CCMA I(maginaries)
==================
Outline
-------
* CCM
* CCMA
* gEI
* notEI
CCM
---
A sort for each compact complex manifold;
relations for analytic subsets (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.
TA and CCMA (5:00)
-----------
T complete theory.
T_\s := Th({ | M |= T, \s : M --> M endomorphism})
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 x 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:
X CCM, f : X --> X holomorphic automorphism
(or bimeromorphism, or self-correspondence);
(X;f)^# := { x \in X | \s(x) = f(x) }
finite-dimensional defble set in CCMA.
Theorem:
Zilber dichotomy for finite-dimensional minimal types in CCMA:
any such is either one-based
or is almost internal to Fix(\s) \cap \P^1.
(Proof: CBP via jet spaces)
Proved for *real* types, but imaginary types can come up in an analysis...
Imaginaries in TA (15:00)
-----------------
Suppose T superstable with EI and TA exists.
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: if acl(b) |= T, 3-uniqueness holds by coheiring.
So e.g. ACFA has EI.
Fact [Hrushovski]: TA has gEI
(imaginaries are interalgebraic with reals)
notEI (25:00)
-----
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)(a_0/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 done 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)
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).