
\documentclass{slides}

\usepackage{amssymb, amsmath, amscd, color, epsfig}
%\usepackage[matrix,arrow]{xy}

\newcommand{\green}{\color[rgb]{0,0.4,0}}
\newcommand{\purple}{\color[rgb]{0.4,0,0.4}}
\newcommand{\red}{\color[rgb]{0.7,0,0}}
\newcommand{\blue}{\color{blue}}

\def\eqref#1{(\ref{#1})}
\newcommand{\goth}{\mathfrak}
\newcommand{\g}{{\frak g}}
\newcommand{\arrow}{{\:\longrightarrow\:}}
\newcommand{\Z}{{\Bbb Z}}
\def\C{{\Bbb C}}
\newcommand{\R}{{\Bbb R}}
\newcommand{\Char}{\operatorname{\sf char}}
\newcommand{\Q}{{\Bbb Q}}
\renewcommand{\H}{{\Bbb H}}
\newcommand{\6}{\partial}
\def\1{\sqrt{-1}\:}
\newcommand{\restrict}[1]{{\left|_{{#1}}\right.}}
\newcommand{\cntrct}                % contraction with a vector field
{\hspace{2pt}\raisebox{1pt}{\text{$\lrcorner$}}\hspace{2pt}}


\def\Bbb#1{\mathbb #1}


\newcommand{\calo}{{\cal O}}
\newcommand{\cac}{{\cal C}}

% Correcting TeX...
%\let\oldtilde=\tilde
%\renewcommand{\tilde}{\widetilde}
\renewcommand{\bar}{\overline}
\renewcommand{\phi}{\varphi}
\renewcommand{\epsilon}{\varepsilon}
\renewcommand{\geq}{\geqslant}
\renewcommand{\leq}{\leqslant}

% Operatornames
\newcommand{\even}{{\rm even}}
\newcommand{\ev}{{\rm even}}
\newcommand{\odd}{{\rm odd}}
\newcommand{\const}{{\it const}}
\newcommand{\fl}{{\rm fl}}
\newcommand{\im}{\operatorname{im}}
\newcommand{\End}{\operatorname{End}}
\newcommand{\Sym}{\operatorname{Sym}}
\newcommand{\Hol}{\operatorname{{\cal H}ol}}
\newcommand{\Tot}{\operatorname{Tot}}
\newcommand{\Id}{\operatorname{Id}}
\newcommand{\id}{\operatorname{\text{\sf id}}}
\newcommand{\Vol}{\operatorname{Vol}}
\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\Aut}{\operatorname{Aut}}
\newcommand{\Alt}{\operatorname{Alt}}
\newcommand{\Iso}{\operatorname{Iso}}
\newcommand{\Sec}{\operatorname{Sec}}
\newcommand{\Can}{\operatorname{Can}}
\newcommand{\Sing}{\operatorname{Sing}}
\newcommand{\Spin}{\operatorname{Spin}}
\newcommand{\codim}{\operatorname{codim}}
\newcommand{\coim}{\operatorname{coim}}

\newcommand{\coker}{\operatorname{coker}}
\newcommand{\slope}{\operatorname{slope}}
\newcommand{\rk}{\operatorname{rk}}
\newcommand{\Def}{\operatorname{Def}}
\newcommand{\Lie}{\operatorname{Lie}}
\newcommand{\Tw}{\operatorname{Tw}}
\newcommand{\Tr}{\operatorname{Tr}}
\newcommand{\Spec}{\operatorname{Spec}}
\newcommand{\Diff}{\operatorname{Diff}}

\renewcommand{\Re}{\operatorname{Re}}
\renewcommand{\Im}{\operatorname{Im}}



\newcommand{\inbfpare}[1]{{%
  \mbox{\tt (}\hspace{-5pt}\mbox{\tt (} #1 % 
  \mbox{\tt )}\hspace{-5pt}\mbox{\tt )}%
}}
\newcommand{\comment}[1]{{}}

