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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Locally conformally K\"ahler manifolds \\[15mm]
\small lecture 15: complex surfaces}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

{\small Misha Verbitsky } 
\\[20mm]

{\tiny\bf HSE and IUM, Moscow
\\[2mm]  June 9, 2014
}
\end{center}

\newpage

{\bf\blue Complex surfaces: Riemann-Roch theorem}

\definition
{\bf \blue A complex surface} is a compact, complex manifold
of complex dimension 2. It is called {\bf \blue minimal}
if it does not contain rational curves with self-intersection -1.

%\claim
%A rational curve with self-intersection -1 is obtained by
%a blow-up of a smooth complex surface.
%
%\claim
%Any complex surface $M$ admits {\bf \blue minimal model},
%which is a minimal complex surface birational to $M$ and
%obtained by blowing down a sequence of $-1$-curves.

\definition
{\bf \blue Holomorphic Euler characteristic}
of a coherent sheaf $F$ is $\sum_i (-1)^i \dim H^i(F)$.

\theorem {\bf \blue (Riemann-Roch-Hirzebruch)}
Let $L$ be a holomorphic line bundle on a complex surface
$X$, and $K_X$ its canonical bundle. 
{\bf \red Then $\chi(L)= \chi(\calo_X)+ \frac{(L-K_X,L)}2$,
and $\chi(\calo_X) = (c_1(X)^2- c_2(X))/12$}.
\endproof

\corollary
Let $L$ be a line bundle on a complex surface, $(L,L)=d>0$.
Then $\dim H^0(L^k)$ or $\dim H^0(L^{-k}\otimes K_X)$
grows at least quadratically with $k$.

{\bf \blue Proof:} $\chi(L^k)= d k^2 + ak + b$,
where $a, b$ are independent from $k$. However,
$\chi(L^k)= h^0(L^k)- h^1(L^k) + h^2(L^k)$, hence
either $h^0(L^k)$ or $h^2(L^k)$ grows quadratically.
Finally, $ h^2(L^k) = h^0(L^{-k}\otimes K_X)$
by Serre's duality. \endproof


\newpage

{\bf\blue Douady space}


\definition Let $M$ be a metric space, and $S, S'\subset M$
two subsets. The {\bf\blue Hausdorff distance} between $S$ and $S'$
is an infimum of all $\epsilon$ such that $S$ lies in $\epsilon$-neighbourhood
of $S'$ and  $S'$ lies in $\epsilon$-neighbourhood
of $S$.

\definition
Given a complex subvariety $S\subset M$, {\bf \blue the Douady space}
of deformations of $S$ in $M$ is the set of all complex subvarieties
in the same cohomology class, equipped with topology induced by
the Hausdorff metric $d_H(S, S')$. 

\claim {\bf \red Douady space is a complex analytic variety.}
\endproof

%\theorem
%Let $C\subset M$ be a complex curve of genus $g$ on a surface. 
%Then the Douady space of deformations of $C$  is at least $d$-dimensional, where
%$d=(C, K_M)+(1-g)$ when $g>0$ and 
%$d=(C, K_M)+(1-g)-3$ if $C$ is a rational curve.
%\endproof

\newpage

{\bf\blue Base point set of a bundle}

\definition Let $L$ be a holomorphic line bundle. Define 
{\bf \blue base point set} $\bps(L)$ as the set of all
$x\in M$ such that any section of $L$ vanishes in $x$.

\definition
A {\bf\blue  movable divisor} is a divisor with 
positive-dimensional Douady set (that is, movable in a family).

\claim
Let $L$ be a holomorphic line bundle, $t$ its non-zero section,
and $D$ its zero divisor. {\bf\purple Then $D=D_0\cup D_1$, where
  $D_0\subset \bps(L)$, and $D_1$ is a union of
movable divisors.}

{\bf\green Proof:} Let $x\in D\backslash \bps(L)$. Then there exists
a continuous family of divisors such that $D_t\not\ni x$, hence the
component of $D$ containing $x$ is movable.
\endproof

\claim Let $C, C'$ be movable divisors without common
components on a surface. {\bf \purple Then $C\cap C'$ is a finite set.}
\endproof



\newpage

{\bf\blue Finite correspondences}

\definition
Let $Z\subset M\times M'$ be an irreducible subvariety. 
Denote by $\pi$, $\pi'$ the corresponding projections.
It is called {\bf \blue a birationally finite correspondence}
if $\pi^{-1}(m)$ and $\pi'{}^{-1}(m')$ is finite
for general $m, m'$.

