Table Of ContentALGEBRAIC AND O-MINIMAL FLOWS ON COMPLEX
AND REAL TORI
7
1
YA’ACOV PETERZIL AND SERGEI STARCHENKO
0
2
n Abstract. We consider the covering map π : Cn T of a com-
a pact complex torus. Given an algebraic variety X→ Cn we de-
J ⊆
scribe the topological closure of π(X) in T. We obtain a similar
1 description when T is a real torus and X Rn is a set definable
3 ⊆
in an o-minimal structure over the reals.
]
G
A
h. Contents
t
a
1. Introduction............................................... 2
m
2. Preliminaries.............................................. 4
[
2.1. Model theoretic preliminaries.......................... 4
1
2.2. Basics on additive subgroups .......................... 6
v
0 3. Valued field structures on R............................... 6
8
3.1. Closure and the standard part map.................... 6
0
9 3.2. The algebraic closure C of R as an ACVF structure.... 7
0 3.3. On µ-stabilizers of types............................... 9
.
1 3.3.1. The o-minimal case................................ 9
0
3.3.2. The algebraic case................................. 10
7
1 4. Affine asymptotes ......................................... 12
: 5. Describing cl(X +Λ) using asymptotic flats ............... 14
v
i 6. Completing the proof in the algebraic case................. 17
X
6.1. On families of affine subspaces......................... 17
r
a 6.2. Proof of the main theorem in the algebraic case........ 18
7. The o-minimal result...................................... 22
7.1. Asymptotic R-flats.................................... 22
7.2. Neat families in the o-minimal context................. 24
7.3. Proof of the main theorem............................. 25
7.3.1. Proof of Clause (i)................................. 26
7.3.2. Proof of Clause (ii)................................ 27
8. An example............................................... 27
References..................................................... 29
BothauthorsthanktheIsrael-USBinationalScienceFoundationforitssupport.
The second author was supported by the NSF researchgrant DMS-1500671.
1
2 YA’ACOVPETERZIL AND SERGEI STARCHENKO
1. Introduction
Let A be a complex abelian variety of dimension n, and let π : Cn
→
A be its covering map. In articles [5] and [6], Ullmo and Yafaev con-
sidered the following question: For an algebraic subvariety X Cn, or
⊆
more generally a definable set in an o-minimal expansion of the reals,
what is the topological closure of π(X) in A?
They prove the following result (see [5, Theorem 2.4]): If X Cn is
⊆
an irreducible algebraic curve then the topological closure of π(X) in
A is
m
cl(π(X)) = π(X) Z ,
k
∪
k=1
[
where each Z is a real weakly special subvariety of A, namely a coset
k
of a real Lie subgroup of A. They conjecture that the same is true for
algebraic subvarieties X Cn of arbitrary dimension.
⊆
In this article we give a full description of cl(π(X)) when X is an
algebraic subvariety of Cn of arbitrary dimension and also when X is
definable in an o-minimal structure over the reals.
Asweshow, theconjecturefrom[5]failsasstatedsincethefrontierof
π(X) may contain an infinite family of real weakly special subvarieties
(see Section 8). We prove a modified version of the conjecture by
showing that the frontier of π(X) consists of finitely many families
of real weakly special subvarieties. Our theorem holds for arbitrary
compact complex tori.
Theorem 1.1. Let π : Cn T be the covering map of a compact
→
complex torus and let X be an algebraic subvariety of Cn. Then there
are finitely many algebraic varieties C ,...,C Cn and finitely many
1 m
⊆
real subtori (i.e real Lie subgroups) T ,...,T T such that
1 m
⊆
m
cl(π(X)) = π(X) (π(C )+T ).
i i
∪
i=1
[
In addition,
(i) If T is maximal with respect to inclusion among the subtori then
i
C is finite.
i
(ii) For every i = 1,...,m, we have dim C < dim X.
C i C
Notice that in general the sets π(X) and π(C ) need neither be
i
closed nor definable in any o-minimal structure. Note also that when
dim X=1 then dim C =0 hence Theorem 1.1 implies the result of
C C i
Ullmo and Yafaev mentioned above.
