\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}