{\bf \green Proposition 1:}
Let $L$ be a holomorphic line bundle on a surface, $c_1(L)^2>0$,
$h^0(L)>2$. Consider a subvariety
$Z\subset M \times {\Bbb P}H^0(L)^*$,
\[ Z= \{x\in M, t \in H^0(L)^*\ \ |\ \ V_x\subset \ker t\},
\]
where $V_x:= \{ h\in H^0(L)\ \ |\ \  h\restrict x=0\}$
is the space of all sections vanishing in $x$. 
{\bf \red Then $Z$ is a birationally finite correspondence}
between $M$ and $\pi_2(Z)$, where 
$\pi_2:\; Z \arrow {\Bbb P}H^0(L)^*$ is a projection.

{\bf \green Proof. Step 1:} 
Let $t\in H^0(L)\backslash 0$, and let $Z_t\subset M$
be the zero divisor of $t$, and $H_t \subset {\Bbb P}H^0(L)^*$
the dual hypersurface. Then $\pi_1(\pi_2^{-1}(H_t))=Z_t$. 
For any $t_1\neq t_2 \in H^0(L)\backslash 0$,
denote by $W_{t_1, t_2}$ the intersection $H_{t_1}\cap H_{t_2}$.
Then $\pi_1(\pi_2^{-1}(W_{t_1,t_2}))=Z_{t_1}\cap Z_{t_2}$.


{\bf \green Step 2:} The intersection
$Z_{t_1}\cap Z_{t_2}$ is a union of base point divisors
and intersection of movable divisors. Since it is
0-dimensional outside of $\bps(L)$, the correspondence 
is finite outside of $\bps(L)$.
\endproof

\newpage

{\bf\blue Finite correspondences (2)}


\theorem
Let $M$ be a complex surface, admitting a birationally
finite correspondence to $M'$. Then {\bf \red $M'$ is K\"ahler (projective) 
if and only if $M$ is K\"ahler (projective).}

The proof is based on currents and Hahn-Banach separation theorem.

\remark This is true {\bf \red only for surfaces}!

\newpage

% {\bf\blue Finite correspondences, projectivity,
% K\"ahlerness and currents}

{\bf\blue Currents}


\definition 
Let $F$ be a Hermitian bundle with connection $\nabla$, 
on a Riemannian manifold $M$ with Levi-Civita connection, and 
\[
\|f\|_{C^k}:=\sup_{x\in M}\left(|f| + |\nabla f| + ... + |\nabla^kf|\right)
\]
the corresponding {\bf \blue $C^k$-norm} 
defined on smooth sections with compact support.
{\bf \purple 
The $C^k$-topology is independent from the choice of connection and metrics.}

\definition
{\bf \blue A generalized function} is a functional on top forms
with compact support, which is continuous in one of $C^i$-topologies.

\definition
{\bf \blue A $k$-current} is a functional on $(\dim M-k)$-forms
with compact support, which is continuous in one of $C^i$-topologies.

\remark Currents are forms with coefficients in generalized functions.


\newpage

{\bf\blue Currents on complex manifolds}

\definition
 The space of currents is equipped with {\bf \blue weak topology}
(a sequence of currents converges if it converges on all
forms with compact support). The space of currents with 
this topology is a {\bf \red Montel space} (barrelled,
locally convex, all bounded subsets are precompact). 
Montel spaces are {\bf \red reflexive} (the map
to its double dual with strong topology is an isomorphism).


\claim De Rham differential is continuous on currents, and 
the Poincare lemma holds. Hence, {\bf \purple
the cohomology of currents are the same
as cohomology of smooth forms.}

\definition
On an complex manifold, 
{\bf \blue $(p,q)$-currents} are $(p,q)$-forms with 
coefficients in generalized functions

\remark {\bf\red In the literature, this is sometimes called
$(n-p,n-q)$-currents}.

\claim The Dolbeault lemma holds on $(p,q)$-currents,
and {\bf \purple the $\bar\6$-cohomology are the same as for forms.}

\newpage


{\bf \blue Positive forms and currents}

\definition
A {\bf \blue weakly positive $(p,p)$-form} is a real $(p,p)$-form 
$\eta$ which satisfies $\eta(x_1,Ix_1,x_2,Ix_2,... x_{p},Ix_p)\geq 0$
for all $x_1, ... n_p \in TM$. 
{\bf \purple The set of weakly positive $(p,p)$-forms 
is a convex cone.} 

\definition
A {\bf\blue weakly positive $(p,p)$-current} is a current taking 
non-negative values in weakly 
positive compactly supported $(n-p,n-p)$-forms.

