2.1 o-minimality

We shall be working throughout in the o-minimal structure $\mathbb {R}_{\mathrm {an},\exp }$ , see [Reference van den Dries29] for background. We shall use the following definable Chow theorem of Peterzil–Starchenko:

For an algebraic variety S with a local system $V_{\mathbb {C}}$ (on $S^{\mathrm {an}}$ ), the total space $\mathbb {V}$ has a natural definable structure (the ‘algebraic definable structure’) coming from its canonical algebraic structure. Also another definable structure on $\mathbb {V}^{\mathrm {an}}$ (the ‘flat definable structure’) is obtained by taking a definable cover of $S^{\mathrm {an}}$ by simply connected open sets and using flat coordinates. In our case, the two are the same by the following:

We shall use the following precise corollary to provide a definable fundamental domain for $\Sigma _{\mathcal {V}}$ .

We fix $\mathcal {F}\subset \widetilde {S^{\mathrm {an}}}$ to be an open definable analytic subspace with simply connected components with a surjective map $\phi :\mathcal {F}\rightarrow S^{\mathrm {an}}$ . Then by Proposition 2.3, we have that $\Sigma _{\phi ^*\mathcal {V}}\subset \phi ^*{\mathbb {I}}$ is definable, and therefore, so is its image $\mathcal {F}_{\mathcal {V}}$ in $\Sigma _{\mathcal {V}}$ . Observe that $\mathcal {F}_{\mathcal {V}}$ surjects onto $S^{\mathrm {an}}$ and thus $\Gamma \cdot \mathcal {F}_{\mathcal {V}} = \Sigma _{\mathcal {V}}$ .

2.2 Point counting on transcendental sets

Recall that the height of a rational point $\frac ab$ with $\gcd (a,b)=1$ is $H(\frac ab):=\max (|a|,|b|)$ . For a point $q=(q_1,\dots ,q_n)\in \mathbb {Q}^n$ , we define $H(q):=\max _i H(q_i)$ . Finally, for any subset $X\subset \mathbb {R}^n$ , we define

$$ \begin{align*}N(X,T):=\#\{q\in X\cap \mathbb{Q}^n\mid H(q)\leq T\}.\end{align*} $$

Given a set $X\subset \mathbb {R}^n$ , we set $X^{\mathrm {alg}}$ to be the union of all connected, positive dimensional, semi-algebraic subsets of X. Then we have the following theorem of Pila–Wilkie:

Theorem 2.4 (Pila–Wilkie [Reference Pila and Wilkie28, Thm 1.8]).

For a set $X\subset \mathbb {R}^n$ definable in an o-minimal structure, and any $\epsilon>0$ , we have

$$ \begin{align*}N(X-X^{\mathrm{alg}},T) = T^{o(1)}.\end{align*} $$

In fact, we shall need the slightly stronger version:

2.3 Mumford–Tate groups

We follow the conventions of [Reference André1]. For a rational mixed Hodge structure $V=(V_{\mathbb {Q}},W_\bullet V,F^\bullet V)$ , recall that the group of weight zero Hodge classes of V is $W_0V\cap F^0V$ . We define the Mumford–Tate group of V to be the subgroup of $\mathbf {GL}(V_{\mathbb {Q}})$ stabilizing each weight zero Hodge tensor in all tensor powers $V^{\otimes m}\otimes (V^{\vee })^{\otimes n}$ .

2.4 Flat torsors

Let $\mathbf {G}$ be a complex algebraic group. Recall that an algebraic $\mathbf {G}$ -torsor over a smooth variety S is an algebraic S-variety $\pi :P\to S$ equipped with an algebraic left action by $\mathbf {G}$ such that the induced map

$$\begin{align*}\mathbf{G}\times_S P\to P\times_S P,\;\;\; (g,x)\mapsto (gx,x)\end{align*}$$

is an isomorphism. This means $\mathbf {G}(\mathbb {C})$ acts simply transitively on the fibers of $P\to S$ .

An algebraic flat connection on P is an algebraic splitting of the extension

$$\begin{align*}0\to T_{P/S}\to T_P\to \pi^*T_S\to 0\end{align*}$$

which is $\mathbf {G}$ -invariant and with the property that the induced map $\pi ^* T_S\to T_P$ is a foliation. The leaves of P are the leaves $\mathcal {L}$ of this foliation.

Given an algebraic flat $\mathbf {G}$ -torsor $\pi :P\to S$ , choosing a point $p_0\in P$ and setting $s_0=\pi (p_0)$ , we obtain a monodromy representation $\rho :\pi _1(S,s_0)\to \mathbf {G}(\mathbb {C})$ by solving the connection. This data determines $\pi : P\to S$ analytically, as we recover

$$\begin{align*}P^{\mathrm{an}}=(\widetilde{S^{\mathrm{an}}}\times \mathbf{G}(\mathbb{C}))/\Gamma,\end{align*}$$

where $\gamma \in \Gamma $ acts via the canonical right action on the first factor and via right multiplication by $\rho (\gamma )$ on the second. Moreover, the constant sections $\widetilde {S^{\mathrm {an}}}\times g$ map to the leaves of P. We define the full algebraic monodromy group (resp. algebraic monodromy group) of P to be the Zariski closure (resp. identity component of the Zariski closure) of the image of $\rho $ in $\mathbf {G}$ .

2.5 Period domains and period maps

We recall some definitions regarding weak Mumford–Tate domains. See [Reference Green, Griffiths and Kerr19, Reference Bakker, Brunebarbe, Klingler and Tsimerman7] for details.

Let $\mathcal {D}_0$ be a period domain of graded-polarized integral mixed Hodge structures with generic Mumford–Tate group $\mathbf {G}_0$ . For any point $p\in \mathcal {D}$ , let $\mathbf {G}$ be a normal $\mathbb {Q}$ -subgroup of its Mumford–Tate group ${\mathbf {MT}}_p$ and $\mathbf {U}$ the unipotent radical of $\mathbf {G}$ . The orbit $\mathcal {D}:=\mathbf {G}(\mathbb {R})\mathbf {U}(\mathbb {C})\cdot p$ is a closed complex subspace of $\mathcal {D}_0$ whose generic Mumford–Tate group is ${\mathbf {MT}}_p$ . We call such a subspace a weak Mumford-Tate (sub)domain of $\mathcal {D}_0$ . Each such $\mathcal {D}$ is naturally contained as a semialgebraic subset in a complex algebraic variety $\check {\mathcal {D}}$ called its dual.

The quotient $\mathbf {G}(\mathbb {Z})\backslash \mathcal {D}$ has the natural structure of a definable analytic variety, and for any period map $\tilde \phi :S^{\mathrm {an}}\to \mathbf {G}_0(\mathbb {Z})\backslash \mathcal {D}_0$ , the inverse image of $\mathbf {G}(\mathbb {Z})\backslash \mathcal {D}$ is an algebraic subvariety of S [Reference Bakker, Brunebarbe, Klingler and Tsimerman7]. We call each component a weak Mumford–Tate subvariety of S. Note that by Theorem 2.6 and the above definitions, we have the following important property:

Suppose $\mathcal {V}=(V_{\mathbb {Z}},W_\bullet V_{\mathbb {Q}},F^\bullet V)$ is an admissible variation of graded-polarizable integral mixed Hodge structures, and let $\phi :\widetilde {S^{\mathrm {an}}}\to \mathcal {D}$ be the associated period map, where $\pi :\widetilde { S^{\mathrm {an}}}\to S$ is the minimal cover trivializing $V_{\mathbb {Z}}$ . Consider the map $\pi \times \phi :\widetilde {S^{\mathrm {an}}}\to S^{\mathrm {an}}\times \mathcal {D}$ , which is a closed embedding of a component of $S^{\mathrm {an}}\times _{\mathbf {G}(\mathbb {Z})\backslash \mathcal {D}}\mathcal {D}$ . For the next section, we observe that $\pi \times \phi |_{\mathcal {F}}$ is definable by [Reference Bakker, Brunebarbe, Klingler and Tsimerman7, Theorem 1.1], where $\mathcal {F}$ is as in §2.1.

3.1 First proof

We first give a proof using the Ax–Schanuel for principal bundles of [Reference Blásquez-Sans, Casales, Freitag and Nagloo11]. For this, we shall need the following definition:

Theorem 3.3 [Reference Blásquez-Sans, Casales, Freitag and Nagloo11, Thm. A].

Suppose $\mathbf {G}$ is sparse. Let $\pi :P\to S$ be an algebraic flat $\mathbf {G}$ -torsor, $W\subset P$ a subvariety, $\mathcal {L}$ a leaf of P, and U a component of $W^{\mathrm {an}}\cap \mathcal {L}$ . If

$$\begin{align*}\operatorname{codim}_W U<\dim \mathbf{G},\end{align*}$$

then $\pi (U)^{\mathrm {Zar}}\subset S$ has algebraic monodromy of strictly smaller dimension than $\mathbf {G}$ .

