
\documentclass{slides}

\usepackage{amsmath, amssymb, amscd, color, epsfig}


\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}}
\newcommand{\C}{{\Bbb C}}
\newcommand{\R}{{\Bbb R}}
\newcommand{\Q}{{\Bbb Q}}
\renewcommand{\H}{{\Bbb H}}
\newcommand{\6}{\partial}
\def\1{\sqrt{-1}\:}
\newcommand{\restrict}[1]{{\left|_{{\phantom{|}\!\!}_{#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{\Diff}{\operatorname{Diff}}
\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{\Lin}{\operatorname{Lin}}
\newcommand{\Hor}{\operatorname{Hor}}
\newcommand{\Def}{\operatorname{Def}}
\newcommand{\Lie}{\operatorname{Lie}}
\newcommand{\Tw}{\operatorname{Tw}}
\newcommand{\Tr}{\operatorname{Tr}}
\newcommand{\Spec}{\operatorname{Spec}}

\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 Principal toric fibrations\hfil
  \tiny M. Verbitsky}}
  \renewcommand{\@evenhead}{\@oddhead}
  \renewcommand{\@oddfoot}{\hfil\thepage\hfil}
  \renewcommand{\@evenfoot}{\@oddfoot}}
 
\pagestyle{verbit}

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

\newcounter{section}
\newcounter{Mycounter}[section]
\newcounter{lemma}[section]
\setcounter{lemma}{0}
\renewcommand{\thelemma}{\noindent{Lemma \thesection.\arabic{lemma}}}
\newcommand{\lemma}{%
     \setcounter{lemma}{\value{Mycounter}}
     \refstepcounter{lemma}
     \stepcounter{Mycounter}
     {\bf \green LEMMA:\ }}

\newcounter{claim}[section]
\setcounter{claim}{0}
\renewcommand{\theclaim}{\noindent{Claim \thesection.\arabic{claim}}}
\newcommand{\claim}{%
     \setcounter{claim}{\value{Mycounter}}
     \refstepcounter{claim}
     \stepcounter{Mycounter}
     {\bf \green CLAIM:\ }}

\newcounter{corollary}[section]
\setcounter{corollary}{0}
\renewcommand{\thecorollary}{\noindent{Corollary \thesection.\arabic{corollary}}}
\newcommand{\corollary}{%
     \setcounter{corollary}{\value{Mycounter}}
     \refstepcounter{corollary}
     \stepcounter{Mycounter}
     {\bf \green COROLLARY:\ }}

\newcounter{theorem}[section]
\setcounter{theorem}{0}
\renewcommand{\thetheorem}{\noindent{Theorem \thesection.\arabic{theorem}}}
\newcommand{\theorem}{%
     \setcounter{theorem}{\value{Mycounter}}
     \refstepcounter{theorem}
     \stepcounter{Mycounter}
     {\bf \green THEOREM:\ }}

\newcounter{conjecture}[section]
\setcounter{conjecture}{0}
\renewcommand{\theconjecture}{\noindent{Conjecture \thesection.\arabic{conjecture}}}
\newcommand{\conjecture}{%
     \setcounter{conjecture}{\value{Mycounter}}
     \refstepcounter{conjecture}
     \stepcounter{Mycounter}
     {\bf \green CONJECTURE:\ }}

\newcounter{proposition}[section]
\setcounter{proposition}{0}
\renewcommand{\theproposition}
       {\noindent{Proposition \thesection.\arabic{proposition}}}
\newcommand{\proposition}{%
     \setcounter{proposition}{\value{Mycounter}}
     \refstepcounter{proposition}
     \stepcounter{Mycounter}
     {\bf \green PROPOSITION:\ }}

\newcounter{definition}[section]
\setcounter{definition}{0}
\renewcommand{\thedefinition}
       {\noindent{Definition~\thesection.\arabic{definition}}}
\newcommand{\definition}{%
     \setcounter{definition}{\value{Mycounter}}
     \refstepcounter{definition}
     \stepcounter{Mycounter}
     {\bf \green DEFINITION:\ }}


\newcounter{example}[section]
\setcounter{example}{0}
\renewcommand{\theexample}{\noindent{Example \thesection.\arabic{example}}}
\newcommand{\example}{%
     \setcounter{example}{\value{Mycounter}}
     \refstepcounter{example}
     \stepcounter{Mycounter}
     {\bf \green EXAMPLE:\ }}

\newcounter{remark}[section]
\setcounter{remark}{0}
\renewcommand{\theremark}{\noindent{Remark \thesection.\arabic{remark}}}
\newcommand{\remark}{%
     \setcounter{remark}{\value{Mycounter}}
     \refstepcounter{remark}
     \stepcounter{Mycounter}
     {\bf \green REMARK:\ }}