\def\blacksquare{\hbox{\vrule width 10pt height 10pt depth 0pt}}
\def\endproof{\blacksquare}
\def\shortdash{\mbox{\vrule width 4.5pt height 0.55ex depth -0.5ex}}


\makeatletter

%\@ifundefined{Bbb}
%     {\newcommand{\Bbb}[1]{{\mathbb #1}}}%
%{}%     {\edef\Bbb#1{{\Bbb #1}}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%       Pagestyle                                %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
\newcommand{\ps@verbit}{%
  \renewcommand{\@oddhead}{%
          \scriptsize {\it \small Stable bundles and foliations \hfil
  \tiny Misha Verbitsky}}
  \renewcommand{\@evenhead}{\@oddhead}
  \renewcommand{\@oddfoot}{\hfil\thepage\hfil}
  \renewcommand{\@evenfoot}{\@oddfoot}}
 
\pagestyle{verbit}


   \setlength\paperheight {10in}%
    \setlength\paperwidth  {13.5in}
\setlength{\textwidth}{0.8\paperwidth}
\setlength{\textheight}{0.8\paperheight}

 \setlength{\pdfpageheight}{\paperheight}
 \setlength{\pdfpagewidth}{\paperwidth}
\addtolength{\topmargin}{-20mm}
\addtolength{\leftmargin}{-25mm}
\addtolength{\rightmargin}{-25mm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Lemma, sublemma, corollary, proposition, theorem,             %
% definition,example defined there:                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\lemma}{%
     {\bf \green LEMMA:\ }}
\newcommand{\claim}{%
     {\bf \green CLAIM:\ }}
\newcommand{\corollary}{%
     {\bf \green COROLLARY:\ }}
\newcommand{\theorem}{%
     {\bf \green THEOREM:\ }}
\newcommand{\conjecture}{%
     {\bf \green CONJECTURE:\ }}
\newcommand{\proposition}{%
     {\bf \green PROPOSITION:\ }}
\newcommand{\definition}{%
     {\bf \green DEFINITION:\ }}
\newcommand{\example}{%
     {\bf \green EXAMPLE:\ }}
\newcommand{\remark}{%
     {\bf \green REMARK:\ }}
\newcommand{\question}{%
     {\bf \green QUESTION:\ }}
\newcommand{\observation}{%
     {\bf \green OBSERVATION:\ }}


\begin{document}
\setcounter{page}{1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Stable bundles on non-K\"ahler manifolds \\[3mm]
with transversally K\"ahler foliations}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

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

{\small\bf
Holomophic foliations and complex dynamics
\\[2mm] \tiny 14 June 2012, Poncelet Laboratory, Moscow, Russia
}
\end{center}

\newpage

{\bf \blue Motivation.}

\question How to study non-K\"ahler complex manifolds?

\theorem (Harvey-Lawson) Let $M$ be a compact
complex $n$-manifold not admitting a K\"ahler structure.
{\bf \purple Then there exists a positive $(n-1,n-1)$-current $T$
which is an $(n-1,n-1)$-part of an exact current.}

\theorem (Harvey-Lawson) Let $M$ be a compact
complex non-K\"ahler surface. {\bf \red Then $M$ admits a
positive, exact (1,1)-current.}

\definition
{\bf \blue K\"ahler rank} of a complex manifold $M$ is maximal rank of
a positive, closed (1,1)-form on $M$.

{\bf \green Marco Brunella:} classification of non-K\"ahler surfaces
according to their K\"ahler rank.

{\bf \blue We are interested} in $n$-manifolds admitting 
positive, exact forms of rank $n-1$, which are transversally
K\"ahler with respect to a 1-dimensional foliation.

\newpage

{\bf \blue Plan.}

{\bf \blue 1. Motivation:} 
Manifolds with transversally K\"ahler foliations
(examples).

{\bf \blue 2. Immediate application:} Classification
of complex subvarieties.

