8.1 Complex manifolds

To begin with, we define a holomorphic (or analytic) map on ${\mathbb{C}}^m$ . A complex-valued function $f:{\mathbb{C}}^m\to \mathbb{C}$ is holomorphic if $f=f_1+\mathrm{i}{f}_2$ satisfies the Cauchy-Riemann relations for each $z^{\mu }=x^{\mu }+\mathrm{i}{y}^{\mu }$ ,

$$\frac{\partial f_1}{\partial x^{\mu }}=\frac{\partial f_2}{\partial y^{\mu }}\frac{\partial f_2}{\partial x^{\mu }}=-\frac{\partial f_1}{\partial y^{\mu }}.\text{\hspace{1em}(8.1)}$$

A map $(f^1,\dots ,f^n):{\mathbb{C}}^m\to {\mathbb{C}}^n$ is called holomorphic if each function $f^{\lambda }$ $(1\le \lambda \le n)$ is holomorphic.

8.1.1 Definitions

Definition 8.1

$M$ is a complex manifold if the following axioms hold,

(i) $M$ is a topological space.
(ii) $M$ is provided with a family of pairs $\{(U_i,{\phi }_i)\}$ .
(iii) $\{U_i\}$ is a family of open sets which covers $M$ . The map ${\phi }_i$ is a homeomorphism from $U_i$ to an open subset $U$ of ${\mathbb{C}}^m$ . [Hence, $M$ is even dimensional.]
(iv) Given $U_i$ and $U_j$ such that $U_i\cap U_j\ne \varnothing$ , the map ${\psi }_{ji}={\phi }_j\circ {\phi }_i^{-1}$ from ${\phi }_i(U_i\cap U_j)$ to ${\phi }_j(U_i\cap U_j)$ is holomorphic.

8

The number $m$ is called the complex dimension of $M$ and is denoted as ${\dim }_{\mathbb{C}}M=m$ . The real dimension $2m$ is denoted either by ${\dim }_{\mathbb{R}}M$ or simply by $\dim M$ . Let $z^{\mu }={\phi }_i(p)$ and $w^{\nu }={\phi }_j(p)$ be the (complex) coordinates of a point $p\in U_i\cap U_j$ in the charts $(U_i,{\phi }_i)$ and $(U_j,{\phi }_j)$ , respectively. Axiom (iv) asserts that the function $w^{\nu }=u^{\nu }+\mathrm{i}{v}^{\nu }$ ( $1\le \nu \le m$ ) is holomorphic in $z^{\mu }=x^{\mu }+\mathrm{i}{y}^{\mu }$ , namely

$$\frac{\partial u^v}{\partial x^v}=\frac{\partial v^v}{\partial y^v}\frac{\partial u^v}{\partial y^v}=-\frac{\partial v^v}{\partial x^v}1\le \mu ,v\le m.$$

These axioms ensure that calculus on complex manifolds can be carried out independently of the special coordinates chosen. For example, ${\mathbb{C}}^n$ is the simplest complex manifold. A single chart covers the whole space and $\phi$ is the identity map.

Let $\{(U_i,{\phi }_i)\}$ and $\{(V_j,{\psi }_j)\}$ be atlases of $M$ . If the union of two atlases is again an atlas which satisfies the axioms of definition 8.1, they are said to define the same complex structure. A complex manifold may carry a number of complex structures (see example 8.2).

8.1.2 Examples

Example 8.1

In exercise 5.1, it was shown that the stereographic coordinates of a point $P(x,y,z)\in S^2-\{\text{North Pole}\}$ projected from the North Pole are

$$(X,Y)=\left(\frac{x}{1-z},\frac{y}{1-z}\right)$$

while those of a point $P(x,y,z)\in S^2-\{\text{South Pole}\}$ projected from the South Pole are

$$(U,V)=\left(\frac{x}{1+z},\frac{-y}{1+z}\right).$$

[Note the orientation of $(U,V)$ in figure 5.5.] Let us define complex coordinates

$$Z=X+\mathrm{i}Y,\overline{Z}=X-\mathrm{i}Y,W=U+\mathrm{i}V,\overline{W}=U-\mathrm{i}V.$$

$W$ is a holomorphic function of $Z$ ,

$$W=\frac{x-\mathrm{i}y}{1+z}=\frac{1-z}{1+z}(X-\mathrm{i}Y)=\frac{X-\mathrm{i}Y}{X^2+Y^2}=\frac{1}{Z}.$$

Thus, $S^2$ is a complex manifold which is identified with the Riemann sphere $\mathbb{C}\cup \{\infty \}$ .

