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;\s> | 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).