Example 3.5. Let $S=A$ be a simple abelian surface and let $\omega $ be a nonzero differential 1-form. Consider the local system with monodromy

$$\begin{align*}\pi_1(A,0)\to\mathbf{G}_m^2,\;\;\;\gamma\mapsto \left(e^{\int_\gamma\omega},e^{\lambda\int_\gamma\omega}\right).\end{align*}$$

The associated $\mathbf {G}_m^2$ -torsor $\pi :P \to A$ (equipped with its canonical algebraic structure) has a section $s:A\to P$ whose lift is given by

$$\begin{align*}\tilde s:\widetilde {A^{\mathrm{an}}}\mapsto \mathbf{G}_m^2,\;\;\; a\mapsto \left(e^{\int_0^a\omega},e^{\lambda\int_0^a\omega}\right),\end{align*}$$

which is algebraic by GAGA. For $\lambda $ irrational, the fibers of $\tilde s$ are one-dimensional and project to the intersections $\mathcal {L}\cap s(A)$ , which are therefore one-dimensional as well. The algebraic monodromy is all of $\mathbf {G}_m^2$ , but if a fiber were not Zariski dense in A, it would necessarily be an elliptic curve factor of A.

3.2 Second proof

In this section, we describe the proof of Theorem 1.3 using o-minimality in the spirit of [Reference Mok, Pila and Tsimerman22, Reference Bakker and Tsimerman9, Reference Chiu13, Reference Gao and Klingler18].

With the notation as in the setup of Theorem 1.3, we prove the conclusion by induction on the triple $(\dim S,\dim W-\dim U,-\dim U)$ with the lexicographical ordering, the base case of $\dim S=0$ being trivial. We thus assume that the theorem is valid for all lexicographically previous triples. Suppose we have a $W\subset \Omega _{\mathcal {V}}$ as in the statement of the theorem and a component U of $W\cap \Sigma _{\mathcal {V}}$ whose projection $\pi (U)$ is not contained in a proper weak Mumford–Tate subvariety of S. Note that $\pi (U)$ is Zariski dense in S by the inductive hypothesis, as otherwise, we could replace S with the Zariski closure $S'$ of $\pi (U)$ and W with the intersection $W\cap \Omega _{\mathcal {V}_{S'}}$ .

Recall that $\mathbf {G}(\mathbb {C})$ acts on $\Omega _{\mathcal {V}}$ (algebraically) by post-composition. Let $\Gamma \subset \mathbf {G}(\mathbb {Q})$ be the image of the monodromy representation of $V_{\mathbb {Z}}$ (possibly after replacing S with a finite cover). As in [Reference Mok, Pila and Tsimerman22, §3], we consider the component M of the Hilbert scheme $\operatorname {Hilb}\left (\overline {\Omega }_{\mathcal {V}}\right )$ containing the closure of W for some equivariant algebraic compactification $\overline {\Omega }_{\mathcal {V}}$ of $\Omega _{\mathcal {V}}$ . Let $\mathcal {W}\subset \Omega _{\mathcal {V}}\times M$ be the universal family and $\mathcal {W}_{\Sigma }$ the intersection with $\Sigma _{\mathcal {V}}\times M^{\mathrm {an}}$ , which is $\Gamma $ -invariant and proper over $\Sigma _{\mathcal {V}}$ . The quotient $\mathcal {U}:=\Gamma \backslash \mathcal {W}_{\Sigma }$ is naturally a definable analytic variety as follows. Letting $\mathcal {F}_{\mathcal {V}}\subset \Sigma _{\mathcal {V}}$ be the subset from §2.1, we set $\mathcal {G}_{\mathcal {V}}= \mathcal {F}_{\mathcal {V}}\times M^{\mathrm {an}}$ , which is an open definable fundamental set for $\Gamma $ on $\Sigma _V\times M^{\mathrm {an}}$ . Let $\sim $ be the induced definable étale equivalence relation on $\mathcal {G}_{\mathcal {V}}$ , and we take the definable structure induced by the identification $\mathcal {U}\cong (\mathcal {W}_{\Sigma }\cap \mathcal {G}_{\mathcal {V}})/\sim $ .

The natural map $\mathcal {U}\to S^{\mathrm {an}}$ is proper definable analytic. We think of $\mathcal {U}$ as parametrizing $\Gamma $ -orbits of pairs $(W',p)$ with $[W']\in M$ and $p\in W^{\prime \mathrm {an}}\cap \Sigma _{\mathcal {V}}$ . There is a closed $\Gamma $ -invariant definable analytic subvariety $A\subset \mathcal {U}$ parametrizing pairs $(W',p)$ with $\dim _p(W^{\prime \mathrm {an}}\cap \Sigma _{\mathcal {V}})\geq \dim U$ . If $A_0$ is an irreducible component of A containing $(W,p)$ for all $p\in U$ , then $A_0$ descends to a closed definable analytic subvariety $B_0\subset \mathcal {U}$ by taking the quotient $B_0:=(A_0\cap \mathcal {G}_{\mathcal {V}})/\sim $ .

We have a natural proper definable analytic map $q:B_0\to S^{\mathrm {an}}$ . By Theorem 2.1, the image is algebraic, and therefore, q is surjective by the inductive hypothesis and the properness of the map. Note that the monodromy $\Gamma _0$ of the pullback $q^*V_{\mathbb {Z}}$ stabilizes $A_0$ . Since $B_0$ surjects onto S, we have that $\Gamma _0\subset \Gamma $ is finite index, so the identity component of the $\mathbb {Q}$ -Zariski closure of $\Gamma _0$ is $\mathbf {G}$ .

Letting $Y\subset M^{\mathrm {an}}$ be the projection of $A_0$ , we let $H_{\mathrm {gen}}$ be the stabilizer of a very general point of Y, and $\mathbf {H}$ the identity component of its $\mathbb {Q}$ -Zariski closure. Note that every point of Y – in particular, $[W]$ – is stabilized by $\mathbf {H}(\mathbb {C})$ . Moreover, since $\Gamma _0$ sends a very general point to a very general point, it follows that $H_{\mathrm {gen}}$ is normalized by $\Gamma _0$ , and hence, $\mathbf {H}\subset \mathbf {G}$ is normal.

Proof. Let $Z=S^{\mathrm {an}}\times _{\mathbf {G}(\mathbb {Z})\backslash \mathcal {D}}\mathcal {D}\subset S^{\mathrm {an}}\times \check {\mathcal {D}}^{\mathrm {an}}$ where $\mathcal {D}$ is the relevant weak Mumford–Tate domain and $\check {\mathcal {D}}$ its dual. Let $F\subset Z$ be a definable fundamental set for the action of $\mathbf {G}(\mathbb {Z})$ . We have the following result of Chiu [Reference Chiu13]:

Proposition 3.7. Let $\mathbf {H}\subset \mathbf {G}$ be a normal $\mathbb {Q}$ -subgroup and $K\subset Z$ a closed irreducible complex analytic subvariety which is stabilized by $\mathbf {H}(\mathbb {Z})$ and for which $K\cap \gamma F$ is definable for any $\gamma \in \mathbf {G}(\mathbb {Z})$ . Let

$$\begin{align*}J=\{\gamma\in\mathbf{G}(\mathbb{Z})\mid K\cap \gamma F\neq\varnothing\}\end{align*}$$

and note that $\mathbf {H}(\mathbb {Z})$ acts on J. Then either $\mathbf {H}(\mathbb {Z})\backslash J$ is finite or has polynomially many integer points.

Proof. This is proven in [Reference Chiu13]. The case that $\mathbf {H}(\mathbb {Z})\backslash J$ has polynomially many points corresponds to cases (1) and (2) in the trichotomy at the end of Section 7 in [Reference Chiu13], and is proven in Sections 8 and 9, respectively. The case that $\mathbf {H}(\mathbb {Z})\backslash J$ is finite is case (3).

Chiu proves it for the specific set U in his notation, but the proof works verbatim for an arbitrary K as in the statement of Proposition 3.7.

We apply this proposition to $\Sigma _{\mathcal {V}}$ , which is identified with a component of Z as in §2.5, using $K=W^{\mathrm {an}}\cap \Sigma _{\mathcal {V}}$ and $F=\mathcal {F}_{\mathcal {V}}$ . Consider

$$\begin{align*}I=\{g\in\mathbf{G}(\mathbb{R})\mid \dim(g^{-1}W^{\mathrm{an}}\cap \mathcal{F}_{\mathcal{V}})=\dim U\}.\end{align*}$$

Note that the J in Proposition 3.7 is a subset of $I(\mathbb {Z})$ . Moreover, I is definable. Thus, by Theorem 2.4, either

1. 1. I contains a semi-algebraic curve C with non-constant image in $\mathbf {H}(\mathbb {R})\backslash I$ , or

2. 2. $\mathbf {H}(\mathbb {Z})\backslash J$ is finite.