\definition A {\bf \blue positive generalized function} is a 
generalized function
taking non-negative values on all positive volume forms. 

\remark Positive generalized functions are $C^0$-continuous.
A positive generalized function multiplied by a positive
volume form {\bf \purple gives a measure on a manifold,} and all measures 
are obtained this way.


\corollary
{\bf \purple A weakly positive $(p,p)$-current is $C^0$-continuous.}

\newpage

{\bf \blue Closed positive currents and psh functions}

\definition
Let $Z\subset M$ be a complex analytic subvariety.
{\bf \blue The current of integration} $[Z]$ is the current 
$\alpha\arrow \int_Z\alpha$. {\bf \purple It is closed and positive} 
(Lelong).

\remark (Poincare-Lelong formula) $\frac\1\pi dd^c\log |\phi|=[Z_\phi]$,
where $Z_\phi$ is a divisor of a holomorphic function $\phi$.

\definition
A locally integrable function 
$f:\; M \arrow [\infty, \infty[$ is called 
{\bf \blue plurisubharmonic} (psh) if $dd^c f$ is a positive current.

\claim (a local $dd^c$-lemma)
{\bf \purple Locally, every positive, closed (1,1)-current is obtained as 
$dd^c f$,} for some psh function $f$.

\exercise
Prove that {\bf \red a locally integrable plurisubharmonic function on
a compact complex manifold is constant.}


\newpage

{\bf \blue Hahn-Banach separation theorem and its
  applications}

\theorem
{\bf \blue (Hahn-Banach separation theorem)}\\
Let $V$ be a locally convex topological vector space,
$A\subset V$ an open convex subset, and $W\subset V$
a closed subspace. Assume that $W\cap A=\emptyset$.
Then there exists a continuous functional $\xi \in V^*$
such that $\xi(W)=0$ and $\xi(A)>0$.
\endproof

\theorem {\bf \blue (Harvey-Lawson)}\\
Let $M$ be a compact non-K\"ahler complex manifold.
{\bf \red Then $M$ admits an exact $2n-2$-current  such that
its $(n-1, n-1)$-part is positive.}

\remark Converse is also true: if $M$ admits such a current,
$M$ is non-K\"ahler {\bf \purple (prove this).}

{\bf \green Proof of Harvey-Lawson theorem. Step 1:}\\
Let $A\subset \Lambda^{1,1} M$ be the set of all 
strictly positive forms, and $W$ the space of all
closed (1,1)-forms. {\bf \purple Hahn-Banach separation theorem
produces a current $\xi^{1,1}\in D^{n-1,n-1}(M)$ 
such that $\xi^{1,1}(A)$ is positive and $\xi^{1,1}(W)=0$.}
Clearly, $\xi^{1,1}(A)>0$ $\Leftrightarrow$ $\xi^{1,1}$ is positive.


\newpage

{\bf \blue Hahn-Banach separation theorem and its
  applications (2)}


\theorem {\bf \blue (Harvey-Lawson)}\\
Let $M$ be a compact non-K\"ahler complex manifold.
{\bf \red Then $M$ admits an exact $2n-2$-current  such that
its $(n-1, n-1)$-part is positive.}


{\bf \green Proof of Harvey-Lawson theorem. Step 1:}\\
Let $A\subset \Lambda^{1,1} M$ be the set of all 
strictly positive forms, and $W$ the space of all
closed (1,1)-forms. {\bf \purple Hahn-Banach separation theorem
produces a current $\xi^{1,1}\in D^{n-1,n-1}(M)$ 
such that $\xi^{1,1}(A)$ is positive and $\xi^{1,1}(W)=0$.}
Clearly, $\xi^{1,1}(A)>0$ $\Leftrightarrow$ $\xi^{1,1}$ is positive.