{\bf \blue 3. Geometry of Vaisman manifolds.} 
Structure theorem and transversal foliations.
Application to complex subvarieties.

{\bf \blue 4. Stable bundles on non-K\"ahler manifolds}. 
Gauduchon metrics and stability.

{\bf \blue 5. Coherent sheaves on Vaisman manifolds}.
Filtered sheaves and applications of Yang-Mills theory.

\newpage

{\bf \blue Transversally K\"ahler forms.}

\definition
{\bf \red A model situation.}
Let $M$ be a compact complex manifold, $\Sigma\subset TM$
a holomorphic 1-dimensional foliation, generated by a nowhere
vanishing holomorphic vector field $s$, and $\omega_0$
a closed semipositive (1,1)-form on $M$.
We say that $\omega_0$ is {\bf \blue transversally
K\"ahler}, if $\Sigma$ is the null-space of $\omega_0$,
{\bf \blue  equivariant}, if $\Lie_s \omega_0=0$, 
and {\bf \blue transversally K\"ahler exact}
if $\omega_0$ is exact: $\omega_0 = d\theta$.

\remark
A manifold admitting a transversally K\"ahler 
exact form {\bf \purple is never K\"ahler}.
Indeed, if $\omega$ is a K\"ahler form,
we would have $\int_M \omega_0 \wedge \omega^{\dim M -1}>0$,
which is impossible by Stokes' theorem, because
$\omega_0 \wedge \omega^{\dim M -1}=d (\theta \wedge \omega^{\dim M -1}).$

\example 
{\bf \blue A classical Hopf surface}
is $H:=\C^2 \backslash 0/ \Z$, where $\Z$ acts as
a multiplication by a complex number $\lambda$, $|\lambda|>1$.

\observation $H$ is diffeomorphic to $S^1 \times S^3$,
and fibered over $\C P^1$ with fiber $\C^*/\langle \lambda \rangle$.

\claim Let $\pi:\; H \arrow \C P^1$ be the standard
projection, and $\omega_0:= \pi^*\omega_{\C P^1}$
be a pullback of the Fubini-Study form. Clearly,
{\bf \purple 
$\omega_0$ is exact, because $H^2(H)=0$} (by K\"unneth formula).
Therefore, {\bf \red $H$ admits a transversally K\"ahler, exact form.}

\newpage

{\bf \blue Locally trivial elliptic fibrations.}


\definition  {\bf \blue A 
principal elliptic fibration } $M$ is a complex
manifold equipped with a free holomorphic action of a
1-dimensional compact complex torus $T$. 

{\red \bf Such a manifold is fibered over
$M/T$, with fiber $T$.} 

\remark
It is a principal $T$-bundle: all fibers are identified with $T$,
with $T$ acting on fibers freely.

\definition
 Let $M\stackrel \pi\arrow X$ be a principal elliptic fibration,
$M$ compact. We say that $M$ is {\bf \green positive elliptic fibration},
if for some K\"ahler class $\omega$ on $X$, $\pi^*\omega$ is exact.
{\blue \bf (``K\"ahler class'' is a cohomology class of a K\"ahler form).}

\example
The classical Hopf surface introduced earlier.

\example 
A more general example is given by
$\Tot(L^*)/\langle\Z\rangle$, where $L$ is an ample line bundle.
Such manifold is called {\bf\blue a regular Vaisman manifold}.
It is positive, because $\pi^*(c_1(L))=0$, and $c_1(L)$
is a K\"ahler class.

\newpage

{\bf \blue Calabi-Eckmann manifolds}

{\bf \green Calabi-Eckmann manifolds.}\\
Fix $\alpha \in \C$, $\alpha$ non-real, $|\alpha|>1$.
Consider a subgroup 
\[ G := \{ e^t\times e^{\alpha t} \subset \C^*\times \C^*,
\ \ t\in \C\}\subset \C^*\times \C^*
\]
within $\C^*\times \C^*$. It is clearly co-compact and closed,
with $\C^*\times \C^*/G$ being an elliptic curve 
$\C^*/\langle\alpha\rangle$. 