\newcounter{observation}[section]
\setcounter{observation}{0}
\renewcommand{\theobservation}{\noindent{Question \thesection.\arabic{observation}}}
\newcommand{\observation}{%
     \setcounter{observation}{\value{Mycounter}}
     \refstepcounter{observation}
     \stepcounter{Mycounter}
     {\bf \green OBSERVATION:\ }}

\newcounter{question}[section]
\setcounter{question}{0}
\renewcommand{\thequestion}{\noindent{Question \thesection.\arabic{question}}}
\newcommand{\question}{%
     \setcounter{question}{\value{Mycounter}}
     \refstepcounter{question}
     \stepcounter{Mycounter}
     {\bf \green QUESTION:\ }}


   \setlength\paperheight {8in}%
    \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}


\begin{document}
\setcounter{page}{1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Complex subvarieties in homogeneous complex manifolds}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

{\small Misha Verbitsky} 
\\[8mm]

{\tiny

{\bf August 10, 2014\\[14mm]

Topology of Torus Actions\\ and Applications to Geometry and Combinatorics,\\
Daejeon, Korea.
}}
\end{center}

\newpage


{\bf \blue Homogeneous complex manifolds}

\definition 
A complex manifold $M$ is called {\bf \blue homogeneous}
if its automorphism group acts transitively.

{\bf \green Examples of compact homogeneous manifolds:}

0. Flag spaces and partial flag spaces.

1. Calabi-Eckmann and Hopf manifolds.

2. Tori.

3. Let $G$ be a compact, even-dimensional Lie group.
Then {\purple\bf $G$ admits a left-invariant complex structure}
(H. Samelson, 1953). 

%\remark Compact homogeneous complex manifolds are usually
%non-K\"ahler (exception: partial flag spaces,
%tori, their products and finite quotients).

\newpage

{\blue \bf Hopf surface}

{\bf \green The (classical) Hopf surface}. Fix
$\alpha\in \C$, $|\alpha|>1$.
Consider the quotient
$H=(\C^2\backslash 0)/\langle \Z\rangle$, with $\Z$ acting on $\C^2$
by $(x, y) \arrow (\alpha x, \alpha y)$. It is called 
{\bf\blue the Hopf surface}. Topologically the Hopf surface
is isomorphic to $S^1\times S^3$ (hence, non-K\"ahler).
The elliptic curve $T^2= \C^*/\langle \alpha \rangle$
acts on $H$ by $t, (x, y) \arrow (tx, ty)$. This action is
free, and its quotient is $\C P^1$. The Hopf surface is
a {\bf\purple principal elliptic fibration.} Topologically,
it's a product of a Hopf fibration $S^3\arrow S^2$ and a circle.

\newpage

{\blue \bf 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.
Then $M$ is called {\bf \blue the Calabi-Eckmann manifold}.
It is diffeomorphic to $S^{2n-1}\times S^{2m-1}$. The group
$U(n)\times U(m)$ acts on $M$ transitively.

{\bf \purple We obtained a homogeneous complex structure on
$S^{2n-1}\times S^{2m-1}$}.

It is non-K\"ahler, because $H^2(M)=0$.


\newpage

{\bf \blue Principal toric fibrations}

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

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

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

{\bf \purple To trivialize a principal group bundle it means to find a section}.


\newpage

{\bf \blue Borel-Remmert-Tits theorem}

{\bf \green Borel-Remmert-Tits theorem:}
Let $M$ be a compact, complex, simply connected 
homogeneous manifold . Then $M$ is a principal toric fibration, with 
a base which is a homogeneous, rational projective manifold.


{\bf\blue Proof:} Let $K^{-1}=\Lambda^{\dim M}_\C(TM)$ be the anticanonical
class of $M$. Since $TM$ is globally generated, the same is true
for $K^{-1}$. This gives a $G$-invariant  morphism
\[
M\stackrel \pi \arrow {\Bbb P} H^0(K^{-1}).
\]
{\purple \bf The fibers $F$ of $\pi$ are homogeneous
with trivial canonical class, and its
base is homogeneous and projective (hence, rational).} 
The fundamental group of $F$ is a quotient of $\pi_2(X)$,
as follows from the long exact sequence of homotopy groups
for a Serre's fibration:
\[
\pi_2(X)\arrow \pi_1(F) \arrow \pi_1(M) =0
\]
Therefore, $\pi_1(F)$ is abelian.
{\bf \green It remains to show that it is a torus.}

\newpage

{\bf \blue Homogeneous manifolds with trivial canonical class}

\lemma
Let $F$ be a compact, complex, homogeneous manifold
with $\pi_1(F)$ abelian and a trivial anticanonical class $K^{-1}$. 
{\bf \red Then $F$ is a torus.}