Suppose first that we are in case (1). Then I contains a semialgebraic curve C. We claim that W is not stabilized by C. If it were, then it would be stabilized by $C\cdot C^{-1}$ , which contains an integer point not in $\mathbf {H}(\mathbb {Z})$ (in fact, we can arrange it to contain arbitrarily many by the conclusion of the stronger 2.5). This is a contradiction by the definition of $\mathbf {H}$ .

Therefore, $c^{-1}W$ varies with $c\in C$ . If $c^{-1}U$ does not vary with $c\in C$ , then we may replace W with $W\cap c^{-1}W$ for a generic element c and obtain a lexicographically smaller counterexample. Else, if $c^{-1}U$ does vary with $c\in C$ , we may replace W with $C^{\mathrm {Zar}}W$ and obtain a lexicographically smaller counterexample.

Thus, we may assume we are in case (2). In this case, it follows that the image $\pi (U)$ of U in S is definable, and hence algebraic by Theorem 2.1. Since the monodromy of $\pi (U)$ and U are the same, if we have $\mathbf {H}\neq \mathbf {G}$ , then $\pi (U)$ would be contained in a proper weakly special subvariety, which contradicts the assumption on U.

We may therefore suppose W is $\mathbf {G}$ -invariant, but then it is obvious that $\operatorname {codim}_WU\geq \dim \mathbf {G}$ , and this contradiction proves the theorem.

4.1 Outline

In this section, we prove Theorem 1.1, whose statement is formulated in terms of functions fields of complex algebraic varieties. Nori’s category of motives is defined only for subfields of $\mathbb {C}$ , and due to its reliance on a Betti realization functor, the generalization of his construction to complex function fields requires a little care. In §4.2, we recall the original construction of Nori, and in §4.3, we precisely state the classical Grothendieck period conjecture and its generalization due to André. In §4.4, we make the necessary modifications to Nori’s construction, and in §4.5, we relate the relative motivic Galois group to the algebraic monodromy. In §4.6, we relate the torsor of comparisons between Betti and de Rham realizations to our period torsor $\Omega _{\mathcal {V}}$ . In §4.7, we show has this perspective gives a simple perspective on the geometric Kontsevich–Zagier conjecture. Finally in §4.8, we formulate and prove the precise version of Theorem 1.1. The reader who is willing to assume a reasonable category of motives over a complex function field together with the statement of Theorem 4.17 can skip directly to the proof.

4.2 Nori motives

In this section, we briefly recall Nori motives, which will provide for us a Tannakian category of motives in both the classical and functional setting, and therefore a motivic Galois group. Ayoub [Reference Ayoub6] (see also [Reference Ayoub4]) takes a slightly different approach, using Voevodsky’s theory to define such a group directly, but they are canonically the same [Reference Choudhury and Gallauer Alves de Souza15]. Essentially, for any subfield $k\subset \mathbb {C}$ , the category of Nori k-motives will be the abelian subcategory of $\mathbb {Q}\mbox {-mod}$ generated by singular cohomology groups $H^i((X_{\mathbb {C}})^{\mathrm {an}},\mathbb {Q})$ of k-varieties X together with all morphisms that can be constructed naturally from maps of k-varieties. The main reference is [Reference Huber and Müller-Stach20].

By a diagram D, we mean a directed graphFootnote ⁴ with the obvious notion of morphism. Note that for any category $\mathcal {C}$ , there is a natural underlying diagram, and for every functor $\mathcal {C}\to \mathcal {C}'$ an underlying morphism of diagrams. A representation $F:D\to \mathcal {C}$ of a diagram D in a category $\mathcal {C}$ is a morphism on the level of diagrams. Concretely, F assigns an object of $\mathcal {C}$ to each vertex of D, and a morphism of $\mathcal {C}$ to each edge of D with the obvious compatibility on the source and target.

Let k be a field with an embedding $\iota :k\to \mathbb {C}$ . The diagram $\mathrm {Pairs}^{\mathrm {eff}}(k,\iota )$ has as vertices triples $(X,Y,i)$ with X an algebraic variety over k, $Y\subset X$ a closed subvariety (defined over k), and $i\in \mathbb {Z}$ . The edges of $\mathrm {Pairs}^{\mathrm {eff}}$ consist of

• • for each morphism $f:X\to X'$ with $f(Y)\subset Y'$ and integer i, there is an edge $f^*:(X',Y',i)\to (X,Y,i)$ ;

• • for each chain $X\supset Y\supset Z$ of varieties (the inclusions being of closed subvarieties) and each integer i, there is an edge $\partial :(Y,Z,i)\to (X,Y,i+1)$ .

Betti cohomology $(X,Y,i)\mapsto H^i((X_{\mathbb {C}})^{\mathrm {an}},(Y_{\mathbb {C}})^{\mathrm {an}},\mathbb {Q})$ defines a natural representation

$$\begin{align*}\mathrm{Betti}_\iota:\mathrm{Pairs}^{\mathrm{eff}}(k)\to \mathbb{Q}\mbox{-mod}\end{align*}$$

where the edge $f^*$ is sent to the pullback via f and $\partial $ is sent to the coboundary map in the long exact sequence of the triple.

Note in particular that we have $H_\iota \circ \mathcal {H}_\iota =\mathrm {Betti}_\iota $ . We refer to $\mathcal {MM}_{\mathrm {Nori}}^{\mathrm {eff}}(k,\iota )$ (resp $\mathcal {MM}_{\mathrm {Nori}}(k,\iota )$ ) as the category of effective Nori $(k,\iota )$ -motives (resp. Nori $(k,\iota )$ -motives).

4.3 Classical period conjectures

It will be useful to have a category of pairs of vector spaces equipped with a comparison over a fixed field extension. Let $\iota :k\to L$ be an embedding of characteristic 0 fields. Define $(k,\mathbb {Q})_\iota \mbox {-mod}$ to be the category of triples $(U,V,\phi )$ of a k-vector space U a $\mathbb {Q}$ -vector space V, and an isomorphism $\phi :U\otimes _k L\to V\otimes _{\mathbb {Q}} L$ with the obvious notion of morphism.

For a field k and an embedding $\iota :k\to \mathbb {C}$ , we have a natural representation

$$\begin{align*}\mathrm{Pairs}^{\mathrm{eff}}(k)\to (k,\mathbb{Q})_\iota\mbox{-mod}\end{align*}$$

given by sending $(X,Y,i)$ to $(H_{DR}^i(X,Y),H^i(X,Y,\mathbb {Q}),\phi _{X,Y,i})$ , where $\phi _{X,Y,i}$ is the natural comparison given by integration. By the universal property, this extends to a functor

$$\begin{align*}\mathcal{MM}_{\mathrm{Nori}}^{\mathrm{eff}}(k,\iota)\to (k,\mathbb{Q})_\iota\mbox{-mod},\;\;\;\;M\mapsto (H_{DR}(M),H_\iota(M),\phi_M),\end{align*}$$

which in turn extends to a functor

$$\begin{align*}\mathcal{MM}_{\mathrm{Nori}}(k,\iota)\to (k,\mathbb{Q})_\iota\mbox{-mod},\;\;\;\;M\mapsto (H_{DR}(M),H_\iota(M),\phi_M).\end{align*}$$

Definition 4.3. For $\iota :k\to \mathbb {C}$ a field embedding and M a Nori $(k,\iota )$ -motive, we define

$$\begin{align*}k(\mathrm{periods}_\iota\;\mathrm{of}\;M)\subset\mathbb{C}\end{align*}$$

to be the field of definition of the comparison $\phi _M$ .

Concretely, $k(\mathrm {periods}_\iota \;\mathrm {of}\;M)$ is obtained by adjoining the periods of k-rational de Rham classes of M to k.

Conjecture 4.1 (Grothendieck period conjecture).

Let $\iota :k\to \mathbb {C}$ be an embedding of a number field and M a Nori $(k,\iota )$ -motive. Then

$$\begin{align*}\operatorname{trdeg}_{\mathbb{Q}} k(\mathrm{periods}_\iota\;\mathrm{of}\;M)=\dim\mathbf{G}_{\mathrm{mot}}(M,\iota).\end{align*}$$

One downside of the Grothendieck period conjecture is that it does not imply the other major conjecture about transcendence: the Schanuel conjecture about exponentials. André proposed the following strengthening to address this:

Conjecture 4.2 (André–Grothendieck period conjecture).

