\documentclass[a4paper,12pt,leqno,oneside]{amsart} \usepackage{microtype} \usepackage[british]{babel} \usepackage[svgnames]{xcolor} \usepackage[all]{xy} \usepackage[bookmarksnumbered,bookmarksdepth=2,bookmarksopen,colorlinks,linkcolor=Maroon,citecolor=DarkGreen,urlcolor=DarkBlue,linktoc=all,pdftitle={Shiing-shen Chern: Complex manifolds without potential theory (with an appendix on the geometry of characteristic classes)}]{hyperref} \usepackage[style=numeric,maxnames=9,sorting=nyt,datamodel=biblatex-zbl]{biblatex} \usepackage[stable]{footmisc} \usepackage{physics} \usepackage{graphicx} \usepackage{tikz-cd} \usepackage{pgfplots} \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsthm} \usepackage{cleveref} \usepackage{bbm} \usepackage{mathtools} \usepackage{csquotes} \usepackage{xltabular} \usepackage{enumitem} \usepackage{fmtcount} \usepackage{etoolbox} \usepackage{cases} \usepackage{chngcntr} \usepackage{nicematrix} \usepackage{datetime} \usepackage{refcount} 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\newcommand{\gb}{{\overline{g}}} \newcommand{\hb}{{\overline{h}}} \newcommand{\Hb}{{\overline{H}}} \newcommand{\Jb}{{\overline{J}}} \newcommand{\lb}{{\overline\lambda}} \newcommand{\ob}{{\overline{\omega}}} \newcommand{\Ob}{{\overline{\Omega}}} \newcommand{\pib}{{\overline{\pi}}} \newcommand{\Pib}{{\overline{\Pi}}} \newcommand{\Rb}{{\overline{R}}} \newcommand{\sib}{{\overline{\sigma}}} \newcommand{\tlb}{{\overline{t}}} \newcommand{\tb}{{\overline{\theta}}} \newcommand{\Tb}{{\overline{T}}} \newcommand{\Vb}{{\overline{V}}} \newcommand{\wb}{{\overline{w}}} \newcommand{\xb}{{\overline{\xi}}} \newcommand{\zb}{{\overline{z}}} \newcommand{\Zb}{{\overline{Z}}} \newcommand{\zeb}{{\overline{\zeta}}} \newcommand{\Hh}{{\widehat{H}}} \newcommand{\ct}{{\widetilde{c}}} \newcommand{\Dt}{{\widetilde{\Delta}}} \newcommand{\Ft}{{\widetilde{F}}} \newcommand{\phit}{{\widetilde{\phi}}} \newcommand{\Phit}{{\widetilde{\Phi}}} \newcommand{\gt}{{\widetilde{g}}} \newcommand{\Gt}{{\widetilde{G}}} \newcommand{\Ht}{{\widetilde{H}}} \newcommand{\ot}{{\widetilde{\omega}}} \newcommand{\Ot}{{\widetilde{\Omega}}} \newcommand{\pt}{{\widetilde{p}}} \newcommand{\Pit}{{\widetilde{\Pi}}} \newcommand{\wt}{{\widetilde{w}}} \newcommand{\zt}{{\widetilde{z}}} \theoremstyle{definition} \newtheorem{defn}{Definition}[section] \newtheorem*{coro}{Corollary} \newtheorem{coron}[defn]{Corollary} \newtheorem*{eg}{Example} \newtheorem{egn}[defn]{Example} \newtheorem*{egs}{Examples} \newtheorem{lem}[defn]{Lemma} \newtheorem{prop}[defn]{Proposition} \newtheorem*{rmk}{Remark} \newtheorem{thm}[defn]{Theorem} \renewcommand{\thedefn}{\thesection(\Alph{defn})} \numberwithin{equation}{section} \title[Complex manifolds without potential theory]{Complex manifolds without potential theory (with an appendix on the geometry of characteristic classes)} \author{Shiing-shen Chern} \begin{document} \sloppy \renewcommand{\abstractname}{Typesetter's note} \begin{abstract} The following is a \LaTeX itification of \textsc{S. Chern.} \textit{Complex manifolds without potential theory (with an appendix on the geometry of characteristic classes)}. 2nd ed. Universitext. New York, NY: Springer, 1979. Zbl: \href{https://zbmath.org/0444.32004}{\texttt{0444.32004}}. We mostly respect the original typography, but we use British spellings and do not capitalise name-derived adjectives, e.g. k\"ahlerian. For convenience, we number the examples, propositions etc. by sections via \texttt{amsthm}. Similarly, we put all references in one section and let \texttt{BibLaTeX} order them, and we add bibliographic information to help the reader navigate to the source. The old page numbers are indicated in the right margin, which are the ones used in the index. We claim no originality except errors. \end{abstract} \maketitle \subsection*{Preface} The main text, on complex manifolds, was the notes from a course with the same title given at UCLA in the fall of 1966. It was written up after each lecture; only minor changes have been made. To the Department of Mathematics at UCLA, and Lowell Paige in particular, I wish to express here my belated thanks. The \hyperref[part:app]{Appendix} was an expanded version of a series of lectures given at a summer seminar of the Canadian Mathematical Congress taken place in Halifax, Nova Scotia in 1971. I wish to thank Professor J. R. Vanstone for his hospitality and kindness. Noteworthy is the treatment of the secondary characteristic classes, which is different from the one given in \cite{chernsimons74} in the Bibliography to the Appendix. Needless to say, my deepest gratitude is to the University of California at Berkeley and the National Science Foundation for their continuous support of my research. \makeatletter \def\l@section{\@tocline{1}{0pt}{0pt}{}{}} \makeatother \begingroup \hypersetup{hidelinks} \tableofcontents \endgroup \section{Introduction and examples} \label{sec:1} \oldpage{1}A complex manifold is a paracompact Hausdorff space which has a covering by neighbourhoods each homeomorphic to an open set in the \(m\)-dimensional complex number space such that where two neighbourhoods overlap the local coordinates transform by a complex analytic transformation. That is, if \(z^1,\ldots,z^m\) are local coordinates in one such neighbourhood and if \(w^1,\ldots,w^m\) are local coordinates in another neighbourhood, then where they are both defined, we have \(w^i=w^i\left(z^1,\ldots,z^m\right)\), where each \(w^i\) is a holomorphic (or analytic) function of the \(z\)'s and the functional determinant \(\frac{\partial\left(w^1,\ldots,w^m\right)}{\partial\left(z^1,\ldots,z^m\right)}\ne 0\). We will give some examples of complex manifolds: \subsection{Example} The complex number space \(\C_m\) whose points are the ordered \(m\)-tuples of complex numbers \(\left(z^1,\ldots,z^m\right)\). \(\C_1\) is called the gaussian plane. \subsection{Example} \label{eg:1.2} The complex projective space \(\bP_m\). To define it, take \(\C_{m+1}-0\), where \(0\) is the point \((0,\ldots,0)\), and identify those points \(\left(z^0,z^1,\ldots,z^m\right)\) which differ from each other by a factor. The resulting quotient space is \(\bP_m\). It can be covered by \(m+1\) open sets \(U_i\) defined respectively by \(z^i\ne 0\), \(0\le i\le m\). In \(U_i\) we have the local coordinates \(_i\zeta^k=\frac{z^k}{z^i}\), \(0\le k\le m\), \(k\ne i\). The transition of local coordinates in \(U_i\cap U_j\) is given by \(_j\zeta^h=\frac{_i\zeta^h}{_i\zeta^j}\), \(0\le h\le m\), \(h\ne j\), which are holomorphic functions. In particular, \(\bP_1\) is the Riemann sphere. By assigning to a point of \(\C_{m+1}-0\) the point it defines in the quotient space, we get a natural projection \(\psi:\C_{m+1}-0\to\bP_m\),\oldpage{2} for which the inverse image of each point is \(\C^\ast=\C_1-0\). This relationship is the first example of the important notion of a holomorphic line bundle and it is justified to enter into some detail. In fact, in \(\psi^{-1}(U_i)\) we can use instead of the coordinates \(\left(z^0,\ldots,z^m\right)\) the coordinates \(_i\zeta^h=\frac{z^h}{z^i}\), \(0\le h\le m\), \(h\ne i\), and \(z^i\). This has the advantage of expressing clearly the fact that \(\psi^{-1}(U_i)\) is a product \(U_i\times\C^\ast\), \(z^i\) being the fibre coordinate (relative to \(U_i\)). In \(\psi^{-1}(U_i\cap U_j)\), the fibre coordinates \(z^i\) and \(z^j\), relative to \(U_i\) and \(U_j\) respectively, are related by \[ z^i=z^j{}_j\zeta^i=\frac{z^j}{_i\zeta^j}. \] Thus the change of fibre coordinates is expressed by the multiplication of a nonzero holomorphic function. The general notion of a holomorphic line bundle, which generalises this example, plays a central role in complex manifolds. To a point \(p\in\bP_m\) the coordinates of a point of \(\psi^{-1}(p)\) are called its homogeneous coordinates. They can be normalised so that \begin{equation} \label{eq:1.1} \sum z^k\zb^k=1. \end{equation} Equation \eqref{eq:1.1} defines a sphere \(S^{2m+1}\) of real dimension \(2m+1\).The restriction of \(\psi\) gives the mapping \(\psi:S^{2m+1}\to\bP_m\), under which the inverse image of each point is a circle. This is called the \textit{Hopf fibring} of \(S^{2m+1}\). Further examples are obtained from submanifolds of \(\bP_m\) and quotient manifolds of \(\C_m\). \subsection{Example} Nonsingular submanifolds of \(\bP_m\), in particular, the nonsingular hyperquadric \begin{equation} (z^0)^2+\cdots+(z^m)^2=0. \end{equation} By a theorem of Chow, every compact submanifold embedded in \(\bP_m\) is an algebraic variety, i.e. it is the locus defined by a finite number of homogeneous polynomial equations \cite[170]{gunningrossi65}. \oldpage{3} It will not be significant to consider compact submanifolds of \(\C_m\), because of the: \begin{thm} A connected compact submanifold of \(\C_m\) is a point. \end{thm} The proof makes use of the lemma: Let \(f\) be a holomorphic function on a complex manifold \(M\). Suppose \(p_0\in M\) is a point such that \(|f(p)|\le\left|f(p_0)\right|\) for all \(p\) in a neighbourhood of \(p_0\). Then \(f(p)=f(p_0)\) in a neighbourhood of \(p_0\). For one variable this follows from the maximum modulus principle. The case of \(m\) variables follows from the consideration of the lines through \(p_0\) and the application of the one variable case to these lines. Now let \(M\) be a connected compact submanifold of \(\C_m\). Each coordinate of \(\C_m\) is a holomorphic function on \(M\). By the lemma, it must be constant on every connected component of \(M\). Since \(M\) is connected, \(M\) is a point. However, various significant examples arise from the quotient manifolds of \(\C_m\). \subsection{Example} \label{eg:1.4} Let \(\Gamma\) be the discontinuous group generated by \(2m\) translations of \(\C_m\), which are linearly independent over the reals. Then \(\C_m/\Gamma\) is called the \textit{complex torus}. If a complex torus can be embedded as a nonsingular submanifold of a projective space of sufficiently high dimension, it is called an \textit{abelian variety}. Let \(\Delta\) be the discontinuous group generated by \(z^k\to 2z^k\), \(1\le k\le m\). The quotient manifold \((\C_m-0)/\Delta\) is called the \textit{Hopf manifold}. It is homeomorphic to \(S^1\times S^{2m-1}\). Consider \(\C_3\) to be the group of all matrices \begin{equation} \begin{pmatrix} 1 & z_1 & z_2 \\ 0 & 1 & z_3 \\ 0 & 0 & 1 \end{pmatrix} \end{equation} \oldpage{4}Let \(D\) be the discrete subgroup consisting of those matrices for which \(z_1,z_2,z_3\) are gaussian integers (i.e. \(z_k=m_k+in_k\), \(1\le k\le 3\) where \(m_k,n_k\) are rational integers). Then \(\C_3/D\) is called an \textit{Iwasawa manifold}. Its fundamental group is isomorphic to \(D\), and hence is not abelian. \subsection{Example} An orientable surface is a complex manifold (of dimension one). We suppose the surface to be \(C^\infty\) and define on it a positive definite riemannian metric. By the theorem of Korn--Lichtenstein there exist local isothermal parameters \(x,y\) so that locally the metric can be written \begin{equation} \rd s^2=\lambda^2\left(\rd x^2+\rd y^2\right),\quad \lambda>0 \end{equation} or \(\rd s^2=\lambda^2 \rd z \rd\zb\), where \(z=x+iy\), the orientation of the manifold being defined by \(\rd dx\wedge\rd y=\frac i2\rd z\wedge\rd\zb\). If \(w\) is another local coordinate we will have \[ \rd s^2=\lambda^2\rd z\ \rd\zb=\mu^2\rd w\ \rd\wb \] because \(\rd s^2\) is globally defined. It follows that \(\rd w\) is a multiple of \(\rd z\) or \(\rd\zb\). If we assume that the complex coordinates \(z\) and \(w\) define the same orientation, then \(\rd w\) must be a multiple of \(\rd z\). This means that \(w\) is a holomorphic function of \(z\), and the surface becomes a complex manifold. A one-dimensional complex manifold is called a \textit{Riemann surface}. \subsection{Example \textmd{(Calabi--Eckmann)}} Let \(S\) and \(S'\) be spheres of dimensions \(2p+1\) and \(2q+1\) respectively, \(p,q>0\). By the Hopf fibring in Ex. \ref{eg:1.2} we have a fibration \[ \pi:S\times S'\to \bP_p\times \bP_q \] with fibre a two (real) dimensional torus. Since both the base space and the fibre are complex manifolds, we would expect that the total\oldpage{5} space could be given a complex structure. This we will prove to be the case as follows: Let \(S\) be the set of all points \(z=\left(z^0,\ldots,z^p\right)\) such that \(\sum_{0\le k\le p}z^k\zb^k=1\), and \(S'\) be the set of all points \(z'=\left(z'^0,\ldots,z'^q\right)\) such that \(\sum_{0\le j\le q}z'^j\zb'^j=1\). We define \[ V_{kj}=\left\{(z,z')\in S\times S':z^kz'^j\ne 0,\ 0\le k\le p,\ 0\le j\le q\right\}. \] Then the sets \(V_{kj}\) form an open covering of \(S\times S'\). Let \(\tau\) be a complex number such that \(\Im(\tau)\ne 0\). In \(V_{kj}\) we introduce the following complex coordinates \begin{equation} \label{eq:1.5} \begin{aligned} _kw^h &=\frac{z^h}{z^k},\ _jw'^\ell=\frac{z'^\ell}{z'^j},\ h\ne k,\ \ell\ne j,\ 0\le h\le p,\ 0\le\ell\le q, \\ t_{kj} &=\frac{1}{2\pi i}\left(\log z^k+\tau\log z'^j\right), \end{aligned} \end{equation} where \(t_{kj}\) is defined mod \(1\) and \(\tau\). Thus \(t_{kj}\) defines a point on the torus \(T_{(1,\tau)}\) which is the quotient of \(\C\) by the translations \(1\) and \(\tau\). In this way we have \(p+q+1\) coordinates in \(V_{kj}\) and these define a map \(V_{kj}\to\C_{p+q}\times T_{(1,\tau)}\). We show that this map is a homeomorphism. It suffices to show that \(_kw^h\), \(_jw'^\ell\) and \(t_{kj}\) determine the \(z\)'s and \(z'\)'s uniquely. Now \[ \sum_{h\ne k}{}_kw^h{}_k\wb^h=\sum_h\frac{z^h\zb^h}{z^k\zb^k}-1=\frac{1}{\left|z^k\right|^2}-1, \] so \(\left|z^k\right|\) is determined. Similarly, \(\left|z'^j\right|\) is determined. By the second equation of \eqref{eq:1.5} we have \[ t_{kj}=\frac{1}{2\pi i}\left(\log \left|z^k\right|+\tau\log\left|z'^j\right|+i\arg z^k+\tau i\arg z'^j\right),\quad\Mod(1,\tau). \] \oldpage{6} Hence \(\arg z^k\), \(\arg z'^j\) are determined mod \(2\pi\). The other \(z\)'s and \(z'\)'s are then determined by the first equations of \eqref{eq:1.5}. This proves that our map is a homeomorphism. In \(V_{kj}\cap V_{rs}\) we have \[ _rw^h=\frac{_kw^h}{_kw^r},\ _sw'^\ell=\frac{_jw'^\ell}{_jw'^s} \] and \[ t_{rs}=t_{kj}+\frac{1}{2\pi i}\left(\log {}_kw^r+\tau\log {}_jw'^s\right),\quad\Mod (1,\tau) \] where we set \(_kw^k=1\) and \(jw'^j=1\). Hence we have defined a complex structure on \(S\times S'\) with the \(_kw^h\), \(_jw'^\ell\), \(t_{kj}\) as