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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}