Now, let $M:= (\C^n\backslash 0) \otimes (\C^m\backslash 0)/G,$
with $G\subset \C^*\times \C^*$ acting on 
$(\C^n\backslash 0) \otimes (\C^m\backslash 0)$
by $(t_1, t_2) (x, y) \arrow (t_1x, t_2 y)$.
Clearly, $M$ is fibered over 
\[ \C P^{n-1} \times \C P^{m-1}=
(\C^n\backslash 0) \otimes (\C^m\backslash 0)/\C^*\times \C^*
\]
with a fiber $\C^*\times \C^*/G$, which is an elliptic curve.
Its total space $M$ {\bf \blue the Calabi-Eckmann manifold}.
It is diffeomorphic to $S^{2n-1}\times S^{2m-1}$.

\remark
The map $M \arrow \C P^{n-1} \times \C P^{m-1}$
{\bf \purple is a principal elliptic fibration}.

\remark
The pullback of a K\"ahler form
from $\C P^{n-1} \times \C P^{m-1}$ to 
$M$ {\bf \red is exact}, because
$H^2(M)=0$ (by K\"unneth formula).


\newpage

{\bf \blue Irregular and quasi-regular foliations}

\definition
A foliation is called {\bf \blue quasi-regular} if all its
leaves are compact. If this is not so, it is called {\bf\blue irregular}.
A foliation is called {\bf \blue regular} if all its leaves are
compact, and the leaf space is smooth.

So far, in our examples all foliations were
quasi-regular.
Let's have some irregular examples.

\remark Calabi-Eckmann manifolds were generalized by
 Lopez de Medrano, Verjovsky and Meersseman.
The complex structure on Calabi-Eckmann can be
deformed together with the foliation, giving
{\bf \purple an equivariant transversally K\"ahler manifold with
a foliation having non-compact leaves}
(``LVM-manifolds'').

\remark 
If $(M,\Sigma, \omega_0)$ is a manifold equipped with 
a transversally K\"ahler, exact form, and
$Z\subset M$ is a complex subvariety, {\bf \purple then
$Z$ is a union of closures of leaves of $\Sigma$.}

For more examples, we define {\bf \blue Vaisman manifolds}.


\newpage

{\bf \blue LCK manifolds}



\definition
Let $M$ be a complex manifold, $\tilde M \arrow M$ its covering,
$\tilde M/\Gamma=M$, and $\Gamma$ the {\bf \blue monodromy group}
freely acting on $\tilde M$. Assume that $\tilde M$ is K\"ahler, and
$\Gamma$ acts on $\tilde M$ by homotheties. Then $M$ is called
{\bf\blue locally conformally K\"ahler} (LCK).

\example {\bf \blue A classical Hopf manifold} 
$\C^n\backslash 0 /\langle A\rangle$,  $A(x) =q x$, $|q|>1$,
is obviously LCK.

\definition Let $M$ be an LCK manifold, $(\tilde M, \tilde \omega)$
its covering. An {\bf \blue LCK metric} on $M$ is a metric
conformal to $\tilde \omega$.


\remark If $\omega = f \tilde \omega$, one has $d\omega = df\wedge \omega'$.
Therefore, {\bf \red an LCK metric satisfies $d\omega=\theta\wedge \omega$,}
for some closed 1-form $\theta$.