local coordinates in \(V_{kj}\). \section{Complex and hermitian structures on a vector space} \label{sec:2} Let \(V\) be a real vector space of dimension \(n\). \(V\) is said to have a complex structure if there exists a linear endomorphism \(J:V\to V\), such that \(J^2=-1\), where \(1\) denotes the identity endomorphism. An eigenvalue of \(J\) is a complex number \(\lambda\) such that the equation \(Jx=\lambda x\) has a nonzero solution \(x\in V\). Applying \(J\) to this equation, we get \[ -x=J^2x=\lambda Jx=\lambda^2 x. \] It follows that \(\lambda^2=-1\) or \(\lambda=\pm i\). Since the complex eigenvalues occur in conjugate pairs, it follows that \(V\) must be of even dimension \(n=2m\). The following relations are immediately verified: \begin{equation} \label{eq:2.1} (J-i)(J+i)=(J+i)(J-i)=0. \end{equation} Let \(V^\ast\) be the dual space of \(V\), i.e., the space of all\oldpage{7} real-valued linear functions over \(V\). We denote the pairing of \(V\) and \(V^\ast\) by \(\lrag{x,y^\ast}\), \(x\in V\), \(y^\ast\in V^\ast\), so that this function is \(\R\)-linear in each of the arguments. Alternatively, we also write \(y^\ast(x)=\lrag{x,y^\ast}\). In addition to \(V^\ast\) we consider \(V^\ast\otimes\C\), i.e., the space of all complex-valued \(\R\)-linear functions over \(V\). Then \(V^\ast\otimes\C\) is a complex vector space of complex dimension \(n\). An element \(f\in V^\ast\otimes\C\) is said to be of type \((1,0)\) (respectively \((0,1)\)) if \begin{equation} \label{eq:2.2} f(Jx)=if(x)\ (\text{resp. } f(Jx)=-if(x)),\quad x\in V. \end{equation} Let \(e^{\ast\alpha}\), \(1\le\alpha\le n\), be a basis of \(V^\ast\). Consider the functions \begin{equation} \label{eq:2.3} \lambda^\alpha(x)=\lrag{(J+i)x,e^{\ast\alpha}}=\lrag{Jx,e^{\ast\alpha}}+i\lrag{x,e^{\ast\alpha}}. \end{equation} Since \(-i\) is an eigenvalue of \(J\) of multiplicity \(m\), exactly \(m\) of these functions are linearly independent with respect to \(\C\). It can be immediately verified that \(\lambda^\alpha(x)\) are of type \((1,0)\), and their complex conjugates \[ \lb^\alpha(x)=\lrag{(J-i)x,e^{\ast\alpha}} \tag{2.3a} \] are of type \((0,1)\). Suppose our basis \(e^{\ast\alpha}\) is so chosen that \(\lambda^k(x)\), \(1\le k\le m\), are linearly independent with respect to \(\C\). We split them into the real and imaginary parts: \begin{equation} \lambda^k(x)=\lambda'^k(x)+i\lambda''^k(x). \end{equation} We wish to show that \(\lambda'^k(x),\lambda''^k(x)\), \(1\le k\le m\), are linearly independent with respect to \(\R\). In fact, suppose that \[ \sum_k r_k\lambda'^k(x)+\sum_k s_k\lambda''^k(x)=0,\quad x\in V, \] \oldpage{8}where \(r_k,s_k\in\R\). This relation can be written as \[ \sum_k\left(r_k-is_k\right)\lambda^k(x)+\sum_k\left(r_k+is_k\right)\lb^k(x)=0. \] Replacing \(x\) by \(Jx\) and using the fact that \(\lambda^k(x)\) and \(\lb^k(x)\) are of types \((1,0)\) and \((0,1)\) respectively, we get \[ \sum_k\left(r_k-is_k\right)\lambda^k(x)-\sum_k\left(r_k+is_k\right)\lb^k(x)=0. \] Adding these two equations, we find \[ \sum\left(r_k-is_k\right)\lambda^k(x)=0, \] which gives \(r_k-is_k=0\), and hence \(r_k=s_k=0\), since \(\lambda^k(x)\) are linearly independent over \(\C\). Using the exterior algebra \(\bigwedge\left(V^\ast\otimes\C\right)\), we can express the fact proved above by \begin{equation} \label{eq:2.5} \left(\frac i2\right)^m\wedge_k\lambda^k\lb^k=\bigwedge_k\lambda'^k\lambda''^k\ne 0. \end{equation} It follows from \eqref{eq:2.5} that \(\lambda^k,\lb^k\) are linearly independent over \(\C\) and that \(V^\ast\otimes\C\) is a direct sum of \(V_\C\oplus\Vb_\C\), where \(V_\C\) (reps. \(\Vb_\C\)) is the space of all elements of \(V^\ast\otimes\C\) of type \((1,0)\) (resp. \((0,1)\)). Conversely, a direct sum decomposition of \(V^\ast\otimes\C\) over \(\C\) into two subspaces, complex conjugate to each other, defines a complex structure on \(V\), if the subspaces are defined to be consisting of the elements of types \((1,0)\) and \((0,1)\) respectively. This follows from the fact that when \(x\) is given the equations in \eqref{eq:2.2} determine the values of the elements of \(V^\ast\otimes\C\) at \(Jx\), whereby \(Jx\) is determined. As an example, let \(e_k,e_{m+k}\) be a dual basis of \(\lambda'^k,\lambda''^k\), so that \begin{equation} \lambda^k(e_h)=\lambda'^k(e_{m+h})=\delta_h^k,\quad 1\le h,\ k\le m, \end{equation} all other pairings being zero. This can be written \begin{equation} \label{eq:2.6a} \lambda^k(e_h)=\frac1i\lambda^k(e_{m+h})=\delta_h^k. \tag{2.6a} \end{equation} \oldpage{9} On the other hand, we have \[ \lambda^k(Jx)=i\lambda^k(x)=-\lambda''^k(x)+i\lambda'^k(x), \] from which it follows that \begin{equation} \label{eq:2.7} \lambda^k(Je_h)=\frac1i\lambda^k(Je_{m+h})=i\delta_h^k. \end{equation} Comparing \eqref{eq:2.6a} and \eqref{eq:2.7}, we get \begin{equation} Je_h=e_{m+h},\quad Je_{m+h}=-e_h. \end{equation} The elements \(\lambda^k\) form a basis of \(V_\C\) over \(\C\). Under a change of basis the real-valued \(2m\)-form in \eqref{eq:2.5} will be multiplied by a positive factor. Hence the complex structure \(J\) in \(V\) defines an orientation of \(V\). If \(J\) defines a complex structure in \(V\), \(-J\) does too. The two complex structures are said to be \textit{conjugate} to each other. A form of type \((1,0)\) (resp. type \((0,1)\)) in the structure \(J\) is a form of type \((0,1)\) (resp. \((1,0)\)) in the structure \(-J\) and vice versa. Suppose \(V\) is provided with a complex structure \(J\). An hermitian structure in \(V\) is a complex-valued function \(H(x,y)\), \(x,y\in V\), which satisfies the following conditions: \begin{enumerate} \item \(H\left(\lambda_1x_1+\lambda_2x_2,y\right)=\lambda_1H(x_1,y)+\lambda_2H(x_2,y)\), \(x_1,x_2,y\in V\), \(\lambda_1,\lambda_2\in\R\); \item \label{item:herm2} \(\overline{H(x,y)}=H(y,x)\); \item \label{item:herm3} \(H(Jx,y)=iH(x,y)\). \end{enumerate} In view of \eqref{item:herm2}, \eqref{item:herm3} is equivalent to the following: \begin{enumerate} \item[(3')] \(H(x,Jy)=-iH(x,y)\). \end{enumerate} We split \(H(x,y)\) into its real and imaginary parts:\oldpage{10} \begin{equation} \label{eq:2.9} H(x,y)=F(x,y)+iG(x,y). \end{equation} Then \eqref{item:herm2} is equivalent to \begin{equation} F(x,y)=F(y,x),\quad G(x,y)=-G(y,x), \end{equation} and \eqref{item:herm3} is equivalent to \begin{equation} \label{eq:2.11} F(x,y)=G(Jx,y),\quad \text{or }G(x,y)=-F(Jx,y). \end{equation} Thus \(H(x,y)\) defines a pair of real-valued bilinear functions, of which one is symmetric and the other antisymmetric, which are related to each other by \eqref{eq:2.11}. Either one of these functions, together with \(J\), determines \(H(x,y)\). The hermitian scalar product \(H(x,y)\) is called \textit{positive definite} if the corresponding real-valued symmetric bilinear function \(F(x,y)\) is positive definite. It is known that the space \(\bigwedge^2(V^\ast)\) of exterior forms of degree two is isomorphic to the space of all antisymmetric bilinear functions over \(V\). The isomorphism is established by the fact that it is a vector space isomorphism and that for \(\xi,\eta\in V^\ast\) the bilinear function corresponding to \(\xi\wedge\eta\) is \begin{equation} \label{eq:2.12} (\xi\wedge\eta)(x,y)=\frac12\left\{\xi(x)\eta(y)-\xi(y)\eta(x)\right\},\quad x,y\in V. \end{equation} By means of this isomorphism there is an exterior form \(\Hh\) of degree two corresponding to the function \(-\frac12 G(x,y)\). \(\Hh\) is called the \textit{K\"ahler form} of the hermitian structure. We wish to express \(H(x,y)\) in terms of the basis \(\lambda^K\) of \(V_\C\). For this purpose let \begin{equation} x=\sum_\alpha x^\alpha e_\alpha,\ y=\sum_\beta y^\beta e_\beta,\quad 1\le\alpha,\beta\le 2m,\ 1\le k,j\le m. \end{equation} Then we have\oldpage{11} \[ \begin{aligned} H(x,y) &= H\left(\sum_k\left(x^ke_k+x^{m+k}e_{m+k}\right),y\right)=H\left(\sum_k\left(x^ke_k+x^{m+k}Je_k\right),y\right) \\ &=\sum_k\left(x^k+ix^{m+k}\right)H(e_k,y) \\ &=\sum_{k,j}\left(x^k+ix^{m+k}\right)\left(y^j-iy^{m+j}\right)H(e_k,e_j) \\ &=\sum_{k,j}\lambda^k(x)\lb^j(y)H(e_k,e_j). \end{aligned} \] It follows that we can write \begin{equation} H=\sum_{k,j}h_{kj}\lambda^k\otimes\lb^j \end{equation} where \begin{equation} h_{kj}=H(e_k,e_j)=\hb_{jk}. \end{equation} To find the expression for the K\"ahler form \(\Hh\), we derive from \eqref{eq:2.9} \[ \begin{aligned} -\frac12(G(x,y)) &=\frac i4\left\{H(x,y)-\Hb(x,y)\right\} \\ &=\frac i4\sum_{k,j}h_{kj}\left\{\lambda^k(x)\lb^j(y)-\lb^j(x)\lambda^k(y)\right\} . \end{aligned} \] By \eqref{eq:2.12} it follows that \begin{equation} \label{eq:2.16} \Hh=\frac i2\sum_{k,j}h_{kj}\lambda^k\wedge\lb^j. \end{equation} If a real vector space has a complex structure and in addition to it an hermitian structure, the exterior algebra has rich properties. In particular, a complex-valued exterior form, i.e., an element of the exterior algebra \(\bigwedge(V^\ast\otimes\C)\), is said to be of type \((p,q)\), if it is a sum of terms each of which contains \(p\) factors \(\lambda^k\) and \(q\) factors \(\lb^h\). A form of degree \(r\) can be written uniquely as a sum \begin{equation} \alpha=\sum_{p+q=r}\alpha_{pq},\quad (p,q)\text{ mutually distinct}, \end{equation} \oldpage{12} where \(\alpha_{pq}\) is of type \((p,q)\). The latter will also be denoted by \begin{equation} \alpha_{pq}=\Pi_{pq}\alpha, \end{equation} whereby the operators \(\Pi_{pq}\) are defined. Another operator, which we will denote by \(L\), is defined by \begin{equation} L\alpha=\Hh\wedge\alpha. \end{equation} \(L\) is a real operator in the sense that it maps a real-valued form into a real-valued form. This operator \(L\) plays an important role in Hodge's work on transcendental methods in algebraic geometry. \section{Almost complex manifolds; integrability conditions} \label{sec:3} Let \(M\) be a \(C^\infty\) manifold of dimension \(n\). To a point \(x\in M\) we will denote by \(T_x\) and \(T_x^\ast\) the tangent and cotangent spaces respectively. An \textit{almost complex structure} on \(M\) is a \(C^\infty\) field of endomorphisms \(J_x:T_x\to T_x\), such that \(J_x^2=-1_x\), where \(1_x\) denotes the identity endomorphism in \(T_x\). A manifold which is given an almost complex structure is called \textit{almost complex}. Not all manifolds have this property. In fact, from the discussions in \secref{sec:2} follows the: \begin{thm} An almost complex manifold is even-dimensional and orientable. \end{thm} \begin{rmk} This condition is not sufficient for a manifold to have an almost complex structure. For instance, it was proved by Ehresmann and Hopf that the \(4\)-sphere \(S^4\) cannot be given an almost complex structure \cite[217]{steenrod51}. \end{rmk} Alternatively, an almost complex structure can be defined by the space \(A\) of its complex-valued \(C^\infty\) forms of type \((1,0)\). If \(\Ab\) denotes the space consisting of forms which are conjugate complex to those of \(A\), then at every \(x\in M\) we have the direct sum\oldpage{13} decomposition \begin{equation} T_x^\ast\otimes\C=A_x\oplus\Ab_x, \end{equation} where \(A_x\) (resp. \(\Ab_x\)) is the space of the forms of \(A\) (resp. \(\Ab\)) at \(x\). To establish the relation between the definitions let \(x^\alpha\), \(1\le\alpha,\beta\le n\), be a local coordinate system. Then a basis in the tangent space \(T_x\) is given by \(\frac{\partial}{\partial x^\alpha}\) and its dual basis \(T_x^\ast\) consists of the differential forms \(\rd x^\beta\). The endomorphism \(J_x\) will be defined by \begin{equation} J_x\left(\sum_\alpha\xi^\alpha\frac{\partial}{\partial x^\alpha}\right)=\sum_{\alpha,\beta}a_\beta^\alpha\xi^\beta\frac{\partial}{\partial x^\alpha}. \end{equation} The condition that \(J_x^2=-1_x\) is expressed by \begin{equation} \sum_\beta a_\beta^\alpha a_\gamma^\beta=-\delta_\gamma^\alpha,\quad 1\le\alpha,\beta,\gamma\le n. \end{equation} At each point \(x\in M\) the discussions of \secref{sec:2} and we see that the forms \begin{equation} \label{eq:3.4} \sum_\beta\left(a_\beta^\alpha+i\delta_\beta^\alpha\right)\rd x^\beta \end{equation} are of type \((1,0)\). They are \(n\) in number and exactly \(m=\frac n2\) of them are linearly independent over the ring of complex-valued \(C^\infty\) functions (cf. \eqref{eq:2.3}). (The situation being local, we restrict ourselves to a sufficiently small neighbourhood. As all our functions are \(C^\infty\) unless otherwise specified, we will later on frequently omit the adjective ``\(C^\infty\)''.) \begin{prop} A complex manifold has an almost complex structure. \end{prop} \oldpage{14}In fact, the complex-valued \(1\)-forms which, in terms of the local coordinates \(z^k\), \(1\le k\le m\), are linear combinations of \(\rd z^k\), are well-defined in a complex manifold \(M\). These we define to be the forms of type \((1,0)\). Since \[ \left(\frac i2\right)^m\wedge_k \rd z^k\wedge\rd\zb^k\ne 0, \] we have defined an almost complex structure on \(M\). To describe \(J\) in terms of the local coordinates \(z^k\) let \begin{equation} \label{eq:3.5} z^k=x^k+iy^k. \end{equation} Then we have, using the fact that \(\rd z^k\) is of type \((1,0)\), \[ \begin{aligned} (\rd z^k)\left(\frac{\partial}{\partial x^j}\right)&=\delta_j^k,\quad (\rd z^k)\left(\frac{\partial}{\partial y^j}\right)= i\delta_j^k, \\ (\rd z^k)\left(J\frac{\partial}{\partial x^j}\right)&= i\delta_j^k,\quad (\rd z^k)\left(J\frac{\partial}{\partial y^j}\right)= -\delta_j^k,\quad 1\le j, k\le m. \end{aligned} \] It follows that \begin{equation} J\left(\frac{\partial}{\partial x^j}\right)=\frac{\partial}{\partial y^j},\quad J\left(\frac{\partial}{\partial y^j}\right)= -\frac{\partial}{\partial x^j}. \end{equation} The question arises whether this is the only way to get an almost complex manifold, i.e., whether every almost complex manifold is complex. This is the case for \(n=2\), but not in general. The question is whether local coordinates \(x^k\), \(y^k\), \(1\le k\le m=\frac n2\), can be introduced such that, if \(z^k\) are defined by \eqref{eq:3.5}, the forms of type \((1,0)\) are linear combinations of \(\rd z^k\). Suppose the almost complex structure is locally defined by the forms \(\theta^k\) of type \((1,0)\) which are linearly independent (over the ring of complex-valued \(C^\infty\) functions). Their exterior derivatives can be written\oldpage{15} \begin{equation} \label{eq:3.7} \rd\theta^k=\frac12\sum_{j,\ell}A_{j\ell}^k\theta^j\wedge\theta^\ell+\sum_{j,\ell}B_{j\ell}^k\theta^j\wedge\tb^\ell+\frac12\sum_{j,\ell}C_{j\ell}^k\tb^j\wedge\tb^\ell \end{equation} where \(A_{j\ell}^k\), \(B_{j\ell}^k\), \(C_{j\ell}^k\) are complex-valued functions satisfying \begin{equation} A_{j\ell}^k+A_{\ell j}^k=C_{j\ell}^k+C_{\ell j}^k=0,\quad 1\le j,k,\ell\le m. \end{equation} The condition \begin{equation} \label{eq:3.9} \rd\theta^k\equiv 0,\quad\Mod\theta^j \end{equation} remains invariant under a linear transformation of the \(\theta^k\). It is satisfied if \(\theta^k=\rd z^k\). Thus it is a necessary condition for an almost complex structure to arise from a complex structure. We will call \eqref{eq:3.9} the \textit{integrability condition}. By \eqref{eq:3.7} it can also be written \begin{equation} \label{eq:3.9a} C_{j\ell}^k=0. \tag{3.9a} \end{equation} Before proceeding, we will express the integrability condition in terms of the tensor field \(a_\beta^\alpha\) which defines the endomorphism \(J_x\). Suppose that our Greek indices range from \(1\) to \(n\): \begin{equation} 1\le\alpha,\beta,\gamma,\lambda,\mu,\rho,\sigma\le n. \end{equation} Then we have: \begin{prop}[Eckmann--Frölicher] \label{thm:3C} Let \begin{equation} \begin{aligned} a_{\beta\gamma}^\alpha &= -a_{\gamma\beta}^\alpha=\frac{\partial a_\beta^\alpha}{\partial x^\gamma}-\frac{\partial a_\gamma^\alpha}{\partial x^\beta}, \\ t_{\beta\gamma}^\alpha &= \sum_\rho\left(a_{\beta\rho}^\alpha a_\gamma^\rho-a_{\gamma\rho}^\alpha a_\beta^\rho\right). \end{aligned} \end{equation} The integrability condition of the almost complex structure defined by the tensor field \(a_\beta^\alpha\) is \(t_{\beta\gamma}^\alpha=0\). \end{prop} Since the forms of type \((1,0)\) are linear combinations of\oldpage{16} those in \eqref{eq:3.4}, the integrability condition can be expressed by \[ \sum_\beta\rd a_\beta^\alpha\wedge dx^\beta \equiv 0,\quad \Mod\sum_\lambda\left(a_\lambda^\gamma+i\delta_\lambda^\gamma\right)\rd x^\lambda , \] or \[ \sum_{\beta,\gamma}a_{\beta\gamma}^\alpha\rd x^\beta\wedge\rd x^\gamma\equiv 0,\quad \Mod\sum_\lambda\left(a_\lambda^\gamma+i\delta_\lambda^\gamma\right)\rd x^\lambda. \] If we equate to zero the forms in \eqref{eq:3.4}, a fundamental system of solutions of the resulting linear equations in \(\rd x^\beta\) can be selected from \(a_\lambda^\gamma-i\delta^\lambda_\gamma\) (cf. \eqref{eq:2.1}). The condition above can therefore be written \[ \sum_{\beta,\gamma}a_{\beta\gamma}^\alpha\left(a_\lambda^\beta-i\delta_\lambda^\beta\right)\left(a_\mu^\gamma-i\delta_\mu^\gamma\right)=0. \] Equating to zero either the real or the imaginary part of this equation, we get \(t_{\beta\gamma}^\alpha=0\). \begin{rmk} It can be verified that \(t_{\beta\gamma}^\alpha\) are the components of a tensor field. \end{rmk} The integrability condition is identically satisfied when \(n=2\), as can be seen from \eqref{eq:3.9a}. For \(n\ge 4\) the condition is clearly nontrivial. An almost complex structure satisfying the integrability condition is called \textit{integrable}, otherwise \textit{nonintegrable}. An almost complex manifold of dimension \(\ge 4\) always has a nonintegrable almost complex structure, for even if the given one is integrable, it can be perturbed slightly to give a nonintegrable one. A significant example of an almost complex manifold is the \(6\)-sphere \(S^6\). From the theory of Lie groups it is known that \(S^6\) can be considered as a coset space \(G_2/\SU(3)\), where \(G_2\) is the exceptional simple Lie group of \(14\) dimensions and \(\SU(3)\) is the special unitary group in three variables. From the definition of \(G_2\) and its structure equations one sees immediately that \(S^6\) has a nonintegrable\oldpage{17} almost complex structure. Suppose that we have an integrable almost complex structure. The condition \eqref{eq:3.9} suggests us to apply the theorem of Frobenius on completely integrable differential systems. Since the forms are complex-valued, it will be necessary to suppose that the almost complex structure is real analytic, i.e., that the functions \(a_\beta^\alpha\) are real analytic. Under this hypothesis it follows from Frobenius's theorem that there exist complex local coordinates \(z^k\) such that the forms of type \((1,0)\) are linear combinations of \(\rd z^k\). In a neighbourhood where two such local coordinate systems \(z^k\) and \(w^j\) are both valid \(\rd w^j\) are linear combinations of \(\rd z^k\), which implies that \(w^j\) are holomorphic functions of \(z^k\). Thus the manifold has a complex structure. This theorem that a complex structure can be introduced in a manifold with an integrable almost complex structure is also true if the latter is \(C^\infty\) or satisfies even weaker smoothness conditions. This was first proved by A. Newlander and L. Nirenberg \cite{newlandernirenberg57}. Subsequent proofs were given by A. Nijenhuis and W. B. Woolf, J. Kohn and L. Hörmander. These proofs are rather difficult. The case \(n=2\) is a classical theorem of Korn and Lichtenstein which asserts that a two-dimensional riemannian metric of class \(C^{1,\alpha}\) (\(0<\alpha<1\)) is locally conformal to a flat metric. Even the proof of the Korn--Lichtenstein theorem is not simple \cite{chern55}. Thus we see that integrable almost complex structures and complex structures are essentially identical. In some of the problems it is not necessary to make use of the local complex coordinates \(z^k\), and the Newlander--Nirenberg theorem will not be needed. But we will not insist on this point. The integrability condition \eqref{eq:3.9} or \(t_{\beta\gamma}^\alpha=0\) (by \ref{thm:3C}) is a criterion for deciding whether a given almost complex structure is integrable. It gives no information on the problem whether an\oldpage{18} almost complex manifold can be given a complex structure, whose underlying almost complex structure may be different from the given one. Recently van de Ven gave examples of compact four-dimensional almost complex manifolds which do not have any complex structure; his proof makes use of the Atiyah--Singer index theorem \cite{vandeven66}. It is an outstanding problem whether \(S^6\) can have a complex structure. Let \(M\) be an almost complex manifold of dimension \(n=2m\). All complex-valued \(C^\infty\) forms of type \((p,q)\) constitute a module \(A_{pq}\) over the ring of complex-valued \(C^\infty\) functions. The following properties are easily verified: \begin{enumerate} \item If \(\alpha\in A_{pq}\), then \(\ab\in A_{qp}\). \item If \(\alpha\in A_{pq}\), \(\beta\in A_{rs}\), then \(\alpha\wedge\beta\in A_{p+r,q+s}\). \item \label{item:Apq3} \(\rd A_{pq}\subset A_{p+2,q-1}+A_{p+1,q}+A_{p,q+1}+A_{p-1,q+2}\). \item \(A_{pq}=0\) if \(p\) or \(q>m\). \end{enumerate} From \eqref{item:Apq3} we define, for \(\alpha\in A_{pq}\), the operators \begin{equation} \partial\alpha=\Pi_{p+1,q}\rd\alpha,\quad\db\alpha=\Pi_{p,q+1}\rd\alpha. \end{equation} If the almost complex structure is integrable, \eqref{item:Apq3} becomes \begin{equation} \rd A_{pq}\subset A_{p+1,q}+A_{p,q+1}, \tag{3I} \end{equation} as follows immediately from \eqref{eq:3.7}. We can then write \begin{equation} \rd=\partial+\db. \end{equation} Since \(\rd^2=0\), we get \[ \partial^2+\partial\db+\db \partial+\db ^2=0. \] Equating to zero the terms of different types, we find \begin{equation} \partial^2=\partial\db+\db\partial=\db^2=0. \end{equation} \oldpage{19}The last condition gives rise to the following form of the integrability condition: \begin{prop} An almost complex structure is integrable if and only if \[ \db^2=0. \] \end{prop} It remains to prove that the integrability condition is satisfied if \(\db ^2=0\). In fact, let \(F\) be a complex-valued \(C^\infty\) function. We write \[ \rd F=\sum_kF_k\theta^k+\sum_kG_k\tb^k. \] Then we have \[ \partial F=\sum_kF_k\theta^k,\quad\db F=\sum_kG_k\tb^k, \] and \[ \begin{aligned} \db^2F &= \Pi_{0,2} \rd\db F=\Pi_{0,2}\rd(\db-\rd)F=-\Pi_{0,2}\rd\partial F \\ &= -\sum_{j,k,\ell} F_kC_{j\ell}^k\tb^j\wedge\tb^\ell. \end{aligned} \] Since this expression is zero for any \(F\), we get \(C_{j\ell}^k=0\), which is the integrability condition \eqref{eq:3.9a}. From now on suppose \(M\) is a complex manifold. A form \(\alpha\in A_{pq}\) is called \(\db\)-closed if \(\db\alpha=0\). Let \(C_{pq}\) be the space of \(\db\)-closed forms of type \((p,q)\). The quotient groups \begin{equation} D_{pq}(M)=C_{pq}/\db A_{p,q-1} \end{equation} are called the \textit{Dolbeault groups} of \(M\). The Dolbeault groups are analogous to the de Rham groups of a real manifold, whose definitions we recall as follows: Let \(A_r\) be the space of real-valued \(C^\infty\) forms of degree \(r\), and \(C_r\) be the subspace of the forms of \(A_r\) which are annihilated by \(\rd\). Then the de Rham groups are\oldpage{20} \begin{equation} \label{eq:3.16} R_r(M)=C_r/\rd A_{r-1}. \end{equation} Both the de Rham groups and the Dolbeault groups are isomorphic to cohomology groups with coefficient sheaves, which will be treated in \secref{sec:4}. Before concluding this section, we will prove an important lemma: \begin{lem}[Dolbeault--Grothendieck] \label{lem:DolGro} In the number space \(\C_m\) with the coordinates \(z^k\), \(1\le k\le m\), let \(D\) be the polydisc \(\left|z^k\right|0,\quad\xi\ne 0. \end{equation} An \textit{hermitian structure} on the complex vector bundle \eqref{eq:5.2} is a \(C^\infty\) field of positive definite hermitian structures in the fibres of \(E\). That is, if \(\xi,\eta\) are two \(C^\infty\) sections of the bundle, \(H(\xi,\eta)\) is a complex-valued \(C^\infty\) function having properties corresponding to \eqref{eq:5.50}. A complex vector bundle with an hermitian structure is called an \textit{hermitian vector bundle}. By a partition of unity argument, it can be shown that every complex vector bundle can be given an hermitian structure. To every frame field \(s\) the hermitian structure defines an hermitian matrix\oldpage{44} \begin{equation} H_s={}^t\Hb_s=\left(H(s_i,s_j)\right),\quad 1\le i, j\le q, \end{equation} and is in turn completely determined by this matrix. Under a change of frame field \eqref{eq:5.20} this matrix is transformed according to \begin{equation} \label{eq:5.53} H_{s'}=gH_s{}^t\gb, \end{equation} where \[ H_{s'}=\left(H(s_i',s_j')\right),\quad 1\le i,j\le q. \] A connection in an hermitian vector bundle is called \textit{admissible}, if \(H(\xi,\eta)\) remains constant when \(\xi,\eta\) are horizontal sections along arbitrary curves. Let \begin{equation} h_{ik}=H(s_i,s_k),\quad 1\le i, j, k\le q \end{equation} and let \[ \xi=\sum_i\xi^is_i,\quad\eta=\sum_j\eta^js_j. \] Then \[ H(\xi,\eta)=\sum_{i,k} h_{ik}\xi^i\eb^k. \] The sections \(\xi,\eta\) being horizontal, we have \eqref{eq:5.19a} and a similar equation for \(\eta^k\). It follows that \[ \rd H(\xi,\eta)=\sum_{i,k}\left(\rd h_{ik}-\sum h_{jk}\omega_i^j-\sum h_{ij}\ob_k^j \right)\xi^i\eb^k. \] Since horizontal sections along curves exist with arbitrary initial values of \(\xi^i,\eta^k\), the condition for an admissible connection becomes \begin{equation} \rd h_{ik}-\sum_jh_{jk}\omega_i^j-\sum_jh_{ij}\ob_k^j=0, \end{equation} or, in matrix notation \begin{equation} \label{eq:5.55a} \rd H=\omega H+H{}^t\ob, \tag{5.55a} \end{equation} \oldpage{45}where the subscript \(s\) is dropped. By an elementary extension argument, it follows from \eqref{eq:5.55a} that an admissible connection always exists in an hermitian vector bundle. By taking the exterior derivative of \eqref{eq:5.55a}, we get \begin{equation} \label{eq:5.56} \Omega H+H{}^t\Ob=0, \end{equation} i.e., \(\Omega H\) is skew-hermitian. A frame field \(s\) of an hermitian vector bundle is called \textit{unitary}, if \(H_s=I\) (= the unit matrix). Relative to a unitary frame field, the equations \eqref{eq:5.55a} and \eqref{eq:5.56} become respectively \begin{align} \omega+{}^t\ob &= 0, \\ \Omega+{}^t\Ob &= 0, \end{align} i.e., the connection and curvature matrices \(\omega\) and \(\Omega\) are both skew-hermitian. It follows from \eqref{eq:5.56} that for an hermitian vector bundle with an admissible connection, we have \begin{equation} \label{eq:5.59} \det(I+\frac{i}{2\pi}\Omega)=\det(I-\frac{i}{2\pi} \Ob)=\overline{\det(I+\frac{i}{2\pi}\Omega)}. \end{equation} For the coefficients \(P_j(A)\) defined in \eqref{eq:5.35} and their polarised polynomials \(P_j(A_1,\ldots, A_j)\) we write \begin{equation} P_j(A_1,\ldots, A_j)=(\Re P_j)(A_1,\ldots, A_j)+i(\Im P_j)(A_1,\ldots, A_j), \end{equation} so that \(\Re P_j\) and \(\Im P_j\) are real-valued and \(\mathbb{R}\)-linear in each of their arguments. Let \[ \begin{aligned} (\Re P_j)(\Omega) &= (\Re P_j)(\Omega,\ldots,\Omega), \\ (\Im P_j)(\Omega) &= (\Im P_j)(\Omega,\ldots,\Omega). \end{aligned} \] Then it follows from \eqref{eq:5.59} that \[ (\Im P_j)(\Omega)=0 \] \oldpage{46}for the curvature matrix \(\Omega\) of an admissible connection of an hermitian structure. It follows from Theorem \ref{thm:5B} that for any connection the element of the de Rham group \(R_{2j}\) determined by \((\Im P_j)(\Omega)\) is zero. On the other hand, we will show later that the element determined by \((\Re P_j)(\Omega)\) is what is called the \(j\)th Chern class of the bundle \(E\) with real or integer coefficients. \section{Holomorphic vector bundles and line bundles} \label{sec:6} Let \(M\) be a complex manifold of dimension \(m\) and let \(\psi:E\to M\) be a complex vector bundle over \(M\) with fibre dimension \(q\). Relative to a covering \(\{U,V,\ldots\}\) of \(M\) let \(g_{UV}\) be the transition functions of \(E\). The bundle is called \textit{holomorphic} if all these functions \(g_{UV}\) are \textit{holomorphic} (i.e., \(g_{UV}\), considered as a nonsingular \((q\times q)\)-matrix, is a matrix of holomorphic functions in \(U\cap V\)). If \(q=1\), \(E\) is called a \textit{holomorphic line bundle}. An example of a holomorphic vector bundle over \(M\) is the tangent bundle of \(M\). Let \(z_U^1,\ldots,z_U^m\) (respectively \(z_V^1,\ldots,z_V^m\)) be the local coordinates in \(U\) (resp. in \(V\)). Then the tangent bundle has as transition functions the jacobian matrices \begin{equation} j_{UV}=\frac{\partial\left(z_U^1,\ldots,z_U^m\right)}{\partial\left(z_V^1,\ldots,z_V^m\right)}. \end{equation} Let \(E\) be a holomorphic bundle. A section \(\gamma\) of \(E\) over a neighbourhood \(U\subset M\) is \textit{holomorphic} if the components of \(\gamma\) relative to a chart are holomorphic functions. A frame field \(s={}^t(s_1,\ldots,s_q)\) is