Let $\iota :k\to \mathbb {C}$ be any field embedding and M a Nori $(k,\iota )$ -motive. Then

$$\begin{align*}\operatorname{trdeg}_{\mathbb{Q}} k(\mathrm{periods}_\iota\;\mathrm{of}\;M)\geq\mathbf{G}_{\mathrm{mot}}(M,\iota).\end{align*}$$

Example 4.4. Let $\alpha _1,\dots ,\alpha _n\in \mathbb {C}^*$ be multiplicatively independent, and consider $X=\mathbf {G}_m$ and $ Y=\{1, \alpha _1,\dots ,\alpha _n\}\subset \mathbf {G}_m$ over $k=\mathbb {Q}(\alpha _1,\dots ,\alpha _n)$ . Then $H^1_{DR}(X,Y)$ is spanned by $\frac {dx}{x}$ , and the differences $[1]^\vee -[\alpha _i]^\vee $ , whereas $H_1(X,Y,\mathbb {Q})$ is spanned by paths between $1$ and the different $\alpha _i$ , and the loop around $0$ . Thus, the integrals are all integers, as well as $2\pi i, \log \alpha _1,\ldots ,\log \alpha _n$ . Hence, in this case, André’s conjecture says that

$$ \begin{align*}\operatorname{trdeg}_{\mathbb{Q}}(2\pi i, \alpha_1,\ldots,\alpha_n,\log \alpha_1,\ldots, \log\alpha_n)\geq \dim \mathbf{G}_{\mathrm{mot}}(\mathcal{H}^1_\iota(X,Y)).\end{align*} $$

Also André shows [Reference André2] that $\mathbf {G}_{\mathrm {mot}}(\mathcal {H}^1_\iota (X,Y))$ is the same as the Mumford–Tate group of the mixed Hodge structure $H^1(X,Y)$ – and this is easily computed to be an extension of $\mathbf {G}_m$ by n copies of $\mathbf {G}_a$ . Thus, André’s conjecture says that

$$ \begin{align*}\operatorname{trdeg}_{\mathbb{Q}}(2\pi i, \alpha_1,\ldots,\alpha_n,\log \alpha_1,\dots, \log\alpha_n)\geq n+1\end{align*} $$

and therefore that

$$ \begin{align*}\operatorname{trdeg}_{\mathbb{Q}}(\alpha_1,\ldots,\alpha_n,\log \alpha_1,\dots, \log\alpha_n)\geq n\end{align*} $$

which is precisely the statement of Schanuel’s conjecture.

4.4 Nori motives in the functional setting

In the functional setting, we would like to replace k with the function field of a complex algebraic variety, in which case the Betti realization of a Nori motive M defined over k should be the Betti cohomology of a generic fiber of M once we spread M out over a model S of k. In this section, we make these ideas precise.

Throughout, we often take $\iota _0:k_0\to \mathbb {C}$ to be a field embedding and $k_0\subset k$ a finitely generated extension. By analytification, we always mean analytification as $k_0$ -varieties, unless otherwise specified.

Note that classical points may be thought of as arc points via the constant maps.

For an arc point of $S^{\mathrm {an}}$ satisfying the conditions in the lemma, we say that the arc point induced on any other model is stable.

In the next section, we will need the following notion:

For X a k-variety, $Y\subset X$ a closed k-subvariety, and $\gamma $ a $(k_0,\iota _0)$ -arc point of k, we define

$$\begin{align*}H^i_\gamma(X,Y):=H^i(X^{\mathrm{an}}_{\gamma},Y^{\mathrm{an}}_\gamma,\mathbb{Q}).\end{align*}$$

Here, we spread out $X,Y$ over a model S of k, shrink S so that $H^i(X^{\mathrm {an}}_s,Y^{\mathrm {an}}_s)$ forms a local system, and represent the arc point by $\gamma :(0,a)\to S^{\mathrm {an}}$ . Then $X^{\mathrm {an}}_\gamma $ is defined to be the inverse image of $\operatorname {img}\gamma $ in $X^{\mathrm {an}}$ . This naturally yields a representation

$$\begin{align*}\mathrm{Betti}_\gamma:\mathrm{Pairs}^{\mathrm{eff}}(k)\to \mathbb{Q}\mbox{-mod},\;\;\;\; (X,Y,i)\mapsto H^i_\gamma(X,Y),\end{align*}$$

where the edge $f^*$ is sent to the pullback via f and $\partial $ is sent to the coboundary map in the long exact sequence of the triple.

Proof. Consider first the case that k (and therefore $k_0$ ) is countable. By Lemma 4.12, any two arc points have a homotopy class of paths between them, so it is sufficient to consider a single point by Lemma 4.13, and this is the case of Theorem 4.1.

Next, consider the general case of part (a). From the diagram category construction of [Reference Huber and Müller-Stach20, §7], we obtain from the representation $H_{\gamma }:\mathrm {Pairs}^{\mathrm {eff}}(k)\to \mathbb {Q}\mbox {-mod}$ a $\mathbb {Q}$ -linear abelian category $\mathcal {MM}_{\mathrm {Nori}}^{\mathrm {eff}}(k,\gamma )$ with representation $\mathcal {H}_{\gamma _k}:\mathrm {Pairs}^{\mathrm {eff}}(k)\to \mathcal {MM}_{\mathrm {Nori}}^{\mathrm {eff}}(k,\gamma )$ and a faithful exact $\mathbb {Q}$ -linear functor $H_{\gamma }:\mathcal {MM}_{\mathrm {Nori}}^{\mathrm {eff}}(k,\gamma )\to \mathbb {Q}\mbox {-mod}$ satisfying property (1) of Theorem 4.1.

Consider the directed set I of countable subfields $\ell \subset k$ and let $\ell _0=\ell \cap k_0$ . The arc point $\gamma $ induces an arc point $\gamma _\ell $ of $\ell $ . For an inclusion $\ell \subset \ell '$ , the natural base-change morphism $\mathrm {Pairs}^{\mathrm {eff}}(\ell )\to \mathrm {Pairs}^{\mathrm {eff}}(\ell ')$ of diagrams yields a base-change functor $\mathcal {MM}_{\mathrm {Nori}}^{\mathrm {eff}}(\ell ,\gamma _\ell )\to \mathcal {MM}_{\mathrm {Nori}}^{\mathrm {eff}}(\ell ',\gamma _{\ell '})$ . As every variety over k is defined over some $\ell $ , we naturally have

$$\begin{align*}\mathrm{Pairs}^{\mathrm{eff}}(k)=\underset{\ell\in I}{\operatorname{colim}}\,\mathrm{Pairs}^{\mathrm{eff}}(\ell)\end{align*}$$

as diagrams, and moreover,

$$\begin{align*}H_{\gamma_k}=\underset{\ell\in I}{\operatorname{colim}}\,H_{\gamma_\ell}\end{align*}$$

as representations. By the universal property, we then have a canonical identification

$$\begin{align*}\mathcal{MM}_{\mathrm{Nori}}^{\mathrm{eff}}(k,\gamma)=2\text{-}\underset{\ell\in I}{\operatorname{colim}}\,\mathcal{MM}_{\mathrm{Nori}}^{\mathrm{eff}}(\ell,\gamma_\ell).\end{align*}$$

Indeed, the diagram category is constructed as a 2-colimit over finite subdiagrams. Properties (2) and (3) then follow. Property (4) is by Tannakian duality, though in this case, we can directly see that (4) holds with $\mathbf {G}_{\mathrm {mot}}(k,\gamma ):=\lim _{\ell \in I}\mathbf {G}_{\mathrm {mot}}(\ell ,\gamma _\ell )$ .

Part (b) again follows from the corresponding statement in the countable case.

It follows naturally from the construction that paths between stable arc points give compatible isomorphisms between the $\pi _1$ groups, the functors $H_\gamma $ , the categories of Nori motives, and the motivic Galois groups.

4.5 The relative motivic Galois group

In this section, we relate the relative motivic Galois group (over a point) to the algebraic monodromy group. For k a subfield of $\mathbb {C}$ , this is a result of Ayoub [Reference Ayoub4] (in a different but equivalent context by [Reference Choudhury and Gallauer Alves de Souza15], as mentioned above) and of Nori (unpublished). The details of the latter argument have recently been worked out by Mostaed [Reference Mostaed23].

As in the previous section, let $k_0\subset k$ be a finitely generated field extension such that $k_0$ is algebraically closed in k and let $\iota _0:k_0\to \mathbb {C}$ be a field embedding. Let $\gamma $ be a $(k_0,\iota _0)$ -arc point of k.

Denote by ${\mathrm {LocSys}}_{\mathbb {Q}}(k,\gamma )$ the category of finite-dimensional $\mathbb {Q}$ -representations of $\pi _1((k/\mathbb {C})^{\mathrm {an}},\gamma )$ , which is equivalently the category of compatible systems of $\mathbb {Q}$ -local systems on sufficiently small models. The category ${\mathrm {LocSys}}_{\mathbb {Q}}(k,\gamma )$ is naturally a neutral Tannakian category, whose fiber functor is the restriction to $\gamma $ . Concretely, the Tannakian group of the subcategory generated by an object L is the Zariski closure of the image of the monodromy representation. We denote the full Tannakian group of ${\mathrm {LocSys}}_{\mathbb {Q}}(k,\gamma )$ by $\Pi _1(k,\gamma )$ .

