!newbb A
!newbb P
!newbb C
!newop Fix
\newcommand{\s}{\sigma}
\newcommand{\Acal}{\mathcal{A}}
\newcommand{\ph}{\phi}
\renewcommand{\O}{\mathcal{O}}

# 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
    *e)*

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 $(X \\ X') \cap \Gamma_\s != 0$
    *e)*

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:
    $(\Acal,\s) |=$ CCMA,
    $X$,$F$ closed irreducible in $\Acal$,
    $F (= X \times X^\sigma$,
    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. $a_i \|/_b a_ja_k$)
	$\acl(a_1a_2) \cap \dcl( \acl(a_0a_1), \acl(a_0a_2) )
	    = \dcl( \acl(a_1), \acl(a_2) )$
    *ee*

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).

    Work in monster model $\A' |= CCM$.

    Let $b \in B$ 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:
\begin{verbatim}
:::	C* === C*
:::	.      |
:::	.      |
:::	v  fin v
:::	X' ..> X
:::	.      |
:::	.      |
:::	v      v
:::	B' --> B
:::	   fin
\end{verbatim}
    **(II) 
	Quotient by action of nth roots of unity on fibres:

\begin{verbatim}
:::	   [n]
:::	C* --> C*
:::	|      .
:::	|      .
:::	v      v
:::	X' ..> X
:::	|      .
:::	|      .
:::	v      v
:::	B  === B
\end{verbatim}
    *ee*

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 $x/E \in \acl^{TA,eq}(b) \\ \acl^{TA}(b)$.