holomorphic if each \(s_i\) is a holomorphic section. When \(s\) and \(s'\) are holomorphic frame fields, the matrix \(g\) in the equation \eqref{eq:5.20} is a matrix of holomorphic functions. A connection such that the connection matrix is a matrix of \(1\)-forms of type \((1,0)\) relative to a holomorphic frame field will be called a \textit{connection of type \((1,0)\)}. \oldpage{47}Suppose an hermitian structure is defined in \(E\). From \eqref{eq:5.55a} it follows that it has a uniquely defined admissible connection of type \((1,0)\). In fact, its connection matrix is \begin{equation} \label{eq:6.2} \omega=\partial H\cdot H^{-1}. \end{equation} From \eqref{eq:6.2} we find that its curvature matrix is \begin{equation} \label{eq:6.3} \Omega=-\partial\db H\cdot H^{-1}+\partial H\cdot H^{-1}\wedge\db H\cdot H^{-1}, \end{equation} so that \(\Omega\) is of type \((1,1)\). (A matrix of differential forms is said to be of type \((p,q)\) if each element is a form of type \((p,q)\).) In case \(q=1\), the matrices in question are \((1\times 1)\)-matrices: \begin{equation} H=(h),\ \Omega=(\Omega),\quad h>0, \end{equation} and \eqref{eq:6.3} can be written \begin{equation} \label{eq:6.5} \Omega=-\partial\db\log h. \end{equation} Notice that for \(q=1\) a tensorial matrix of the adjoint type is a form in \(M\), so that \(\Omega\) is globally defined in \(M\). We call \(\frac{i}{2\pi}\Omega\) the \textit{curvature form} of the connection. On \(M\) let \(\cO\) be the sheaf of germs of holomorphic functions and \(\cO^\ast\) be the sheaf of germs of nowhere zero holomorphic functions, the latter with multiplication as the group operation. A meromorphic function is locally the ratio of two holomorphic functions. Let \(\mu\) be the sheaf of germs of meromorphic functions. Then \(\cO^\ast\) is a subsheaf of \(\mu\) and the quotient sheaf \(\cD\) is by definition a sheaf of germs of divisors. The latter is locally represented by a meromorphic function defined up to the multiplication by a nowhere zero holomorphic function. We have the exact sequence \begin{equation} 0\to\cO^\ast\xrightarrow{i}\mu\xrightarrow{k}\cD\to 0. \end{equation} Its induced exact cohomology sequence has the part\oldpage{48} \begin{equation} 0\to H^0(M,\cO^\ast)\xrightarrow{i^0}H^0(M,\mu)\xrightarrow{k^0}H^0(M,\cD)\xrightarrow{\delta^0}H^1(M,\cO^\ast)\to\cdots. \end{equation} From the exactness it follows that the quotient group \begin{equation} \label{eq:6.8} H^0(M,\cD)/k^0H^0(M,\mu) \end{equation} is isomorphic to a subgroup of \(H^1(M,\cO^\ast)\). An element of \(H^0(M,\cD)\) is called a \textit{divisor}. Two divisors are called \textit{linearly equivalent} if they differ from each other (multiplicatively) by a meromorphic function in \(M\). Thus the group \eqref{eq:6.8} is the group of divisor-classes with respect to linear equivalence. On the other hand, \(H^1(M,\cO^\ast)\) is the group of all holomorphic line bundles over \(M\), the group operation being defined by tensor product. Kodaira and Spencer proved that if \(M\) is a nonsingular projective variety the group \eqref{eq:6.8} is isomorphic to \(H^1(M,\cO^\ast)\), \cite{kodairaspencer53}. We wish to study the group \(H^1(M,\cO^\ast)\). For this purpose we consider the exact sequence of sheaves: \begin{equation} \label{eq:6.9} 0\to\Z\xrightarrow{i}\cO\xrightarrow{e}\cO^\ast\to 0, \end{equation} where \(e\) is defined by \begin{equation} e(f(x))=2\pi i\exp(f(x)),\quad f(x)\in\cO. \end{equation} The sequence \eqref{eq:6.9} leads to the homomorphism \begin{equation} \label{eq:6.11} \delta^1:H^1(M,\cO^\ast)\to H^2 (M,\Z), \end{equation} and we wish to describe the image of \(\delta^1\). Let \(\cA_\R^r\) be the sheaf of germs of real-valued \(C^\infty\) \(r\)-forms in \(M\) and \(\cC_\R^r\) be the subsheaf of \(\cA_\R^r\) consisting of those germs which are closed under \(\rd\), so that \(\cC_\R^0\) is the constant sheaf \(\R\). Then we have the exact sequence \begin{equation} 0\to\cC_\R^r\xrightarrow{\ell^r}\cA_\R^r\xrightarrow{\rd}\cC_\R^{r+1}\to 0, \end{equation} \oldpage{49}whose induced exact cohomology sequence is \begin{equation} \label{eq:6.13} \vcenter{\vbox{\xymatrix@R=.1pc{ \cdots\ar[r] & H^p(M,\cC_\R^r)\ar[r]^-{\ell^{p,r}} & H^p(M,\cA_\R^r)\ar[r]^-\rd & H^p(M,\cC_\R^{r+1}) \\ & \phantom{H^p(M,\cC_\R^r)}\ar[r]^-{\partial^{p,r}} & H_\R^{p+1}(M,\cC_\R^r) & \cdots . }}} \end{equation} We note that \(\cA_\R^r\) is a fine sheaf. We now write the following diagram of cohomology groups of \(M\) connected by homomorphisms: \begin{equation} \vcenter{\vbox{\xymatrix@R=1.8pc{ & & H^1(\cA_\R^0)=0 \ar[d]^-\rd \\ H^0(\cA_\R^1)\ar[r]^-\rd & H^0(\cC_\R^2)\ar[r]^-{\partial^{0,1}} & H^1(\cC_\R^1)\ar[r]^-{\ell^{1,1}} \ar[d]^-{\partial^{1,0}} & H^1(\cA_\R^1)=0 \\ H^1(\cO^\ast)\ar[r]^-{\delta^1} & H^2(\Z)\ar[r]^-j & H^2(\R)\ar[d]^{\ell^{2,0}} \\ & & H^2(\cA_\R^0)=0 }}} \end{equation} In this diagram the manifold \(M\) is omitted in the notation of the cohomology groups for the sake of simplicity. The vertical sequence and the top horizontal sequence are exact, being parts of the sequence \eqref{eq:6.13}. The homomorphism \(j\) in the second horizontal sequence is induced by the coefficient homomorphism \(j:\Z\to\R\). For an hermitian line bundle \(E\in H^1(M,\cO^\ast)\) we wish to determine \begin{equation} \label{eq:6.15} \left(\partial^{0,1}\right)^{-1}\circ\left(\partial^{1,0}\right)^{-1}\circ j\circ\delta^1E, \end{equation} which is a real-valued closed \(2\)-form in \(M\). In fact, we wish to show that the negative of the form \eqref{eq:6.15} is the curvature form \(\frac{i}{2\pi}\Omega\), up to an additive term \(\rd\alpha\), where \(\alpha\) is a real-valued \(1\)-form in \(M\). This ``diagram chasing'' is not difficult, the main point being to remember the definitions of the homomorphisms in question. Before proceeding we will give some more discussion of the hermitian structure\oldpage{50} of a holomorphic line bundle \(E\) and its curvature form. In fact, let \(\cU=\{U,V,W,\ldots\}\) be an open covering of \(M\) which is sufficiently fine. Let \(s_U\) be a holomorphic frame field over \(U\). The fibre dimension being one, \(s_U\) is given by a nowhere zero holomorphic function in \(U\). Let \begin{equation} \label{eq:6.16} h_U=H(s_U,s_U)>0. \end{equation} The change of frame field in \(U\cap V\) is given by \begin{equation} \label{eq:6.17} s_Ug_{UV}=s_V, \end{equation} where \(g_{UV}\) is a nowhere zero holomorphic function in \(U\cap V\). From \eqref{eq:6.16} and \eqref{eq:6.17} we derive \begin{equation} \label{eq:6.18} h_U\left|g_{UV}\right|^2=h_V\quad \text{in }U\cap V. \end{equation} It follows that \begin{equation} \partial\db\log h_U=\partial\db\log h_V\quad \text{in }U\cap V, \end{equation} which gives a verification of the remark following formula \eqref{eq:6.5}. Suppose \(M\) is equipped with a riemannian metric and that the members of the covering \(\cU\) are convex. Then the intersection of any number of the members of the covering, if nonempty, is convex. In \(U\cap V\) (\(\ne\varnothing\)) we construct the holomorphic function \(f_{UV}\) satisfying \begin{equation} \label{eq:6.20} g_{UV}=\exp(2\pi if_{UV}). \end{equation} In \(U\cap V\cap W\ne\varnothing\) let \begin{equation} c_{UVW}=f_{UV}+f_{VW}+f_{WU}. \end{equation} Then \[ \exp(2\pi i c_{UVW})=g_{UV}g_{VW}g_{WU}=1, \] so that \(c_{UVW}\) is an integer. The two-cochain of the nerve \(N(\cU)\) of the covering \(\cU\) defined by assigning to the simplex \(UVW\) the integer\oldpage{51} \(c_{UVW}\) is a two-cocycle and defines a representative of \(\delta^1E\) and hence of \(j\circ\delta^1E\). Next we wish to find a representative of the element of \(H^1(\cC_\R^1)\) which is mapped by \(\partial^{1,0}\) to \(j\circ\delta^1E\). This will be given by a real-valued closed \(1\)-form in every \(U\cap V\ne\varnothing\), and we see that the form \(\frac12\rd\left(f_{UV}+\fb_{UV}\right)\) has the desired property. In fact, by \eqref{eq:6.20} and \eqref{eq:6.18} we have \[ \begin{aligned} \frac12\rd\left(f_{UV}+\fb_{UV}\right) &= \frac{1}{4\pi i} \left\{\partial\log g_{UV}-\db\log\gb_{UV}\right\} \\ &= \frac{1}{4\pi i}(\partial-\db)\log\left|g_{UV}\right|^2 \\ &= \frac{1}{4\pi i}(\partial-\db)(-\log h_U+\log h_V). \end{aligned} \] By the definition of \(\partial^{0,1}\) we get a representative of \eqref{eq:6.15} as \[ \frac{1}{4\pi i}\rd(\partial-\db) \log h_U=\frac{i}{2\pi} \partial\db \log h_U=-\frac{i}{2\pi}\Omega. \] Thus we have the: \begin{thm} Let \(E\in H^1(M,\cO^\ast)\) be a holomorphic line bundle over a complex manifold \(M\). Let \(c(E)=-\delta^1E\in H^2(M,\Z)\) be its Chern class. Suppose that \(E\) has an hermitian structure with the curvature form \(\frac{i}{2\pi}\Omega\). Then the element of the de Rham group \(R_2(M)\) defined by \(\frac{i}{2\pi}\Omega\) corresponds to the element \(jc(E)\in H^2(M,\R)\) via the de Rham isomorphism \eqref{eq:4.15}, \(j\) being induced by the coefficient homomorphism \(j:\Z\to\R\). \end{thm} Consider the de Rham isomorphism \begin{equation} R_2(M) \xrightarrow{\rho} H^2(M,\R). \tag{4.15} \end{equation} There is a subgroup \(R_{11}(M)\) of \(R_2(M)\), whose elements have as representatives forms of type \((1,1)\). (Recall that an element \(\gamma\) of \(R_2(M)\) is a class of forms \(\alpha+\rd\beta\), where \(\alpha\) is a given real-valued closed \(2\)-form in \(M\) and \(\beta\) runs over all real-valued \(1\)-forms\oldpage{52} in \(M\). Any such form \(\alpha+\rd\beta\) is called a representative of \(\gamma\).) Let \begin{equation} \begin{aligned} \rho R_{11}(M) &= H_{(1,1)}^2(M,\R), \\ H_{(1,1)}^2(M,\Z) &= j^{-1}H_{(1,1)}^2(M,\R). \end{aligned} \end{equation} Then we have the: \begin{thm} \label{thm:6B} The image of the homomorphism \(\delta^1\) in \eqref{eq:6.11} is \(H_{(1,1)}^2(M,\Z)\). \end{thm} Since the curvature form \(\frac{i}{2\pi}\Omega\) is of type \((1,1)\), we have proved \[ \delta^1H^1(M,\cO^\ast)\subset H_{(1,1)}^2(M,\Z). \] To prove inclusion in the other direction, consider the following exact sequence induced by \eqref{eq:6.9}: \begin{equation} \label{eq:6.23} \vcenter{\vbox{\xymatrix@R=.1pc{ \cdots\ar[r] & H^1(M,\Z) \ar[r]^-{i^1} & H^1(M,\cO)\ar[r]^{e^1} & H^1(M,\cO^\ast)\\ & \phantom{H^1(M,\Z)}\ar[r]^-{\delta^1} & H^2(M,\Z)\ar[r]^-{i^2} & H^2(M,\cO)\ar[r] & \cdots }}} \end{equation} It suffices to prove that \(i^2H_{(1,1)}^2(M,\Z)=0\). As previously let \(\cA^r\) be the sheaf of germs of complex-valued \(C^\infty\) \(r\)-forms and \(\cC^r\) be the subsheaf of the germs of \(\cA^r\) closed under \(\rd\). Also let \(\cA^{pq}\) be the sheaf of germs of \(C^\infty\) forms of type \((p,q)\) and \(\cC^{pq}\) be the subsheaf of germs of \(\cA^{pq}\) closed under \(\db\). Thus by definition \(\cC^0=\C\) and \(\cC^{00}=\cO\). We have the diagram\oldpage{53} \begin{equation} \vcenter{\vbox{\xymatrix{ 0\ar[r] & \cC^r\ar[r]^-k\ar[d]^-{\Pi_{0r}} & \cA^r\ar[r]^-\rd\ar[d]^-{\Pi_{0r}} & \cC^{r+1}\ar[r]\ar[d]^-{\Pi_{0,r+1}} & 0 \\ 0\ar[r] & \cC^{0r}\ar[r]^-{k'} & \cA^{0r}\ar[r]^-\db & \cC^{0,r+1}\ar[r] & 0 }}} \end{equation} where \(k\) and \(k'\) are inclusions. This diagram is clearly commutative. Moreover both horizontal sequences are exact. The above diagram implies the following commutative diagram of cohomology groups, where the manifold \(M\) is omitted in the notation: \begin{equation} \label{eq:6.25} \vcenter{\vbox{\xymatrix{ H^0(\cC^2)/\rd H^0(\cA^1)\ar[d]^-{\Pi_{02}}\ar[r]^-{\Delta^0} & H^1(\cC^1)\ar[r]^-{\Delta^1}\ar[d]^-{\Pi_{01}} & H^2(\cC)\ar[d]^-{\Pi_{00}} \\ H^0(\cC^{02})/\db H^0(\cA^{01})\ar[r]^-{\Dt^0} & H^1(\cC^{01})\ar[r]^-{\Dt^1} & H^2(\cO) }}} \end{equation} Moreover, \(\Delta^0\), \(\Delta^1\), \(\Dt^0\), \(\Dt^1\) are isomorphisms (cf. \secref{sec:4}, in particular \eqref{eq:4.11}). We decompose the inclusion \(i\) in \eqref{eq:6.9} by \begin{equation} \Z\xrightarrow{h}\C\xrightarrow{\Pi_{00}}\cO, \end{equation} so that \(i=\Pi_{00}\circ h\). For any \(\beta\in H^2(\Z)\) we have then \[ i^2\beta=\Pi_{00}h\beta=\Dt^1\Dt^0\Pi_{02}(\Delta^0)^{-1}(\Delta^1)^{-1}h\beta, \] by the commutativity of the diagram \eqref{eq:6.25}. If \(\beta\in H^2_{(1,1)}(M,\Z)\) we have \[ \Pi_{02}(\Delta^0)^{-1}(\Delta^1)^{-1}h\beta=0, \] so that \(i^2\beta=0\). This completes the proof of \ref{thm:6B}. To study \(H^1(M,\cO^\ast)\) the next step is to consider the subgroup of all \(E\in H^1(M,\cO^\ast)\) such that \(c(E)=0\). By the exactness of the sequence \eqref{eq:6.23} this is isomorphic to \oldpage{54} \begin{equation} \label{eq:6.27} H^1(M,\cO)/i^1H^1(M,\Z). \end{equation} For a nonsingular projective variety \(M\) the group \eqref{eq:6.27} is compact and is called the \textit{Picard variety} of \(M\), \cite{kodairaspencer53}. The following are some important examples of holomorphic line bundles. \subsection{Example} The determinant bundle \(\bigwedge^q(E)\) of a holomorphic vector bundle \(E\) of fibre dimension \(q\). If \(g_{UV}\) are the transition functions of \(E\), so that \(g_{UV}\) are nonsingular \((q\times q)\)-matrices with elements which are holomorphic functions in \(U\cap V\), the bundle \(\bigwedge^q(E)\) is defined by the transition functions \(\det g_{UV}\). If \(T^\ast\) is the cotangent bundle of \(M\) and \(\dim M=m\), then \(\bigwedge^m(T^\ast)\) is called the \textit{canonical bundle} of \(M\); it will be denoted as \(K(M)\). If \(z_U^1,\ldots,z_U^m\) and \(z_V^1,\ldots,z_V^m\) are the local coordinates in \(U\) and \(V\) respectively, the transition functions of \(K(M)\) are the jacobian determinants \begin{equation} k_{UV}=\frac{\partial\left(z_U^1,\ldots,z_U^m\right)}{\partial\left(z_V^1,\ldots,z_V^m\right)} \end{equation} \subsection{Example} \label{eg:6.2} Consider the line bundle in Example \ref{eg:1.2}. We will call it the \textit{universal line bundle} over \(\bP_m\). Here the base space \(\bP_m\) has the covering \(\{U_i\}\), and the bundle has the transition functions \(g_{ij}={}_j\xi^i=\frac{z^i}{z^j}\), \(0\le i,j\le m\), \(i\ne j\). The linear form \(\sum a_iz^i\) in \(\C_{m+1}-0\), where the \(a\)'s are constants, has in the local coordinates in \(\psi^{-1}(U_i)\) the expression \[ \sum_ja_jz^j=z^i\left(a_{0_i}\xi_0+\cdots+\underset{i\text{th}}{1}+\cdots+a_{m_i}\xi_m\right). \] The expression in parentheses, which is essentially the linear form at the left-hand side in ``nonhomogeneous'' coordinates in \(U_i\), defines a section in the line bundle whose transition functions are\oldpage{55} \(g_{ij}'=\frac{z^j}{z^i}=\left({}_j\zeta^i\right)^{-1}\). Because