We have a natural sequence of functors of neutral Tannakian categories (that is, tensor functors respecting the fiber functor)

$$\begin{align*}\mathcal{MM}_{\mathrm{Nori}}^{\mathrm{eff}}(\mathbb{C},\operatorname{id})\to\mathcal{MM}_{\mathrm{Nori}}^{\mathrm{eff}}(k,\gamma)\xrightarrow{\mathscr{H}_\gamma} {\mathrm{LocSys}}_{\mathbb{Q}}(k,\gamma)\end{align*}$$

the first given by base-change from $\mathbb {C}$ to k as in Proposition 4.14 and the second the functor associated via the universal property to the representation of $\mathrm {Pairs}^{\mathrm {eff}}(k,\gamma )$ which sends $(X,Y,i)$ to the local system $\mathscr {H}^i_\gamma (X,Y)$ whose fiber over s is $H^i(X_s,Y_s,\mathbb {Q})$ , for a sufficiently small model of S. This is a easily checked to be a tensor functor.

Theorem 4.16 (Ayoub [Reference Ayoub4, Théorème 2.57], Nori, Mostaed [Reference Mostaed23]).

The resulting sequence of pro-algebraic groups

$$\begin{align*}\Pi_1(k,\gamma)\to\mathbf{G}_{\mathrm{mot}}(k,\gamma)\to\mathbf{G}_{\mathrm{mot}}(k_0,\iota_0)\to 1\end{align*}$$

is exact.

4.6 The comparison torsor

From now on, we take $k_0=\mathbb {C}$ and let k be a finite generated extension of $\mathbb {C}$ , and $\gamma $ a $(\mathbb {C},\operatorname {id})$ -arc point of k. In this section, we review the construction of the torsor of comparisons between the Betti and de Rham fiber functors in the context of function fields over $\mathbb {C}$ as in [Reference Huber and Müller-Stach20]. We then identify the comparison torsor with the torsor $\Omega _{\mathcal {V}}$ constructed in the introduction.

Our comparision map between Betti and de Rham cohomology will no longer be over $\mathbb {C}$ but instead over a larger field of germs of meromorphic functions. We therefore make the following definition:

Definition 4.18. Let $\mathcal {S}$ be a complex manifold and $\gamma $ an arc point of $\mathcal {S}$ . We define the localization $\mathcal {O}_{\mathcal {S},\gamma }$ to be

$$ \begin{align*}\mathcal{O}_{\mathcal{S},\gamma}:=\lim_{\gamma\subset U} \mathcal{O}_{\mathcal{S}}(U),\end{align*} $$

where the limit is over all open subsets through which the arc point factors.

We define $k^{\mathrm {an}}_{\gamma }$ to be the fraction field of $\mathcal {O}_{S^{\mathrm {an}},\gamma }$ for any model S of k. It is immediate that this is independent of the model.

For any vertex $(X,Y,i)$ of $\mathrm {Pairs}^{\mathrm {eff}}(K)$ , we claim there is a natural comparison

(1) $$ \begin{align}\phi_{X,Y,i}:H^i_{DR}(X,Y)\otimes_k k_\gamma^{\mathrm{an}}\to H^i_{\gamma}(X,Y)\otimes_{\mathbb{Q}} k_\gamma^{\mathrm{an}}.\end{align} $$

By spreading out X and Y and possibly shrinking S, we may think of $X,Y$ as varieties over S such that $H^i(X_t,Y_t,\mathbb {Q})$ forms a local system over $S^{\mathrm {an}}$ , and $H^i_{DR}(X,Y)$ is naturally the associated algebraic flat vector bundle. The comparison (1) is then the analytic comparison over $S^{\mathrm {an}}$ via fiberwise integration.

This naturally yields a representation of $\mathrm {Pairs}^{\mathrm {eff}}(k)$ , and as above, we therefore have a functor

$$\begin{align*}\mathcal{MM}_{\mathrm{Nori}}(k)\to (k,\mathbb{Q})_{k_\gamma^{\mathrm{an}}}\mbox{-mod},\;\;\;\; M\mapsto (H_{DR}(M),H_{\gamma}(M),\phi_M)\end{align*}$$

and therefore a faithful exact functor $H_{DR}:\mathcal {MM}_{\mathrm {Nori}}(k)\to k\mbox {-mod}$ . As in [Reference Huber and Müller-Stach20, §8.4], there is an affine k-pro-scheme $\mathcal {X}$ whose points over a k-algebra R are the isomorphisms of fiber functors

$$\begin{align*}H_{DR}\otimes_k R\to H_\gamma\otimes_{\mathbb{Q}} R.\end{align*}$$

Moreover, $\mathcal {X}(k,\gamma )$ is naturally a torsor for $\mathbf {G}_{\mathrm {mot}}(k,\gamma )_k$ . Likewise, for any Nori motive M over k, there is an affine k-scheme $\mathcal {X}(M,\gamma )$ of such isomorphisms of the restrictions to the tensor category $\langle M\rangle $ generated by M, and it is a torsor for $\mathbf {G}_{\mathrm {mot}}(M,\gamma )_k$ .

Let $\mathcal {X}(k/\mathbb {C},\gamma )\subset \mathcal {X}(k,\gamma )$ be the closed sub-pro-scheme of isomorphisms which restrict to the canonical comparison (fiberwise integration) on constant motives, which is naturally a torsor for $\mathbf {G}_{\mathrm {mot}}(k/\mathbb {C},\gamma )$ , and likewise define $\mathcal {X}(M/\mathbb {C},\gamma )\subset \mathcal {X}(M,\gamma )$ , which is a torsor for $\mathbf {G}_{\mathrm {mot}}(M/\mathbb {C},\gamma )$ .

Choose a model S for k such that M is in the diagram category generated by pairs with models over S whose cohomologies are local systems over $S^{\mathrm {an}}$ . With the notation as in the introduction and taking $\mathcal {V}$ to be the variation of Hodge structures over $S^{\mathrm {an}}$ with underlying local system $\mathscr {H}_\gamma (M)$ , there is a natural algebraic closed embedding $\mathcal {X}(M/\mathbb {C},\gamma )\to \mathbb {I}_k$ by evaluating on M, as an isomorphism of fiber functors on $\langle M\rangle $ is determined by its value on M. Moreover, analytic continuation of the canonical comparison yields a point of $\mathcal {X}(M/\mathbb {C},\gamma )$ over $k^{\mathrm {an}}_\gamma $ , so $\mathcal {X}(M/\mathbb {C},\gamma )$ contains the germ of $\Sigma _{\mathcal {V}}$ , and hence, $\Omega _{\mathscr {H}_\gamma (M)}:=(\Omega _{\mathcal {V}})_k$ . Also both $(\Omega _{\mathcal {V}})_k$ and $\mathcal {X}(M/\mathbb {C},\gamma )$ are torsors for $\mathbf {G}_{\mathrm {mot}}(M/\mathbb {C},\gamma )$ by Corollary 4.17.

Thus, we deduce the following, which is essentially Nori’s proof of the Grothendieck period conjecture:

4.7 The geometric Kontsevich–Zagier conjecture

We begin by defining the ring of formal periods in our setting. It shall be convenient to work with relative homology classes, so we define $H_{n,\gamma }(X,Y)$ in precisely the same way we did for cohomology, as a limit along a path up to equivalence.

We turn $\tilde {\mathbb {P}}^{\mathrm {eff}}(k)$ into an algebra by setting

$$ \begin{align*}[X,Y,\omega,\ell]\cdot [X',Y',\omega',\ell']:=[X\times X',Y\times Y', \omega\wedge\omega',\ell\times\ell'].\end{align*} $$

That multiplication is well defined is a standard check; see [Reference Huber and Müller-Stach20, 13.1.3]. Finally, we define the ring of formal periods $\tilde {\mathbb {P}}(k)$ to be the localization of $\tilde {\mathbb {P}}^{\mathrm {eff}}(k)$ at $[\mathbb {G}_m,\{1\},\frac {dx}{x},S^1]$ .

Note that there is a natural evaluation map $\textrm {ev}_k:\tilde {\mathbb {P}}(k)\rightarrow k$ given by fiber-wise integration [Reference Huber and Müller-Stach20, Def 5.4.1]. Concretely, if $X,Y$ are smooth and a class $[\omega ]\in H^i_{\operatorname {DR}}(X,Y)$ is represented by a closed differential form $\omega $ whose restriction to Y vanishes, then

$$ \begin{align*}\textrm{ev}_k(X,Y,[\omega],\ell):=\int_\ell \omega.\end{align*} $$

Next, we define the ring of relative formal periods. The idea is that for constant families (i.e., base-changed from $\mathbb {C}$ ), we want to identify the formal period with the actual complex number it evaluates to.