{\bf \green Step 2:} Consider the space
$W_1\subset \Lambda^2(M)$ generated by all closed furms
and all (1,1)-forms. Extend $\xi^{1,1}$ to $W_1$ by
taking $\xi^{1,1}(v)=0$ for all closed $v$. Since $\xi^{1,1}$ vanishes on closed
(1,1)-forms, it is well defined on $W_1$, and can
be extended to a continuous functional on $\Lambda^2$
(Hahn-Banach extension theorem). We obtain {\bf \purple a
$(2n-2)$-current $\xi$ vanishing on closed forms and
with positive (1,1)-part.}

{\bf \green Step 3:} It remains to prove that $\xi$ is
exact. Since $\langle \xi, d\alpha\rangle= 
\pm \langle d\xi, \alpha\rangle= 0$ for all $\alpha$,
the current $\xi$ is closed. However, a pairing of $\xi$
with any closed form vanishes, hence {\bf \purple $\xi$ is exact by
Poincare duality.}
\endproof

\newpage

 {\bf\blue Gauduchon metrics and Hahn-Banach theorem}

\definition
A Hermitian metric $\omega$ on complex $n$-manifold is called 
{\bf\blue Gauduchon} if $dd^c\omega^{n-1}=0$.

\theorem
{\bf \red Any compact complex manifold admits a Gauduchon metric.}

{\bf \green Step 1:} 
Any strictly positive $(n-1, n-1)$-form is $(n-1)$-th
power of a Hermitian form. Therefore, to construct a Gauduchon metric,
it suffices to find a $dd^c$-closed strictly positive $(n-1, n-1)$-form.

{\bf \green Step 2:} Let $A\subset \Lambda^{n-1,n-1}(M)$ be
the cone of strictly positive $(n-1, n-1)$-forms, and $W:= \ker dd^c$.
If these sets don't intersect, we can find $\xi \in \Lambda^{1,1}(M)$
which is positive and satisfies  $\langle\xi, \ker dd^c\rangle =0$.

\newpage

 {\bf\blue Gauduchon metrics and Hahn-Banach theorem (2)}

\theorem
{\bf \red Any compact complex manifold admits a Gauduchon metric.}

{\bf \green Step 2:} Let $A\subset \Lambda^{n-1,n-1}(M)$ be
the cone of strictly positive $(n-1, n-1)$-forms, and $W:= \ker dd^c$.
If these sets don't intersect, we can find $\xi \in \Lambda^{1,1}(M)$
which is positive and satisfies  $\langle\xi, \ker dd^c\rangle =0$.


{\bf \green Step 3:} Since $\im d \subset \ker \ker dd^c$,
this gives $\langle\xi, \im d \rangle =0$, hence $\xi$ is closed.
 Define {\bf \blue Aeppli cohomology}
as 
\[ 
 H^{p,q}_{AE}(M):= \frac {\ker dd^c\restrict
   {\Lambda^{p,q}(M)}} {\im d+ \im d^c}.
\]
The Poincare pairing $H^{p,q}_{AE}(M)\times
H^{n-p,n-q}_{BC}(M)\arrow \C$, where $H^{*,*}_{BC}(M)$ is
Bott-Chern cohomology, is non-degenerate
{\bf \red (prove this)!} Since $\xi$ is orthogonal to
$H^{p,q}_{AE}(M)$ under this pairing, its class in $H^{1,1}_{BC}(M)$
vanishes, hence $\xi = dd^c f\geq 0$.


{\bf \green Step 4:} There are no globally plurisubharmonic 
generalized functions on a compact manifold {\bf \purple (prove it!)}.
\endproof




\remark In fact, {\bf \red there exists a Gauduchon metric
in each conformal class, and it is unique up to a constant.}
This is proven using elliptic equations and E. Hopf's strict
maximum principle.






\newpage

 {\bf\blue Finite correspondences and K\"ahlerness}



\theorem
Let $M_1$ be a complex surface, admitting a birationally
finite correspondence to $M_2$. Then {\bf \red $M_1$ is K\"ahler
if and only if $M_2$ is K\"ahler.}

{\bf \green Proof. Step 1:} 
Let $\pi_1, \pi_2:\; Z \arrow M_i$ be the projection maps, 
$\omega_2$ be a K\"ahler form on $M_2$, and
$\zeta:= \pi_1{}_*\pi_2^*\omega$. Then $\zeta$
is a positive, closed (1,1)-current which is smooth and
strictly positive at each point $z\in M_1$ where the
correspondence $Z$ is finite. 

{\bf \green  Step 2:} Since $\zeta$ is infinite
only around the points $z\in M_1$ where $\pi_1^{-1}(z)$
is positive-dimensional, $\zeta$ is smooth outside
of a closed, finite set $\sing(\zeta)$.

