Last time we discussed how we can consider o-minimal structures and enjoy the flexibility of real analytic functions. We will now apply this to some period domains to profit from the theory.
Definable manifolds#
We first need a theory for globalizing this theory. Let us fix a o-minimal structure $\mathcal{S}$ containing $\mathbb{R}_ \mathrm{alg}$ at least. We make the following definition.
Definition 1. Let $M$ be a second countable Hausdorff topological space. An $\mathcal{S}$-definable atlas is a finite cover $M = \bigcup_i S_i$ together with a collection of open embeddings $\phi_i \colon U_i \cong \mathbb{R}^n$ where
- $\phi_i(U_i \cap U_j)$ are definable open subsets of $\mathbb{R}^n$ (including $i = j$),
- the transition maps $\phi_i(U_i \cap U_j) \cong \phi_j(U_i \cap U_j)$ are definable.
We say that two atlases $\lbrace \phi_i \rbrace$ and $\lbrace \psi_j \rbrace$ are equivalent when their union is an atlas. An $\mathcal{S}$-definable manifold is a second countable Hausdorff topological space with an equivalence class of atlases.
Here it’s really important that we only allow a finite number of charts. This is because in first-order logic we can do only finitary case work. But at the end, it is going to be true that
- equivalence of atlases is indeed an equivalence relation,
- there is a well-defined notion of a definable subset inside a definable manifold,
- products of definable manifolds are definable manifolds, hence we can make sense of a definable map,
- a surjective definable map between definable manifolds has a definable section.
Proposition 2. Let $X/\mathbb{R}$ be a quasi-compact smooth scheme. Then $X(\mathbb{R})$ with its natural topology can be upgraded uniquely to a $\mathbb{R}_ \mathrm{alg}$-definable manifold, satisfying the property that every Zariski open subset is definable and algebraic functions on such are also definable.
We can reduce to the case when $X$ is irreducible and has an $\mathbb{R}$-point (in which case it is geometrically irreducible as well). Then
Remark 3. There is also van den Dries–Miller’s definition.
Symmetric domains#
Let $\mathbb{G}/\mathbb{Q}$ be a semisimple linear algebraic group. We will be interested in the real Lie group $$ G = \mathbb{G}(\mathbb{R})^+, $$ where $+$ just denotes the connected component of the identity. This inherits a $\mathbb{R}_ \mathrm{alg}$-manifold structure from the one on $G$.
Remark 4. By some sort of cell decomposition fact, $\mathcal{S}$-definable subsets of $\mathbb{R}^n$ have finitely many connected components, and they are also all $\mathcal{S}$-definable.
We now let $M \subseteq G$ be a compact connected subgroup. By Cartan’s theorem, $M$ is a compact Lie group, and there is the following theorem.
Theorem 5 (Chevalley). The category of connected compact Lie groups is equivalent to the category of $\mathbb{R}$-anisotropic connected reductive $\mathbb{R}$-groups, where the one functor is $\mathbb{G} \mapsto \mathbb{G}(\mathbb{R})$. Moreover, every real finite-dimensional representation of $\mathbb{G}(\mathbb{R})$ comes from an algebraic representation of $\mathbb{G}$.
This means that $M$ can be canonically promoted to the $\mathbb{R}$-points of an algebraic group $\mathbb{M}$, and hence there is a canonical $\mathbb{R}_\mathrm{alg}$-structure on $M$. Moreover, the inclusion $M \hookrightarrow G$ can be realized as a map of algebraic groups $\mathbb{M} \hookrightarrow \mathbb{G}$. This implies that $M$ is indeed a definable subset in $G$.
Proposition 6. There uniquely exists a $\mathbb{R}_\mathrm{alg}$-structure on the manifold $G/M$ for which the projection map $G \to G/M$ is definable. Moreover, the left translation action $G \times G/M \to G/M$ is definable.
Uniqueness is easy. The definable structure on a manifold $X$ is determined by the collection of continuous definable functions $X \to \mathbb{R}$. Now once there is a definable projection map $G \to G/M$, a function $G/M \to \mathbb{R}$ will be definable if and only if $G \to G/M \to \mathbb{R}$ is. This is because $G \to G/M$ will have a definable section.