of this origin the latter bundle, to be denoted by \(H\), is called the \textit{hyperplane section bundle} of \(\bP_m\); it is the negative or dual of the universal line bundle. Moreover, a holomorphically immersed submanifold \(f:M\to\bP_m\) has an induced bundle \(f^\ast H\), called the \textit{hyperplane section bundle} of \(M\). \section{Hermitian geometry and k\"ahlerian geometry} Let \(M\) be a complex manifold of dimension \(m\). \(M\) is called \textit{hermitian} if an hermitian structure \(H\) is given in its tangent bundle \(T(M)\). With the local coordinates \(z^1,\ldots,z^m\) a natural frame field is given by \begin{equation} \label{eq:7.1} s_i=\frac{\partial}{\partial z^i},\quad 1\le i,j,k,\ell\le m, \end{equation} and this frame is holomorphic. Let \begin{equation} h_{ik}=H\left( \frac{\partial}{\partial z^i}, \frac{\partial}{\partial z^k} \right)=\hb_{ki}. \end{equation} Then the matrix \begin{equation} H={}^t\Hb=(h_{ik}) \end{equation} is positive definite hermitian. There are special features arising from the fact that the bundle in question is the tangent bundle. First there is the K\"ahler form (cf. \eqref{eq:2.16}) \begin{equation} \label{eq:7.4} \Hh=\frac i2\sum_{j,k}h_{jk}\rd z^j\wedge\rd\zb^k, \end{equation} which is a real-valued form of type \((1,1)\). An hermitian manifold is called \textit{k\"ahlerian} if its K\"ahler form is closed: \begin{equation} \label{eq:7.5} \rd\Hh=0. \end{equation} \oldpage{56}Secondly, let \(s\) be a local frame field, holomorphic or not. To \(s\) there is uniquely determined a coframe field \(\sigma=(\sigma^1,\ldots,\sigma^m)\) such that at every point \(x\in M\) the sections \(s_1(x),\ldots,s_m(x)\) of \(s\) and the sections \(\sigma^1(x),\ldots,\sigma^m(x)\) of \(\sigma\) in the cotangent bundle are dual bases. The sections \(\sigma^i\), being in the cotangent bundle, are complex-valued \(1\)-forms, and they are everywhere linearly independent. Let \(s'\) be a new frame field, related to \(s\) by \eqref{eq:5.20}, and let \(\sigma'\) be its dual coframe field. Write \begin{equation} \begin{aligned} ^ts &= (s_1,\ldots,s_m),\quad s'=(s_1',\ldots,s_m'), \\ \sigma &= (\sigma'^1,\ldots,\sigma'^m). \end{aligned} \end{equation} If \(T_x\) and \(T^\ast_x\) are respectively the tangent and cotangent spaces at \(x\), we denote their pairing by \begin{equation} \lrag{\xi,\omega},\quad\xi\in T_x,\ \omega\in T^\ast_x. \end{equation} Then we have \begin{equation} \label{eq:7.8} \lrag{s_i,\sigma^k}=\lrag{s_i',\sigma'^k}=\delta_i^k. \end{equation} Equation \eqref{eq:5.20} can be written \begin{equation} s_i'=\sum_jg_i^js_j,\quad g=(g_i^j). \end{equation} In view of \eqref{eq:7.8} we have \begin{equation} \sigma^i=\sum_jg_j^i\sigma'^j \end{equation} or, in matrix notation, \begin{equation} \label{eq:7.10a} \sigma=\sigma'g. \tag{7.10a} \end{equation} By taking the exterior derivative of \eqref{eq:7.10a} and using \eqref{eq:5.22}, we get\oldpage{57} \begin{equation} \label{eq:7.11} (\rd\sigma'-\sigma'\wedge\omega')g=\rd\sigma-\sigma\wedge\omega. \end{equation} We will call the \((1\times q)\)-matrix \begin{equation} \label{eq:7.12} \tau=\rd\sigma-\sigma\wedge\omega \end{equation} the \textit{torsion matrix}. It is a matrix of complex-valued two-forms and follows the transformation law \eqref{eq:7.11} under a change of the frame field, holomorphic or not. \begin{prop} \label{prop:7A} Let \(M\) be an hermitian manifold. A connection in its tangent bundle is of type \((1,0)\) if and only if its torsion matrix is of type \((2,0)\). \end{prop} To prove this let \(\sigma\) be the dual coframe field of a holomorphic frame field \(s\). The connection matrix \(\omega\) relative to \(s\) can be written in a unique way as \[ \omega=\omega_1+\omega_2, \] where \(\omega_1\) and \(\omega_2\) are matrices of \(1\)-forms of types \((1,0)\) and \((0,1)\) respectively. \(\sigma\) is a matrix of forms of type \((1,0)\) with holomorphic coefficients. The torsion matrix \(\tau\) in \eqref{eq:7.12} is of type \((2,0)\) if and only if \[ \sigma\wedge\omega_2=0. \] Let \[ \sigma=(\sigma^1,\ldots,\sigma^m),\ \omega_2=(\theta_i^k),\quad 1\le i,j,k\le m. \] Then the above relation can be written explicitly as \[ \sum_i\sigma^i\wedge\theta_i^k=0. \] Since \(\sigma^i\) are linearly independent, we have \[ \theta_i^k=\sum_ja_{ij}^k\sigma^j, \] \oldpage{58}where \(a_{ij}^k\) are functions. But \(\sigma^j\) is of type \((1,0)\), while \(\theta_i^k\) is of type \((0,1)\) by our hypothesis. It follows that the above relation is equivalent to \[ \theta_i^k=0\quad\text{or}\quad\omega_2=0. \] This proves \ref{prop:7A}. The criterion expressed by \ref{prop:7A} has the advantage that, unlike the notion of a connection of type \((1,0)\) which is defined in terms of holomorphic frame fields, it has a meaning for \(C^\infty\) frame fields. In the study of hermitian manifolds it is desirable to use \(C^\infty\) frame fields, for example, the unitary frame fields. It follows from \secref{sec:6} and \ref{prop:7A} that an hermitian manifold has a uniquely determined admissible connection in its tangent bundle whose torsion matrix is of type \((2,0)\). When we speak of the connection in an hermitian manifold, this will be the connection meant. It is to be noticed that the curvature matrix of this connection is of type \((1,1)\) (relative to \(C^\infty\) frame fields). \begin{prop} An hermitian manifold is k\"ahlerian if and only if the torsion matrix of its connection is zero. \end{prop} Since both properties are independent of the choice of a frame field, it suffices to verify this theorem by using the natural frame field \eqref{eq:7.1}. Its dual coframe field is \[ \sigma=(dz^1,\ldots, dz^m), \] so that \(\rd\sigma=0\). By \eqref{eq:6.2} the vanishing of the torsion matrix can be written \[ \sigma\wedge\partial H=0, \] or, in expanded form, \[ \sum_{i,j}\frac{\partial h_{ik}}{\partial z^j}\rd z^j\wedge\rd z^i=0,\quad 1\le i,j,k\le m. \] The latter is equivalent to the conditions\oldpage{59} \begin{equation} \label{eq:7.13} \frac{\partial h_{ik}}{\partial z^j}-\frac{\partial h_{jk}}{\partial z^i}=0. \end{equation} One sees directly that \eqref{eq:7.5} and \eqref{eq:7.13} are equivalent. \begin{prop} An hermitian manifold is k\"ahlerian if and only if there exists locally a real-valued \(C^\infty\) function \(u\), such that its K\"ahler form can be written \begin{equation} \label{eq:7.14} \Hh=i\partial\db u. \end{equation} \end{prop} It suffices to prove that a k\"ahlerian manifold has the property stated in the theorem, for the form \eqref{eq:7.14} is clearly closed. Suppose therefore that \(\Hh\) is closed. There exists locally a real-valued \(1\)-form \(\omega\) such that \[ \Hh=\rd\omega. \] We can write \[ \omega=\alpha+\ab, \] where \[\ alpha=\Pi_{1,0}\omega,\quad \ab=\Pi_{0,1}\omega. \] Then \[ \rd\omega=\partial\alpha+(\db\alpha+\partial\ab)+\db\ab, \] where the terms are of types \((2,0)\), \((1,1)\), \((0,2)\) respectively. Since \(\rd\omega\) is of type \((1,1)\), we have \[ \db\ab=0. \] It follows from the \hyperref[lem:DolGro]{Dolbeault--Grothendieck lemma} that there exists a complex-valued \(C^\infty\)-function \(F\) such that \[ \ab=\db F. \] \oldpage{60}Then \[ \Hh=\rd\omega=\partial\db(F-\Fb). \] The theorem follows by setting \(u=-i(F-\Fb)\). The most important local properties of an hermitian manifold arise from its curvature matrix. The latter is defined in terms of a frame field. To have the situation under control we list together the formulas giving the effect from a change of the frame field on the various matrices we have introduced (formulas \eqref{eq:5.20}, \eqref{eq:5.23}, \eqref{eq:5.53}, \eqref{eq:7.10a}): \begin{equation} \label{eq:7.15} \begin{aligned} s' &= gs, \\ \Omega'g &= g\Omega, \\ H' &= gH{}^t\gb, \\ \sigma &=\sigma'g. \end{aligned} \end{equation} From the second and third formulas of \eqref{eq:7.15} we get \begin{equation} \label{eq:7.16} \Omega'H'=g\Omega H{}^t\gb. \end{equation} We note that \(\Omega H\) is skew-hermitian (cf. \eqref{eq:5.56}). Since \(\Omega H\) is of type \((1,1)\), we set \begin{equation} \begin{aligned} \Omega H &= (\Omega_{ik}), \\ \Omega_{ik} &= \sum_{j,\ell} R_{ikj\ell}\sigma^j\wedge\sib^\ell. \end{aligned} \end{equation} The skew-hermitian property of \(\Omega H\) is then expressed by \begin{equation} \label{eq:7.18} R_{ikj\ell}=\Rb_{ki\ell j}. \end{equation} Throughout this part of our discussion we suppose as usual that our small Latin indices have the range from 1 to \(m\): \begin{equation} 1\le i,j,k,\ell,p,q,u,v\le m. \end{equation} The fourth equation of \eqref{eq:7.15} and the equation \eqref{eq:7.16} can be written out in detail as follows:\oldpage{61} \[ \begin{aligned} \sigma^i &= \sum_jg_j^i\sigma'^j, \\ \sum_{j,\ell}R'_{ikj\ell}\sigma'^j\wedge\sib'^\ell &= \sum_{p,q,u,v}g_i^p\gb_k^qR_{pquv}\sigma^u\wedge\sib^v, \end{aligned} \] where the left-hand sides of the second equation are the entries in the matrix \(\Omega'H'\). It follows that \begin{equation} \label{eq:7.20} R'_{ikj\ell}=\sum_{p,q,u,v}g_i^p\gb_k^qg_j^u\gb_\ell^vR_{pquv}. \end{equation} Let \begin{equation} \label{eq:7.21} \xi=\sum_i\xi^is_i=\sum_j\xi'^js'_j \end{equation} be a vector at \(x\in M\). The \(\xi^i\) and \(\xi'^j\) in \eqref{eq:7.21} are the components of the vector relative to the frames \(s\) and \(s'\) respectively. Between them we have the relation \begin{equation} \label{eq:7.22} \xi^i=\sum_jg_j^i\xi'^j. \end{equation} From \eqref{eq:7.20} and \eqref{eq:7.22} we get \[ \sum_{i,\ldots,\ell}R'_{ikj\ell}\xi'^i\xb'^k\xi'^j\xb'^\ell=\sum_{i,\ldots,\ell}R_{ikj\ell}\xi^i\xb^k\xi^j\xb^\ell, \] so that the common expression is independent of the choice of the frame field. If \(\xi\ne 0\), we define \begin{equation} R(x,\xi)=2\frac{\sum_{i,\ldots,\ell}R_{ikj\ell}\xi^i\xb^k\xi^j\xb^\ell}{\left(\sum_{i,k}h_{ik}\xi^i\xb^k\right)^2} \end{equation} to be the \textit{holomorphic sectional curvature} at \((x,\xi)\). From the second equation of \eqref{eq:7.15} we get \begin{equation} \Tr\Omega'=\Tr\Omega=\phi\quad \text{(say).} \end{equation} \oldpage{62}\(\phi\) is a form of type \((1,1)\) and is called the \textit{Ricci form} of the hermitian metric. The metric is called \textit{hermitian-einsteinian} if the Ricci form is a multiple of the K\"ahler form. Let \(h^{ik}\) be the elements of the matrix \(H^{-1}\). By the symmetry relations \eqref{eq:7.18} we see that \begin{equation} R=\sum_{i,\ldots,\ell}R_{ikj\ell}h^{ik}h^{j\ell} \end{equation} is real; it is also independent of the choice of the frame field. This quantity \(R\) is called the \textit{scalar curvature}. Compact k\"ahlerian manifolds have strong topological restrictions. Perhaps the simplest among them is the following: \begin{prop} \label{prop:7D} The second Betti number of a compact k\"ahlerian manifold is positive. \end{prop} \begin{coro} The Hopf and Calabi--Eckmann manifolds \(S^{2p+1}\times S^{2q+1}\), \(p\ge 0\), \(q\ge 1\), cannot be given a k\"ahlerian structure. (Cf. \secref{sec:1}.) \end{coro} Since \(\Hh\) is closed, it determines by de Rham's theorem an element \(u\in H^2(M,\R)\). To prove \ref{prop:7D} we make use of the fact that the \(2m\)-form \[ \Hh^m=\Hh\wedge\cdots\wedge\Hh,\quad m\text{ times} \] determines the element \(u^m=u\cup\cdots\cup u\) (cup product \(m\) times) of \(H^{2m}(M,\R)\). Using the local expression \eqref{eq:7.4}, we find \begin{equation} \Hh^m=\left(\frac i2\right)^m m! (\det H) \bigwedge_j\rd z^j\wedge\rd\zb^j. \end{equation} Since the matrix \(H\) is positive definite, \(\det H>0\). It follows that \[ \int_M \Hh^m>0, \] \oldpage{63}and \(u^m\ne 0\). Therefore \(u\ne 0\). Let \(M,N\) be complex manifolds, of dimensions \(m,n\) respectively. A continuous mapping \(f:M\to N\) is called \textit{holomorphic}, if locally it is defined by expressing the coordinates of the image point as holomorphic functions of those of the original point. \(f\) is called an \textit{immersion}, if \(m\le n\) and if the jacobian matrix is of rank \(m\) everywhere. An immersion \(f\) is called an \textit{embedding}, if it is one-one, i.e., if \(f(x)=f(y)\), \(x,y\in M\), implies \(x=y\). The following is immediate: \begin{thm} \label{thm:7E} Let \(N\) be a k\"ahlerian manifold and let \(f:M\to N\) be a holomorphic immersion. Then \(M\) has a k\"ahlerian structure. \end{thm} Consider the complex projection space \(\bP_n\) of dimension \(n\). It is known that (cf. \secref{sec:8}) \begin{equation} \begin{aligned} H^{2i}(\bP_n,\Z) &\cong\Z,\quad 0\le i\le n, \\ H^k(\bP_n,\Z) &= 0,\quad k\text{ odd}. \end{aligned} \end{equation} Moreover we will show in \secref{sec:8} that \(\bP_n\) is k\"ahlerian. In this case, however, there is an additional important fact: The cohomology group \(H^2(\bP_n,\R)\) is a real vector space of real dimension \(1\) and is isomorphic to \(jH^2(\bP_n,\Z)\otimes\R\), where \(j\) is induced by the coefficient homomorphism \(j:\Z\to\R\). In other words, if \(\zeta\) denotes a generator of \(H^2(\bP_n,\Z)\), \(j\zeta\) generates \(H^2(\bP_n,\R)\) over \(\R\). By the multiplication of a constant factor when necessary, we can define on \(\bP_n\) a k\"ahlerian metric such that the cohomology class \(u\in H^2(\bP_n,\R)\) determined by the K\"ahler form belongs to \(jH^2(\bP_n,\Z)\). A k\"ahlerian manifold with this property is said to be \textit{of restricted type}. Under the conditions of Theorem \ref{thm:7E} we have the commutative diagram\oldpage{64} \[ \xymatrix@R=1.7pc{ H^2(N,\Z)\ar[r]^-j \ar[d]^-{f^\ast} & H^2(N,\R)\ar[d]^-{f^\ast} \\ H^2(M,\Z)\ar[r]^-j & H^2(M,\R) } \] where \(f^\ast\) is induced by the mapping \(f\) and \(j\) is induced by the coefficient homomorphism. Theorem \ref{thm:7E} has the following complement: \addtocounter{defn}{-1}\renewcommand{\thedefn}{\thesection(\Alph{defn}')} \begin{thm} Let \(N\) be a k\"ahlerian manifold of restricted type and let \(f:M\to N\) be a holomorphic immersion. Then \(M\) is a k\"ahlerian manifold of restricted type. \end{thm} \renewcommand{\thedefn}{\thesection(\Alph{defn})} A theorem of Chow says that a compact complex manifold holomorphically embedded in \(\bP_n\) is an algebraic variety, i.e., its locus is defined by a finite number of polynomial equations. The embedding theorem of Kodaira says that a compact k\"ahlerian manifold of restricted type can be holomorphically embedded in a projective space, \cite{kodaira54}. As an example we will study the conditions that the complex torus \(\Theta=\C_m/\Gamma\) (Ex. \ref{eg:1.4}) can be given a