\remark {\bf \purple If $\theta=df$, the form $e^{-f} \omega$ is K\"ahler.}

\newpage

{\bf \blue Hermitian metrics on LCK manifolds}

\claim If $(M, I, \omega)$ is a Hermitian manifold
with $d\omega=\theta\wedge \omega$, 
for some closed 1-form $\theta$, {\bf \red then $(M,I)$ is LCK.}

{\bf \green Proof:} For some covering of $M$, the pullback of $\theta$
is exact, and then {\bf \purple the pullback of $\omega$ is conformal to
a K\"ahler form: $\theta=df$, then $d(e^{-f}\omega)=e^{-f}\omega\wedge\theta-e^{-f}\omega\wedge\theta=0$. } \endproof 

\definition A Hermitian manifold $(M, I, \omega)$ 
is called {\bf \blue LCK} if $d\omega=\theta\wedge \omega$, 
for some closed 1-form $\theta$.

\definition 
The form $\theta$ is called {\bf\blue the Lee form} of an LCK-manifold.
The dual vector field $\theta^\sharp$ -- {\bf \blue the Lee field}.

\newpage

{\bf \blue Vaisman manifolds}

\definition
An LCK manifold $(M,\omega, \theta)$ is called {\bf \blue Vaisman}
(``generalized Hopf'') if $\nabla_{LC}\theta=0$, where $\nabla_{LC}$ is the
Levi-Civita connection.

\theorem
(Kamishima-Ornea) Let $M$
be an LCK manifold equipped with conformal a holomorphic flow.
Assume that this flow acts by non-isometric homotheties on the K\"ahler
covering. {\bf \red Then $M$ is Vaisman.}

\example 
The classical Hopf manifolds are obviously Vaisman.

\theorem
Let $(M,\omega, \theta)$ be a Vaisman manifold.
Then {\bf \purple the form $\omega_0:=\omega - \theta\wedge I(\theta)$
is semi-positive and exact: $\omega_0=d (I\theta)$.}
Moreover, {\bf \purple the foliation $\Sigma$ generated by $\theta^\sharp$
is holomorphic,} and $\omega_0$ is equivariant and transversally K\"ahler
with respect to $\Sigma$.

\remark 
{\bf \blue Vaisman manifolds are build from Sasakian manifolds;}
there is an inexhaustible supply of those. More of this
later in this talk.


\newpage

{\bf \blue Oeljeklaus-Toma manifolds}


Let $K$ be a number field which has $2t$
complex embedding denoted $\tau_i, \bar \tau_i$ and 
$s$ real ones denoted $\sigma_i$,  $s>0$, $t>0$.

Let $\calo_K^{*,+}:= \calo_K^*\cap \bigcap_i \sigma^{-1}_i(\R^{>0})$.
Choose in $\calo_K^{*,+}$
a free abelian subgroup $\calo_K^{*,U}$ of rank
$s$ such that the quotient
$\R^s/\calo_K^{*,U}$ is compact, where
$\calo_K^{*,U}$ is mapped to $\R^t$ as
$\xi \arrow \big(\log(\sigma_1(\xi)), ..., \log(\sigma_t(\xi))\big).$
Let $\Gamma:= \calo^+_K\rtimes \calo_K^{*,U}$.


\definition
{\bf \blue An Oeljeklaus-Toma manifold} is a quotient
$\C^t \times H^s/\Gamma$, where
$\calo^+_K$ acts on $\C^t \times H^t$
as 
\[ \zeta(x_1,..., x_t, y_1, ..., y_s) = (x_1+
   \tau_1(\zeta), ..., x_t + \tau_t(\zeta), 
   y_1+\sigma_1(\zeta), ..., y_s+\sigma_s(\zeta)),
\]
and $\calo_K^{*,U}$ as $\xi(x_1,..., x_t, y_1, ..., y_s)=
  (x_1,..., x_t,\sigma_1(\xi) y_1,  ..., \sigma_t(\xi) y_t)$

\theorem (Oeljeklaus-Toma) 
The OT-manifold  $M:=\C^t \times H^s/\Gamma$
 {\bf \red is a compact complex manifold,} without any non-constant
meromorphic functions. When $t=1$, it is locally
conformally K\"ahler. When $s=1, t=1$, it is an Inoue surface
of class $S^0$.