FLOWS ON TORI 3
In fact, as we show, the choice of C depends only on X (and not
i
on T) and furthermore, each subtorus T T in the above description
i
⊆
is of the form cl(π(V )) with V Cn a complex linear subspace which
i i
⊆
also depends only on X.
Let Λ = kerπ be the corresponding lattice in Cn. We find it more
convenient to work with the topological closure of X+Λ in Cn instead
of cl(π(X)). It is easy to see that cl(π(X)) = π(cl(X +Λ)) hence the
above theorem can be deduced from our analysis of cl(X +Λ) in Cn.
We first introduce some notation: Let V be a complex or real linear
subspace of Cn. We will denote by V⊥ the orthogonal compliment of V
with respect to the standard inner product on Cn. Also for a lattice Λ
in Cn we denote by VΛ the smallest R-linear subspace of Cn containing
Λ with a basis in Λ (equivalently, this is the connected component of
0 in the real Lie group cl(V +Λ)).
The following is the main result of the first part of this article.
Main Theorem (algebraic case). [see Theorem 6.3] Let X Cn be
⊆
an algebraic subvariety. There are linear C-subspaces V ,...,V Cn
1 m
⊆
of positive dimension and algebraic subvarieties C V⊥,...,C
1 ⊆ 1 m ⊆
V⊥ such that for any lattice Λ < Cn we have
m
m
cl(X +Λ) = (x+Λ) (C +VΛ +Λ).
∪ i i
i=1
[
In addition,
(i) For each V that is maximal among V ,...,V , the set C Cn
i 1 m i
⊆
is finite.
(ii) For each i = 1,...,m, we have dim C < dim X.
C i C
Notice that Theorem 1.1 is an immediate corollary.
In the second part of the article we obtain a similar result in the o-
minimal setting, and disprove the analogouso-minimal conjecture from
[6]. In order to formulate the theorem, we fix an o-minimal expansion
R of the real field.
om
Theorem 1.2. Let π : Rn T be the covering map of a compact real
→
torus and let X Rn be a definable set in R . There are finitely
om
⊆
many definable closed sets C ,...,C Rn and finitely many real
1 m
⊆
subtori T ,...,T T such that
1 m
⊆
m
cl(π(X)) = π(X) (π(C )+T ).
i i
∪
i=1
[
In addition,
4 YA’ACOVPETERZIL AND SERGEI STARCHENKO
(i) If T is maximal with respect to inclusion among T ,...,T then
i 1 m
C is bounded in Cn and in particular π(C ) is closed.
i i
(ii) For every i = 1,...,m, we have dim C < dim X (where dim
R i R R
is the o-minimal dimension).
As in the algebraic case, the above result follows from a theorem on
the closure of X +Λ in Rn, for Λ = kerπ.
Main Theorem (o-minimal case). [see Theorem 7.8] Let X Rn
⊆
be a definable closed set in an o-minimal expansion R of the real field.
om
There are linear R-subspaces V ,...,V Rn of positive dimension,
1 m
⊆
and for each i = 1,...,m a definable closed C V⊥, such that for
i ⊆ i
any lattice Λ < Rn we have
m
cl(X +Λ) = (X +Λ) (C +VΛ +Λ).
∪ i i
i=1
[
In addition,
(i) For each V that is maximal among V ,...,V , the set C Rn
i 1 m i
⊆
is bounded, and in particular C +VΛ +Λ is a closed set.
i i
(ii) For each i = 1,...,m, we have dim C < dim X.
R i R
All of the above theorems are based on analysis of types on X in
various model theoretic settings. In the o-minimal setting we consider
types over R in the language of R , while in the algebraic case, we
om
work with types over C in the language of valued fields. We introduce
these notions and necessary preliminaries in Section 2 and Section 3.
One of the main tools that we are using in both cases is the theory of
stabilizers of µ-types as was developed in [3] (see Section 3.3 below).