{\bf \green  Step 3:} Let $S_\epsilon$ be an epsilon-neighbourhood of
$\sing(\zeta)$. Using the local $dd^c$-lemma in 
$S_\epsilon$, we could write  $\zeta = dd^c f$.
Since a maximum of plurisubharmonic functions is
plurisubharmonic, we can replace
$\zeta$ by a current which is equal to $\zeta$ outside 
$S_\epsilon$ and equal to $dd^c\max(f, -C)$ in $S_\epsilon$,
where $-C < f\restrict{\6 S_\epsilon}$.
The new $\zeta$ is positive, closed, and
equal to $dd^c(\phi)$ on $S_\epsilon$.

\newpage

 {\bf\blue Finite correspondences and K\"ahlerness (2)}

\theorem
Let $M_1$ be a complex surface, admitting a birationally
finite correspondence to $M_2$. Then {\bf \red $M_1$ is K\"ahler
if and only if $M_2$ is K\"ahler.}


{\bf \green  Step 3:} Let $S_\epsilon$ be an epsilon-neighbourhood of
$\sing(\zeta)$. Using the local $dd^c$-lemma in 
$S_\epsilon$, we could write  $\zeta = dd^c f$.
Since a maximum of plurisubharmonic functions is
plurisubharmonic, we can replace
$\zeta$ by a current which is equal to $\zeta$ outside 
$S_\epsilon$ and equal to $dd^c\max(f, -C)$ in $S_\epsilon$,
where $-C < f\restrict{\6 S_\epsilon}$.
The new $\zeta$ is positive, closed, and
equal to $dd^c(\phi)$ on $S_\epsilon$.


{\bf \green  Step 4:} Consider a smooth convex
function $\max_\epsilon:\; \R^2 \arrow \R$
equal to $\max(x,y)$ for $|x-y|>\epsilon$ and
monotonous in each argument. Using $\max_\epsilon$ 
instead of $\max$ in Step 3, we may assume that
$\zeta$ is actually smooth. 


{\bf \green  Step 5:} 
Suppose that $M_1$ is non-K\"ahler. Then 
Harvey-Lawson bring an exact current $\xi=d\alpha$ with positive
(1,1)-part. The current $\zeta\wedge \xi$ 
is positive; on the other hand, it 
is exact, giving $0=\int_M \zeta\wedge \xi>0$ -- a contradiction.
\endproof

\remark
{\bf \red In dimension $>2$, this theorem is false;}
it should be clear where the argument fails.

\newpage

 {\bf\blue K\"ahler cone and
   Nakai-Moishezon-Demailly-Paun theorem}

\theorem {\bf \blue (Nakai-Moishezon-Demailly-Paun)}\\
Let $M$ be a K\"ahler surface, and 
\[K:=\{\alpha\in H^{1,1}(M)\ \ |\ \ \alpha^2 > 0,
\ \ \alpha\restrict{\text{all complex curves}}>0\}.
\]
{\bf \red Then $K$ is a K\"ahler cone of $M$ unless $M$ has no 
complex curves.} In the latter case, K\"ahler cone of $M$ is
one of two connected components of $K$.
\endproof

\theorem
Let $M_1$ be a complex surface, admitting a birationally
finite correspondence to $M_2$. Then {\bf \red $M_1$ is projective
if and only if $M_2$ is projective.}

{\bf \green Proof. Step 1:} Using the Harvey-Lawson
argument such as above, we prove that {\bf \purple $M_1$ is K\"ahler.}
Let $\omega$ be a rational K\"ahler
form on $M_2$, and $\zeta:= \pi_1{}_*\pi_2^*\omega$.
{\bf \purple Then $\zeta^2>0$} by the same argument (with removal
of isolated singularities) as used above.