\theorem
(Ornea-V.) Let $M$ be an OT-manifold,
$t=1$. {\bf \purple Then $M$ is equipped with a holomorphic
1-dimensional foliation and an equivariant, 
transversally K\"ahler, exact form.}


\newpage

{\bf \blue Complex subvarieties in transversally K\"ahler
  manifolds}

\theorem
Let $(M,\Sigma, \omega_0)$ be a manifold, equipped with
a transversally K\"ahler, exact form, and $Z\subset M$
a complex subvariety. {\bf \red Then $Z$ is a union of leaves of $\Sigma$.}

{\bf \green Proof:}
Suppose that $Z$ is transversal to $\Sigma$
at a point $z$. Then all eigenvalues of $\omega_0\restrict Z$
at $z$ are positive, hence $\int_Z \omega_0^k>0$, where $k:=\dim Z$.
This is impossible by Stokes' theorem, because
$\omega_0^k$ is exact.
\endproof

\corollary
Let $M\stackrel \pi\arrow X$ be a positive, principal elliptic fibration
(such as Calabi-Eckmann, or Hopf manifold). {\bf \purple Then
all positive-dimensional subvarieties of $M$ are
of form $\pi^{-1}(Z)$,} for some complex subvariety
$Z\subset M$. \endproof

\theorem
(Ornea-V.) 
{\bf \red 
An OT-manifold for $t=1$ has no non-trivial complex subvarieties.}

The proof of this theorem requires a bit of number theory.



\newpage


{\bf\blue Sasakian manifolds.}


{\bf \green Definition:} Let $M$ be a smooth manifold, $\dim M=2n-1$,
and $(\omega,I)$ a K\"ahler structure on $M \times \R^{>0}$.
Suppose that $\omega$ is {\bf \red homogeneous}:
$\Psi_q^*\omega=q^2 g$, where $\Psi_q(m,t)= (m, qt)$,
and $I$ is $\Psi_q$-invariant. Then $M$ is called {\bf\blue Sasakian},
and $M \times \R^{>0}$ its {\bf \blue K\"ahler cone}.


{\bf \purple Sasakian geometry is an odd-dimensional
counterpart to K\"ahler geometry}


{\bf \green Example:} {\bf \red An odd-dimensional sphere $S^{2n-1}$ 
is Sasakian.} Indeed, its cone $S^{2n-1}\times \R^{>0} = \C^{n}\backslash 0$
has the standard K\"ahler form $\1\sum_{i=1}^n dz_{i}\wedge d\bar z_i$
which is obviously homogeneous.


{\it \small S. Sasaki, "On differentiable manifolds with certain
structures which are closely related to almost contact
structure", Tohoku Math. J. 2 (1960), 459-476.}


\newpage

{\bf\blue Quasiregular Sasakian manifolds.}

{\bf \green Definition:}
Given a contact manifold $(M,\theta)$ with
a Riemannian structure $g$, the dual vector field $\theta^\sharp$
is called {\bf \blue the Reeb field} of $(M,\theta, g)$.

{\bf \green Remark:} For any Sasakian manifold,
{\bf \purple the Reeb field generates a flow of diffeomorphisms acting
on $M$ by contact isometries}. This is obvious
from the definition, because the Reeb field 
$\theta^\sharp = I t \frac d {dt}$
acts by holomorphic isometries on the K\"ahler cone.

{\bf \green Definition:} A Sasakian manifold $M$ is called
{\bf \blue quasiregular} if all orbits of the Reeb flow are 
compact. The space of orbits of the Reeb flow is a
complex orbifold. {\bf \red Every quasiregular Sasakian manifold
is a total space of $S^1$-bundle over a complex orbifold.}

This is easy to see, because the quotient of $M$ over the Reeb flow
is the same as the quotient of $CM$ over its complexification,
generated by $\theta^\sharp$ and $I\theta^\sharp$.