Althoughthearticleisnotformulatedinthatlanguage, ourapproach
is influenced by van den Dries work on various notions of limits of
definable families and their connection to model theory (see [8]).
2. Preliminaries
2.1. Model theoretic preliminaries. We first introduce the basic
model theoretic settings in which we will be working. We refer to [2]
for basics on model theory and to [7] and [9] for basics on o-minimality.
We denote by the “algebraic” language, i.e. the language of rings
a
L
= +, , ,0,1 , and view the field C as an -structure.
a a
L h − · i L
Working with the field R and semialgebraic sets we use the “semial-
gebraic” language = +, , ,<,0,1 .
sa
L h − · i
FLOWS ON TORI 5
For the algebraic case we also need a language for valued field. We
use the language = +, , , ,0,1 , where is a unary predicate
val
L h − · O i O
with an intended use for the valuation ring.
For the o-minimal case we fix an o-minimal expansion R of the
om
field R and denote its language by . Notice that .
om a sa om
L L ⊆ L ⊆ L
We also work in expansions of R and R by various additive sub-
om
groups Λ Rn. To avoid different expansions it is more convenient
≤
to treat them all at once. Thus we consider the expansion of R by all
subsets of Rn, for all n. The language for this structure is denoted by
, and we have in it a predicate for each subset of Rn. We let R
full full
L
be the associated -structure on R.
full
L
We choose a cardinal κ > 2ω and fix a κ-saturated elementary exten-
sion R of R . We will denote by R the underlying real closed field
full full
and by R the o-minimal reduct R ↾ . Notice that since both
om full om
L
the real closed field R and the o-minimal structure R are reducts of
om
R they are both κ-saturated.
full
To distinguish between subsets of R and R we will use the following
convention: we use roman lettes X,Y,Z etc. to denote subsets of Rn
and script letters , , etc. for subsets of Rn.
X Y Z
Also if X Rn, then we can view X as an -definable set and
full
⊆ L
denote by X♯ the set X(R) of realizations of X in R .
full
Our model theoretic terminology is standard. By definable we mean
definable with parameters. We use -definable and semialgebraic
sa
L
interchangeably.
If is one of our languages, then an -type p(x) (over a set A) is a
• •
L L
collection of -formulas (over A) with free variables x. We identify an
•
L
-type p(x) with the collection of subsets of R|x| defined by formulas
•
L
in p(x). We do not assume that a type is complete, but always assume
it is consistent. Thus an -type p(x) over A is a collection subsets of
•
L
R|x| such that each subset is -definable over A and p(x) satisfies the
•
L
finite intersection property (namely the intersection of a every finite
subcollection of p is nonempty). Given a type p(x) we denote by p(R)
the set . Two -types p(x) and q(x) are called equivalent if
X∈pX L•
for every finite p (x) p(x) there is finite q (x) q(x) with q (R)
0 0 0
T ⊆ ⊆ ⊆
p (R), and vise versa. It follows that p(R) = q(R).
0
Definition 2.1. A subset Rn is called pro-semialgebraic (over
X ⊆
A R) if there is A R (A A) with A < κ and a semialgebraic
0 0 0
⊆ ⊆ ⊆ | |
type p(x) over A such that = p(R).
0
X
Notice that if = q(R) for another semialgebraic type q(x) over A
0
X
then by κ-saturation of R the types p(x) and q(x) are equivalent.
6 YA’ACOVPETERZIL AND SERGEI STARCHENKO
2.2. Basics on additive subgroups. Let W be a finite dimensional
R-vector space and Λ be an additive subgroup of W whose R-span is
the whole W. We do not assume that Λ is a lattice or even finitely
generated.
We say that an R-subspace V W is defined over Λ if it has a basis
⊆
consisting of elements of Λ.
It is not hard to see that the family of subspaces defined over Λ is
closed under arbitrary intersections and finite sums.
ForasubspaceH W wewilldenotebyHΛ thesmallestR-subspace
⊆
of W containing H and defined over Λ.
We will need the following well-known fact.