Example 8.2

Take a complex plane $\mathbb{C}$ and define a lattice $L({\omega }_1,{\omega }_2)\equiv \{{\omega }_{1}m+{\omega }_{2}n|m,n\in \mathbb{Z}\}$ where ${\omega }_1$ and ${\omega }_2$ are two non-vanishing complex numbers such

that ${\omega }_2/{\omega }_1\notin \mathbb{R}$ ; see figure 8.1. Without loss of generality, we may take $\mathrm{Im}({\omega }_2/{\omega }_1)>0$ . The manifold $\mathbb{C}/L({\omega }_1,{\omega }_2)$ is obtained by identifying the points $z_1,z_2\in \mathbb{C}$ such that $z_1-z_2={\omega }_{1}m+{\omega }_{2}n$ for some $m,n\in \mathbb{Z}$ . Since the opposite sides of the shaded area of figure 8.1 are identified, $\mathbb{C}/L({\omega }_1,{\omega }_2)$ is homeomorphic to the torus $T^2$ . The complex structure of $\mathbb{C}$ naturally induces that of $\mathbb{C}/L({\omega }_1,{\omega }_2)$ . We say that the pair $({\omega }_1,{\omega }_2)$ defines a complex structure on $T^2$ . There are many pairs $({\omega }_1,{\omega }_2)$ which give the same complex structure on $T^2$ .

Figure 8.1. Two complex numbers ${\omega }_1$ and ${\omega }_2$ define a lattice $L({\omega }_1,{\omega }_2)$ in the complex plane. $\mathbb{C}/L({\omega }_1,{\omega }_2)$ is homeomorphic to the torus (the shaded area).

When do pairs $({\omega }_1,{\omega }_2)$ and $({\omega }_1^{\prime },{\omega }_2^{\prime })$ $(\mathrm{Im}({\omega }_2/{\omega }_1)>0,\mathrm{Im}({\omega }_2^{\prime }/{\omega }_1^{\prime })>0)$ define the same complex structure? We first note that two lattices $L({\omega }_1,{\omega }_2)$ and $L({\omega }_1^{\prime },{\omega }_2^{\prime })$ coincide if and only if there exists a matrix

$$\left(\begin{array}{ll}a& b\\ c& d\end{array}\right)\in \mathrm{PSL}(2,\mathbb{Z})\equiv \mathrm{SL}(2,\mathbb{Z})/{\mathbb{Z}}_2$$

such that

$$\left(\begin{array}{l}{\omega }_1^{\prime }\\ {\omega }_2^{\prime }\end{array}\right)=\left(\begin{array}{ll}a& b\\ c& d\end{array}\right)\left(\begin{array}{l}{\omega }_1\\ {\omega }_2\end{array}\right).\text{\hspace{1em}(8.2)}$$

This statement is proved as follows.

Suppose

$$\left(\begin{array}{c}{\omega }_1^{\prime }\\ {\omega }_2^{\prime }\end{array}\right)=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\left(\begin{array}{c}{\omega }_1\\ {\omega }_2\end{array}\right)\text{w h e r e}\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in \mathrm{SL}(2,\mathbb{Z}).$$

1 The group $\mathrm{SL}(2,\mathbb{Z})$ has been defined in (2.4). Two matrices $A$ and $-A$ are identified in $\mathrm{PSL}(2,\mathbb{Z})$ .

Since ${\omega }_1^{\prime },{\omega }_2^{\prime }\in L({\omega }_1,{\omega }_2)$ , we find $L({\omega }_1^{\prime },{\omega }_2^{\prime })\subset L({\omega }_1,{\omega }_2)$ . From

$$\left(\begin{array}{c}{\omega }_1\\ {\omega }_2\end{array}\right)=\left(\begin{array}{cc}d& -b\\ -c& a\end{array}\right)\left(\begin{array}{c}{\omega }_1^{\prime }\\ {\omega }_2^{\prime }\end{array}\right)$$

we also find $L({\omega }_1,{\omega }_2)\subset L({\omega }_1^{\prime },{\omega }_2^{\prime })$ . Thus, $L({\omega }_1,{\omega }_2)=L({\omega }_1^{\prime },{\omega }_2^{\prime })$ . Conversely, if $L({\omega }_1,{\omega }_2)=L({\omega }_1^{\prime },{\omega }_2^{\prime })$ , ${\omega }_1^{\prime }$ and ${\omega }_2^{\prime }$ are lattice points of $L({\omega }_1,{\omega }_2)$ and can be written as ${\omega }_1^{\prime }=d{\omega }_1+c{\omega }_2$ and ${\omega }_2^{\prime }=b{\omega }_1+a{\omega }_2$ where $a,b,c,d\in \mathbb{Z}$ . Also ${\omega }_1$ and ${\omega }_2$ may be expressed as ${\omega }_1=d^{\prime }{\omega }_1^{\prime }+c^{\prime }{\omega }_2^{\prime }$ and ${\omega }_2=b^{\prime }{\omega }_1^{\prime }+a^{\prime }{\omega }_2^{\prime }$ where $a^{\prime },b^{\prime },c^{\prime },d^{\prime }\in \mathbb{Z}$ . Then we have

