Algebraic and o-minimal flows beyond the cocompact case

Let $V$ be a vector space over $\mathbb{R}$ or $\mathbb{C}$, $\Lambda\subset V$ a discrete subgroup, and $X\subset V$ a closed subset. Let $\pi\colon V\to V/\Lambda$ denote the projection map. Ullmo-Yafaev [5] observed that as $R\to\infty$, the closures

form what we may call a “flow” of sets, which converge to a closed limiting set ${\operatorname{Fl}}(X)$ exactly containing those cosets $v+\Lambda$ which an unbounded sequence of points $x_{i}\in X$ can approach in Euclidean distance. Evidently we have

In the case where $X\subset\mathbb{C}^{n}$ is a complex curve and $\Lambda\subset\mathbb{C}^{n}$ is cocompact, [5, Theorem 2.4] shows ${\operatorname{Fl}}(X)$ consists of finitely many translates of real subtori $\mathbb{T}\subset\mathbb{C}^{n}/\Lambda$. This was later extended to arbitrary discrete subgroups by Dinh-Vu [2].

When $X$ is a higher-dimensional variety, an example of Peterzil-Starchenko [4, Section 8] shows this description of ${\operatorname{Fl}}(X)$ doesn’t hold, even if $\Lambda$ is cocompact. However, they show that in this cocompact case, replacing translated subtori by algebraic families of translated subtori is enough [4, Theorem 1.1].

Our main result shows the cocompactness restriction can be removed when the real and complex spans of $\Lambda$ agree. For $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$ and $S$ a subset of an $\mathbb{F}$-vector space $V$, let $\mathbb{F}S$ denote its $\mathbb{F}$-linear span.

When $\Lambda$ is cocompact, [4] also shows that we may assume $C_{i}$ is a finite collection of points if $\mathbb{T}_{i}$ is inclusion-maximal among $\mathbb{T}_{1},\ldots,\mathbb{T}_{m}$, but this is false in this more general setting even for $X=\mathbb{C}^{2}$ and $\Lambda=(\mathbb{Z}\oplus i\mathbb{Z})\times\{0\}$.

The condition that the $\mathbb{R}$-span and $\mathbb{C}$-span of $\Lambda$ are equal is essential, even without the dimension conditions, as the following example of [2] shows.

Gallinaro [3, Question 6.3.2], in work on Zilber’s exponential-algebraic closedness conjecture (an equivalent formulation of Zilber’s quasi-minimality conjecture [6] as established by Bays-Kirby [1, Theorem 1.5]), has conjectured that such a decomposition should be possible if the $C_{i}$ are taken semi-algebraic. Our second main result, an o-minimal version of 1.1 with no restrictions on $\Lambda$ generalizing [4, Theorem 1.2], confirms this conjecture.

In the cocompact case [4] also shows that we may take $C_{i}$ to be compact if $\mathbb{T}_{i}$ is inclusion-maximal amongst $\mathbb{T}_{1},\ldots,\mathbb{T}_{m}$, but this is false in this more general setting even for $X=\mathbb{R}^{2}$ and $\Lambda=\mathbb{Z}\times\{0\}$.

To prove the cocompact case, the main technical tool introduced in [4] is a definable collection of “asymptotic flats” $\mathcal{A}^{\mathbb{R}}(X)$ (or $\mathcal{A}^{\mathbb{C}}(X)$) associated to $X$ which encode the linear behaviour of sequences of points of $X$. Roughly speaking, limiting sequences of points are encoded by realizations of $X$ in a saturated extension $\mathcal{R}$ of $\mathbb{R}$ (or in $\mathcal{C}:=\mathcal{R}\oplus i\mathcal{R}$), and we associate to such a point the smallest flat defined over $\mathbb{R}$ (or $\mathbb{C}$) which is infinitesimally close to this realization.

The key idea in our proof is that if we decompose our ambient vector space as $L\oplus L^{\perp}$, then a sequence of points $x_{1},x_{2},\ldots$ limiting to a coset of $\Lambda$ has convergent $L^{\perp}$-component, and the corresponding asymptotic flat would then have linear part contained in $L\oplus\{0\}$. This motivates us to consider just the sub-family of flats of $\mathcal{A}(X)$ with no linear part in the $L^{\perp}$ direction, and by doing so we can carry out a very similar argument in both the complex algebraic and o-minimal settings.