k\"ahlerian structure of restricted type. Suppose that such a k\"ahlerian structure exists on \(\Theta\). The latter being a compact connected Lie group, we can integrate the k\"ahlerian metric over \(\Theta\). The resulting metric will define a k\"ahlerian structure of restricted type which is invariant under the action of \(\Theta\). Let \(z^1,\ldots,z^m\) be the coordinates in \(\C_m\), and let \(\Gamma\) be generated by the vectors \begin{equation} \label{eq:7.28} \pi_\lambda=\left(\pi_\lambda^1,\ldots,\pi_\lambda^m\right),\quad 1\le\lambda,\mu\le 2m, \end{equation} which are linearly independent over \(\R\). Let the k\"ahlerian structure of restricted type be given by \begin{equation} \label{eq:7.29} \rd s^2=\sum_{i,k}h_{ik}\rd z^i\rd\zb^k,\quad 1\le i,j,k\le m, \end{equation} \oldpage{65}where \(h_{ik}\) are \(C^\infty\) functions in \(\Theta\). If the structure is invariant, as we are allowed to assume, \(h_{ik}\) are constants. A homology basis for the two-dimensional cycles of \(\Theta\) is formed by the two-dimensional tori \(\tau_{\lambda\mu}\), \(\lambda<\mu\); \(\tau_{\lambda\mu}\) is the quotient of the space spanned by \(\pi_\lambda\), \(\pi_\mu\) divided by the discrete group generated by \(\pi_\lambda\), \(\pi_\mu\). It follows that the metric \eqref{eq:7.29} is of restricted type, if and only if \begin{equation} \label{eq:7.30} g_{\lambda\mu}=-g_{\mu\lambda}=i\sum_{j,k}h_{jk}\left(\pi_\lambda^j\pib_\mu^k-\pi_\mu^j\pib_\lambda^k\right) \end{equation} are integers. We introduce the matrices \begin{equation} G=\left(g_{\lambda\mu}\right),\quad H=\left(h_{ik}\right),\quad \Pi=\left(\pi_\lambda^k\right), \end{equation} so that their orders are \((2m\times 2m)\), \((m\times m)\), and \((2m\times m)\) respectively. Then \eqref{eq:7.30} can be written \begin{equation} \begin{aligned} G &= \sqrt{-1}\left(\Pi H{}^t\Pib-\Pib H{}^t\Pi\right) \\ &= (\Pi,\Pib)\begin{pmatrix} \sqrt{-1}H & 0 \\ 0 & -\sqrt{-1} \ \Hb \end{pmatrix} \begin{pmatrix} ^t\Pib \\ ^t\Pi \end{pmatrix}. \end{aligned} \tag{7.30a} \end{equation} Taking the inverse matrix of this equation, we get \[ \begin{pmatrix} ^t\Pib \\ ^t\Pi \end{pmatrix} G^{-1} (\Pi,\Pib)=\begin{pmatrix}-\sqrt{-1}H^{-1} & 0 \\ 0 & \sqrt{-1}\ \Hb^{-1} \end{pmatrix}. \] Therefore we have the: \begin{thm} A necessary and sufficient condition for the torus \(\Theta=\C_m/\Gamma\) to have a k\"ahlerian metric of restricted type is that there exists a skew-symmetric matrix \(G\) with integral elements such that\oldpage{66} \begin{equation} \label{eq:7.32} \begin{aligned} i{}^t\Pib G^{-1}\Pi &> 0, \\ ^t\Pi G^{-1}\Pi &= 0. \end{aligned} \end{equation} \end{thm} The first condition in \eqref{eq:7.32} means that the hermitian matrix at the left-hand side is positive definite. The conditions \eqref{eq:7.32} were first given by Riemann. We wish to simplify the Riemann conditions \eqref{eq:7.32} by proper choices of the basis vectors of \(\C_m\) and of \(\Gamma\). Let \[ (z^1,\ldots,z^m)\to (\zt^1,\ldots,\zt^m)=(z^1,\ldots, z^m)T \] be a change of coordinates in \(\C_m\), where \(T\) is a nonsingular \((m\times m)\)-matrix with complex elements. Meanwhile, under a change of basis of \(\Gamma\) the matrix \(\Pi\) is transformed according to \[ \Pi\to\Pit=U\Pi, \] where \(U\) is a unimodular integral matrix. The combined effect of these changes on the matrices is given by \begin{equation} \begin{aligned} \Pi &\to\Pit=U\Pi T, \\ H &\to\Ht=T^{-1} H{}^t\Tb^{-1}, \\ G &\to\Gt=UG{}^tU. \end{aligned} \end{equation} It is known in the theory of matrices that \(U\) can be so chosen that \begin{equation} \Gt=\begin{pmatrix} 0 & D \\ -D & 0 \end{pmatrix} \end{equation} where\oldpage{67} \begin{equation} D=\begin{pmatrix} d_1 & & 0 \\ & \ddots & \\ 0 & & d_m \end{pmatrix},\quad d_i\in\Z. \end{equation} We can then choose \(T\), so that \begin{equation} \Pit=\begin{pmatrix} I \\ \Sigma \end{pmatrix}. \end{equation} With \(\Gt\), \(\Pit\) in place of \(G\), \(\Pi\), the conditions \eqref{eq:7.32} become \begin{equation} \label{eq:7.37} ^tZ=Z,\quad i(\Zb-Z)>0, \end{equation} where \begin{equation} Z=\Sigma D. \end{equation} The domain defined by \eqref{eq:7.37} is of dimension \(\frac{m(m+1)}{2}\); it is called the \textit{Siegel upper half plane} and is known to be biholomorphically equivalent to one of the bounded symmetric domains of \'Elie Cartan. For \(m=1\) it is the Poincar\'e half-plane. An example of a torus not satisfying the Riemann conditions is given in the case \(m=2\) by \begin{equation} \Sigma=\begin{pmatrix} \sqrt{-2} & \sqrt{-5} \\ \sqrt{-3} & \sqrt{-7} \end{pmatrix}. \end{equation} Then \[ \Sigma D=\begin{pmatrix} \sqrt{-2}d_1 & \sqrt{-5}d_2 \\ \sqrt{-3}d_1 & \sqrt{-7}d_2 \end{pmatrix},\quad d_i\in\Z. \] For this matrix to be symmetric we must have \(d_1=d_2=0\). Then the second condition in \eqref{eq:7.37} will not be fulfilled. Thus the corresponding torus will not have a k\"ahlerian structure of restricted type. \oldpage{68}In the general case a meromorphic function on the complex torus \(\Theta\) is identical to a \(2m\)-ply periodic meromorphic function in \(\C_m\) with the period vectors \eqref{eq:7.28}. All the meromorphic functions on \(\Theta\) form a field. It follows from Kodaira's embedding theorem that a complex torus satisfying the conditions \eqref{eq:7.32} can be holomorphically embedded in a projective space and is thus by definition an \textit{abelian variety}. At every point \(x\) of an abelian variety \(\Theta\) of dimension \(m\) there are \(m\) meromorphic functions on \(\Theta\), which are functionally independent at \(x\). On the other hand, there exist complex tori on which every meromorphic function is a constant. \section{The Grassmann manifold} \label{sec:8} Let \begin{equation} \C_{N+1}=\C\times\cdots\times\C,\quad N+1\text{ factors} \end{equation} be the complex number space of \(N+1\) dimensions. Let \(\GL(N+1,\C)\) be the general linear group in \(N+1\) complex variables, which we identify with the group of all \((N+1)\times (N+1)\) nonsingular matrices with complex elements. Suppose \(\GL(N+1,\C)\) acts on \(\C_{N+1}\) to the right, as described by \begin{equation} (z^0,\ldots, z^N)\to (z^0,\ldots, z^N)g,\quad g\in\GL(N+1,\C). \end{equation} Among the subgroups of \(\GL(N+1,\C)\) are: \begin{enumerate} \item the unitary group \(\U(N+1)\), which consists of all matrices \(g\) satisfying \begin{equation} ^tg\gb=I, \end{equation} where \(I\) is the identity matrix; \item the group \(\GL(k+1, N-k,\C)\), consisting of all nonsingular matrices of the form\oldpage{69} \begin{equation} \begin{pNiceMatrix}[last-col,last-row,xdots/horizontal-labels] \hspace{1ex}\ast\hspace{1ex} & \hspace{1ex}0\hspace{1ex} & \Vbrace{1}{k+1} \\ \hspace{1ex}\ast\hspace{1ex} & \hspace{1ex}\ast\hspace{1ex} & \Vbrace{1}{N-k} \\ \Hbrace{1}{k+1} & \Hbrace{1}{N-k} \end{pNiceMatrix} \end{equation} where the elements at the upper-right corner are zero. The group \(\GL(k+1, N-k,\C)\) is the subgroup of all elements of \(\GL(N+1,\C)\) leaving fixed the \((k+1)\)-dimensional subspace of \(C_{N+1}\) spanned by the first \(k+1\) coordinate vectors. \end{enumerate} The space of all \((k+1)\)-dimensional linear subspaces of \(\C_{N+1}\), \(k\ge 0\), is called a \textit{Grassmann manifold}, to be denoted by \(\Gr(N,k)\). Using the projection \begin{equation} \psi:\C_{N+1}-0\to\bP_N \end{equation} in Example \ref{eg:1.2}, the notation suggests that it is also the space of all \(k\)-dimensional linear (projective) subspaces in \(\bP_N\). From the above discussion \(\Gr(N,k)\) can be represented as a right coset space in two different ways: \begin{equation} \Gr(N,k)=\frac{\GL(N+1,\C)}{\GL(k+1,N-k,\C)}=\frac{\U(N+1)}{\U(k+1)\times\U(N-k)}. \end{equation} The first representation shows that it is a complex manifold of dimension \((k+1)(N-k)\). The second representation shows that it is compact. An element of \(\Gr(N,k)\) can be given by a nonzero decomposable \((k+1)\)-vector \begin{equation} \label{eq:8.7} \Lambda=X_0\wedge X_1\wedge\cdots\wedge X_k\ne 0 \end{equation} defined up to a constant factor. If \(e_0^0,\ldots,e_N^0\) denote a fixed frame in \(\C_{N+1}\), we can write \begin{equation} \Lambda=\sum_\alpha P_{\alpha_0,\ldots,\alpha_k} e_{\alpha_0}^0\wedge\cdots\wedge e_{\alpha_k}^0,\quad 0\le\alpha_0,\ldots,\alpha_k\le N, \end{equation} \oldpage{70}where the \(P\)'s are skew-symmetric in their indices. The \(P_{\alpha_0\cdots\alpha_k}\) are called the \textit{Cayley--Pl\"ucker--Grassmann} coordinates in \(\Gr(N,k)\). By considering \(P_{\alpha_0\cdots\alpha_k}\) as the homogeneous coordinates of a projective space of dimension \(\binom{N+1}{k+1}-1\), we get an embedding of \(\Gr(N,k)\) in the latter. We propose to study the topological properties of \(\Gr(N,k)\). Our main step is to obtain a cell decomposition of \(\Gr(N,k)\) by means of the \textit{Schubert varieties}. This was first accomplished by C. Ehresmann in 1934. Let \begin{equation} \label{eq:8.9} 0\le a_0\le a_1\le\cdots\le a_k\le N-k \end{equation} be a sequence of integers, and let \begin{equation} L_{a_0}\subset L_{a_1+1}\subset\cdots\subset L_{a_k+k}\subset\bP_N \end{equation} be a nested sequence of linear spaces whose dimensions are given by the subscripts. (We will deal with linear spaces of \(\bP_N\); their images under \(\psi^{-1}\) will have one dimension higher.) A \textit{Schubert variety} \((a_0a_1\cdots a_k)\) is the set of all \(k\)-dimensional linear spaces \(X\in\Gr(N,k)\) such that \begin{equation} \dim\left(X\cap L_{a_j+j}\right)\ge j,\quad 0\le j\le k. \end{equation} By definition, \((a_0a_1\cdots a_k)\) is a closed subset of \(\Gr(N,k)\) and is determined by the \(a\)'s up to a projective collineation. Its (complex) dimension is \begin{equation} \dim(a_0a_1\cdots a_k)=a_0+a_1+\cdots+a_k. \end{equation} \begin{egs} \begin{enumerate} \item\oldpage{71} \((N-k\cdots N-k)=\Gr(N,k)\); \item \((0\cdots 0)=L_k\); \item \((0\ldots 0 1\ldots 1)\) (\(r\) ones) is the set of all \(X\) satisfying the condition \begin{equation} L_{k-r}\subset X\subset L_{k+1}, \end{equation} where \(L_{k-r}\) and \(L_{k+1}\) are fixed linear spaces of dimensions \(k-r\) and \(k+1\) respectively. \end{enumerate} \end{egs} We take a fixed sequence of linear spaces in \(\bP_N\): \begin{equation} \label{eq:8.14} L_0\subset L_1\subset\cdots\subset L_{N-1}\subset\bP_N \end{equation} and suppose the Schubert varieties are constructed from the linear spaces of this sequence. Put \begin{equation} \label{eq:8.15} (a_0\cdots a_k)^\ast=(a_0\cdots a_k)-\sum_{a_{j-1}0\); the treatment of the case \(a_0=0\) requires only slight modifications. We take a hyperplane \(\pi\) in \(\bP_N\) such that \[ \begin{aligned} \pi\cap L_{a_0} &= L_{a_0-1}, \\ \pi\cap L_q &= L'_{q-1},\quad a_01\}. \end{equation} In other words, \(\C_0\) is the complex line with a unit disk removed. As its exhaustion function we take \(\tau=\log r>0\), \(z=re^{i\theta}\). We suppose that \(f\) is not a constant map. We will derive from \eqref{eq:9.22} the theorem that \(f(\C_0)\) is dense in \(\bP_1\). In fact, suppose that \(\bP_1-\overline{f(\C_0)}\ne\varnothing\). Let \(\rd B\) be the element of area in \(P_1\) so normalised that \begin{equation} \int_{\bP_1}\rd B=1. \end{equation} Let \(A\) be the area of \(\overline{f(\C_0)}\), so that \(A<1\). We integrate the inequality \eqref{eq:9.22} with respect to \(\rd B\) over \(\overline{f(\C_0)}\). Since \(v(u)\) is\oldpage{90} the area of the image \(f(D_u)\) (cf. \eqref{eq:9.15}), we have \[ \int_{\overline{f(\C_0)}}n(u,B)\rd B=v(u), \] and the integration gives \[ T(u)1, \end{equation} with the exception of a subset \(E\) of \(u\in\R^+\) such that \(\int_E\rd u<\infty\). \end{lem} To prove this let \(F(r),G(r)\) be positive-valued functions such that \(F(r)\) is of class \(C^1\) and increasing. Suppose that \[ F'(r)>F(r)^\kappa G(r), \] where \(\kappa>1\) is a constant. Integration gives \[ \frac{1}{\kappa-1}\left(F^{-\kappa+1}(1)-F^{-\kappa+1}(r)\right)>\int_1^rG(r)\rd r, \] so that \begin{equation} \label{eq:9.30} \int_1^\infty G(r)\rd r<+\infty. \end{equation} Thus with the exception of a set of \(r\)-intervals for which \eqref{eq:9.30} holds we have \[ F'(r)\le F^\kappa(r)G(r), \]and hence \[ \log F'\le\kappa\log F+\log G. \] Successive applications of this formula give respectively\oldpage{92} \[ \begin{aligned} \log\int_0^{2\pi}\rho\sigma^2\rd\theta+\log r &\le \kappa\log\left\{\int_1^r r\rd r\int_0^{2\pi}\rho\sigma^2\rd\theta\right\}+\log G, \\ \log\left\{\int_1^r r\rd r\int_0^{2\pi}\rho\sigma^2\rd\theta\right\}-\log r &\le \kappa\log T(u)+O(1)+\log G. \end{aligned} \] Combining them we get \[ \log\int_0^{2\pi}\rho\sigma^2\rd\theta\le\kappa^2\log T(u)+(\kappa-1)\log r+(\kappa+1)\log G+O(1). \] Choosing \(G(r)=\frac1r\) and using the concavity of the logarithmic function \begin{equation} \frac{1}{2\pi}\int_0^{2\pi}\log(\rho\sigma^2)\rd\theta\le\log\left\{\frac{1}{2\pi}\int_0^{2\pi}\rho\sigma^2\rd\theta\right\}, \end{equation} we get \eqref{eq:9.29}. To draw meaningful conclusions from the inequality \eqref{eq:9.29} we need an estimate for the integral \[ \frac{1}{2\pi}\int_0^{2\pi}\log\sigma\rd\theta. \] For such an estimate consider the projective line \(\bP_1=\Gr(1,0)\). Let \(\Lambda=(1,t)\) be its homogeneous coordinate vector so that \(0\le|t|\le\infty\), including infinity. By \eqref{eq:8.43} its metric is \begin{equation} \rd s^2=\frac{\rd t\rd\tlb}{(1+t\tlb)^2} \end{equation} By \eqref{eq:8.46} its K\"ahler form is \[ \Hh_0=\frac i2\frac{\rd t\wedge\rd\tlb}{(1+t\tlb)^2}=\frac i2\partial\db\log(1+t\tlb), \] \oldpage{93}as can be directly verified. A direct computation gives \begin{equation} \int_{\bP_1}\Hh_0=\pi \end{equation} so that \begin{equation} \rd B=\frac{i}{2\pi}\partial\db\log(1+t\tlb). \end{equation} Under the mapping \(f\), \(t\) is a holomorphic function of \(z\) and we have by definition \[ \sigma=(1+t\tlb)^{-1}\pi^{-\frac12}\left|\frac{\partial t}{\partial z}\right| \] It follows that \[ f^\ast\rd B=-\frac{1}{4\pi i}\rd(\partial-\db)\log\sigma,\quad \sigma\ne 0. \] Applying Stokes' theorem to the domain \(D_u\) and supposing that \(\partial D_u\) contains no branch point, we get \begin{equation} -2v(u)=\frac{1}{2\pi i}\int_{\partial D_u}(\partial-\db)\log\sigma-w(u), \end{equation} where \(w(u)\) is the sum of the orders of the branch points in \(D_u\), necessarily finite in number (\(D_u\) being compact). Integrating with respect to \(u\), we get \begin{equation} -2T(u)=\frac{1}{2\pi}\int_0^{2\pi}\log\sigma\rd\sigma-W(u)+\const. \end{equation} where \begin{equation} W(u)=\int_0^u w(u)\rd u. \end{equation} We have thus the inequality\oldpage{94} \begin{equation} \label{eq:9.38} \frac{1}{2\pi}\int_0^{2\pi}\log\sigma\rd\theta>-2T(u)+\const. \end{equation} By \eqref{eq:9.25}, \eqref{eq:9.20}, and \eqref{eq:9.38} we get \begin{equation} \label{eq:9.39} \frac{1}{2\pi}\int_0^{2\pi}\log(\rho\sigma^2)\rd\theta>\const.