Fact 2.2. Let W be a finite dimensional R-vector space and Λ W
≤
an additive subgroup whose R-span is the whole W. If H W is an
⊆
R-subspace then HΛ+L cl(H+Λ) (with equality when Λ is a lattice
⊆
in W).
Remark 2.3. For a C-subspace H Cn and an additive subgroup
⊆
Λ Cn whose R-span is the whole Cn, we still denote by HΛ the
⊆
smallest R-subspace of V containing H having an R-basis in Λ. Thus
HΛ need not be a C-linear space.
3. Valued field structures on R
We denote by the convex hull of R in R. It is a valuation ring of
R
O
R, and we let µ be its maximal ideal. The set µ is the intersection of
R R
allopenintervals( 1/n,1/n)forn N>0, henceitispro-semialgebraic
− ∈
over ∅.
As an additive group is the direct sum = R µ , hence for
R R R
O O ⊕
α there is unique r R with α r + µ . This r is called
R α α R α
∈ O ∈ ∈
the standard part of α and we denote it by st(α). Thus we have the
standard part map st: R. Slightly abusing notations, we use
R
O →
st(x) also to denote the map from n to Rn, and for a subset Rn
R
O X ⊆
we write st( ) for the set st( n).
R
X X ∩O
3.1. Closure and the standard part map. We need the following
claim that relates the topological closure and the standard part map.
It follows from the saturation assumption on R . As usual for ,
full
X Y ⊆
Rn we write + for the set x+y: x ,y .
X Y { ∈ X ∈ Y}
Claim 3.1. (1) For X Rn we have cl(X) = st(X♯). In particular,
⊆
for X,Y Rn we have cl(X +Y) = st(X♯ +Y♯).
⊆
FLOWS ON TORI 7
(2) Let Σ be a collection of subsets of Rn. Then
st X♯ = st(X♯).
!
X∈Σ X∈Σ
\ \
In particular the set st X♯ is closed.
X∈Σ
3.2. The algebraic closur(cid:0)eTC of R(cid:1)as an ACVF structure.
Let C = R+iR, where i = √ 1. It is an algebraically closed field
−
containing C. As usual we identify the underlying set of C with R2 and
the underlying set of C with R2.
Let = + i . It is a valuation ring of C with the maximal
C R R
O O O
ideal µ = µ + iµ . Again we have that = C µ , and we let
C R R C C
O ⊕
st: C be the standard part map.
C
O →
Remark 3.2. We can identify C with the residue field k = /µ so
C C
O
that the residue map res: k is the same as the standard part
C
O →
map st: C.
C
O →
We denote by C the -structure (C;+, , ,0,1).
val val C
L − ·O
Proposition 3.3. If Cn is an -definable set then Cn is
val
X ⊆ L X ∩
-definable in the field C, i.e. it is a constructible subset of Cn.
a
L
If is an -definable family of subsets of Cn then the family F
val
F L { ∩
Cn: F is constructible.
∈ F}
Proof. By Remark 3.2, the field C together with a predicate for C
and the map st(x) is an algebraically closed field with an embedded
residue field. By [1, Lemma 6.3] the embedded residue field C is stably
embedded, i.e. for every -definable subset Cn the set Cn
val
L X ⊆ X ∩
is -definable in the field of complex numbers.
a
L
The second part of the proposition is not stated in [1, Lemma 6.3],
but it easily follows: By quantifier elimination from [1, Lemma 6.3],
the theory of algebraically closed valued fields of characteristic zero
with an embedded residue field is complete, and we can use a standard
compactness argument. (cid:3)
Notice that the structure C is not a reduct of R (e.g. is not
val full C
O
definable in R ), so C does not have to be κ-saturated, and in fact
full val
it is not. For example the set x x / (c + µ ): c C is an
C C
{ ∈ O ∧ ∈ ∈ }
-type over C but has no realization in C.
val
L
However, as we will see below, -types over C that are realized in
val
L
C can be viewed as pro-semialgebraic objects, and working with them
we will make use of the saturation of the field R.