Definition 4.22. Since $\mathbb {C}\subset k$ , there is a natural map $\tilde {\mathbb {P}}(\mathbb {C})\rightarrow \tilde {\mathbb {P}}(k)$ . We define

$$ \begin{align*}\tilde{\mathbb{P}}(k/\mathbb{C}):=\tilde{\mathbb{P}}(k)\times_{\tilde{\mathbb{P}}(\mathbb{C})}\mathbb{C},\end{align*} $$

where we view $\mathbb {C}$ as a $\tilde {\mathbb {P}}(\mathbb {C})$ -algebra via the period map.

We now come to our main statement, the integrality of the relative period ring:

4.8 The geometric André–Grothendieck period conjecture

Let $k\subset K\subset k_\gamma ^{\mathrm {an}}$ be such that $K/k$ is a finitely generated extension and $\tau ^*:K\to k_\gamma ^{\mathrm {an}}$ a k-embedding. For any models S (resp. T) of k (resp. K), we then obtain a rational map $f:T\to S$ and a meromorphic section $\tau :B\to T^{\mathrm {an}}$ with Zariski dense image for an open, simply connected set $B\subset S$ . The composition $\tau \circ \gamma $ is therefore an arc point of K.

As in §4.6, for any vertex $(X,Y,i)$ of $\mathrm {Pairs}^{\mathrm {eff}}(K)$ , there is a natural comparison

$$\begin{align*}\phi_{X,Y,i}:H^i_{DR}(X,Y)\otimes_k k_\gamma^{\mathrm{an}}\to H^i_{\tau\circ\gamma}(X,Y)\otimes_{\mathbb{Q}} k_\gamma^{\mathrm{an}}\end{align*}$$

by pulling back the comparison (1) over $T^{\mathrm {an}}$ along $\tau $ . We therefore obtain a functor

$$\begin{align*}\mathcal{MM}_{\mathrm{Nori}}(K)\to (K,\mathbb{Q})_{k_\gamma^{\mathrm{an}}}\mbox{-mod},\;\;\;\; M\mapsto (H_{DR,\tau}(M),H_{\tau\circ\gamma}(M),\phi_M),\end{align*}$$

and we define

$$\begin{align*}K(\mathrm{periods}_{\tau^*}\;\mathrm{of}\; M)\subset k_\gamma^{\mathrm{an}}\end{align*}$$

to be the field of definition of $\phi _M$ . Concretely, this is the field extension obtained by adjoining the pullbacks via $\tau $ of flat coordinates of algebraic sections of $H_{DR}(M)$ over T.

Theorem 4.25 (Geometric André–Grothendieck period conjecture).

Let $k\subset K$ be finitely generated complex fields, $\gamma $ a $(\mathbb {C},\operatorname {id})$ -arc point of k, and $\tau ^*: K\to k_\gamma ^{\mathrm {an}} $ an embedding of k-extensions. For any Nori $(K,\tau \circ \gamma )$ -motive M, we have

$$\begin{align*}\operatorname{trdeg}_k K(\mathrm{periods}_{\tau^*}\;\mathrm{of}\; M)\geq \dim \mathbf{G}_{\mathrm{mot}}(M/\mathbb{C},\gamma).\end{align*}$$

Proof. Note that there is a functor $MHS:\mathcal {MM}_{\mathrm {Nori}}({K},\tau \circ \gamma )\rightarrow MHS(K)$ where $MHS(K)$ denotes the direct-limit category of admissible variations of graded-polarized integral mixed Hodge structures defined on some model of K. The above functor $\mathcal {MM}_{\mathrm {Nori}}(K,\tau \circ \gamma )\to (K,\mathbb {Q})_{k_\gamma ^{\mathrm {an}}}\mbox {-mod}$ factors through $MHS $ .

Using the above notation, for an admissible variation of graded-polarizable integral mixed Hodge structures $\mathcal {V}$ over T, observe that since B is simply connected, the map $\tau $ lifts to a map $B\rightarrow \widetilde {T^{\mathrm {an}}}$ , and therefore, $\sigma _{\mathcal {V}}\circ \tau :B\rightarrow \Sigma _{\mathcal {V}}$ gives a well-defined map. We have

$$\begin{align*}\dim (\operatorname{img} \sigma_{\mathcal{V}}\circ \tau)^{\mathrm{Zar}}-\dim S=\operatorname{trdeg}_kK(\mathrm{periods}_{\tau^*}\;\mathrm{of}\; \mathcal{V}).\end{align*}$$

Thus, by Theorem 4.17, it suffices to prove the following:

Claim 4.26. For an admissible variation of graded-polarizable integral mixed Hodge structures $\mathcal {V}$ over T,

(2) $$ \begin{align}\dim (\operatorname{img} \sigma_{\mathcal{V}}\circ \tau)^{\mathrm{Zar}} -\dim S\geq\dim\mathbf{G}.\end{align} $$

As the intersection $(\operatorname {img}\sigma _{\mathcal {V}}\circ \tau )^{\mathrm {Zar}}\cap \Sigma _{\mathcal {V}}$ obviously contains $\operatorname {img}\sigma _{\mathcal {V}}\circ \tau $ and $\operatorname {img} \sigma _{\mathcal {V}}\circ \tau $ projects to $\operatorname {img} \tau $ in T which is Zariski dense, the claim is immediate from Theorem 1.3.

Remark 4.27. The geometric André–Grothendieck period conjecture is almost equivalent to Theorem 1.3, the only issue being that some variations may not come from geometry.Footnote ⁵

In fact, the natural generalization of the geometric André–Grothendieck period conjecture to variations of mixed Hodge structures in the form of Claim 4.26 is equivalent to Theorem 1.3. Indeed, the backward implication is used in the proof. For the forward implication, let $\pi (U)$ be the projection of U to S and take $S'=\pi (U)^{\mathrm {Zar}}$ . Take an algebraic projection $S'\to \mathbb {A}^{\dim U}$ which is generically finite on $\pi (U)$ , and take $A=\mathbb {A}^{\dim U}$ . The map $\pi (U)\to A^{\mathrm {an}}$ is generically an isomorphism, and therefore, we obtain a local section $\tau $ of $S^{\prime \mathrm {an}}\to A^{\mathrm {an}}$ with Zariski dense image. Applying (2) (with $(A,S')$ in the place of $(S,T)$ ) yields

$$\begin{align*}\dim\mathbf{G}>\operatorname{codim}_W U\geq \dim U^{\mathrm{Zar}}-\dim U\geq\dim\mathbf{G}',\end{align*}$$

where $\mathbf {G}'$ is the algebraic monodromy of the restriction of $\mathcal {V}$ to $S'$ , so $S'$ is contained in a weak Mumford–Tate subvariety.

5.1 Elliptic curves

Let S be a smooth irreducible variety of dimension m. Let $E_1,\dots ,E_n$ be non-isotrivial, pairwise non-isogenous elliptic curves over S and $f_1,\dots ,f_n$ sections of $E_1,\dots ,E_n$ over S. We therefore obtain a section $f:=(f_1,\ldots , f_n)$ of $E:=E_1\times _S\cdots \times _S E_n$ . Let $\omega _1,\dots ,\omega _n$ be corresponding relative differentials, that is, sections of $H^0(\pi _*\omega _{E_i/S})$ , where $\pi :E\to S$ is the projection. We assume the $f_i$ and $\omega _i$ to be nowhere vanishing, which can always be arranged by shrinking S. Finally, let $B\subset S^{\mathrm {an}}$ be an open ball over which we can trivialize the homology of $E_1,\dots ,E_n$ . Then by picking generators $\alpha _i,\beta _i$ of the first homology and a path $\gamma _i$ from $0$ to $f_i$ , we obtain $3n$ functions by integrating the differentials along the relative homology classes $\alpha _i,\beta _i,\gamma _i$ , and thus, we obtain a map $F:B\rightarrow \mathbb {C}^{3n}$ .

Proof. First, note that over S for each pair $(E_i,f_i)$ , we have an admissible variation of graded-polarizable integral mixed Hodge structures $\mathcal {V}_i=((V_i)_{\mathbb {Z}},W_\bullet V_i,F^\bullet V_i)$ given by assigning to $s\in S$ the relative cohomology group $H^1(E_{i,s},\{0,f_i(s)\},\mathbb {Z})$ . Note that this is an extension of the form

(3) $$ \begin{align}0\to \mathbb{Z}(0)\to H^1(E_{i,s},\{0,f_i(s)\},\mathbb{Z})\to H^1(E_i(x),\mathbb{Z})\to0.\end{align} $$

Let $\mathcal {V}=\bigoplus _i \mathcal {V}_i$ . Note that the $\omega _i$ are algebraic sections of $V_{\mathcal {O}}=H^1_{DR}(E/S)$ over S. By thinking of $f\in \mathbb {E}$ above $s\in S$ as $f=\oplus f_i$ where $f_i:\mathcal {V}_{i,s}\to V_{\mathbb {C},0}$ , the evaluation map $(f_i)\mapsto (f_i(\omega _i))$ gives an algebraic map $g:\Omega _{\mathcal {V}}\to V_{\mathbb {C},0}$ . Then $g\circ \sigma _{\mathcal {V}}:\widetilde {S^{\mathrm {an}}}\to V_{\mathbb {C},0}$ is the map which associates to a point s together with a homotopy class of path to $s_0$ the cohomology class $([\omega _i])\in \bigoplus _i H^1(E_{i,s},\{0_s,f_i(s)\},\mathbb {C})$ , flatly continued to $s_0$ via the path. In particular, using the basis $\alpha _i^\vee ,\beta _i^\vee ,\gamma _i^\vee $ on $V_{\mathbb {C},0}$ , this map agrees with F on a lift of B.

