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\subsection*{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.
\paragraph{Corollary:}
\mbox{}\\    The structure induced on $\A^1 := \P^1 \setminus {\infty}$ is precisely that of the
    complex field, with constants for complex points.


\subsection*{TA and CCMA}

Let $T$ be a complete theory.

$T_\s := \operatorname{Th}( \{ \left<{M;\s}\right> \;|\; M \models  T, \s : M \rightarrow  M automorphism\} )$

$TA :=$ model companion = theory of existentially closed models of $T_\s$
if such exists.

\paragraph{Facts: Suppose TA exists. Then:}
\begin{itemize}\item $\acl^{TA}(C) = \acl_\s(C) := \acl\widehat{T}(\Cup_{i\in\Z})\s^i(C))$
\item QE: for $C=\acl_\s(C)$, $\operatorname{qftp}^{TA}(C) \models  tp^{TA}(C)$ 
\item T (super)stable $=>$ TA (super)simple
\end{itemize}

CCMA exists,

\paragraph{SKIPME:}
\mbox{}\\    axiomatised by:
\begin{itemize}\item $CCM_\s$;
\item given $X \subseteq  Y \times Y^\s$,
\\	$X$ and $Y$ closed irreducible,	<--- Fact: this is 1st order.
\\	co-ord projections dominant,
\\	$X' \subseteq  X$ proper closed,
\\	then $(X \setminus X') \cap \Gamma_\s \neq  0$
\end{itemize}

\paragraph{Holomorphic dynamics and finite-dimensional types:}
\mbox{}\\    $X$ CCM, $f : X \rightarrow  X$ holomorphic automorphism,
\\    $(X,f)^\sharp := \{ x \in X \;|\; \s(x) = f(x) \}$
	finite-dimensional definable set in CCMA.

    More generally:
\\    $(\Acal,\s) \models $ CCMA,
\\    $X$,$F$ closed irreducible in $\Acal$,
\\    $F \subseteq  X \times X^\sigma$,
\\    projections dominant with finite fibres.
\\    Then $(X,F)^\sharp := \{ x \in X \;|\; (x,\s(x)) \in F \}$.

    The \underline{finite-dimensional} types are the generic types
    of such $(X,F)^\sharp$.

\paragraph{Theorem [Trichotomy]:}
\mbox{}\\    Let $p$ be a finite-dimensional minimal type.

    If $p$ is not one-based,
\\    it is non-orthogonal to $(\P^1,\id)^\sharp$.

    If $p$ is one-based non-trivial,
\\    it is non-orthogonal to some $(X,F)^\sharp$ 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...


\subsection*{Imaginaries in TA}

Suppose $T$ superstable with EI and TA exists.

Fact [Hrushovski]: TA has gEI
    (imaginaries are interalgebraic with reals)

\paragraph{Example:}
\mbox{}\\    $T :=$ theory of a connected groupoid with $\pi_1 = Z/2Z$.
\\    In TA,
\\    $X :=$ fixed objects,
\\    $E(x,y) \leftrightarrow  Mor(x,y)$ fixed pointwise by $\sigma$;
\\    then $X/E$ is not eliminable.

\paragraph{Theorem [Hrushovski]: TFAE:}
\begin{enumerate}[(i)]\item TA has EI
\item T "eliminates finite groupoid imaginaries"
\item T has "3-uniqueness":
\\	Given $b$,
\\	and ${a_0,a_1,a_2}$ independent over $b\in\acl(a_i)$,
\\	    (i.e. $a_i \ind \underline{b a}ja_k$)
	$\acl(a_1a_2) \cap \dcl( \acl(a_0a_1), \acl(a_0a_2) )
	    = \dcl( \acl(a_1), \acl(a_2) )$
\end{enumerate}

Remark: $\acl(b) \models  T => 3$-uniqueness (by coheiring).
\\    So e.g. ACFA has EI.


\paragraph{Theorem:}
\mbox{}\\    CCM does not have 3-uniqueness,
\\    so CCMA does not have EI.

\paragraph{Idea:}
\mbox{}\\    Let $X \rightarrow  B$ be a principal $\C^*$-bundle
	(so have definable principal action of $\C^*$ on fibres).

    Work in monster model $\A' \models  CCM$.

    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 $X \rightarrow  B$ defble $\C^*$-bundle s.t. $a \in *X$ generic
	$=> \acl(a) = \dcl(a)$;
\\    i.e. any dominant generically finite $X' \rightarrow  X$ has a generic section.

\paragraph{Finite covers of $\C^*$-bundles:}
\begin{enumerate}[(I)]\item
	Base change:
\\\begin{verbatim}
	C* === C*
	.      |
	.      |
	v  fin v
	X' ..> X
	.      |
	.      |
	v      v
	B' --> B
	   fin
\end{verbatim}
\item 
	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}
\end{enumerate}

\paragraph{Fact:}
\mbox{}\\    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.

\paragraph{Fact:}
\mbox{}\\    Exists simply connected strongly minimal smooth compact (K3) surface $B$ 
    with $H^1(\O_B^*) \cong  \Z$

Let $X \rightarrow  B$ correspond to generator of $H^1(\O_B^*)$.

B s.c. s.m. $=>$ no non-trivial finite $B' \rightarrow  B$;
\\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.


\paragraph{Explicitly, the following imaginary is not eliminable:}
\mbox{}\\    $X \cap \Fix(\s)$
	with $x E x'$ iff
	    $\pi(x) = \pi(x')$
	    and ($x'/x)^{1/2} \subseteq  \Fix(\s)$
\paragraph{Proof:}
\mbox{}\\    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 $x/E \in \acl^{TA,eq}(b) \setminus \acl^{TA}(b)$.
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