Using the identification of Cn with R2n, we say that a subset Cn
X ⊆
is semialgebraic (over A R) if it is semialgebraic (over A) as a subset
⊆
8 YA’ACOVPETERZIL AND SERGEI STARCHENKO
of R2n. Similarly we say that a set Cn is pro-semialgebraic if it is
X ⊆
pro-semialgebraic as a subset of R2n.
For example, every constructible subset of Cn is also semialgebraic,
and for every n N the set µn is pro-semialgebraic over ∅. However
C
∈
the set is not pro-semialgebraic.
C
O
Given an element α Cn we will consider its -type over C and
val
∈ L
alsoitssemialgebraictypeover R. Todistinguish these typeswe denote
by tp (α/C) the complete -type of α over C, i.e.
val Lval
tp (α/C) = Cn: α , is -definable over C ,
val {X ⊆ ∈ X X Lval }
and by tp (α/R) the semialgebraic type of α over R, i.e.
sa
tp (α/R) = Cn: α , is semialgebraic over R .
sa {X ⊆ ∈ X X }
Notice that tp (α/R) can be also written as
sa
X♯ Cn: α X♯, X Cn is semialgebraic .
{ ⊆ ∈ ⊆ }
The following theorem plays an essential role in this paper.
Theorem 3.4. Let α C and p(x) = tp (α/C). There is an -type
∈ val Lval
s(x) over C that is equivalent to p(x) and such that every s(x) is
X ∈
pro-semialgebraic over R.
Proof. By Robinson’s quantifier elimination (see [4]), if Cn is -
val
X ⊆ L
definable over C then it is a finite Boolean combination of sets of the
form X♯ where X Cn is a constructible set, and sets of the form
⊆
z Cn: h(z) q(z) n where h,q C[x].
C
{ ∈ ∈ O } ∈
Since the complement of a constructible set is constructible, and,
for h,q C[x], the complement of the set z Cn: h(z) q(z) n
C
∈ { ∈ ∈ O }
is z Cn: h(z) = 0,q(z)/h(z) µn , every -definable over C set
C val
{ ∈ 6 ∈ } L
Cn isafinitepositiveBooleancombinationofsetsofthefollowing
X ⊆
three kinds:
1. X♯, where X Cn is a constructible set.
⊆
2. z Cn: h(z) = 0, q(z)/h(z) µn , where h,q C[x].
C
{ ∈ 6 ∈ } ∈
3. z Cn: h(z) q(z) n , where h,q C[x].
C
{ ∈ ∈ O } ∈
Notice that sets of all three kinds are -definable over C. Every
val
L
set of the first kind is also a semialgebraic set defined over R, and
every set of the second kind is pro-semialgebraic over R (since µ is
C
pro-semialgebraic).
Let p(x) be of the third kind, i.e.
X ∈
= z Cn: h(z) q(z) n .
C
X { ∈ ∈ O }
Since α , we have h(α) q(α) n. Then either q(α) = 0 (and
C
∈ X ∈ O
hence h(α) = 0) or, for c = st(h(α)/q(α)), we have h(α)/q(α) c µn.
C
− ∈
FLOWS ON TORI 9
In either case we get a set of the first or second kind with α
Y ∈ Y
and .
Y ⊆ X
Thus if we take s(x) to be the set of all p of first and second
X ∈
kinds, then s(x) is equivalent to p(x) and consists of pro-semialgebraic
over R sets. (cid:3)
Corollary 3.5. Let α Cn and p(x) = (α/C). There is an -
val sa
∈ L L
type r (x) over R such that
α
(1) r (C) = p(C).
α
(2) For every finite r′(x) r (x) there is p(x) with r′(x).
α
⊆ X ∈ X ⊆
Remark 3.6. Notice that unless α Cn the semialgebraic type r (x)
α
∈
is different from the semialgebraic type p (x) = tp (α/R) and we only
sa sa
have strict inclusion p (C) r (C) = p(C).
sa α
⊂
Using the κ-saturation of the field R and Corollary 3.5 we obtain
the following.
