\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}} \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|_{{#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}} \newcommand{\2}{{/\!\!/}} % 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{\St}{\operatorname{St}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\Pic}{\operatorname{Pic}} \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{\diam}{\operatorname{\sf diam}} \renewcommand{\max}{{\operatorname{max}}} \newcommand{\slope}{\operatorname{slope}} \newcommand{\rk}{\operatorname{rk}} \newcommand{\Def}{\operatorname{Def}} \newcommand{\Lie}{\operatorname{Lie}} \newcommand{\Kah}{\operatorname{Kah}} \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 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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: % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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:\ }} \newcommand{\question}{% {\bf \green QUESTION:\ }} \newcommand{\problem}{% {\bf \green PROBLEM:\ }} \newcommand{\proof}{{\bf \green Proof:\ }} \newcommand{\pstep}{{\bf \green Proof. Step 1:\ }} \newcommand{\ps@verbit}{% \renewcommand{\@oddhead}{% \scriptsize {\it \small Hyperk\"ahler geometry and hyperbolic gometry \hfil \tiny M. Verbitsky }} \renewcommand{\@evenhead}{\@oddhead} \renewcommand{\@oddfoot}{\hfil\thepage\hfil} \renewcommand{\@evenfoot}{\@oddfoot}} \pagestyle{verbit} \begin{document} \setcounter{page}{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{center} {\Large\bf Hyperbolic geometry and the proof of Morrison-Kawamata cone conjecture (1)} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\scriptsize Misha Verbitsky } \\[20mm] {\tiny\bf Complex Geometry: discussion meeting \\[3mm] 20 March 2017 to 25 March 2017\\[3mm] Ramanujan Lecture Hall, ICTS, Bengaluru } \end{center} \newpage {\bf \blue K\"ahler manifolds} {\bf\green DEFINITION:} An Riemannian metric $g$ on an almost complex manifiold $M$ is called {\bf \blue Hermitian} if $g(Ix, Iy)= g(x,y)$. In this case, $g(x, Iy)= g(Ix, I^2y) = - g(y, Ix)$, hence $\omega(x,y):= g(x, Iy)$ is skew-symmetric. {\bf\green DEFINITION:} The differential form $\omega\in \Lambda^{1,1}(M)$ is called {\bf \blue the Hermitian form} of $(M,I,g)$. \remark It is $U(1)$-invariant, hence {\bf \purple of Hodge type (1,1)}. {\bf\green DEFINITION:} A complex Hermitian manifold $(M,I,\omega)$ is called {\bf \blue K\"ahler} if $d\omega=0$. The cohomology class $[\omega]\in H^2(M)$ of a form $\omega$ is called {\bf \blue the K\"ahler class} of $M$, and $\omega$ {\bf \blue the K\"ahler form}. The set of all K\"ahler classes is called {\bf\blue K\"ahler cone}. \newpage {\bf \blue Hyperk\"ahler manifolds} \definition {\bf \blue (E. Calabi, 1978)}\\ Let $(M, g)$ be a Riemannian manifold equipped with three complex structure operators $I, J, K:\; TM\arrow TM$, satisfying the quaternionic relation $I^2=J^2=K^2=IJK=-\Id.$ Suppose that $I$, $J$, $K$ are K\"ahler. Then $(M, I, J, K, g)$ is called {\bf \blue hyperk\"ahler}. \claim A hyperk\"ahler manifold $(M,I,J,K)$ is {\bf \red holomorphically symplectic} (equipped with a holomorphic, non-degenerate 2-form). Then $M$ is equipped with 3 symplectic forms $\omega_I$, $\omega_J$, $\omega_K$. \lemma The form $\Omega:= \omega_J+\1\omega_K$ {\bf \purple is a holomorphic symplectic 2-form on $(M,I)$.