every  $i$ ,  $2u_{i}+v_{i}=N+1$ . This gives in particular a geometric interpretation of the weight drop in the Ball–Rivoal integrals [Reference RivoalRiv00, Reference Ball and RivoalBR01], which comes from the representations as hypergeometric series (42). Note that a careful analysis of the degree of the polynomial  $P(t)$ in Lemma 5.7 can lead to sufficient conditions for the vanishing of the subleading coefficients.