1.3 The bi-algebraic setting
1.3.2 Algebraicity in the uniformizing space
1. Ye is called an irreducible algebraic subsetof X+ if it is a complex
analytic irreducible component of the intersection of its Zariski closure inX∨ and
X+;
2. Ye is calledalgebraicif it is a finite union of irreducible algebraic subsets ofX+.
In view of Definition 1.3.5, we are in the following bi-algebraic situation: both X+ areS are algebraic, but unif :
X+
→S is transcendental. Hence a priori there is no relation between the algebraic structures on X+ and on S.
Therefore a natural question arises: what are the bi-algebraic objects? This question will be answered in the following sections. We state the result here:
Theorem 1.3.6. A subset Y ⊂S is weakly special iffYe (a complex analytic irreducible component ofunif−1(Y)) is algebraic inX+andY is an irreducible
subvariety of S.
Remark 1.3.7. Recall the following result of Pila-Tsimerman [49, Lemma 4.1]: maximal connected irreducible semi-algebraic subsets of X+ which are con-
tained in a complex analytic subset of X+ are all algebraic (see the paragraph
before Theorem 3.1.2 for the definition of “connected irreducible semi-algebraic subsets”). Hence an equivalent way to restate Theorem 1.3.6 is to replace “Ye is algebraic in X+” by “Ye is a semi-algebraic subset of
X+”.
A more refined version as well as the proof of this theorem will be given in Corollary 2.3.3. Here we only prove the easy part of the theorem, which is:
Lemma 1.3.8. Any weakly special subset of X+ is irreducible algebraic.
Proof. Suppose that Ze is a weakly special subset of X+. Use the notation
of Definition 1.2.2 and assume that i and ϕ satisfy the properties in Propo- sition 1.2.4. Let N := Ker(Q → Q′) and let y be a point of the weakly
special subset, then Ze = N(R)+U
N(C)y is complex analytic irreducible by
Remark 1.2.3.2. But N(R)+U
N(C)y =N(C)y∩ X+ andN(C)y is algebraic,
CHAPTER 1. PRELIMINARIES 63
We finish this section by the functoriality of algebraicity:
Lemma 1.3.9(functoriality of algebraicity). Let f: (Q,Y+)
→(P,X+)be a
Shimura morphism. Then there exists a unique morphism f∨:Y∨ → X∨ of
algebraic varieties such that the diagram commutes:
Y+ ⊂- Y∨ X+ f ? ⊂- X∨ f∨ ? .
Furthermore, for any irreducible algebraic subset Ze of Y+, the closure in the
archimedean topology off(Ze)is irreducible algebraic inX+andf(Ze)contains
a dense open subset of this closure.
In particular, if f is an embedding, then an irreducible algebraic subset of
Y+ is an irreducible component of the intersection of an irreducible algebraic
subset of X+ with Y+.
Proof. Fix a point x0∈ Y+, then we have Y+=Q(R)+U Q(C)/C(x0)⊂ - Y∨=Q(C)/Fx00Q(C) X+=P(R)+UP(C)/C(f(x0)) f ? ⊂- X∨=P(C)/F0 f(x0)P(C) f∨ ? ,
where C(x0) (resp. C(f(x0))) denotes the stabilizer of x0 (resp. f(x0)) in
Q(R)UQ(C)(resp. P(R)UP(C)). The mapf∨is unique sinceQ(R)UQ(C)/C(x0)
is dense in Y∨.
To prove the second statement, it is enough to prove the result forf∨(ZeZar) ⊂ X∨ where ZeZar is the Zariski closure of Ze in
Y∨. This is then an algebro-
geometric result, which follows easily from Chevalley’s Theorem ([22, Chapitre IV, 1.8.4]) and [41, I.10, Theorem 1].
Chapter 2
Ax’s theorem of log type
2.1
Results for the unipotent part
Given a connected mixed Shimura varietyS, letSGbe its pure part. We have a
projectionS−−→[π] SG. For any pointb∈SG, denote byEthe fiberSb. Suppose
thatSis associated with the mixed Shimura datum(P,X+), which can be fur-
ther assumed to satisfyP = MT(X+)by Proposition 1.1.19. Letunif : X+→
S = Γ\X+ be the uniformization. NowE =S
b ≃ΓW\W(R)U(C) with the
complex structure determined by b ∈ SG (E = Sb = ΓW\W(C)/Fb0W(C)),
where ΓW := Γ∩W(Q). WriteT := ΓU\U(C)and A:= ΓA\V(C)/Fb0V(C)
whereΓU := Γ∩U(Q)andΓV := ΓW/ΓU, thenAis a complex abelian variety
and Eis an algebraic torus overAwhose fibers are isomorphic toT.
Lemma 2.1.1. IfE admits a structure of algebraic group whose group law is compatible with the group law of W, then W (hence E) is commutative. In this case E is a semi-abelian variety.
Proof. IfEis an algebraic group, thenT is a normal subgroup ofE. HenceE
acts onT by conjugation, and this action factors viaA, and then it is trivial by [13, 8.10 Proposition]. ThereforeT is in the center ofE. Now consider the commutator pairingE×E→E. This factors through a morphismA×A−→f T. But as a morphism from an abelian variety to an algebraic torus overC,f is then constant. So the commutator pairingE×E→Eis trivial, and hence E
is commutative.
The commutator pairingW×W →W induces an alternating formΨ :V× V →U (see §1.1.2.5) which induces the morphism f above. We have proved in the last paragraph that Ψ(V(R), V(R)) ⊂ ΓU with ΓU := Γ ∩U(Q).
But Ψ(V(R), v) is continuous for any v ∈ V(R) and Ψ(0, V(R)) = 0, so
Ψ(V(R), V(R)) = 0. Hence the commutator pairing W ×W → W is triv- ial, and thereforeW is commutative.