\newpage

{\bf\blue Examples of Sasakian manifolds.}

{\bf \green Example:} Let $X\subset \C P^n$ be a complex submanifold,
and $CX \subset \C^{n+1}\backslash 0$ the corresponding cone.
The cone $CX$ is obviously K\"ahler and homogeneous, hence {\bf \red the
intersection $CX\cap S^{2n-1}$ is Sasakian.} This
intersection  is an $S^1$-bundle over $X$. This construction
gives many interesting contact manifolds,
including Milnor's exotic 7-spheres, which happen to be Sasakian.

\remark In other words, 
{\bf \green a link of a homogeneous singularity is
always Sasakian.}

\remark {\bf \red Every quasiregular Sasakian manifold is obtained
this way,} for some  K\"ahler metric on $\C^{n+1}$ (Ornea-V., 
arXiv:math/0609617 ).

\remark All 3-dimensional Sasakian manifolds are quasiregular
(H. Geiges, 1997, F. Belgun, 2000).

\remark{\bf \purple Every Sasakian manifold is diffeomorphic
to a quasiregular one} (Ornea-V., arXiv:math/0306077)

\newpage

{\bf\blue Structure theorem for Vaisman manifolds}


\theorem
(Ornea-V.)
{\bf \red Every Vaisman manifold is 
obtained as $C(X)/\Z$,} where $X$ is Sasakian, 
$\Z= \bigg\langle (x, t) \mapsto (\phi(x), q t)\bigg\rangle$, $q>1$,
and $\phi$ is a Sasakian automorphism of $X$.
Moreover, the triple $(X, \phi, q)$ is unique.

\remark This is equivalent to the {\bf \purple existence of a Riemannian
submersion $M \arrow S^1$,} with Sasakian fibers.

\remark {\bf \blue This construction gives an equivalence} between 
the category of Vaisman manifolds, and the category
of triples $(X, \phi, q)$  (Sasakian manifold, a Sasakian
automorphism, number).

\newpage

{\bf \blue Gauduchon metrics}

\definition
A Hermitian metric $\omega$ on a complex $n$ manifold is 
called {\bf\blue Gauduchon} if $\6\bar\6\omega^{n-1}=0$.

\theorem
(P. Gauduchon, 1978) Let $M$ be a compact, complex 
manifold, and $h$ a Hermitian form. {\bf \red Then there exists a 
Gauduchon metric conformally equivalent} to $h$, and
it is unique, up to a constant multiplier.

\remark 
If $\omega$ is Gauduchon, then (by Stokes' theorem)
$\int_M \omega^{n-1} \6\bar\6f=0$ for any $f$.
The curvature $\Theta_L$ of a holomorphic line bundle $L$ is well-defined up
to $\6\bar\6\log |h|$, where $h$ is a conformal factor. Therefore,
{\bf \purple for any line bundle $L$, the quantity 
$\deg_\omega L:=\int_M\omega^{n-1} \wedge \Theta_L$
is well defined.}

\remark Unlike the K\"ahler case, $\deg_\omega L$ is a
holomorphic invariant of $L$, and {\bf \red not topological.}

\definition
Given a torsion-free coferent sheaf $F$ of rank $r$, let
$\det F:= \Lambda^r F^{**}$. From algebraic geometry it is
known that $\det F$ is a line bundle. Define {\bf \blue the degree}
$\deg_\omega F:= \deg_\omega\det F$.

\newpage

{\bf \blue Kobayashi-Hitchin correspondence}


{\bf \green DEFINITION:} Let $F$ be a coherent sheaf over
an $n$-dimensional Gauduchon manifold $(M, \omega)$. Let
\[ 
   \text{slope}(F):=\frac{\deg_\omega F}{\text{rank}(F)} 
\]
A torsion-free sheaf $F$ is called {\bf \blue stable}
if for all subsheaves $F'\subset F$ one has
$\text{slope}(F')<\text{slope}(F)$. If $F$ is  
a direct sum of stable sheaves of the same slope, 
$F$ is called {\bf\blue polystable}.


