Algebraic and holomorphic flows in the bi-algebraic context Emmanuel - - PowerPoint PPT Presentation

Algebraic and holomorphic flows in the bi-algebraic context

Emmanuel Ullmo, IHES joint work with Andrei Yafaev. Cetraro, July 14, 2017.

Hermitian Locally Symmetric Spaces-bi-Algebraic point of view.

The following transcendental maps relates "algebraic objects"

◮

expg := (exp, .., exp) : Cg → (C∗)g. (1)

◮

π : Cg → A(C) = Γ\Cg with A an abelian variety. (2)

◮ For D a bounded symmetric domain and Γ a torsion free lattice in

Aut(D) π : D → S = Γ\D. (3)

Hermitian Locally Symmetric Spaces-bi-Algebraic point of view.

◮ If Γ is an arithmetic lattice, S = Γ\D is a quasi-projective variety

(Baily-Borel).

◮ If Γ is irreducible of rank ≥ 2, Γ is arithmetic (Margulis). ◮ When Γ is of rank 1, so D is the unit ball in Cn, Mok proved that

the minimal compactification of S is projective.

◮ D ⊂ p = Cn is semi-algebraic and complex analytic.

Bi-algebraic varieties and weakly special varieties.

Definition 1

◮ Let π : V → W a transcendental map relating 2 algebraic objects.

An algebraic subvariety Y of W is "bi-algebraic" if a component of π−1(Y ) is algebraic.

◮ Let D ⊂ p = Cn be a bounded symmetric domain. An irreducible

algebraic subvariety Θ of D is a component of D ∩ ˜ Θ for an algebraic subvariety ˜ Θ of p. In this situation Θ is semi-algebraic and complex analytic.

Proposition

A subvariety Y of an abelian variety A is bi-algebraic if and only if Y = B + P for an abelian subvariety B of A and a point P. This is equivalent to saying that Y is a totally geodesic subvariety of A.

Bi-algebraic varieties and weakly special varieties.

Definition 2

◮ Let S = Γ\D be an hermitian locally symmetric space and

π : D → S be the uniformizing map. A special subvariety S′ of S is a variety of the form S′ = Γ′\D′ where D′ is a bounded hermitian symmetric subspace of D and where Γ′ := Γ ∩ Aut(D′) is a lattice in D′.

◮ A weakly special subvariety V of S is either special or there exists a

special subvariety S′ = S′

1 × S′ 2 = Γ′ 1\D1 × Γ′ 2\D2 of S and a point

P of S2 such that V = S′

1 × {P} ⊂ S′ 1 × S′ 2 = S′.

Proposition 1 (U-Yafaev)

Assume that Γ is arithmetic. A subvariety V of S = Γ\D is bi-algebraic ⇐ ⇒ V is totally geodesic in S ⇐ ⇒ V is weakly special.

Hyperbolic Ax-Lindemann conjecture.

Theorem 1 (Abelian Ax-Lindemann) Ax, Pila-Zannier

Let π : Cg → A = Γ\Cg and let V be a irreducible algebraic subvariety of

• Cg. Then the Zariski closure W of π(V ) is bi-algebraic (i.e W = B + P).

Theorem 2 (Hyperbolic Ax-Lindemann)

Assume that Γ is an arithmetic lattice. Let π : D → S = Γ\D and let Y be a irreducible algebraic subvariety of Cg. Then the Zariski closure V of π(Y ) is weakly special (i.e totally geodesic or bi-algebraic). The proof is due to U-Yafaev when Γ is cocompact, Pila-Tsimerman for Ag, Klingler-U-Yafaev for a general Shimura variety.

Bloch-Ochiai theorem

Theorem 3 (Bloch-Ochiai)

Let π : Cg → A = Γ\Cg. Let f : C → Cg be a holomorphic map and V = f (C). Then the Zariski closure W of π(V ) is bi-algebraic (i.e W = B + P).

◮ The proof uses mainly Nevalinna theory. ◮ Generalization of this result and of Abelian Ax-Lidemann by

Paun-Sibony for holomorphic maps from a subset of C to Cn with a growth estimate.