We closely follow the setup of [4], which we recall briefly. Fix a cardinal $\kappa>2^{\omega}$. $\mathbb{R}$ will be considered as an ordered field over the language $\langle 0,1,+,\times,<\rangle$. Let $\mathbb{R}_{full}$ be the expansion of the field $\mathbb{R}$ by all subsets of $\mathbb{R}^{n}$ for all $n$, $\mathcal{L}_{full}$ the corresponding language, and $\mathcal{R}_{full}$ a $\kappa$-saturated extension. Denote by $\mathcal{R}$ the underlying field. Fix an o-minimal expansion $\mathbb{R}_{om}$ of $\mathbb{R}$ with corresponding language $\mathcal{L}_{om}$, and let $\mathcal{R}_{om}$ be the corresponding reduct $\mathcal{R}_{full}\upharpoonright\mathcal{L}_{om}$ of $\mathcal{R}_{full}$, which is also $\kappa$-saturated.

We denote $\mu_{\mathcal{R}}\subset\mathcal{R}$ for the set of infinitesimal elements and $\mathcal{O}_{\mathcal{R}}=\mathcal{R}\oplus\mu_{\mathcal{R}}$ the set of bounded elements (neither of which are definable in $\mathcal{R}_{om}$). We denote $\operatorname{st}:\mathcal{O}_{\mathcal{R}}\to\mathbb{R}$ the standard part map, and for a subset $S\subset\mathcal{R}^{n}$ we denote

The complex numbers $\mathbb{C}=\mathbb{R}\oplus i\mathbb{R}$ will be considered as a structure with signature $\langle 0,1,+,\times\rangle$. We denote by $\mathcal{C}=\mathcal{R}\oplus i\mathcal{R}$, the set of infinitesimals $\mu_{\mathcal{C}}=\mu_{\mathcal{R}}\oplus i\mu_{\mathcal{R}}$, and the set of bounded elements $\mathcal{O}_{\mathcal{C}}=\mathcal{O}_{\mathcal{R}}\oplus i\mathcal{O}_{\mathcal{R}}$. With these identifications, we may similarly define $\operatorname{st}:\mathcal{O}_{\mathcal{C}}\to\mathcal{C}$ and $\operatorname{st}(S)=\operatorname{st}(S\cap\mathcal{O}_{\mathcal{C}}^{n})$ for a subset $S\subset\mathcal{C}^{n}$. Through the identification $\mathcal{C}=\mathcal{R}\oplus i\mathcal{R}$, we may consider $\mathcal{C}_{full}$ the expansion of $\mathbb{C}$ by all subsets of $\mathbb{C}^{n}$ for all $n$.

Denote $\mathcal{L}_{val}=\langle 0,1,+,\cdot,\mathcal{O}\rangle$ the language of a valued field, with $\mathcal{O}$ a unary predicate for the valuation ring, and $\mathcal{C}_{val}$ the $\mathcal{L}_{val}$-structure of the valued field $\mathcal{C}$ augmented with the set $\mathcal{O}_{\mathcal{C}}$ of bounded elements. Note also that $\mu_{\mathcal{C}}$ is definable in $\mathcal{C}_{val}$ by the formula $x=0\wedge x^{-1}\not\in\mathcal{O}_{\mathcal{C}}$. Note that $\mathcal{C}_{val}$ is not a reduct of $\mathcal{C}_{full}$ and in particular not $\kappa$-saturated, since for example the finitely-satisfiable collection of $2^{\omega}$ formulas $x\in\mathcal{O}_{\mathcal{C}}\cup\bigcup_{c\in\mathbb{C}}\{x\not\in c+\mu_{\mathcal{C}}\}$ is not satisfiable.

We will denote $(\mathbb{F},\mathcal{F})$ to be either $(\mathbb{R},\mathcal{R})$ or $(\mathbb{C},\mathcal{C})$.

If $\mathcal{L}$ is one of the languages $\mathcal{L}_{om},\mathcal{L}_{full}$ when working in the real setting or $\mathcal{L}_{val},\mathcal{L}_{full}$ when working in the complex setting, then an $\mathcal{L}$-definable partial type $p(x_{1},\ldots,x_{n})$ is a finitely satisfiable collection of formulas over the language $\mathcal{L}$ with free variables $x_{1},\ldots,x_{n}$ and constants in $A$. We denote by $p(\mathcal{F})\subset\mathcal{F}^{n}$ the set of realizations of $p$ over $\mathcal{F}$.