Let us now prove existence. Bakker–Klinger–Tsimerman gives two proofs. In the first proof, we use a theorem of Chevalley that says that there exists an algebraic representation $\mathbb{G} \to \mathrm{GL}_ \mathbb{R}(W)$ and a line $\ell \subseteq W$ for which $\mathbb{M}$ is the stabilizer of $\ell$. But because $M$ is anisotropic, the only map $\mathbb{M} \to \mathbb{G}_ {m,\mathbb{R}}$ is trivial. So upon fixing a nonzero vector $v \in \ell$, we obtain an orbit map $$ G/M \to W; \quad gM \mapsto gv. $$ The image $\mathbb{G}/\mathbb{M}$ does define a locally closed subvariety of the affine space $\Spec \mathrm{Sym}^\bullet W^\vee$, moreover the image of $$ \mathbb{G}(\mathbb{R}) \to (\mathbb{G}/\mathbb{M})(\mathbb{R})$$ contains the connected component of the identity because it is open and stable under the $\mathbb{G}(\mathbb{R})$-action. This shows that $G/M$ is identified with the connected component $(\mathbb{G}/\mathbb{M})(\mathbb{R})^0$. This is now $\mathbb{R}_ \mathrm{alg}$-definable.
Remark 7. There also seems to be another proof based on o-minimality. One can define the notion of a definable equivalence relation $R = G \times M \rightrightarrows G$. Once both projection maps are proper, it seems that the quotient $G/R = G/M$ automatically has a definable structure.
Arithmetic quotients#
If one wants to construct the moduli of polarized Hodge structures, for instance of weights $(1,0)$ and $(0,1)$, once we trivialize the $\mathbb{Z}$-lattice of rank $2$, the resulting space looks like $G / M$ (in our case $\mathrm{SL}_ 2(\mathbb{R}) / \mathrm{SO}_ 2(\mathbb{R}) = \mathcal{H}$). But then, we also need to look at ways of trivializing the $\mathbb{Z}$-lattice to begin with. This motivates us to look at the space $$ \mathrm{SL}_ 2(\mathbb{Z}) \backslash \mathrm{SL}_ 2(\mathbb{R}) / \mathrm{SO}_ 2(\mathbb{R}), $$ which is the modular curve. Oftentimes, to get rid of automorphism, we quotient by a finite index subgroup of $\Gamma \subseteq \mathrm{SL}_ 2(\mathbb{Z})$.
Definition 8. Let $\mathbb{G}/\mathbb{Q}$ be a reductive group. Fix an embedding $\mathbb{G} \hookrightarrow \mathrm{GL}_ {n,\mathbb{Q}}$ and consider $$ K = \mathbb{G}(\mathbb{Q}) \cap \mathrm{GL}_ n(\mathbb{Z}). $$ We say that a subgroup $\Gamma \subseteq \mathbb{G}(\mathbb{Q})$ is an arithmetic subgroup when $K \cap \Gamma$ has finite index in both $K$ and $\Gamma$.
This notion turns out to be independent of the choice of the embedding $\mathbb{G} \hookrightarrow \mathrm{GL}_ {n,\mathbb{Q}}$. We now state the main theorem for the first part of this seminar.
Theorem 9 (Bakker–Klinger–Tsimerman, 1.1). Let $\mathbb{G}/\mathbb{Q}$ be a connected semisimple group, let $G = \mathbb{G}(\mathbb{R})^+$, and let $M \subseteq G$ be a connected compact subgroup. Let $\Gamma \subseteq G$ be a torsion-free arithmetic subgroup.
- There exists a canonical $\mathbb{R}_ \mathrm{alg}$-structure on $\Gamma \backslash G / M$, and moreover there exists a $\mathbb{R}_ \mathrm{alg}$-definable fundamental domain $\mathcal{F} \subseteq G/M$ such that $\pi \vert_\mathcal{F} \colon \mathcal{F} \to \Gamma \backslash G / M$ is $\mathbb{R}_ \mathrm{alg}$-definable.
- Let $\mathbb{G} \to \mathbb{G}^\prime$ be a map of semisimple groups over $\mathbb{Q}$, and let $M \to M^\prime$. Assume that for some $g \in G$, we have $g \Gamma g^{-1} \to \Gamma^\prime$. Then the induced map $$ \Gamma \backslash G / M \to \Gamma^\prime \backslash G^\prime / M^\prime; \quad \Gamma x M \mapsto g \Gamma x M $$ is $\mathbb{R}_ \mathrm{alg}$-definable.
- The induced $\mathbb{R}_ \mathrm{an}$-structure on $\Gamma \backslash G / M$ is the one induced from the Borel–Serre compactification, which has a natural $\mathbb{R}_ \mathrm{an}$-structure.
Remark 10. It is a fact that when $\Gamma$ is torsion-free, then $\Gamma$ acts freely and discontinuously on $G / M$.
Remark 11. Vaughan pointed out that there is an erratum for the above theorem, and it should be fixed. In particular, the $\mathbb{R}_ \mathrm{alg}$-structure on the arithmetic quotient $\Gamma \backslash G / M$ depends on a choice of a maximal compact $K$ containing $M$.