◮ For a unbounded real analytic subet, V ⊂ Cg, definable in some

• -minimal structure the Zariski closure of π(V ) is also bi-algebraic.

(U-Yafaev).

Hyperbolic Bloch-Ochiai theorem

Theorem 4 (Hyperbolic Bloch-Ochiai) U-Yafaev

Let D ⊂ p = Cg be an hermitian bounded symmetric domain and Γ be an arithmetic and cocompact lattice of D. Let π : D → S = Γ\D. Let f : C → Cg be a holomorphic map and V = f (C) ∩ D. Then the connected components of the Zariski closure W of π(V ) are weakly special (i.e totally geodesic or bi-algebraic).

◮ The proof uses mainly hyperbolic geometry, o-minimal theory, the

hyperbolic Ax-Lindemann theorem and some of its consequences and a little bit of Nevanlinna theory. We don’t k how to adapt this proof for the usual Bloch-Ochiai theorem.

◮ The case of Ag or of general Shimura varieties is open.

What about the topological closure ?

Let π : X − → Y be a transcendental map between two "algebraic

• bjects". We saw several natural examples of such maps and subsets Θ of

X such that the Zariski closure of π(Θ) in Y , is bi-algebraic.

Question

In this situation, what can be said about the topological closure π(Θ) of π(Θ) ?

Real weakly special subvarieties

Let A = Cg/Γ be a complex abelian variety.

Definition

Let W ⊂ Cg be a R-vector space such that ΓW := Γ ∩ W is a lattice in

• W. Then W /ΓW is a real torus and is a closed real analytic subset of A.

A real analytic subvariety V of A is said to be real weakly special if V = P + W /ΓW for a point P and a real subtorus W /ΓW of A.

Mumford-Tate tori

Definition

Let Θ be an irreducible algebraic subvariety of Cg containing the origin O

• f Cg. The Mumford-Tate group MT(Θ) of Θ is defined as the smallest

Q-vector subspace W of Γ ⊗ Q such that Θ ⊂ W ⊗ R. More generally, if P ∈ Θ. Then we define MT(Θ) as MT(Θ − P). One can check that the definition is independent of the choice of P ∈ Θ. Let WΘ := MT(Θ) ⊗ R. We denote by TΘ the real weakly-special subvariety of A TΘ = π(P) + WΘ/Γ ∩ WΘ. Then TΘ is independent of P and TΘ is the smallest real weakly special subvariety of A containing π(Θ). We say that TΘ is the Mumford-Tate torus associated to Θ. We write µΘ for µTΘ.

Remark

Let Θ be an irreducible complex algebraic subvariety of Cg. Then π(Θ) ⊂ TΘ. When do we have π(Θ) = TΘ ?

Asymptotic Mumford-Tate tori.

Let C be a curve in Cg. Let C ∗ be the Zariski closure of C in P1(C)g. Then C ∗ − C is a finite set of points {P1, . . . , Ps}. Let Cα be a branch of C near a point Pi. There exists a smallest real affine subspace Qα + Wα such that Wα ∩ Γ is a lattice in Wα and such that Cα is asymptotic to Qα + Wα.

Definition

Let T′

α := Wα/Γ ∩ Wα and Tα := π(Qα) + T′ α. We say that Tα is the

asymptotic Mumford-Tate torus associated to Cα

Topological closure of an algebraic flow.

Theorem 5 (U-Yafaev)

Let C be a curve in Cg. Let C1, . . . , Cr be the set of all branches of C through all points at infinity. For all α ∈ {1, . . . , r} let Tα be the associated asymptotic Mumford-Tate torus. Then π(C) = π(C) ∪

r

• α=1

Tα. The theorem has a version in terms of measures. The proof uses the Weyl criterion, explicit computations of the character groups of the asymptotic Mumford-Tate tori and harmonic analysis in particular some results on

• scillatory integrals (Van der Corput lemma).

Example : The linear case.

The case of W a complex linear subspace of Cg is a simple application of Weyl’s criterion. In this case π(W ) = TW and µZ,R → µW .