} \endproof \theorem (Calabi-Yau, 1978) Let $M$ be a compact, holomorphically symplectic K\"ahler manifold. Then {\bf \red $M$ admits a hyperk\"ahler metric,} which is uniquely determined by the cohomology class of its K\"ahler form $\omega_I$. \newpage \begin{center} {\epsfig{file=Calabi.jpg,width=0.40\linewidth}} \\ {\it\green Eugenio Calabi, \\ born 11 May 1923} \end{center} \newpage {\bf \blue Levi-Civita connection and K\"ahler geometry} \definition Let $(M, g)$ be a Riemannian manifold. A connection $\nabla$ is called {\bf \blue orthogonal} if $\nabla(g) =0$. It is called {\bf \blue Levi-Civita} if it is torsion-free. \theorem (``the main theorem of differential geometry'')\\ {\bf \red For any Riemannian manifold, the Levi-Civita connection exists,\\ and it is unique}. {\bf \green THEOREM:} Let $(M,I,g)$ be an almost complex Hermitian manifold. {\bf \purple Then the following conditions are equivalent.} (i) {\bf \red $(M,I,g)$ is K\"ahler} (ii) One has {\red $\nabla(I)=0$,} where $\nabla$ is the Levi-Civita connection. \newpage {\bf \blue Holonomy group} \definition (Cartan, 1923) Let $(B,\nabla)$ be a vector bundle with connection over $M$. For each loop $\gamma$ based in $x\in M$, let $V_{\gamma, \nabla}:\; B\restrict x \arrow B\restrict x$ be the corresponding parallel transport along the connection. The {\bf \blue holonomy group} of $(B,\nabla)$ is a group generated by $V_{\gamma, \nabla}$, for all loops $\gamma$. If one takes all contractible loops instead, $V_{\gamma, \nabla}$ generates {\bf \blue the local holonomy}, or {\bf \blue the restricted holonomy} group. \remark A bundle is {\bf \blue flat} (has vanishing curvature) {\bf\purple if and only if its restricted holonomy vanishes.} \remark If $\nabla(\phi)=0$ for some tensor $\phi\in B^{\otimes i}\otimes (B^*)^{\otimes j}$, {\bf \red the holonomy group preserves $\phi$.} \definition {\bf \blue Holonomy of a Riemannian manifold} is holonomy of its Levi-Civita connection. \example Holonomy of a Riemannian manifold lies in $O(T_x M, g\restrict x)=O(n)$. \example Holonomy of a K\"ahler manifold lies in $U(T_x M, g\restrict x, I \restrict x)=U(n)$. \remark The holonomy group {\bf \red does not depend on the choice of a point $x\in M$.} \newpage {\bf \blue The Berger's list} \theorem {\bf \blue(de Rham)}\\ A complete, simply connected Riemannian manifold with non-irreducible holonomy {\bf \red splits as a Riemannian product.} \theorem {\bf \blue (Berger's classification of holonomies, 1955)}\\ Let $G$ be an irreducible holonomy group of a Riemannian manifold which is not locally symmetric. {\bf \red Then $G$ belongs to the Berger's list:} { \begin{tabular}{|l|l|} \hline \multicolumn{2}{|c|}{\bf \color[rgb]{0,0,0.6}Berger's list}\\[1mm] \hline \it Holonomy & \it Geometry\\[1mm] \hline $SO(n)$ acting on $\R^n$ & Riemannian manifolds\\[1mm] \hline $U(n)$ acting on $\R^{2n}$ & K\"ahler manifolds\\[1mm] \hline $SU(n)$ acting on $\R^{2n}$, $n>2$ & Calabi-Yau manifolds\\[1mm] \hline $Sp(n)$ acting on $\R^{4n}$ & hyperk\"ahler manifolds\\[1mm] \hline $Sp(n)\times Sp(1)/\{\pm 1\}$ & quaternionic-K\"ahler\\[1mm] acting on $\R^{4n}$, $n>1$ & manifolds\\[1mm] \hline $G_2$ acting on $\R^7$ & $G_2$-manifolds \\[1mm] \hline $Spin(7)$ acting on $\R^8$ & $Spin(7)$-manifolds\\[1mm] \hline \end{tabular} } \newpage \begin{center} {\epsfig{file=berger.jpg,width=0.40\linewidth}} \\ {\it\green Marcel Berger, \\ 14 April 1927 -- 15 October 2016} \end{center} \newpage {\bf \blue Subject of these lectures} 1. Determine the shape of the K\"ahler cone of a hyperk\"ahler manifold. It turns out that it is determined by a quadratic inequality and a set of linear inequaities associated with the rational curves. 2. Interpret various quantities associated with a hyperk\"aher manifold, such as its automorphism group and its moduli space, im terms of the shape of the K\"ahler cone. 3. Associate a hyperbolic manifold to each hyperk\"ahler manifold. Interpret the statement about the shape of the K\"ahler cone as a statement about this manifold. 