Manin kernels and exponentials

Martin Bays and Anand Pillay

Tue 17 Jun 16:14:04 EDT 2014

Let A be an abelian variety over the differential function field K of a complex algebraic curve S with a rational vector field.

Recall the “Manin kernel”, a DCF-definable/K subset A^{#} of A admitting a number of descriptions, one being that it is the smallest Zariski-dense DCFdefinable subset of A.

Let T be the theory of A^{#} with all induced structure (over K). This is a rigid divisible commutative group of finite Morley rank. Let T̂ be the theory of the universal cover of A^{#}, in the sense of [covers-

fRM]. The aim of this note is to exhibit a natural analytic model of T̂. We first describe this model.

Let

\xymatrix{0 \ar[r] & H \ar[r] & G \ar[r]^{p} & A \ar[r] & 0}

be the universal vector extension of A.

G is equipped with a canonical D-structure, i.e. a rational section

s_{G} : G → τG

where τG → G is the twisted tangent bundle (aka first prolongation).

Let G^{δ} be the subgroup of “horizontal” points, i.e. for a differential field extension K′ ≥ K,

G^{δ}(K′) = {x ∈ G(K′) | (x, δx) = s_{G}(x)}.

Then for U ⊧ DCF, A^{#}(U) = p(G^{δ}(U)) [Marker-maninKernels]. Taking the Lie algebras of these algebraic groups, we have

\xymatrix{0 \ar[r] & LH \ar[r] & LG \ar[r]^{Lp} & LA \ar[r] & 0}

Now LG also has a natural D-structure induced from that on G,

Ls_{G} : LG → LτG ≅ τLG

(c.f. [BP]), so we have subgroups LG^{δ} and LA^{#} := Lp(LG^{δ}).

Now let S′ ⊆ S be a disk (or in fact any simply-connected domain) in S, and let L ≥ K be the differential field of meromorphic functions on S′. We consider L-points, where we define

A^{#}(L) := p(G^{δ}(L))
LA^{#}(L) := p(LG^{δ}(L)).

As in [BP Appendix], we have relative exponential maps

\xymatrix{LG(L) \ar[d]_{_{exp_{G}}} \ar@{->>}[r] & LA(L) \ar[d]_{_{exp_{A}}} \\ G(L) \ar@{->>}[r] & A(L)}

which respect the D-structures, hence restrict to

\xymatrix{LG^{δ}(L) \ar[d]_{_{exp_{G}}} \ar@{->>}[r] & LA^{#}(L) \ar[d]_{_{exp_{A}}} \\ G^{δ}(L) \ar@{->>}[r] & A^{#}(L)} .

Now our claim is that,

exp_{A} : LA^{#}(L) → A^{#}(L),

when considered as a structure in the language of T̂ (we discuss below exactly how it may be so considered), is a model of T̂.

Let us remark that in the constant case, i.e. when A is over ℂ, the Dstructures on G and LG are trivial, and A^{#}(L) = A(ℂ), and LA^{#}(L) = LA^{#}(ℂ), and exp_{A} is the usual complex exponential map; so we are reduced to the case of [covers-fRM Corollary 4.2.1].

We begin collecting some facts, (I)-(III) below, which we will need in the proof our structure satisfies the axioms of T̂.

By a remark credited to Hamm ([BuiumDiffAlgDiophGeom p.143]), over S′, G analytically descends to the constants. In terms of L-points, this has the following consequence:

Fact 0.1. Let s_{0} ∈ S′. Let G_{0} := G_{s_{0}} be the fibre of G over s_{0}, a complex Lie group, and let

exp_{G_{0}} : LG_{0} → G_{0}

be its exponential map. There exists an isomorphism

θ_{G} : G(L) → G_{0}(L)

and a corresponding ℂ-linear isomorphism

Lθ_{G} : LG(L) → LG_{0}(L)

such that θ_{G}(G^{δ}(L)) = G_{0}(ℂ), and Lθ_{G}(LG^{δ}(L)) = LG_{0}(ℂ), and exp_{G} ∘Lθ_{G} = θ_{G} ∘ exp_{G_{0}}.