{\bf \green Step 2:} For any curve $C\subset M_1$,
let $C_Z$ be its proper preimage in $Z$. Then
$\langle \xi, C\rangle= \int_{C_Z}\pi_2^*\omega>0$.
Then by Nakai-Moishezon-Demailly-Paun, 
{\bf \purple  $[\xi]$ is a K\"ahler class.}

{\bf \green Step 3:} This class is rational by
construction, hence {\bf \purple $M_1$ is projective by Kodaira.}
\endproof

\newpage

 {\bf\blue Non-projective surfaces and the intersection form}

\theorem
Let $M$ be a complex surface, and $NS(M)$ the group of
all integer cohomology classes represented by closed $(1,1)$-currents.
{\bf \red Then $M$ is non-projective if and only if the intersection
form on $NS(M)$ is non-positive.}

{\bf \green Proof. Step 1:} Suppose that $[\xi]\in NS(M)$
is an integer class with $[\xi]^2=0$, and $L$ the line
bundle with its curvature equal to $\xi$ (it exists by
$dd^c$-lemma). Then either $h^0(L^k)$ or $h^0(L^k \otimes
K_M)$ grows quadratically with $k$ by Riemann-Roch.

{\bf \green Step 2:} Proposition 1 implies that
$M$ admits a birationally finite correspondence with
a projective manifold, hence $M$ is projective as shown
above.

{\bf \green Step 3:} Conversely, if $M$ is projective, 
it admits a curve with positive self-intersection, namely,
the hyperplane section.
\endproof

\definition
{\bf \blue Elliptic surface} is a complex surface $M$ equipped with
a holomorphic map $M\arrow S$, with generic fiber a curve
of genus 1.


\newpage

 {\bf\blue Curves on non-projective surfaces}

\theorem
Let $M$ be a non-projective surface. Then {\bf \red either all curves
on $M$ are isolated, or $M$ admits an elliptic fibration
$\pi:\; M\arrow S$,} with all curves belonging to the fibers of $\pi$.

{\bf \green Step 1:} Since the intersection form on $NS(M)$
is non-positive, for any two distinct irreducible curves
$C, C'$ in the same cohomology class, the intersection 
$C\cap C'$ is empty. Therefore, all non-isolated curves
$C$ satisfy $C\cdot C=0$. 

{\bf \green Step 2:} Given a family 
$C_t$ of non-intersecting curves parametrized by $t\in D$, 
consider the corresponding fibration from a subset
$M_0:=\bigcup_{t\in D} C_t\subset M$ to $D$. Since
$NC_t= \pi^* TD\restrict{C_t}$, the normal bundle to
each $C_t$ is trivial. 


{\bf \green Step 3:} Adjunction formula gives 
$TC_t \otimes NC_t= \Lambda^2 TM\restrict{C_t}$, hence
$\Omega^1C_t = K_M \restrict{C_t}$, where $K_M$ is a
canonical bundle. If $(K_M, C_t)=l\neq 0$,
one would have $(K_M+ tC_t,K_M+ tC_t)= (K_M,K_M)+ 2tl>0$
for appropriate choice of $t$. Then $M$ would be
projective. Therefore, $(K_M, C_t)= 0 = \deg \Omega^1C_t$.
This implies that all smooth fibers of $\pi$ are elliptic
curves. 


{\bf \green Step 5:}
Existence of elliptic fibration $M\arrow B$ would follow
if we show that the deformation space of $C_t$ is compact
for any curve on a complex surface (see the next slide).



\newpage

{\bf\blue Gromov's compactness theorem}

{\bf \green Step 6:}
The same argument as in Step 4 is used to show
that $(C_t, v)=0$ for any $v\in NS(M)$,
hence any irreducible curve belongs to a fiber of $\pi$.
\endproof

\theorem {\bf \blue (Gromov's compactness theorem)}\\
Let $(M,I, \omega)$ be a compact (almost) complex Hermitian manifold,
${\goth D}$ the space of all (pseudo-) holomorphic curves
on $M$, with topology induced by the Hausdorff metric,
$p>0$ a real number, and ${\goth D}_p\subset {\goth D}$ the space of all 
curves $S$ with $\Vol(S):=\int_S \omega\leq p$. Then
${\goth D}_p$ is compact.


\newpage

{\bf\blue Moduli of curves on complex surfaces}

\theorem
Let $M$ be a compact complex surface, $C\subset M$ 
a curve, and $B$ a connected component of  
its Douady space. {\bf \red Then $B$ is compact.}

{\bf \green Proof. Step 1:} Fix a Hermitian metric
$\omega$ on $M$. By Gromov's theorem, the space of curves of bounded 
volume is compact. {\bf \purple It remains to show that the volume
stays bounded on each connected component of the Douady
space,} for appropriate choice of $\omega$.
Notice that this is vacuously true when $\omega$ is
K\"ahler, because then the volume is a cohomological
invariant.