{\bf \green DEFINITION:} A Hermitian metric on a 
holomorphic vector bundle $B$ is called {\bf \blue Yang-Mills}
(Hermitian-Einstein) if 
$\Theta_B \wedge \omega^{n-1} = \text{slope}(F) \cdot
\Id_B\cdot \omega^n,$
where $\Theta_B$ is its curvature.


\theorem ({\bf \blue Kobayashi-Hitchin correspondence}; Donaldson,
Buchsdahl, Uhlenbeck-Yau, Li-Yau, L\"ubke-Teleman):
Let $B$ be a holomorphic vector bundle. {\bf \red Then $B$ admits
Yang-Mills metric if and only if $B$ is polystable.}

{\bf \blue COROLLARY:} Any tensor product of polystable
bundles is polystable. 

\remark
This result {\bf \purple was generalized to coherent sheaves} by Bando
and Siu.


\newpage

{\bf \blue Stability and transversally K\"ahler
  foliations}

\theorem
Let $(M,\Sigma, \omega_0)$ be a compact, complex
manifold, $\dim M >2$, and $\omega_0$ a transversally K\"ahler, exact,
equivariant form. Consider a vector bundle $B$ with a Yang-Mills
metric, $\deg_\omega B=0$, and let $\nabla$ denote
the Yang-Mills connection. {\bf \red Then $\nabla$ is flat
along the leaves of $\Sigma$, and equivariant with respect to 
$s\in \Gamma_M(\Sigma)$. }

\remark
When $\Sigma$ is quasiregular, {\bf \purple $M$ is equipped with
a holomorphic projection to the leaf space,} 
$\pi:\; M\arrow X=M/\Sigma$. In this situation,
the category of coherent sheaves can be described
explicitly, in terms of a projective orbifold $M/\Sigma$.

\theorem
Let $F$ be a stable coherent sheaf on a be a compact, complex
manifold $(M,\Sigma, \omega_0)$, with a transversally K\"ahler, exact
form, $\dim M >2$. Assume that $\Sigma$ is quasiregular,  and
let $\pi:\; M\arrow X=M/\Sigma$ be the projection map.
{\bf \red Then $F=\pi^* F_0$, for some coherent sheaf $F_0$ on 
$M/\Sigma$.}

\corollary
In these assumptions, {\bf \red any coherent sheaf on
$M$ is } {\bf \blue filtrable}, that is, admits a 
filtration with rank 1 quotient sheaves.

\remark 
Filtrability is a very strong property!
{\bf \purple It fails on almost all non-algebraic surfaces.}

\newpage

{\bf \blue Open questions}

\question
{\bf \blue What happens when $\Sigma$ is irregular?}

{\bf \green Conjecture 1:}
{\bf \purple Any coherent sheaf on $(M,\Sigma, \omega_0)$
is filtrable,} even if $\Sigma$ is irrecular.

{\bf \green Conjecture 2:}
Let $M$ be a Vaisman manifold, obtained
as a quotient $C(X)/\Z$ from a pair
$(X, \phi)$, where $X$ is Sasakian, and
$\phi$ is a Sasakian automorphism. 
Consider the Lie group $G\subset \Aut(S)$
obtained as a closure of $\Z=\langle\phi^n\rangle$.
Then $M$ is equipped with a natural projection
$\pi:\; M\arrow X/G$, with $X/G$ parametrizing the 
closure of the appropriate leaves.
Let $(B,\nabla)$ be a Hermitian vector bundle
with connection which is flat on $\Sigma$ and equivariant. 
{\bf \red Then $(B,\nabla)= \pi^* (B_0, \nabla_0)$,
for some $(B_0, \nabla_0)$ on $X$. }

\remark
Conjecture 2 implies Conjecture 1. Also, Conjectures 1 and
2 are proven when $M = \C^n/ \langle A\rangle$, where
$A$ is a linear map (``linear Hopf manifold'').




\end{document}