Real tori are needed

Let V be a C-vector space of dimension 2 and (e1, e2) be a C-basis of V . Let Γ be the lattice Γ := Ze1 ⊕ Z √ −1e1 ⊕ Ze2 ⊕ Z √ −5e2 Then A := A/Γ is an abelian variety of dimension 2. Let W := C(e1 + e2) of V . MT(W ) = Q(e1 + e2) + Q √ −1e1 + Q √ −5e2 and MT(W ) ⊗ R = R(e1 + e2) + R √ −1e1 + R √ −5e2. As a consequence MT(W ) ⊗ R/Γ ∩ MT(W ) ⊗ R is a real torus of real dimension 3. This shows that we can’t expect that in the conjecture 2 that the analytic closure of π(W ) has a complex structure.

Instructing example 1

Proposition

Let n ≥ 3 be an integer. Let C ∈ C2 be the hyperelliptic curve with equation Z 2

2 = Z n 1 + an−1Z n−1 1

+ · · · + a0. Then for any abelian surface A = C2/Γ we have π(C) = A and µC,R → µA as R → ∞. In this case TC = A = Tα for all infinite branches Cα of C.

Instructing example 2

Let C be the hyperbole Z1Z2 = 1 in C2.

case 1.

Let Γ = Z[√−1] ⊕ Z[√−1] ⊂ C2 and A = E × E = C2/Γ. Then π(C) = π(C) ∪ E × {0} ∪ {0} × E, and µC,R → 1 2(µE×{0} + µ{0}×E). In this case TC = A, with two branches C1 near (0, ∞) and C2 near (∞, 0). Then T1 = {0} × E and T2 = E × {0}.

case 2.

If Γ ⊂ C2 is such that the dual lattice Γ of Γ contains no element of the form (0, b) or of the form (a, 0), then π(C) = C2/Γ = A and µC,R → µA. In this case TC = T1 = T2 = A.

The results of Peterzil-Starchenko

Theorem 6 (Peterzil-Starchenko)

Let Θ be a algebraic subvariety of Cg. There exists finitely many algebraic subvarieties C1, . . . , Cm of Cg, finitely many complex vector subspaces V1, . . . , Vm depending only on Θ such that π(Θ) = π(Θ) ∪ ∪m

i=1(π(Ci) + Ti)

where Ti = TVi is the Mumford-Tate torus of Vi. Moreover dim(Ci) < dim(Θ).

Remark

◮ If dim(Θ) = 1 they give a new proof of the theorem 5. ◮ CI and Vi are independent of Γ but Ti = TVi depends on Γ. ◮ π(Θ) and π(Ci) are in general neither closed nor definable in a

• -minimal structure.

◮ I don’t know what should be the measure theoretic version of the

theorem

◮ The proof uses many inputs from Model theory and o-minimal

theory.

Algebraic flows on hermitian locally symmetric spaces

Let π : D − → S = Γ\D be the uniformizing map of a Shimura variety.

Definition

A real weakly special subvariety of S is a real analytic subset of S of the form Z = Γ ∩ H(R)+\H(R)+.x where H is an algebraic subgroup of G such that the radical of H is unipotent and the real points of the Q-simple factors of a Levi of H are not compact and x ∈ D.

Theorem

Let V be a complex totally geodesic subspace of D. Then π(V ) is real weakly special. Proof : ergodic theory (Ratner’s theorem).

Example

Let G = SL2 × SL2, X + = H × H and for some g ∈ SL2(R) Z = {(τ, gτ), τ ∈ H}. Let Γ = SL2(Z) × SL2(Z) and π: H × H − → Γ\X + = Y0(1) × Y0(1).

◮ If g ∈ G(Q), the closure of π(Z) is a special subvariety Y0(n) for

some n ∈ N.

◮ If g /

∈ G(Q), then π(Z) is dense in Γ\X +.

Question

Question

Let Θ be an algebraic subvariety of X +. Describe π(Θ) ⊂ S in term of real weakly special subvarieties.

Example

Let C ′ be an algebraic curve in C2. Let C = C ′ ∩ H × H. Let π = j × j : H × H − → Y0(1) × Y0(1). Describe π(C).

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