{\bf \green Step 2:} Consider the incidence variety
$Z\subset M\times B$ consisting of pairs 
$C\in B, x\in C\subset M$, and let $\pi_1:\; Z \arrow M$,
$\pi_2:\; Z \arrow B$ be the standard projections. 
Denote by $\Vol:\; B \arrow \R^{>0}$ the volume function,
$\Vol(C)= \int_C \omega$. 
Then $\Vol= \pi_2{}_* \pi_1^* \omega$.

{\bf \green Step 3:} Choose now $\omega$ Gauduchon.
Then $dd^c\Vol= \pi_2{}* \pi_1^* dd^c\omega=0$.
Therefore, $\Vol$ has no local minima or local maxima
on $B$. However, $\Vol$ is a proper function by Gromov's
theorem, hence it has to reach minimum somewhere.
{\bf \purple Therefore, $\Vol=\const$.} Now, $B$ is compact
by Gromov's theorem. \endproof


\newpage

{\bf\blue Class VII surfaces (a survey)}

\definition
Define the {\bf \blue Kodaira dimension} of a manifold $M$ as
\\ $\kappa(M):=\lim\sup_n \frac {\log h^0(K^n_M)}{n}$.

\definition
A {\bf \blue Kodaira class VII surface}
is a surface with $\kappa(M)=-\infty$ and $b_1(M)=1$.

\example All Hopf surfaces and all Inoue surfaces are Kodaira class VII.

\definition
{\bf \blue Kato surface} is a surface $M$ which contains a 3-dimensional
sphere $S^3$ such that $M\backslash S^3$ is connected, and
a neighbourhood of $S^3$ is biholomorphic to a neighbourhood of
standard $S^3$ in $\C^2$ (``global spherical shell'').

\theorem
All Kato surfaces are class VII.

\newpage

{\bf\blue The rest of classification theorems (a survey)}

\theorem
Let $M$ be a non-projective K\"ahler minimal surface.
Then $M$ is elliptic, or isomorphic to a K3 or a torus.

\theorem
Let $M$ be a non-K\"ahler elliptic
minimal surface. Then $M$ is isotrivial
(all fibers are isomorphic) and Vaisman.

\theorem
Let $M$ be a non-K\"ahler non-elliptic
surface. Then $M$ is class VII.

\theorem
(Bogomolov)
All class VII surfaces with $b_2=0$ are Inoue or Hopf.

\theorem (Andrei Teleman)
All class VII surfaces with $b_2=1$ are Kato.

\conjecture (GSS conjecture) All 
minimal class VII surfaces with $b_2>0$
are Kato.

\theorem  (Brunella) All Kato surfaces are LCK.









%{\bf \green Step 2:} Since $\xi$ is non-negative on closure of 
%$A$, it is non-negative on all (2,0) and (0,2)-forms. Then
%$\xi$ is of type (n-1, n-1).

%{\bf \green Step 3:} Since $(\xi, dd^cf)=0$ for each $f\in C^\infty M$,
%this gives $dd^c\xi =0$. Define {\bf \blue Aeppli cohomology}
%as 
%\[ 
% H^{p,q}_{AE}(M):= \frac {\ker dd^c\restrict
%   \Lambda^{p,q}(M)} {\im d+ \im d^c}.
%\]
%The natural pairing $H^{p,q}_{AE}(M)\times
%H^{n-p,n-q}_{BC}(M)\arrow \C$ is non-degenerate
%{\bf \red (prove this)!} This gives a commutative diagram
%with horizontal arrows induced by the de Rham
%differential, and vertical arrows by duality.
%\[\begin{CD}
%\bigoplus_{p+q=m} H^{p,q}_{AE}(M)@>d>>
%\bigoplus_{p+q=m+1} H^{p,q}_{BC}(M) \\
%@VVV @VVV\\
%\bigoplus_{p+q=2n-m} H^{p,q}_{AE}(M)^*@>{d^*}>>
%\bigoplus_{p+q=2n-m-1} H^{p,q}_{BC}(M).
%\end{CD}
%\]
%From this diagram, we obtain that $\ker d^*\subset H^*_{AE}(M)^*$ is 
%identified with $\im d\subset  H^*_{AE}(M)$.
%Since $\langle \xi, \ker d^*\rangle=0$,
%this diagram gives $\xi\in d(H^{2n-3}_{AE}(M))$,
%hence $\xi


\end{document}