Suppose we are in one of the $\kappa$-saturated structures $\mathcal{R}_{om},\mathcal{R}_{full}$, or $\mathcal{C}_{full}$, and consider types $p$ defined over a set of constants $A$ with $|A|<\kappa$. Two such types $p,q$ are considered equivalent if for every finite $p_{0}\subset p$ there exists a finite $q_{0}\subset q$ such that $q_{0}(\mathcal{F})\subset p_{0}(\mathcal{F})$ and vice versa. By $\kappa$-saturation, two such types $p,q$ are equivalent if and only if $p(\mathcal{F})=q(\mathcal{F})$. If $p,q$ are any two types, then the sum $p+q$ is defined by

By $\kappa$-saturation, for two such types we have $(p+q)(\mathcal{F})=p(\mathcal{F})+q(\mathcal{F})$, and consequently for three such types $p,q,r$, the types $(p+q)+r$ and $p+(q+r)$ are equivalent.

For $S\subset\mathcal{F}^{n}$, we can consider $S$ as an $\mathcal{F}_{full}$-definable subset of $\mathcal{F}^{n}$. By abuse of notation we also use $S$ to denote the partial type of all $\mathcal{F}_{full}$-formulas with constants in $\mathbb{F}$ satisfied by $S$, and we write $S^{\sharp}:=S(\mathcal{F})$ for the set of realizations of $S$ over $\mathcal{F}$. Also by abuse of notation, we identify $\mu_{\mathcal{R}}$ with the partial type $\{-r<x<r\}_{r\in\mathbb{R}_{>0}}$ in $\mathcal{R}_{om}$ or $\mathcal{R}_{full}$ and $\mu_{\mathcal{C}}$ with the partial type $\{-r<Re(x),Im(x)<r\}_{r\in\mathbb{R}_{>0}}$ in $\mathcal{C}_{full}$, so that $\mu_{\mathcal{R}}(\mathcal{R})=\mu_{\mathcal{R}}$ and $\mu_{\mathcal{C}}(\mathcal{C})=\mu_{\mathcal{C}}$. These abuses of notation are consistent with the sum notation for types, as by $\kappa$-saturatedness for $S,T\subset\mathcal{F}^{n}$ we have

Finally, by $\kappa$-saturatedness, for any $S\subset\mathbb{F}^{n}$ we have

where $\overline{S}\subset\mathbb{F}^{n}$ is the topological closure.

Note that

We will give an alternate description of $\pi^{-1}{\operatorname{Fl}}(X)$ via non-standard realizations.

Let $\operatorname{GrAff}^{\mathbb{F}}_{i}(\mathbb{F}^{n})$ be the Grassmannian of affine $i$-dimensional flats of $\mathbb{F}^{n}$, and let

Note that $\operatorname{GrAff}^{\mathbb{F}}_{0}(\mathbb{F}^{n})=\mathbb{F}^{n}$ and $\operatorname{GrAff}^{\mathbb{F}}_{n}(\mathbb{F}^{n})=\{\{\mathbb{F}^{n}\}\}$.

For $\alpha\in\mathcal{F}^{n}$, we denote by $A_{\alpha}^{\mathbb{F}}$ the minimal $\mathbb{F}$-flat under inclusion such that $\alpha\in(A_{\alpha}^{\mathbb{F}})^{\#}+\mu_{\mathcal{F}}^{n}$ (that this exists is [4, Proposition 4.2] when $\mathbb{F}=\mathbb{C}$ and as noted in [4, Section 7.1], the proof also works for $\mathbb{F}=\mathbb{R}$). For $0\leq i\leq n$ define $\mathcal{A}_{i}^{\mathbb{F}}(X):=\{A_{\alpha}^{\mathbb{F}}:\alpha\in X^{\sharp}\text{ and }\dim_{\mathbb{F}}(A_{\alpha}^{\mathbb{F}})=i\}\subset{\operatorname{GrAff}}_{i}^{\mathbb{F}}(\mathbb{F}^{n})$ and

By [4, Theorems 4.5 and 7.1] this is a definable subset of $\operatorname{GrAff}^{\mathbb{F}}(\mathbb{F}^{n})$, and by definition $A_{\alpha}^{\mathbb{F}}\in\mathcal{A}_{0}^{\mathbb{F}}(X)$ if and only if $\alpha\in\mathcal{O}^{n}$, in which case $A_{\alpha}^{\mathbb{F}}=\operatorname{st}(\alpha)$ (see [4, Lemma 4.8]). In particular $\mathcal{A}_{0}^{\mathbb{F}}(X)=\overline{X}$. Write $\mathcal{A}_{pos}^{\mathbb{F}}(X):=\bigsqcup_{i=1}^{n}\mathcal{A}^{\mathbb{F}}_{i}(X)$.