We therefore let $W\subset \Omega _{\mathcal {V}}$ be $W=g^{-1}(T)$ , and observe that R lifts to the intersection $W\cap \Sigma _{\mathcal {V}}$ . To apply Theorem 1.3 we must

1. (a) compute the algebraic monodromy group of $\mathcal {V}$ ;

2. (b) compute the dimension of W.

Proposition 5.2. The algebraic monodromy group $\mathbf {G}$ of $\mathcal {V}$ is $(\mathbf {G}_a^2\rtimes \mathbf {SL}_2)^n$ .

Proof. That $\mathbf {G}$ surjects onto $\mathbf {SL}_2^n$ follows from the fact that the elliptic curves are non-isogenous and non-isotrivial, together with the classification of weakly-special subvarieties of $X(1)^n$ (see [Reference Edixhoven17, Proposition 2.1]). Since $\mathbf {SL}_2$ acts irreducibly on its standard representation, we claim that it is sufficient to show that none of the algebraic monodromy groups $\mathbf {G}_i$ of any of the factors $\mathcal {V}_i$ is $\mathbf {SL}_2$ . Indeed, if this is the case, then the unipotent radical is a sum of $(\mathbf {G}_a^2)^n$ which surjects to each factor and is invariant under $\mathbf {SL}_2^n$ . Since the irreducible constituents are simply the fibers and they are mutually non-isomorphic, the claim follows.

To see that none of the $\mathbf {G}_i$ is $\mathbf {SL}_2$ , we first note by Theorem 2.6 that the algebraic monodromy group is normal in the derived subgroup of the generic Mumford-Tate group, and thus, it is sufficient to show that the generic Mumford-Tate group of each $\mathcal {V}_i$ is maximal.

Lemma 5.3. Let E be a mixed Hodge structure of the form (3) and suppose $\operatorname {gr}^W_1 E$ is Mumford–Tate general. Then the Mumford–Tate group of E is $\mathbf {GL}_2$ if and only if the extension of mixed Hodge structures

(4) $$ \begin{align}0\to \mathbb{Z}(0)\to E\to \operatorname{gr}^W_1E\to0\end{align} $$

is $\mathbb {Q}$ -split.

Proof. Recall (see §2.3) that the Mumford–Tate group of E is the stabilizer of all Hodge classes in all tensors $E^{\otimes m}\otimes (E^\vee )^{\otimes n}$ . If the Mumford–Tate group of E is $\mathbf {GL}_2$ , then there is a fixed vector in $E_{\mathbb {Q}}$ which therefore splits (4), and the converse is obvious.

Now it remains to note that the space of extensions (4) up to integral isomorphism is $(F^1\operatorname {gr}_1^WE)^\vee /(\operatorname {gr}_1^WE)_{\mathbb {Z}}^\vee \cong (\operatorname {gr}_1^WE)_{\mathbb {C}}/F^0\operatorname {gr}_1^WE+(\operatorname {gr}_1^WE)_{\mathbb {Z}}$ which is just the elliptic curve corresponding to $\operatorname {gr}_1^WE$ , and the $\mathbb {Q}$ -split points are the torsion points.

We now compute the dimension of W. Note that $g:\Omega _{\mathcal {V}}\to V_{\mathbb {C},0}$ is equivariant with respect to the action of $\mathbf {G}(\mathbb {C})$ . Moreover, the class $0\neq [\omega _i]\in F^1V_i$ is not contained in $W_0V_i$ for any i at any point. Thus, the orbit of any point in the image of g is an open subset of $V_{\mathbb {C},0}$ , and in particular of dimension $3n$ . Thus, the fibers of g all have the same dimension $3n$ , and $\operatorname {codim}_{\Omega _{\mathcal {V}}} W=k$ . We then have $\operatorname {codim}_W U<\dim \mathbf {G}$ , and it follows from Theorem 1.3 that R is contained in a proper weak Mumford–Tate subvariety. In particular, it is not Zariksi dense, and R must be contained in either:

1. 1. the locus where some $E_i$ becomes isotrivial, corresponding to the algebraic monodromy group of the restriction of $\mathcal {V}_i$ being contained in $\mathbf {G}_a^2$ ;

2. 2. the locus where some $E_i,E_j$ for $i\neq j$ become isogenous, corresponding to the algebraic monodromy group of the restriction of $\mathcal {V}_i\oplus \mathcal {V}_j$ being contained in the preimage of (a conjugate of) the diagonal under $(\mathbf {G}_a^2\rtimes \mathbf {SL}_2)^2\to \mathbf {SL}_2^2$ ;

3. 3. the locus where some section $f_i$ of some $E_i$ is torsion, corresponding to the algebraic monodromy group of the restriction of $\mathcal {V}_i$ being contained in (a lift of) $\mathbf {SL}_2$ .

To complete the proof, we just need to show that if we are only in the last case, then at least two sections become torsion. Assume that only one section (without loss of generality $f_1$ ) becomes torsion. Consider the variation $\mathcal {V}':=\operatorname {gr}^W_1\mathcal {V}_1\oplus \bigoplus _{i>1}\mathcal {V}_i$ , which is a quotient of $\mathcal {V}$ . Let $V_{\mathbb {C},0}\to V^{\prime }_{\mathbb {C},0}$ be the corresponding quotient and $T'$ the image of T. Now $\operatorname {codim} T'\geq k-1$ and $\operatorname {codim}_{R^{\mathrm {Zar}}} R<k-1$ , so applying the same analysis as above, we obtain a contradiction.

5.2 Ax–Lindemann for abelian differentials

In this section, we prove a recent conjecture of Klingler–Lerer [Reference Klingler and Lerer21]. We first briefly recall strata of abelian differentials.

Let $g> 0$ be an integer, $\alpha =(\alpha _i)$ a partition of $2g-2$ , and $S=S_\alpha $ the moduli space of pairs $(C,\omega )$ where C is a genus g curve and $\omega $ is a regular 1-form on C whose zero divisor $Z(\omega )$ has type $\alpha $ , meaning it is of the form $\sum \alpha _ip_i$ for distinct points $p_i$ on C. There is a natural variation of mixed Hodge structures over $S_\alpha $ whose fiber over $(C,\omega )$ is the relative cohomology group $H^1(C^{\mathrm {an}},Z(\omega ),\mathbb {Z})$ .

Fixing a basepoint $(C_0,\omega _0)$ , let $\pi :\widetilde {S^{\mathrm {an}}}\to S^{\mathrm {an}}$ be the universal cover. The map $\phi :\widetilde {S^{\mathrm {an}}}\to V_0:=H^1(C_0,Z(\omega _0),\mathbb {C})$ mapping $(C,\omega )$ to the image of the class $[\omega ]\in H^1(C,Z(\omega ),\mathbb {C})$ under the flat trivialization is a local isomorphism by a theorem of Veech [Reference Veech30, Thm. 7.15].

Following [Reference Klingler and Lerer21], we say an (irreducible) algebraic subvariety $W\subset S$ is bialgebraic if the Zariski closure of $\phi (W_0)$ in $V_0$ has dimension $\dim W$ for some (hence any) component $W_0$ of $\pi ^{-1}(W)$ . We likewise say $W\subset V_0$ is bialgebraic if the Zariski closure of $\pi (W_0)$ has dimension $\dim W$ for some (hence any) component $W_0$ of $\phi ^{-1}(W)$ . The following is the Ax–Lindemann conjecture of [Reference Klingler and Lerer21].

Proof. Let X be the Zariski closure of $\pi (W_0)$ , define $\widetilde {X^{\mathrm {an}}}$ to be the universal cover of $X^{\mathrm {an}}$ and let $\mathcal {V}$ be the above variation of mixed Hodge structures restricted to X. The class $[\omega ]$ gives a section s of $\mathcal {V}$ , and there is a natural algebraic map $r:\Omega _{\mathcal {V}}\to V_0$ by evaluating a comparison on s. It follows that $\phi |_{\widetilde {X^{\mathrm {an}}}}=r\circ \sigma _{\mathcal {V}}$ .