$$\left(\begin{array}{l}{\omega }_1\\ {\omega }_2\end{array}\right)=\left(\begin{array}{ll}{a}^{\prime }& b^{\prime }\\ c^{\prime }& d^{\prime }\end{array}\right)\left(\begin{array}{l}{\omega }_1^{\prime }\\ {\omega }_2^{\prime }\end{array}\right)=\left(\begin{array}{ll}{a}^{\prime }& b^{\prime }\\ c^{\prime }& d^{\prime }\end{array}\right)\left(\begin{array}{ll}a& b\\ c& d\end{array}\right)\left(\begin{array}{l}{\omega }_1\\ {\omega }_2\end{array}\right)$$

from which we find

$$\left(\begin{array}{ll}{a}^{\prime }& b^{\prime }\\ c^{\prime }& d^{\prime }\end{array}\right)\left(\begin{array}{ll}a& b\\ c& d\end{array}\right)=\left(\begin{array}{ll}1& 0\\ 0& 1\end{array}\right).$$

Equating the determinants of both sides, we have $(a^{\prime }{d}^{\prime }-b^{\prime }{c}^{\prime })(ad-bc)=1$ . All the entries being integers, this is possible only when $ad-bc=\pm 1$ . Since

$$\mathrm{Im}\left(\frac{{\omega }_2^{\prime }}{{\omega }_1^{\prime }}\right)=\mathrm{Im}\left(\frac{b{\omega }_1+a{\omega }_2}{d{\omega }_1+c{\omega }_2}\right)=\frac{ad-bc}{|c({\omega }_2/{\omega }_1)+d{|}^2}\mathrm{Im}\left(\frac{{\omega }_2^{\prime }}{{\omega }_1^{\prime }}\right)>0$$

we must have $ad-bc>0$ , that is,

$$\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in \mathrm{SL}(2,\mathbb{Z}).$$

In fact, it is clear that

$$\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in \mathrm{SL}(2,\mathbb{Z})$$

defines the same lattice as

$$-\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$$

and we have to identify those matrices of $\mathrm{SL}(2,\mathbb{Z})$ which differ only by their overall signature. Thus, two lattices agree if they are related by $\mathrm{PSL}(2,\mathbb{Z})\equiv$ $\mathrm{SL}(2,\mathbb{Z})/{\mathbb{Z}}_2$

Assume that there exists a one-to-one holomorphic map $h$ of $\mathbb{C}/L({\omega }_1,{\omega }_2)$ onto $\mathbb{C}/L({\tilde{\omega }}_1,{\tilde{\omega }}_2)$ where $\mathrm{Im}({\omega }_2/{\omega }_1)>0$ , $\mathrm{Im}({\tilde{\omega }}_2/{\tilde{\omega }}_1)>0$ . Let $p:\mathbb{C}\to \mathbb{C}/L({\omega }_1,{\omega }_2)$ and $\tilde{p}:\mathbb{C}\to \mathbb{C}/L({\tilde{\omega }}_1,{\tilde{\omega }}_2)$ be the natural projections. For example, $p$ maps a point in $\mathbb{C}$ to an equivalent point in $\mathbb{C}/L({\omega }_1,{\omega }_2)$ . Choose the origin 0 and define $h_{\ast }(0)$ to be a point such that $\tilde{p}\circ h_{\ast }(0)=h\circ p(0)$ (figure 8.2),

Figure 8.2. A holomorphic bijection $h:\mathbb{C}/L({\omega }_1,{\omega }_2)\to \mathbb{C}/L({\tilde{\omega }}_1,{\tilde{\omega }}_2)$ and the natural projections $p:\mathbb{C}\to \mathbb{C}/L({\omega }_1,{\omega }_2)$ , $\tilde{p}:\mathbb{C}\to \mathbb{C}/L({\tilde{\omega }}_1,{\tilde{\omega }}_2)$ define a holomorphic bijection $h_{\ast }:\mathbb{C}\to \mathbb{C}$ .

$$\begin{array}{ccc}\mathbb{C}& \stackrel{h_{\ast }}{\to }& \mathbb{C}\\ p\downarrow & & \downarrow \tilde{p}\\ \mathbb{C}/L\left({\omega }_1,{\omega }_2\right)& \stackrel{h}{\to }& \mathbb{C}/L\left({\tilde{\omega }}_1,{\tilde{\omega }}_2\right).\end{array}\text{\hspace{1em}(8.3)}$$

