# CCM
A sort for each compact complex manifold,
and for each reduced irreducible compact analytic space;
relations for analytic subsets of products of the sorts
(locally defined by vanishing of holomorphic \C-valued functions on
polydiscs).
QE, EI; tt, fRM, Z.
Chow: analytic subsets of \P^n(\C) are precisely the Zariski closed sets in the
sense of AG.
Corollary:
The structure induced on \A^1 := \P^1 \\ {\infty} is precisely that of the
complex field, with constants for complex points.
# TA and CCMA
Let T be a complete theory.
T_\s := \|Th({ <{ M;\s }> | M |= T, \s : M --> M automorphism})
TA := model companion = theory of existentially closed models of T_\s
if such exists.
Facts: Suppose TA exists. Then:
* \acl^{TA}(C) = \acl_\s(C) := \acl^T(\Cup_{i\in\Z})\s^i(C))
* QE: for C=\acl_\s(C), qftp^{TA}(C) |= tp^{TA}(C)
* T (super)stable => TA (super)simple
CCMA exists,
SKIPME:
axiomatised by:
* CCM_\s;
* given X (= Y \times Y^\s,
X and Y closed irreducible, <--- Fact: this is 1st order.
co-ord projections dominant,
X' (= X proper closed,
then (X \\ X') \cap \Gamma_\s != \0
Holomorphic dynamics and finite-dimensional types:
X CCM, f : X --> X holomorphic automorphism,
(X,f)^# := \{ x \in X | \s(x) = f(x) \}
finite-dimensional definable set in CCMA.
More generally:
(\A,\s) |= CCMA,
X,F closed irreducible in \A,
F (= X \times X^\sigma,
projections dominant with finite fibres.
Then (X,F)^# := \{ x \in X | (x,\s(x)) \in F \}.
The _finite-dimensional_ types are the generic types
of such (X,F)^#.
Theorem [Trichotomy]:
Let p be a finite-dimensional minimal type.
If p is not one-based,
it is non-orthogonal to (\P^1,\id)^#.
If p is one-based non-trivial,
it is non-orthogonal to some (X,F)^# with X and F definable groups.
(Proof of Zilber Dichotomy: CBP via jet spaces)
This is for *real* types, but imaginary types can come up in an analysis...
# Imaginaries in TA
Suppose T superstable with EI and TA exists.
Fact [Hrushovski]: TA has gEI
(imaginaries are interalgebraic with reals)
Example:
T := theory of a connected groupoid with \pi_1 = Z/2Z.
In TA,
X := fixed objects,
E(x,y) <=> Mor(x,y) fixed pointwise by \sigma;
then X/E is not eliminable.
Theorem [Hrushovski]: TFAE:
(i) TA has EI
(ii) T "eliminates finite groupoid imaginaries"
(iii) T has "3-uniqueness":
Given b,
and {a_0,a_1,a_2} independent over b\in\acl(a_i),
(i.e. a_i \|/_b a_ja_k)
\acl(a_1a_2) \cap \dcl( \acl(a_0a_1), \acl(a_0a_2) )
= \dcl( \acl(a_1), \acl(a_2) )
Remark: \acl(b) |= T => 3-uniqueness (by coheiring).
So e.g. ACFA has EI.
Theorem:
CCM does not have 3-uniqueness,
so CCMA does not have EI.
Idea:
Let X --> B be a principal \C^*-bundle
(so have definable principal action of \C^* on fibres).
In monster model:
b \in *B generic;
a_0,a_1,a_2 \in *X_b generic independent / b;
let \ph \in (a_2/a_1)^{1/n} \in *\C^*.
Now (a_2/a_1)^{1/n}
= (a_2/a_0)^{1/n} * (a_0/a_1)^{1/n},
so \ph \in \dcl( \acl(a_0a_1), \acl(a_0a_2) ).
So this shows non-3-uniqueness unless
\ph \in \dcl( \acl(a_1), \acl(a_2) ).
Now \ph \notin \dcl(a_1,a_2) = \dcl(a_1,\ph^n)
since \ph \notin \dcl(\ph^n).
So STS \acl(a_i) = \dcl(a_i).
So want X --> B defble \C^*-bundle s.t. a \in *X generic
=> \acl(a) = \dcl(a);
i.e. any dominant generically finite X' --> X has a generic section.
Finite covers of \C^*-bundles:
(I)
Base change:
C* === C*
. |
. |
v fin v
X' ..> X
. |
. |
v v
B' --> B
fin
(II)
Quotient by action of nth roots of unity on fibres:
[n]
C* --> C*
| .
| .
v v
X' ..> X
| .
| .
v v
B === B
Fact:
Holomorphic \C^*-bundles over B are classified by
first cohomology group of sheaf of local holomorphic \C^*-valued functions,
H^1(\O_B^*),
and (II) corresponds to multiplication by n in this group.
Fact:
Exists simply connected strongly minimal smooth compact (K3) surface B
with H^1(\O_B^*) ~= \Z
Let X --> B correspond to generator of H^1(\O_B^*).
B s.c. s.m. => no non-trivial finite B' --> B;
B s.m. => X has no ramified finite covers,
=> any cover has to be as in (II)
but no such exist since [X] \in H^1(\O_B^*) not divisible.
Explicitly, the following imaginary is not eliminable:
X \cap \Fix(\s)
with x E x' iff
\pi(x) = \pi(x')
and (x'/x)^{1/2} (= \Fix(\s)
Proof:
Let b \in *B generic with \s(b)=b;
let x \in *X_b generic with \s(x)=x.
Since \acl_\s(x) = \acl(x) = \dcl(x)
and \acl_\s(b) = \acl(b) = \dcl(b),
by the QE, tp(x/\acl_\s(b)) is determined by
"x is generic in *X_b and \s(x) = x".
So x/E \in \acl^{TA}^{eq}(b) \\ \acl^{TA}(b).