We claim that $r(\Omega _{\mathcal {V}})$ and $\mathbf {G}(\mathbb {C})W$ have the same Zariski closure. Now $Y:=\overline {\mathbf {G}(\mathbb {C})W}$ is certainly contained in $\overline {r(\Omega _{\mathcal {V}})}$ . Also $Y $ is $\mathbf {G}(\mathbb {C})$ -invariant, so the pullback to $\widetilde {X^{\mathrm {an}}}$ descends to an algebraic subvariety of X by definable Chow 2.1 and contains $\pi (W_0)$ ; hence, it must be all of X. Thus, $\phi (\widetilde {X^{\mathrm {an}}})$ is contained in Y, as therefore is its Zariski closure $\overline {r(\Omega _{\mathcal {V}})}$ , so we have the inclusion in the reverse directions.

As $\phi (\widetilde {X^{\mathrm {an}}})\subset Y$ , for X to be bialgebraic, it suffices to show $\dim X\geq \dim Y$ . Let $W'$ be the pullback of W to $\Omega _{\mathcal {V}}$ . The dimension of $W'$ is $\dim F+\dim W$ where F is the generic fiber of $r:\Omega _{\mathcal {V}}\to V_0$ over W. By the above, $r(\Omega _{\mathcal {V}})$ and $\mathbf {G}(\mathbb {C})W$ have the same Zariski closure, and since r is $\mathbf {G}(\mathbb {C})$ -equivariant, it follows that the generic fiber dimension of r over its image is equal to the generic fiber dimension over W. Thus,

$$ \begin{align*} \dim W'&= \dim \Omega_{\mathcal{V}}-\dim Y+\dim W\\ &=\dim \mathbf{G}+\dim X-\dim Y+\dim W.\end{align*} $$

Now $\pi (W_0)$ is Zariski dense in X but also lifts to the intersection of $W'$ with a leaf, so by Theorem 1.3, we must have

$$\begin{align*}\dim \mathbf{G}\leq \dim W'-\dim W=\dim\mathbf{G}+\dim X-\dim Y,\end{align*}$$

and therefore, $\dim X\geq \dim Y$ , as desired.

5.3 Twisting by the period torsor

In this section, we explain formally how one may deduce many of the previous Ax-Schanuel theorems from Theorem 1.3.

In applications, one often has a variety M with an algebraic (left) $\mathbf {G}(\mathbb {C})$ -action and an equivariant algebraic map $g:\Omega _{\mathcal {V}}\to M$ , as in the last subsection. In this case, Theorem 1.3 is readily applied.

Another common situation is to have an algebraic variety $\mathbb {P}\to S$ over S, an algebraic variety P equipped with a (left) $\mathbf {G}(\mathbb {C})$ -action, and a $\mathbf {G}(\mathbb {C})$ -equivariant map

$$\begin{align*}\langle\;,\;\rangle:\Omega_{\mathcal{V}}\times_S\mathbb{P}\to P \end{align*}$$

which we call a twisting map. Such a map yields a map $\langle \sigma _{\mathcal {V}},\;\rangle :\widetilde {S^{\mathrm {an}}}\times _{S^{\mathrm {an}}}\mathbb {P}^{\mathrm {an}}\to P^{\mathrm {an}}$ on the base-change to the universal cover. Note that these two setups are equivalent, as we may take $\langle \;,\;\rangle $ to be the $\mathbf {G}(\mathbb {C})$ -equivariant map

$$\begin{align*}g_{\langle\;,\;\rangle}:\Omega_{\mathcal{V}}\to \mathrm{Hom}_S(\mathbb{P},P_S),\end{align*}$$

where $P_S=P\times S$ , and in the other direction we may take $\mathbb {P}=S$ and ${\langle \;,\;\rangle }_g=g$ .

Examples of twisting maps include the following:

• • $\mathbb {P}=\mathbb {V}$ and $P=V_{\mathbb {C},0}$ and $\langle \;,\;\rangle $ the obvious evaluation. The map $\langle \sigma _{\mathcal {V}},\;\rangle $ is then the flat trivialization.

• • $\mathbb {P}=$ the relative flag variety of $V_{\mathcal {O}}$ for which the Hodge filtration yields a section, $P=$ the flag variety of $V_{\mathbb {C},0}$ containing the relevant period domain (that is, its dual), and $\langle f, F^\bullet V_s \rangle =f(F^\bullet V_s)$ . The map $\langle \sigma _{\mathcal {V}},\;\rangle $ is then the period map.

• • For any artinian ring A and any twisting map $\langle \;,\;\rangle :\Omega _{\mathcal {V}}\times _S\mathbb {P}\to P$ , we get a map on A-jet spaces

  $$\begin{align*}J_A\Omega_{\mathcal{V}}\times_{J_AS}J_A\mathbb{P}\to J_AP.\end{align*}$$
  The horizontal jets yield a natural subspace $\Omega _{\mathcal {V}}\times _SJ_AS\subset J_A\Omega _{\mathcal {V}} $ which is preserved by the $\mathbf {G}(\mathbb {C})$ action. We therefore obtain a twisting map $\langle \;,\;\rangle _A:\Omega _{\mathcal {V}}\times _SJ_A\mathbb {P}\to J_AP$ , and the map $\langle \sigma _{\mathcal {V}},\;\rangle _A$ is then the map on jet spaces induced by $\langle \sigma _{\mathcal {V}},\;\rangle $ . In this way, we may access transcendence statements for the derivatives of $\langle \sigma _{\mathcal {V}},\;\rangle $ , as in [Reference Mok, Pila and Tsimerman22].

• • By taking $\mathbb {P}$ to be $S\times X$ and $P=\Omega _{\mathcal {V}}\times X$ for a variety X, we obtain the Ax-Schanuel result ‘in families’, or ‘relative Ax-Schanuel’ as it has been called in the literature.

Given a twisting map, we define

$$\begin{align*}\mu:=\langle \;,\;\rangle\times \pi_2 :\Omega_{\mathcal{V}}\times_S\mathbb{P}\to P\times \mathbb{P}.\end{align*}$$

We say the twisting map is balanced if the fibers of $\mu $ all have the same dimension. Note that the fiber over $(p',p)$ is identified with the stabilizer $\operatorname {Stab}_{\mathbf {G}(\mathbb {C})}(p)$ . In practice, given a twisting map, we can always assume it is balanced by passing to a Zariski open subset.

Proposition 5.5. Let $\langle \;,\;\rangle $ be a balanced twisting map as above and let $\Delta \subset \operatorname {img}\mu $ be the image of $\Sigma _{\mathcal {V}}\times _{S^{\mathrm {an}}}\mathbb {P}^{\mathrm {an}}$ under $\mu ^{\mathrm {an}}$ . Let $W\subset \operatorname {img}\mu $ be an algebraic variety and U a component of $W^{\mathrm {an}}\cap \Delta $ such that

$$\begin{align*}\operatorname{codim}_\Delta U<\operatorname{codim}_{\operatorname{img} \mu} W.\end{align*}$$

Then the projection of U to $S^{\mathrm {an}}$ is contained in a weak Mumford–Tate subvariety.

Proof. Let k be the dimension of the fibers of $\mu $ . First, U naturally lifts to $\Sigma _{\mathcal {V}}\times _{S^{\mathrm {an}}}\mathbb {P}^{\mathrm {an}}\subset \Omega _{\mathcal {V}}^{\mathrm {an}}\times _{S^{\mathrm {an}}} \mathbb {P}^{\mathrm {an}}$ . Call this lift $U'$ . Let $W'$ be the component of the preimage of W under $\mu $ which contains $U'$ , and let $W"\subset \Omega _{\mathcal {V}}$ (resp. $U"\subset \Sigma _{\mathcal {V}}$ ) be the image of $W'$ (resp. $U'$ ) under the first projection. Clearly, $W"\cap \Sigma _{\mathcal {V}}$ contains $U"$ , and we also have that $\dim W'=\dim W+k$ . The fibers of $W'\to W"$ are the intersections of W with subvarieties of the form $p'\times \mathbb {P}_t$ , and these are the same fibers as $U'\to U"$ over $U"$ . Up to replacing W with an algebraic subvariety for which the generic fiber of $W'\to W"$ has the same size as the generic fiber of $U'\to U"$ (and without changing U), we therefore have

$$ \begin{align*} \operatorname{codim}_{W"} U"&=\operatorname{codim}_{W'} U' \\ &=\operatorname{codim}_W U+k\\ &=\operatorname{codim}_{\Delta}U-\operatorname{codim}_{\operatorname{img}\mu}W+(k+\dim\operatorname{img}\mu-\dim\Delta)\\ &<\dim\mathbf{G}. \end{align*} $$

Applying Theorem 1.3, the result follows.

Note that in the context of the proposition, $\operatorname {img}\mu $ is the Zariski closure of $\Delta $ in $P\times \mathbb {P}$ .