4. Using ergodic theory and hyperbolic geometry, prove that the group of holomorphic automorphisms of a hyperk\"ahler manifold acts on the polyhedral faces of its K\"ahler cone with finitely many orbits (``Morrison-Kawamata cone conjecture''). {\bf \red The results are obtained in a serie of joint papers with Ekaterina Amerik.} \newpage {\bf \blue Calabi-Yau manifolds} \definition\\ {\bf\blue A Calabi-Yau manifold} is a compact Kaehler manifold with $c_1(M,\Z)=0$. \newcommand{\Ric}{\operatorname{Ric}} \definition Let $(M,I, \omega)$ be a Kaehler $n$-manifold, and $K(M):= \Lambda^{n,0}(M)$ its {\bf \blue canonical bundle.} We consider $K(M)$ as a holomorphic line bundle, $K(M)= \Omega^n M$. Denote by $\Theta_K$ the curvature of the connection on $K(M)$ induced by Levi-Civita connection. The {\bf\blue Ricci curvature} $\Ric$ of $M$ is a symmetric 2-form $\Ric(x,y)= \Theta_K(x, Iy)$. \definition A K\"ahler manifold is called {\bf \blue Ricci-flat} if its Ricci curvature vanishes. \theorem (Calabi-Yau) \\ Let $(M, I, g)$ be Calabi-Yau manifold. {\bf \red Then there exists a unique Ricci-flat Kaehler metric in any given Kaehler class.} \remark Converse is also true: {\bf \purple any Ricci-flat K\"ahler manifold has a finite covering which is Calabi-Yau.} This is due to Bogomolov. \newpage {\bf \blue Bochner's vanishing} \theorem (Bochner vanishing theorem) On a compact Ricci-flat Calabi-Yau manifold, {\bf \red any holomorphic $p$-form $\eta$ is parallel} with respect to the Levi-Civita connection: $\nabla(\eta)=0$. \remark Its proof is based on spinors: $\eta$ gives a harmonic spinor, and {\bf \purple on a Ricci-flat Riemannian spin manifold, any harmonic spinor is parallel.} \definition A {\bf \blue holomorphic symplectic manifold} is a manifold admitting a non-degenerate, holomorphic symplectic form. \remark A holomorphic symplectic manifold is Calabi-Yau. The top exterior power of a holomorphic symplectic form {\bf \purple is a non-degenerate section of canonical bundle.} \newpage {\bf \blue Hyperk\"ahler manifold} \remark Due to Bochner's vanishing, {\bf \red holonomy of Ricci-flat Calabi-Yau manifold lies in $SU(n)$}, and {\bf \red holonomy of Ricci-flat holomorphically symplectic manifold lies in $Sp(n)$} (a group of complex unitary matrices preserving a complex-linear symplectic form). \definition A holomorphically symplectic K\"ahler manifold with a Calabi-Yau metric is called {\bf \blue hyperk\"ahler}. \remark Since $Sp(n)=SU({\Bbb H}, n)$, a {\bf \purple hyperk\"ahler manifold admits quaternionic action in its tangent bundle.} \newpage {\bf \blue EXAMPLES.} \example An even-dimensional complex vector space. \example An even-dimensional complex torus. \example {\bf \purple A non-compact example:} $T^* \C P^n$ (Calabi). \remark $T^*\C P^1$ {\bf \blue is a resolution of a singularity $\C^2/{\pm1}$.} \remark Let $M$ be a 2-dimensional complex manifold with holomorphic symplectic form outside of singularities, which are all of form $\C^2/{\pm1}$. Then {\bf \purple its resolution is also holomorphically symplectic.} \example Take a 2-dimensional complex torus $T$, then all the singularities of $T/{\pm1}$ are of this form. Its resolution $\widetilde {T/{\pm1}}$ is called {\bf \green a Kummer surface}. {\bf \red It is holomorphically symplectic}. \remark Take a symmetric square $\Sym^2 T$, with a natural action of $T$, and let $T^{[2]}$ be a blow-up of a singular divisor. {\bf \purple Then $T^{[2]}$ is naturally isomorphic to the Kummer surface $\widetilde {T/{\pm1}}$.} \newpage {\bf \blue K3 surfaces} \definition {\bf \blue A K3-surface} is a deformation of a Kummer surface. {\bf \red ``K3: Kummer, K\"ahler, Kodaira''} (a name is due to A. Weil). \begin{center}\epsfig{file=Broad_Peak8051m.jpg,width=0.5\linewidth} {\it\color{blue} ``Faichan Kangri (K3) is the 12th highest mountain on Earth.''