A slight refinement of the arguments in [4] shows that if $\Lambda$ is cocompact in $\mathbb{F}^{n}$, then

We will adapt this now to lattices $\Lambda$ with $\mathbb{R}\Lambda=L$ an $\mathbb{F}$-linear subspace of $\mathbb{F}^{n}$.

Let $\operatorname{GrAff}_{i}^{\mathbb{F}}(\mathbb{F}^{n};L)\subset\operatorname{GrAff}_{i}^{\mathbb{F}}(\mathbb{F}^{n})$ be the subset of affine $i$-dimensional flats $A$ of $\mathbb{F}^{n}$ such that the linear part of $A$ is contained in $L$, and

For $0\leq i\leq\dim_{\mathbb{F}}(L)$, we define $\mathcal{A}^{\mathbb{F}}_{i}(X;L):=\mathcal{A}^{\mathbb{F}}_{i}(X)\cap\operatorname{GrAff}_{i}^{\mathbb{F}}(\mathbb{F}^{n};L)$ and

Clearly this is a definable subset, and $\mathcal{A}^{\mathbb{F}}(X;L)=X$. Write $\mathcal{A}_{pos}^{\mathbb{F}}(X;L):=\mathcal{A}^{\mathbb{F}}_{pos}(X)\cap\operatorname{GrAff}^{\mathbb{F}}(\mathbb{F}^{n};L)$.

Before proving the theorem, we start by proving some lemmas.

In this section we take “definable” to mean defianble either over $\mathbb{R}_{om}$ in the language of ordered fields, or over $\mathbb{C}$ in the language of fields, according to whether the setting is real or complex. By Chevalley’s theorem, in the latter case the definable sets are exactly constructible sets, i.e., Boolean combinations of algebraic sets.

For a flat $A\in\operatorname{GrAff}^{\mathbb{F}}(\mathbb{F}^{n})$, we write $L(A)\subset\mathbb{F}^{n}$ for the linear part of $A$ (the $\mathbb{F}$-subspace such that we can write $A=L(A)+c$ for some $c$), and for a subset $T\subset\operatorname{GrAff}^{\mathbb{F}}(\mathbb{F}^{n})$, we define $L(T)\subset\mathbb{F}^{n}$ to be the smallest $\mathbb{F}$-linear space containing $L(A)$ for all $A\in T$.

By this theorem, we may break up the definable family of flats as $\mathcal{A}_{pos}^{\mathbb{F}}(X;L)=T_{1}\cup\ldots\cup T_{m}$ where each $T_{i}$ is a neat family in $\operatorname{GrAff}_{j}(\mathbb{F}^{n};L)$ with $1\leq j\leq\dim(L)$. We need the following proposition, the analogue of [4, Proposition 6.2 and Proposition 7.6] for $\operatorname{GrAff}_{j}(\mathbb{F}^{n};L)$.

Now the proofs of 1.1 and 1.4 proceed identical to [4, Theorem 7.8]. We sketch how this works. For each $i$, let $L(T_{i})^{\perp}$ be a complementary $\mathbb{F}$-subspace to $L(T_{i})$. For $A\in T_{i}$ the intersection $(A+L(T_{i}))\cap L(T_{i})^{\perp}$ is a single point $c_{A,i}$, and we let $C_{i}^{\prime}=\bigcup_{A\in T_{i}}c_{i,A}\subset L(T_{i})^{\perp}$ be the closure of the definable set of such points as we range over $A\in T$. Letting $C_{i}=\overline{C_{i}^{\prime}}$, we have $\dim_{\mathbb{F}}C_{i}=\dim_{\mathbb{F}}C_{i}^{\prime}$, and [4, Section 6.2 and 7.3.1, Proofs of Clause (i)] applied to $C_{i}^{\prime}$ shows that with this construction, $\dim_{\mathbb{F}}C_{i}=\dim_{\mathbb{F}}C_{i}^{\prime}<\dim_{\mathbb{F}}X$. Let $V_{i}=L(T_{i})$ and $\mathbb{T}_{i}=\overline{\pi(V_{i})}$. Then we have (by 4.1 for the first equality, by 5.3 and the fact that ${\operatorname{Fl}}(X)$ is closed for the second equality, that $A+L(T_{i})=c_{A,i}+V_{i}$ for the third equality, and that ${\operatorname{Fl}}(X)$ is closed for the fourth equality) that