+2\lambda\sum_i m(u,A_i)-4T(u). \end{equation} We introduce the defect of the point \(A\in\bP_1\) by \begin{equation} \delta(A)=\liminf\frac{m(u,A)}{T(u)}\quad \text{as }u\to\infty. \end{equation} Thus \(\delta(A)=1\) if \(A\notin f(\C_0)\) (cf. \eqref{eq:9.21}). By letting \(\lambda\to 1\) in \eqref{eq:9.39}, we immediately get, by using \eqref{eq:9.29}, the: \begin{thm}[Nevanlinna's defect relation] Let \(f:\C_0\to\bP_1\) be a nonconstant holomorphic mapping. Let \(A_i\), \(1\le i\le s\), be a set of mutually distinct points of \(\bP_1\). Then \begin{equation} \sum_i\delta(A_i)\le 2. \end{equation} \end{thm} \begin{coro}[Picard's theorem] A nonconstant meromorphic function in \(\C\) omits at most two values. \end{coro} \part{Geometry of characteristic classes\footnote{Reprinted by permission from Proc. 13th Biennial Seminar, Canadian Math. Congress, 1972.}} \label{part:app} \counterwithout{equation}{section} \setcounter{equation}{0} \renewcommand{\thedefn}{\thesection.\arabic{defn}} \section{Historical remarks and examples} \label{sec:A1} \oldpage{97}The last few decades have seen the development, in different branches of mathematics, of the notion of a local product structure, i.e., fibre spaces and their generalisations. Characteristic classes are the simplest global invariants which measure the deviation of a local product structure from a product structure. They are intimately related to the notion of curvature in differential geometry. In fact, a real characteristic class is a ``total curvature'', according to a well-defined relationship. We will give in this paper an exposition of the relations between characteristic classes and curvature and discuss some of their applications. The simplest characteristic class is the Euler characteristic. If \(M\) is a finite cell complex, its Euler characteristic is defined by \begin{equation} \label{eq:A1} \chi(M)=\sum_k(-1)^ka_k=\sum_k(-1)^kb_k, \end{equation} where \(a_k\) is the number of \(k\)-cells and \(b_k\) is the \(k\)-dimensional Betti number of \(M\). The equality of the last two expressions in \eqref{eq:A1} is known as the Euler--Poincar\'e formula. Now let \(M\) be a compact oriented differentiable manifold of dimension \(n\) and let \(\xi\) be a smooth vector field on \(M\) with isolated zeroes. Each zero can be assigned a multiplicity. In his dissertation (1927) H. Hopf proved that\oldpage{98} \begin{equation} \label{eq:A2} \chi(M)=\sum\text{zeroes of }\xi. \end{equation} This gives a differential topological meaning to \(\chi(M)\). This idea can be immediately generalised. Instead of one vector field we consider \(k\) smooth vector fields \(\xi_1,\ldots,\xi_k\). In the generic case the points on \(M\) where the exterior product \(\xi_1\wedge\cdots\wedge\xi_k=0\), i.e., where the vectors are linearly dependent, form a \((k-1)\)-dimensional submanifold. Depending on the parity of \(n-k\), this defines a \((k-1)\)-dimensional cycle, with integer coefficients \(\Z\) or with coefficients \(\Z_2\), whose homology class, and in particular the homology class mod \(2\) in all cases, is independent of the choice of the \(k\) vector fields. Because the linear dependence of vector fields is expressed by ``conditions'', it is more proper to define the differential topological invariants so obtained as cohomology classes. This leads to the Stiefel--Whitney cohomology classes \(w^i\in H^i(M,\Z_2)\), \(1\le i\le n-1\), \(i=n-k+1\). The \(n\)th Stiefel--Whitney class corresponding to \(k=1\) or the Euler class has integer coefficients \(w^n\in H^n(M,\Z)\). It is related to \(\chi(M)\) by \begin{equation} \label{eq:A3} \chi(M)=\int_M w^n, \end{equation} where we write the pairing of homology and cohomology by an integral. Whitney went much farther. He saw the great generality of the notion of a vector bundle over an arbitrary topological space \(M\). (Actually Whitney considered sphere bundles, thus gaining the advantage that the fibres are compact but losing the linear structure on the fibres. He was not concerned with the latter, as he was only interested in topological problems.) He also saw the effectiveness of the principal bundles and the fact that the universal principal bundle \begin{equation} \label{eq:A4} \oO(q+N)/\oO(N)\to\oO(q+N)/\{\oO(q)\times\oO(N)\}=G(q,N) \end{equation} say, has the property\oldpage{99} \begin{equation} \pi_i(\oO(q+N)/\oO(N))=0,\quad 0\le in-k\), the form \(F(\Phi)=0\). Hence every Pontrjagin class, being a polynomial of the Pontrjagin classes defined in \secref{sec:A1}, of the bundle \(TM/W\) is zero, if its dimension is \(>2(n-k)\). We state this theorem of Bott as follows: Let \(M\) be a compact manifold of dimension \(n\), which has a foliation \(W\) of dimension \(k\). Then every Pontrjagin class (with real coefficients) of dimension \(>2(n-k)\) of the quotient bundle \(TM/W\) is zero. The theorem is remarkable because the integrability of \(W\) involves differential conditions, so that it cannot be proved by standard methods in fibre bundles. The necessary condition asserted in the theorem is not vacuous. For example, there are real codimension two subbundles in the complex projective space \(\bP_5(\C)\) of complex dimension \(5\), which do not satisfy the above condition. \section[Secondary invariants]{Secondary invariants\footnote{The results in this section are taken from joint work with James Simons, cf. \cite{chernsimons71}.}} When the characteristic classes are given representatives by differential forms, the vanishing of the forms leads to further invariants which deserve investigation. We follow the notations of the last section and consider the formula \eqref{eq:A79}. If \(F(\Phi)=0\),\oldpage{120} the form \(TF(\phi)\) is closed and defines an element of \(H^{2h-1}(P,\R)\). The latter depends on the connection \(\phi\) and it is desirable to study the effect on \(TF(\phi)\) under a change of the connection. This is given by the following: \begin{lem} Let \(\phi(\tau)\) be a family of connections in \(P\) depending on a parameter \(\tau\). Let \(\psi(\tau)=\frac{\partial\phi}{\partial\tau}\) and \begin{equation} \begin{aligned} V(\tau)=\int_0^1 &t^{h-1}F\left(\psi,\phi(\tau),\rd\phi(\tau)-\frac12t[\phi(\tau),\phi(\tau)],\right. \\ &\left.\ldots,\rd\phi(\tau)-\frac12t[\phi(\tau),\phi(\tau)]\right)\rd t. \end{aligned} \end{equation} Then \begin{equation} \label{eq:A88} \frac{\partial}{\partial\tau}TF(\phi(\tau))=h(h-1)\rd V(\tau)+F(\psi,\Phi(\tau),\ldots,\Phi(\tau)). \end{equation} \end{lem} To avoid long expressions we will use the convention that if \(F\) contains fewer than \(h\) arguments the last one is to be repeated a number of times so as to make \(F\) a function of \(h\) arguments. Thus \begin{equation} F(X)=F(X,\ldots,X),\quad F(X,Y)=F(X,\underbrace{Y,\ldots,Y}_{h-1}),\quad \text{etc.} \end{equation} With this convention in mind we find \[ \begin{aligned} &\rd F\left(\psi,\phi(\tau),\rd\phi(\tau)-\frac12t[\phi(\tau),\phi(\tau)]\right)= \\ &F\left(\rd\psi,\phi(\tau),\rd\phi(\tau)-\frac12t[\phi(\tau),\phi(\tau)]\right)-F\left(\psi,\rd\phi(\tau),\rd\phi(\tau)-\frac12t[\phi(\tau),\phi(\tau)]\right) \\ &-(h-2)tF\left(\psi,\phi(\tau),\left[\rd\phi(\tau)-\frac12t[\phi(\tau),\phi(\tau)],\phi(\tau)\right],\rd\phi(\tau)-\frac12t[\phi(\tau),\phi(\tau)]\right). \end{aligned} \] By \eqref{eq:A66} the last term is equal to \[ \begin{aligned} &-tF\left([\phi(\tau),\psi],\phi(\tau),\rd\phi(\tau)-\frac12t[\phi(\tau),\phi(\tau)]\right)\\&+tF\left(\psi,[\phi(\tau),\phi(\tau)],\rd\phi(\tau)-\frac12t[\phi(\tau),\phi(\tau)]\right). \end{aligned} \] \oldpage{121}We have therefore \[ \begin{aligned} &\rd F\left(\psi,\phi(\tau),\rd\phi(\tau)-\frac12t[\phi(\tau),\phi(\tau)]\right) \\ &=F\left(\rd\psi-t([\phi(\tau),\psi],\phi(\tau),\rd\phi(\tau)-\frac12t[\phi(\tau),\phi(\tau)]\right) \\ &-F\left(\psi,\rd\phi(\tau)-t[\phi(\tau),\phi(\tau)],\rd\phi(\tau)-\frac12t[\phi(\tau),\phi(\tau)]\right). \end{aligned} \] On the other hand, we have \[ \begin{aligned} &h^{-1}\frac{\partial}{\partial\tau}TF(\phi(\tau)) \\ &=\int_0^1 t^{h-1}F\left(\psi,\rd\phi(\tau)-\frac12t[\phi(\tau),\phi(\tau)]\right)\rd t \\ &+(h-1)\int_0^1 t^{h-1}F\left(\phi(\tau),\rd\psi-t[\phi(\tau),\psi],\rd\phi(\tau)-\frac12t[\phi(\tau),\phi(\tau)]\right)\rd t. \end{aligned} \] It follows that \[ \begin{aligned} &h^{-1}\frac{\partial}{\partial\tau}TF(\phi(\tau))-(h-1)\rd V(\tau) \\ &=h\int_0^1 t^{h-1}F\left(\psi,\rd\phi(\tau)-\frac{2h-1}{2h}t[\phi(\tau),\phi(\tau)],\rd\phi(\tau)-\frac12t[\phi(\tau),\phi(\tau)]\right)\rd t. \end{aligned} \] To simplify the last integral we introduce the curvature form \begin{equation} \Phi(\tau)=\rd\phi(\tau)-\frac12[\phi(\tau),\phi(\tau)] \end{equation} of the connection \(\phi(\tau)\). Putting \[ a=\frac{2h-1}{h}, \] the integrand above, up to the factor \(t^{h-1}\), can be expanded:\oldpage{122} \[ \begin{aligned} &F\left(\psi,\rd\phi(\tau)-\frac12at[\phi(\tau),\phi(\tau)],\rd\phi(\tau)-\frac12t[\phi(\tau),\phi(\tau)]\right) \\ &=F\left(\psi,\Phi(\tau)+\frac12(1-at)[\phi(\tau),\phi(\tau)],\Phi(\tau)+\frac12(1-t)[\phi(\tau),\phi(\tau)]\right) \\ &=F(\psi,\phi(\tau)) \\ &+\sum_{1\le r\le h-1} c_r(t)F\left(\psi,\underbrace{\phi(\tau),\ldots,\phi(\tau)}_{h-1-r},\underbrace{\frac12[\phi(\tau),\phi(\tau)],\ldots,\frac12[\phi(\tau),\phi(\tau)]}_{r}\right) \end{aligned} \] where \[ c_r(t)=\binom{h-2}{r}(1-t)^r+\binom{h-2}{r-1}(1-t)^{r-1}(1-at). \] From elementary calculus we know \[ \int_0^1 t^m(1-t)^n\rd t=\frac{m!n!}{(m+n+1)!},\quad m\ge 0,\ n\ge 0. \] Using this it is immediately verified that \[ \int_0^1 t^{h-1}c_r(t)\rd t=0,\quad 1\le r\le h-1. \] This proves the lemma. From the lemma follows immediately the: \begin{thm} Let \(\pi: P\to M\) be a principal bundle with the group \(G\), and let \(F\in I(G)\) be an invariant polynomial. Let \(\phi(\tau)\) be a family of commuations, with the curvature form \(\Phi(\tau)\), which satisfy the conditions \begin{equation} \label{eq:A91} \begin{aligned} F\left(\frac{\partial\phi(\tau)}{\partial\tau},\Phi(\tau),\ldots,\Phi(\tau)\right)&=0, \\ F(\Phi(\tau),\ldots,\Phi(\tau))&=0. \end{aligned} \end{equation} Then the cohomology class \(\{TF(\phi(\tau))\}\) is independent of \(\tau\). \end{thm} As \(h\) is the degree of \(F\), the conditions \eqref{eq:A91} are automatically satisfied when \(2h>\dim M+1\). Equivalently the conditions \eqref{eq:A91} can be written in terms of a\oldpage{123} local chart. By \eqref{eq:A46} and using the fact that \(F\) is an invariant polynomial, we can write \eqref{eq:A91} as \begin{equation} \begin{aligned} F\left(\frac{\partial\theta_U(\tau)}{\partial\tau},\Theta_U(\tau),\ldots,\Theta_U(\tau)\right)&=0, \\ F(\Theta_U(\tau),\ldots,\Theta_U(\tau))&=0. \end{aligned} \end{equation} We now apply these results to the principal bundle of the tangent bundle of a manifold \(M\) of dimension \(n\), so that the structure group is \(G=\GL(n;\R)\). The connection will be the Levi-Civita connection of a riemannian metric in \(M\) and it will have special properties. We use a local chart with the coordinates \(x^i\), \(1\le i,j,k,\ell\le n\), and we will omit the subscript \(U\) in our notations. The riemannian metric is given by the scalar products \begin{equation} h_{ij}=h_{ji}=\left(\frac{\partial}{\partial x^i},\frac{\partial}{\partial x^j}\right), \end{equation} which are the elements of a positive definite symmetric matrix: \begin{equation} H={}^tH=(h_{ij})>0. \end{equation} Let the connection matrix be \begin{equation} \theta=(\theta_i^j),\quad \theta_i^j=\sum_k\Gamma_{ik}^j\rd x^k. \end{equation} It is determined by the conditions \begin{equation} \label{eq:A96} \rd H-\theta H-H{}^t\theta=0 ,\quad\Gamma_{ik}^j=\Gamma_{ki}^j. \end{equation} The curvature form is given by \begin{equation} \Theta=\rd\theta-\theta\wedge\theta. \end{equation} Exterior differentiation of \eqref{eq:A96} gives \begin{equation} \label{eq:A98} \Theta H+H{}^t\Theta=0, \end{equation} i.e., the matrix \(\Theta H\) is antisymmetric. \begin{lem} \label{lem:A4.2} Let \(F\) be an invariant polynomial of odd degree \(h\), and \(\Theta\) the curvature matrix of the Levi-Civita connection of a riemannian metric. Then \begin{equation} \label{eq:A99} F(\Theta)=0. \end{equation} \end{lem} To prove this, notice that \(F\) clearly has the property: \begin{equation} F(\Theta)=F({}^t\Theta). \end{equation} By \eqref{eq:A98} this is equal to \[ F(-H^{-1}\Theta H)=(-1)^hF(\Theta). \] Hence we have \eqref{eq:A99} when \(h\) is odd. We write \begin{equation} \Theta=(\Theta_i^j), \end{equation} where we set \begin{equation} \Theta_i^j=-\frac12\sum_{k,\ell}R_{ik\ell}^j\rd x^k\wedge\rd x^\ell,\quad R_{ik\ell}^j+R_{i\ell k}^j=0. \end{equation} The \(R_{ik\ell }^j\) define the Riemann curvature tensor and satisfy the symmetry relations \begin{equation} \begin{aligned} R_{ik\ell}^j+R_{i\ell k}^j&=0, \\ R_{ik\ell}^j+R_{k\ell i}^j+R_{\ell ik}^j&=0. \end{aligned} \end{equation} The last relation can also be written \begin{equation} \label{eq:A104} \sum_i\rd x^i\wedge\Theta_i^j=0. \end{equation} We will consider the case of a conformal family of riemannian metrics, given by the matrix \(H(\tau)=\exp(2\sigma\tau)H\), where \(\sigma\) is a scalar function and \(\tau\) is the parameter. Then we have \cite[89]{eisenhart49} the matrix equation\oldpage{125} \begin{equation} \frac1\tau(\theta(\tau)-\theta(0))=\rd\sigma I+\alpha+\beta, \end{equation} where \(I\) is the unit matrix and \begin{equation} \begin{aligned} \alpha=\left(\frac{\partial\sigma}{\partial x^i}\rd x^j\right),\quad \beta&=-\left(\sum_k h_{ik}\rd x^k\sum_\ell\frac{\partial\sigma}{\partial x^\ell}h^{j\ell}\right) \\ H^{-1}&=(h^{ij}). \end{aligned} \end{equation} \begin{lem} Let \(F\) be an invariant polynomial of even degree 2s. For the Levi-Civita connections of a conformal family of riemannian metrics we have \begin{equation} F\left(\frac{\partial\theta(\tau)}{\partial\tau},\Theta(\tau)\right)=0. \end{equation} \end{lem} In fact, the left-hand side of this equation is equal to \[ \rd\sigma F(I,\Theta(\tau))+F(\alpha,\Theta(\tau))+F(\beta,\Theta(\tau)). \] By Lemma \ref{lem:A4.2} the first term is