{\bf \green Proof:} The sheaf of holomorphic vector fields
on $M$ is globally generated. Taking a vector field $v_1$
and multiplying it by general vector fields
$v_2, ... v_n$, we obtain a section of $K^{-1}$,
which is non-zero for general $v_i$, and 
therefore non-degenerate. We obtain that $v_i$ are
linearly independent everywhere. Taking the corresponding
flows of diffeomorphisms, we obtain that $F$ is a quotient
of a holomorphic Lie group $G$ by a cocompact lattice.
{\bf\purple Since $\pi_1(F)$ is abelian, $G$ is commutative, and $T$
is a torus.} \endproof


\newpage

{\bf \blue Positive elliptic fibrations}

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

{\bf\blue Examples:}\\
1. {\bf\purple Hopf manifold}, $H^2(M)=0$, hence positive. \\
2. {\bf\purple Calabi-Eckmann manifold} (same). \\
3. {\bf\purple $SU(3)$} is elliptically 
fibered over the flag manifold $F(2,3)$, also $H^2(M)=0$.

\newpage

{\bf \blue Subvarieties of positive elliptic fibrations}

{\bf \green Theorem:} Let $M\stackrel \pi\arrow X$ 
be a positive elliptic $T$ fibration, and $Z\subset M$ be a
subvariety, of positive dimension $m$. {\bf \red Then $Z$ is $T$-invariant.}

{\bf\green Proof:} Let $\omega_0=\pi^*\omega$ be a pullback of a K\"ahler form
which is exact. Then \[ \int_Z \omega_0^m=0. \]
On the other hand, all eigenvalues of $\omega_0\restrict Z$
are non-negative, and all are positive, unless $Z$ is tangent
to the action of $T$. In a point where $Z$ is not tangent to $T$,
the form $\omega_0^m$ is positive, and 
{\bf\purple in this case the integral $\int_Z \omega_0^m$
is also positive.}

\newpage

{\bf\blue Positive toric fibrations}

\definition Let $M\stackrel \pi\arrow X$
be a complex principal toric fibration, $M$ compact,
with fiber $T$. We say that $\pi$ is {\bf \blue convex} if 
$\pi^*\omega$ is exact for some K\"ahler form $\omega$.
We say that  $\pi$ is {\bf \blue positive} if
for any proper complex subtorus $T'\subset T$, the
corresponding quotient fibration $M/T'\arrow X$ is convex.

\example
Let $M$ be a complex, compact 
homogeneous manifold with \\$H^2(M)=0$ (e.g. a Lie group),
and $M\stackrel \pi \arrow X$ the Borel-\-Remmert-\-Tits toric fibration.
Assume that the fiber of $\pi$ have no proper subtori
(easy to insure by taking a generic invariant complex structure).
{\bf \purple Then $M$ is positive.}

\newpage

{\bf \blue Subvarieties in principal toric fibrations}

\theorem
Consider an irreducible complex subvariety $Z\subset M$
of a positive principal toric fibration $M\stackrel \pi\arrow X$, 
with fiber $T$. {\bf \red Then $Z$ is $T$-invariant, or is 
contained in a fiber of $\pi$.}

{\bf \green Proof:} 1. For any positive-dimensional subvariety $Z_0\subset X$,
the restriction of $\pi$ to $Z_0$ {\bf\purple has no multisections}
(because $\int_Z \omega_0^m$ must vanish).

2. Given a K\"ahler manifold $A$ 
with an action of $T$, consider an associated
fiber bundle $M\times_T A$ over $X$. Unless 
$T$ acts on $A$ trivially, $M\times_T A$ 
is also convex, hence {\bf\purple admits no multisections.}

3. If $Z\subset M$ is not $T$-invariant, it provides 
us with a multisection from $X$ to $M\times_T A$,
where $A$ is the space of deformations of the fiber
$Z \cap \pi^{-1} (t_0)$. It is convex (step 2). 
{\bf\purple Cannot have multisections! }
Contradiction. \endproof

\newpage

{\bf \blue Open questions}

1. 
\theorem
Let $M\stackrel \pi\arrow X$,
$\dim_\C X >1$, 
be a positive elliptic $T$ fibration, and $F$ a stable
reflexive sheaf on $M$. {\bf \red Then $F \cong L \otimes \pi^*
F_0$, where $L$ is a line bundle, and $F_0$ a stable
coherent sheaf on $X$.}

{\bf \purple Is there a similar result for positive torus fibrations?}

2. Is it possible to define positivity for $\C^n$-fibrations?
For $\C$-fibrations it is possible in terms of curvature;
all subvarieties would be also $\C$-invariant.


\end{document}