Corollary 3.7. Let α Cn, p(x) = tp (α/C) and Cn a pro-
∈ val Z ⊆
semialgebraic set. If = ∅ for every p(x) then p(C) = ∅.
Z∩X 6 X ∈ Z∩ 6
Corollary 3.8. Let h(x) be a polynomial over C, α Cn, α = h(α),
1
∈
p(x) = tp (α/C) and p (x) = tp (α /C). Then h(x) maps p(C) onto
val 1 val 1
p (C).
1
Proof. Let s(x) be an -type over C equivalent to p(x) consisting of
val
L
pro-semialgebraic sets, as in Theorem 3.4.
First notice that p (x) is equivalent to the type
1 val
L
h(p′(C)): p′(x) p(x) is finite .
{ ⊆ }
Let s (x) = h(s′(C)): s′(x) s(x) is finite . It is easy to see that
1
{ ⊆ }
s (x) consists of pro-semialgebraic sets and since s(x) and p(x) are
1
equivalent, s (x) and p (x) are equivalent as well.
1 1
As h(x) is a polynomial over C it is also a semialgebraic map, and
by the κ-saturation of R we have h(s(C)) = s (C). Since s(C) = p(C)
1
and s (C) = p (C) the result follows. (cid:3)
1 1
3.3. On µ-stabilizers of types.
3.3.1. The o-minimal case.
We review briefly µ-stabilizers of -types over R and refer to [3]
om
L
for more details.
Let α Rn and p(x) = tp (α/R). We define the µ-stabilizer of p
∈ om
as
(3.1) Stabµ (p) = v Rn: v +(p(R)+µn) = p(R)+µn ,
om { ∈ R R}
10 YA’ACOVPETERZIL AND SERGEI STARCHENKO
we also denote it by Stabµ (α/R).
om
The fact below follows from the main results of [3] (see Proposition
2.17 and Theorem 2.10 there).
Fact 3.9. Let α Rn.
∈
(1) Stabµ (α/R) is an -definable subgroup of (Rn,+).
(2) If α oismunbounded, iL.eo.mα / n, then Stabµ (α/R) is infinite.
∈ OR om
3.3.2. The algebraic case.
In order to define the µ-stabilizer in the algebraic case we need some
preliminaries.
Definition 3.10. For α Cn we let µ = p(C) + µC, where p(x) =
∈ Pα n
tp (α/C).
val
Notice that α µ and µ depends only on the -type of α over
∈ Pα Pα Lval
C.
Lemma 3.11. Let α Cn.
∈
(1) If v Cn then v + µ = µ .
∈ Pα Pv+α
(2) If Cn is -definable over C then
val
X ⊆ L
(a) α +µ implies µ +µ; and
∈ X Pα ⊆ X
(b) α / +µ implies µ ( +µ) = ∅.
∈ X Pα ∩ X
Proof. Let p(x) = tp (α/C).
val
(1). Let β = v + α and q(x) = tp(β/C). It is easy to see that
v+p(C) = q(C), hence v +p(C)+µC = q(C)+µC = Pµ .
n n v+α
(2). Assume α +µ. Then p(C) +µ and p(C)+µ +µ.
∈ X ⊆ X ⊆ X
(3). Assume α / +µ. Let = Cn ( + µ). Then + µ =
∈ X Y \ X Y Y
and α . By the previous clause µ .
∈ Y Pα ⊆ Y
(cid:3)
Proposition 3.12. Let α Cn and p(x) = tp (α/C). Then
∈ val
µ = ( +µn)
Pα X C
X+µC∈p(x)
\n
Proof. It follows from Lemma 3.11(2) that
µ ( +µn),
Pα ⊆ X C
X+µ\Cn∈p(x)
and we only need to show the inclusion
( +µn) p(C)+µn.
C C
X ⊆
X+µC∈p(x)
\n
Let β ( + µn). We need to show β p(C) + µn, or
∈ X+µCn∈p(x) X C ∈ C
equivalently β + µn p(C) = ∅. Let p(x). Then +µC p(x)
T C ∩ 6 X ∈ X n ∈