It follows that LG^{δ}(L) is a 2g-dimensional ℂ-vector space, and ker exp_{G} ≤ LG^{δ}(L), and hence

ker exp_{A} ≤ LA^{#}  (I)

Since exp_{G_{0}} : LG_{0}(ℂ) ↠ G_{0}(ℂ), it also follows that exp_{G} : LG^{δ}(L) ↠ G^{δ}(L), and it follows by diagram-chase that

exp_{A} : LA^{#}(L) ↠ A^{#}(L).  (II)

So we have:

\xymatrix{LG^{δ}(L) \ar@{->>}[d]_{_{exp_{G}}} \ar@{->>}[r] & LA^{#}(L) \ar@{->>}[d]_{_{exp_{A}}} \\ G^{δ}(L) \ar@{->>}[r] & A^{#}(L)} .

Lemma 0.2.

A^{#}(L^{diff}) = A^{#}(L).  (III)

Proof. We first show LG^{δ}(L^{diff}) = LG^{δ}(L).

Let X be a ℂ-basis of LG^{δ}. For any subdisc S′′ ⊆ S′ and corresponding field L′ ≥ L of meromorphic functions, by the above Fact, LG^{δ}(L′) is still a 2g-dimensional ℂ-vector space, so LG^{δ}(L′) = LG^{δ}(L) = ⟨X⟩_{ℂ}.

So by the (proof of) the Seidenberg Embedding Theorem [MarkerDCF Lemma A.1], for any y ∈ LG^{δ}(L^{diff}), we have y ∈ ⟨X⟩_{ℂ(L^{diff} )} = ⟨X⟩_{ℂ} = LG^{δ}(L).

So LG^{δ}(L^{diff}) = LG^{δ}(L), and hence LA^{#}(L^{diff}) = LA^{#}(L).

Now G^{δ}(L′) = exp_{G}(LG^{δ}(L′)) = exp_{G}(LG^{δ}(L)) = G^{δ}(L). By another Seidenberg argument applied to algebraic extensions, we therefore have G^{δ}(L^{alg}) = G^{δ}(L).

Hence A^{#}(L^{alg}) = A^{#}(L). But it follows from [Wagner FieldsFRM] that A^{#}(L^{diff}) = A^{#}(L^{alg}); this is discussed in [BBP-MLMM], Corollary 1.11 and proof of Theorem 1.1(i).

So A^{#}(L) ⊧ T.

From now on, we mostly omit explicit mention of L, writing A^{#} for A^{#}(L) and so on.

To make

exp_{A} : LA^{#} → A^{#},

a structure in the language of T̂, it remains to define Ĥ ≤ (LA^{#})^{n} for each connected definable subgroup H ≤ (A^{#})^{n}.

If B is a connected algebraic subgroup of A, by [BBP 4.9] we have

A^{#} ∩ B = B^{#}.  (*)

Now LB^{#} ⊆ LA^{#}, and by (I) and (*) we have

LA^{#} ∩ LB = LB^{#}.  (IV)

Now if H is a connected definable subgroup of A^{#}, then H = B^{#} where B is the Zariski closure of H; indeed, by (*) we have H ≤ A^{#} ∩ B = B^{#}, and meanwhile B^{#} ≤ H since B^{#} is the smallest Zariski-dense definable subgroup of B.

Note that (A^{n})^{#} = (A^{#})^{n}, and L(A^{n})^{#} = (LA^{#})^{n}.

So for B a connected algebraic subgroup of A^{n}, we interpret B̂^{#} as LB^{#}.

Proposition 0.3. With the structure described above,

exp_{A} : LA^{#}(L) → A^{#}(L)

is a model of T̂.

Proof. (A1) is by (I).

(A2)-(A5) are clear from the setup and (IV). (A6) is by (II). (A7) and (A8) follow from (I). (A9)(I): by (IV), the exact sequence

0 → L(K^{o}) → LG → LH → 0

remains exact on applying (·^{#}). (A9)(II) is by (I).

References