} \end{center} \theorem Any complex compact surface with $c_1(M)=1$ and $H^1(M)=0$ {\bf \purple is isomorphic to K3.} Moreover, {\bf \blue it is hyperk\"ahler.} \newpage {\bf \blue Hilbert schemes} \remark {\bf\blue A complex surface} is a 2-dimensional complex manifold. \definition A {\bf\blue Hilbert scheme} $M^{[n]}$ of a complex surface $M$ is a classifying space of all ideal sheaves $I\subset \calo_M$ for which the quotient $\calo_M/I$ has dimension $n$ over $\C$. \remark A Hilbert scheme {\bf \purple is obtained as a resolution of singularities} of the symmetric power $\Sym^n M$. \theorem (Fujiki, Beauville) {\bf \red A Hilbert scheme of a hyperk\"ahler surface is hyperk\"ahler.} \example {\bf\blue A Hilbert scheme of K3}. \example Let $T$ is a torus. Then it acts on its Hilbert scheme freely and properly by translations. For $n=2$, the quotient $T^{[n]}/T$ is a Kummer K3-surface. For $n>2$, it is called {\bf \blue a generalized Kummer variety}. \remark There are 2 more ``sporadic'' examples of compact hyperk\"ahler manifolds, constructed by K. O'Grady. {\bf \purple All known compact hyperkaehler manifolds of maximal holonomy are these 2 and the three series:} tori, Hilbert schemes of K3, and generalized Kummer. \newpage {\bf \blue Bogomolov's decomposition theorem} \theorem {\bf \blue (Cheeger-Gromoll)} Let $M$ be a compact Ricci-flat Riemannian manifold with $\pi_1(M)$ infinite. {\bf \red Then a universal covering of $M$ is a product of $\R$ and a Ricci-flat manifold.} \corollary A fundamental group of a compact Ricci-flat Riemannian manifold is {\bf \blue ``virtually polycyclic'':} {\bf \purple it is projected to a free abelian subgroup with finite kernel.} \remark This is equivalent to any compact Ricci-flat manifold having a finite covering which has free abelian fundamental group. \remark This statement contains the Bieberbach's solution of Hilbert's 18-th problem on classification of crystallographic groups. \theorem {\bf \blue (Bogomolov's decomposition)} Let $M$ be a compact, Ricci-flat Kaehler manifold. {\bf \red Then there exists a finite covering $\tilde M$ of $M$ which is a product of Kaehler manifolds of the following form:} \[ \tilde M = T \times M_1 \times ... \times M_i \times K_1 \times ... \times K_j, \] with all $M_i$, $K_i$ simply connected, $T$ a torus, and $\Hol(M_l) = Sp(n_l)$, $\Hol(K_l)=SU(m_l)$ \newpage {\bf \blue Holomorphic Euler characteristic} \definition {\bf\blue A holomorphic Euler characteristic} $\chi(M)$ of a K\"ahler manifold is a sum $\sum(-1)^p\dim H^{p,0}(M)$. \theorem (Riemann-Roch-Hirzebruch) For an $n$-fold, {\bf \red $\chi(M)$ can be expressed as a polynomial expressions of the Chern classes,} $\chi(M)=td_{n}$ where $td_n$ is an $n$-th component of the Todd polynomial, {\small \[ td(M) = 1 + \frac1 {2}c_1 + \frac{1}{12}(c_1^2+c_2) + \frac{1}{24}c_1c_2 + \frac1{720}(-c_1^4 + 4c_1^2c_2 + c_1c_3 + 3c_2^22 - c_4) + ... \]} \vspace{-10mm} \remark The Chern classes are obtained as polynomial expression of the curvature (Gauss-Bonnet). Therefore {\bf \purple $\chi(\tilde M)= p\chi(M)$ for any unramified $p$-fold covering $\tilde M \arrow M$.} \remark Bochner's vanishing and the classical theory of invariants imply: \\ \phantom{XXX} 1. When $\Hol(M) = SU(n)$, we have {\bf \red $\dim H^{p,0}(M)= 1$ for $p=1,n$, and 0 otherwise. } In this case, $\chi(M)=2$ for even $n$ and 0 for odd. \\ \phantom{XXX} 2. When $\Hol(M) = Sp(n)$,we have {\bf \red $\dim H^{p,0}(M)= 1$ for even $p$ with $0\leq p\leq 2n$, and 0 otherwise.} In this case, $\chi(M)=n+1$. \corollary $\pi_1(M)=0$ if $\Hol(M) = Sp(n)$, or $\Hol(M) = SU(2n)$. If $\Hol(M) = SU(2n+1)$, {\bf \purple $\pi_1(M)$ is finite by Cheeger-Gromoll, but can be non-trivial. } \end{document}