zero. By the fundamental theorem on vector invariants every term in \(F(\alpha,\Theta(\tau))\) contains a factor \(\sum\rd x^j\wedge\Theta_j^i(\tau)\), which is zero by \eqref{eq:A104}. Similarly, every term in \(F(\beta,\Theta(\tau))\) contains a factor \(\sum_{i,k}h_{ik}\rd x^k\wedge\theta_j^i(\tau)\), which is also seen to be zero. Thus the lemma is proved. This lemma, together with the formula \eqref{eq:A88} gives the: \begin{thm} Let \(\pi:P\to M\) be the principal bundle of the tangent bundle of a manifold \(M\) of dimension \(n\). Let \(F\in I(\GL(n;\R))\) be an invariant polynomial of degree \(2s\). Let \(\phi\) and \(\phi^\ast\) be the connection forms of the Levi-Civita connections of two riemannian metrics on \(M\), which are conformal to each other. Then there exists a form \(W\) of degree \(4s-2\) in \(P\), such that \begin{equation} TF(\phi^\ast)-TF(\phi)=\rd W. \end{equation} \end{thm} \begin{coron} The form \(F(\phi)\) remains invariant under a conformal transformation of the riemannian metric. \end{coron} \oldpage{126}More precisely, A. Avez \cite{avez70} expressed \(F(\phi)\) in terms of the Weyl conformal tensor of the riemannian metric. \begin{coron} \label{coro:A4.2} If \(F(\phi)=0\), then \(TF(\phi)\) defines an element \(\{TF(\phi)\}\in H^{4s-1}(P,\R)\), which depends only on the underlying conformal structure of the riemannian manifold \(M\). \end{coron} In exactly the same way one can establish results concerning a projective transformation of the riemannian metric, i.e., a change of the riemannian metric which leaves the geodesics invariant. Such a change is described by a form \(\lambda=\sum_i a_i(x)\rd x^i\) and the connection forms are related by \cite[132]{eisenhart49} \begin{equation} \theta^\ast-\theta=\lambda I+\alpha, \end{equation} where \begin{equation} \alpha=\left(a_i\rd x^j\right). \end{equation} The connections \(\theta\) and \(\theta^\ast\) can be joined by the family \(\theta(\tau)\), \(0\le\tau\le 1\), given by \begin{equation} \frac1\tau(\theta(\tau)-\theta)=\lambda I+\alpha. \end{equation} The above arguments apply and we conclude that \(F(\phi)\) is a projective invariant and that, if \(F(\phi)=0\), the cohomology class \(\{TF(\phi)\}\) in \(P\) is also a projective invariant. In order to utilise our secondary invariants we look for cases where \(F(\phi)=0\). One is the situation which occurs in Bott's theorem on foliations discussed above. Recently this has given rise to an active development in the works of Bott, Haefliger, etc. \cite{botthaefliger72}. Another case concerns with immersed submanifolds \(M\) in the euclidean space \(E^N\) of dimension \(N=n+h\), which we will discuss in some detail. The basic fact is the commutative diagram\oldpage{127} \begin{equation} \vcenter{\vbox{\xymatrix{ P\ar[r]^-\gt \ar[d]_-\pi & \oO(n+h)/\oO(h)\ar[d]^-{\pi_0} \\ M\ar[r]^-g & \oO(n+h)/\{\oO(n)\times\oO(h)\} }}} \end{equation} Here \(P\) is the bundle of orthonormal frames over \(M\) and \(g\) and \(\gt\) are the Gauss mappings defined by parallelisms in the ambient euclidean space. The bundle at the right-hand side of the diagram has a canonical connection described at the end of \secref{sec:A2}. We denote its connection and curvature forms by \(\phit\) and \(\Phit\) respectively; they are therefore antisymmetric matrices of forms. Then the Levi-Civita connection is given by \(\phi=\gt^\ast \phit\) and its curvature form is \(\Phi=\gt^\ast\Phit\). In fact, this was the original definition of Levi-Civita of his connection, generalising a classical construction for surfaces in \(E^3\). We put \begin{equation} \label{eq:A113} P_k(\Phi)=E_{2k}(\Phi), \end{equation} where the latter are defined in \eqref{eq:A84}; these will be called the \textit{Pontrjagin forms}. The dual Pontrjagin forms are introduced by the equation \begin{equation} \sum_{k\ge 0}P_k(\Phi)\sum_{j\ge 0}P_j^\perp(\Phi)=1,\quad P_0=P_0^\perp=1, \end{equation} and are uniquely determined. By the duality theorem on Pontrjagin classes, the cohomology classes \(\left\{P_j^\perp(\Phi)\right\}\in H^{4j}(M,\R)\) are the Pontrjagin classes of the normal bundle of \(M\) in \(E^N\). Since the normal bundle has fibre dimension \(h\), we have \begin{equation} \left\{P_j^\perp(\Phi)\right\}=0,\quad \left[\frac h2\right]+1\le j. \end{equation} We will show that the forms \(P_j^\perp(\Phi)\) themselves are zero. In fact, we have \[ \left\{P_j^\perp(\Phit)\right\}=0,\quad \left[\frac h2\right]+1\le j. \] \oldpage{128}But the Grassmann manifold is a symmetric riemannian manifold and the form \(P_j^\perp(\Phit)\) is invariant under the action of the group \(\oO(n+h)\). Hence, in the range of \(j\) described above, we have \(P_j^\perp(\Phit)\) and therefore \[ P_j^\perp(\Phi)=\gt^\ast P_j(\Phit)=0. \] It is thus possible to apply the construction of secondary invariants to the invariant polynomials \(P_j^\perp\). We will state our general theorem as follows: \begin{thm} \label{thm:A4.3} Let \(M\) be a compact manifold of dimension \(n\), with a riemannian metric \(\rd s^2\). Necessary conditions for its conformal immersion in \(E^{n+h}\) are: \begin{align} P_j^\perp(\rd s^2)=0,\quad &\left[\frac h2\right]+1\le j, \label{eq:A116} \\ \left\{\frac12TP_j^\perp(\rd s^2)\right\}\in H^{4j-1}(P,\Z),\quad &\left[\frac h2\right]+1\le j\le \left[\frac{n-1}{2}\right], \label{eq:A117} \end{align} where we use the argument \(\rd s^2\) to replace its Levi-Civita connection in the notation. \end{thm} Conditions \eqref{eq:A116} follows from the above discussions. The proof of \eqref{eq:A117} is lengthy and can be found in \cite{chernsimons74}. We will carry out our construction for \(P_1=-P_1^\perp\). By \eqref{eq:A113} and \eqref{eq:A84} we have \begin{equation} P_1(\Phi)=\frac{1}{8\pi^2}\sum_{i,j}(\Phi_i^i\Phi_j^j-\Phi_i^j\Phi_j^i). \end{equation} In the notation of \eqref{eq:A72} we introduce the form \begin{equation} \Phi_t=t(\rd\phi-t\phi\wedge\phi)=t\{\Phi+(1-t)\phi\wedge\phi\}, \end{equation} so that we find the polarised form\oldpage{129} \begin{equation} P_1(\phi, \Phi_t)=\frac{t}{8\pi^2}\sum_{i,j,k}\left\{\phi_i^j\wedge\Phi_j^j-\phi_i^j\wedge\Phi_j^i-(1-t)\phi_i^j\wedge\phi_j^k\wedge\phi_k^i\right\}. \end{equation} It follows that \begin{equation} \label{eq:A121} \begin{aligned} TP_1(\phi)&=2\int_0^1 P_1(\phi,\Phi_t)\rd t \\ &=\frac{1}{8\pi^2}\sum_{i,j,k}\left\{ \phi_i^i\wedge\Phi_j^j-\phi_i^j\wedge\Phi_j^i-\frac13\phi_i^j\wedge\phi_j^k\wedge\phi_k^i\right\}. \end{aligned} \end{equation} When restricted to orthonormal frames the matrices \[ \phi=(\phi_{ij}),\quad \Phi=(\Phi_{ij}) \] are antisymmetric and \eqref{eq:A121} simplifies to \begin{equation} TP_1(\phi)=\frac{1}{8\pi^2}\left\{\sum_{i,j}\phi_{ij}\wedge\Phi_{ij}-\frac13\sum_{i,j,k}\phi_{ij}\wedge\phi_{jk}\wedge\phi_{ki}\right\} \end{equation} When \(M\) is of dimension \(3\), \(P_1(\Phi)\) vanishes for dimension reasons and we get a closed form \(TP_1(\phi)\) in the bundle \(P\) of orthonormal frames. In view of Theorem \ref{thm:A4.3} we write \begin{equation} \frac12 TP_1(\phi)=\frac{1}{8\pi^2}\sum_{1\le i0\) in each \(U\), such that \begin{equation} \label{eq:A172} \nm{y}^2=h_U^{-1}|y_U|^2=h_V^{-1}|y_V|^2\quad\text{in }U\cap V. \end{equation} Equation \eqref{eq:A172} is equivalent to \begin{equation} h_U|g_{UV}|^2=h_V \tag{172a} \end{equation} It follows that \begin{equation} \label{eq:A173} \Omega=\frac{i}{2\pi}\partial\db\log h_U \end{equation} is independent of \(U\). \(\Omega\) defines a closed form of bidegree \((1,1)\) in \(M\), the curvature form of the hermitian line bundle \(E\), whose cohomology class \(c_1(E)\) is the first Chern class of \(E\). It is desirable to allow the hermitian structure to have singularity on a divisor. If \(\phi_U=0\) is the local representation of a divisor, we suppose \begin{equation} h_U=|\phi_U|^{2s}h_U',\quad h_U'>0, \end{equation} where \(s\) is an integer. This generalised structure will be called \textit{semi-hermitian}. If \(M\) is one-dimensional, the singularities of \(h_U\) are isolated and, relative to a suitable local coordinate \(\zeta\), \(h_U\) will be of the form\oldpage{142} \begin{equation} h_U=|\zeta|^{2s}h_U',\quad h_U'>0. \end{equation} The integer \(s\) is called the order of the singularity. An application of Stokes' Theorem to \eqref{eq:A173} gives the theorem \cite{chern72}: \begin{thm}[Gauss--Bonnet] \label{thm:A6.1} Let \(\pi:E\to M\) be a semi-hermitian holomorphic line bundle over a one-dimensional complex manifold \(M\). Let \(D\) be a compact domain of \(M\) with smooth boundary \(\partial D\) and \(s:D\to E\) be a holomorphic section such that \begin{enumerate} \item \(\partial D\) contains no singularity of the hermitian structure; \item \(s(\partial D)\) does not meet the zero section of the bundle. \end{enumerate} Then \begin{equation} n(s)-n(h)=-\int_D\Omega+\frac{1}{2\pi}\int_{\partial D}\rd^C\log\nm{s},\quad \rd^C=i(\db-\partial), \end{equation} where \(n(s)\) is the number of zeroes of the section in \(D\), \(n(h)\) is the number of singularities of the semi-hermitian structure in \(D\), and \(\nm{s}\) is the hermitian norm of the section on the boundary. \end{thm} We consider the complex projective space \(\bP_n(\C)\) of dimension \(n\). To define it we take the complex vector space \(\C_{n+1}\) of dimension \(n+1\) and identify its nonzero vectors which differ from each other by a factor. The identification \begin{equation}\label{eq:A177} \pi:\C_{n+1}-\{0\}\to\bP_n(\C) \end{equation} defines a holomorphic line bundle over \(\bP_n(\C)\). To \(x\in\bP_n(\C)\), \(\pi^{-1}(x)\) is a nonzero vector \(Z=(z_0,\ldots,z_n)\) of \(\C_{n+1}\), determined up to a factor; \(Z\) will be called a homogeneous coordinate vector of \(x\). The geometry in \(\bP_n(\C)\) arises from the hermitian scalar product \begin{equation} (Z,W)=z_0\wb_0+\cdots+z_n\wb_n,\quad W=(w_0,\ldots,w_n) \end{equation} \oldpage{143}in \(\C_{n+1}\). We set \begin{equation} |Z,W|=|(Z,W)|,\quad |Z|^2=(Z,Z). \end{equation} Then \(|Z|\) defines an hermitian norm in the bundle \eqref{eq:A177}. By \eqref{eq:A173} the Chern class of its dual bundle, the hyperplane section bundle, is represented by the curvature form \begin{equation} \Omega=\frac i\pi\partial\db\log|Z|. \end{equation} This form has the further property that it is positive definite, in the following sense: The complex structure on \(\bP_n(\C)\) sets up a one-one correspondence between real forms of bidegree \((1,1)\) and the hermitian differential forms; the hermitian form corresponding to \(\Omega\) is positive definite. We can therefore use it to define an hermitian structure on \(\bP_n(\C)\), which gives the classical Fubini--Study metric (cf. \cite{chern67}). It can be verified that \begin{equation} \int_{\bP_1}\Omega=1, \end{equation} so that \(\{\Omega\}\), the cohomology class represented by \(\Omega\), is a generator of \(H^2(\bP_n(\C),\Z)\). It follows that \(\Omega^n\) is a volume element of \(\bP_n(\C)\), with total volume equal to \(1\). Consider an algebraic curve \begin{equation} f:M\to\bP_n(\C), \end{equation} where \(M\) is a compact one-dimensional complex manifold without boundary and \(f\) is holomorphic. The above discussion identifies the area of the curve with its order and we have the formula of Wirtinger: \begin{equation} \label{eq:A183} A(M)=\int_{f(M)}\Omega=n(f(M)\cap\alpha)=v(M). \end{equation} Here \(A(M)\) is the area, \(v(M)\) is the order of the curve, and \(n(f(M)\cap\alpha)\) is the number of common points of \(f(M)\) with any\oldpage{144} hyperplane \(\alpha\); the equalities at the two ends of \eqref{eq:A183} are definitions. The Gauss--Bonnet theorem \ref{thm:A6.1} extends this relationship to a compact domain \(D\) with boundary and we have the: \begin{thm}[Unintegrated first main theorem] \label{thm:A6.2} Let \(M\) be a one-dimensional complex manifold and \(f: M\to\bP_n(\C)\) be a holomorphic mapping. Let \(D\) be a compact domain of \(M\) with smooth boundary \(\partial D\), such that \(f(\partial D)\) does not meet a hyperplane \(\alpha\). Then \begin{equation} \label{eq:A184} n(D,\alpha)-A(D)=\frac{1}{2\pi}\int_{f(\partial D)}\rd^C\log\frac{|Z,\alpha^\perp|}{|Z|\cdot|\alpha^\perp|}, \end{equation} where \(n(D,\alpha)\) is the number of points in \(f(D)\cap\alpha\) and \(\alpha^\perp\) is the ``pole'' of \(\alpha\), i.e., the point orthogonal to all points of \(\alpha\). \end{thm} Thus, while the formula \eqref{eq:A183} is not valid for a domain \(D\) with boundary, Theorem \ref{thm:A6.2} gives a useful expression for the difference \(n(D,\alpha)-A(D)\). When \(M\) is noncompact and \(D\) exhausts \(M\), each of \(n(D,\alpha)\) and \(A(D)\) could become infinite and our main concern is to estimate their relative growth and the growth of other geometrical quantities which eventually enter into play. The quantity against which the growths are measured is the \textit{exhaustion function}. It is by definition a smooth function \(\tau:M\to\R^+\) satisfying the conditions: \begin{enumerate} \item The mapping \(\tau\) is proper, i.e., the inverse of a compact set is compact; \item \label{term:exhstfun2} The critical points are isolated. \end{enumerate} An example of a function satisfying \eqref{term:exhstfun2} is a real harmonic function. Suppose \(\tau\) is a harmonic exhaustion function of \(M\). Let \begin{equation} D_u=\{\zeta\in M:\tau(\zeta)\le u\}. \end{equation} For simplicity we write \begin{equation} n(D_u,\alpha)=n(u,\alpha),\quad A(D_u)=v_0(u). \end{equation} \oldpage{145}Then \eqref{eq:A184} can be written \begin{equation} \label{eq:A187} n(u,\alpha)-v_0(u)=\frac{1}{2\pi}\int_{f(\partial D_u)}\rd^C\log\frac{|Z(\zeta),\alpha^\perp|}{|Z(\zeta)|\cdot|\alpha^\perp|},\quad\zeta\in\partial D_u. \end{equation} By a standard argument the integration and the differential operator \(\rd^C\) can be interchanged and we can integrate the above formula with respect to \(u\). We put \begin{equation} N(u,\alpha)=\int_0^u n(t,\alpha)\rd t ,\quad T_0(u)=\int_0^u v_0(t)\rd t \end{equation} and \begin{equation} m(u,\alpha)=\frac{1}{2\pi}\int_{\partial D_u}\log\frac{|Z(\zeta),\alpha^\perp|}{|Z(\zeta)|\cdot|\alpha^\perp|}\rd^C\tau\ge 0,\quad\zeta\in\partial D_u. \end{equation} Then the integration of \eqref{eq:A187} gives \begin{equation} \label{eq:A190} N(u,\alpha)+m(u,\alpha)=T_0(u)+m(0,\alpha). \end{equation} This is the \textit{integrated form of the first main theorem}. As a corollary we have the \textit{fundamental inequality} \begin{equation} \label{eq:A191} N(u,\alpha)