Then by analytic continuation from the origin, we obtain a one-to-one holomorphic map $h_{\ast }$ of $\mathbb{C}$ onto itself satisfying

$$\tilde{p}\circ h_{\ast }(z)=h\circ p(z)\text{f o r a l l}z\in \mathbb{C}\text{\hspace{1em}(8.4)}$$

so that the diagram (8.3) commutes. It is known that a one-to-one holomorphic map of $\mathbb{C}$ onto itself must be of the form $z\to h_{\ast }(z)=az+b$ , where $a,b\in \mathbb{C}$ and $a\ne 0$ . We then have $h_{\ast }({\omega }_1)-h_{\ast }(0)=a{\omega }_1$ and $h_{\ast }({\omega }_2)-h_{\ast }(0)=a{\omega }_2$ . For $h$ to be well defined as a map of $\mathbb{C}/L({\omega }_1,{\omega }_2)$ onto $\mathbb{C}/L({\tilde{\omega }}_1,{\tilde{\omega }}_2)$ , we must have $a{\omega }_1,a{\omega }_2\in L({\tilde{\omega }}_1,{\tilde{\omega }}_2)$ , see figure 8.2. By changing the roles of $({\omega }_1,{\omega }_2)$ and $({\omega }_1^{\prime },{\omega }_2^{\prime })$ , we have $\tilde{a}{\tilde{\omega }}_1,\tilde{a}{\tilde{\omega }}_1\in L({\omega }_1,{\omega }_2)$ where $\tilde{a}\ne 0$ is a complex number. Hence, we conclude that if $\mathbb{C}/L({\omega }_1,{\omega }_2)$ , $\mathbb{C}/L({\tilde{\omega }}_1,{\tilde{\omega }}_2)$ have the same complex structure, there must be a matrix $M\in \mathrm{SL}(2,\mathbb{Z})$ and a complex number $\lambda (={\tilde{a}}^{-1})$ such that

$$\left(\begin{array}{c}{\tilde{\omega }}_1\\ {\tilde{\omega }}_2\end{array}\right)=\lambda M\left(\begin{array}{c}{\omega }_1\\ {\omega }_2\end{array}\right).\text{\hspace{1em}(8.5)}$$

Conversely, we verify that $({\omega }_1,{\omega }_2)$ and $({\omega }_1^{\prime },{\omega }_2^{\prime })$ related by (8.5) define the same complex structure. In fact,

$$\left(\begin{array}{c}{\omega }_1\\ {\omega }_2\end{array}\right)\text{a n d}M\left(\begin{array}{c}{\omega }_1\\ {\omega }_2\end{array}\right)$$

define the same lattice (modulo translation) and we may take $h_{\ast }:\mathbb{C}\to \mathbb{C}$ to be $z\mapsto z+b$ . $L({\omega }_1,{\omega }_2)$ and $L(\lambda {\omega }_1,\lambda {\omega }_2)$ also define the same complex structure. We take, in this case, $h_{\ast }:z\mapsto \lambda z+b$ .

We have shown that the complex structure on $T^2$ is defined by a pair of complex numbers $({\omega }_1,{\omega }_2)$ modulo a constant factor and $\mathrm{PSL}(2,\mathbb{Z})$ . To get rid of the constant factor, we introduce the modular parameter $\tau \equiv {\omega }_2/{\omega }_1\in \mathrm{H}\equiv \{z\in \mathbb{C}|\mathrm{Im}z>0\}$ , to specify the complex structure of $T^2$ . Without loss of generality, we take 1 and $\tau$ to be the generators of a lattice. Note, however, that not all of $\tau \in \mathrm{H}$ are independent modular parameters. As was shown previously, $\tau$ and ${\tau }^{\prime }=(a\tau +b)/(c\tau +d)$ define the same complex structure if

$$\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in \mathrm{PSL}(2,\mathbb{Z}).$$

The quotient space $\mathrm{H}/\mathrm{PSL}(2,\mathbb{Z})$ is shown in figure 8.3, the derivation of which can be found in Koblitz (1984) p 100, and Gunning (1962) p 4.

Figure 8.3. The quotient space $H/\mathrm{PSL}(2,\mathbb{Z})$ .

The change $\tau \to {\tau }^{\prime }$ is called the modular transformation and is generated by $\tau \to \tau +1$ and $\tau \to -1/\tau$ . The transformation $\tau \to \tau +1$ generates a Dehn twist along the meridian $m$ as follows (figure 8.4(a)). (i) First, cut a torus along $m$ . (ii) Then take one of the lips of the cut and rotate it by $2\pi$ with the other lip kept fixed. (iii) Then glue the lips together again. The other transformation $\tau \to -1/\tau$ corresponds to changing the roles of the longitude $l$ and the meridian $m$ (figure 8.4(b)).

(a)

(b)

Figure 8.4. (a) Dehn twists generate modular transformations. (b) $\tau \to -1/\tau$ changes the roles of $l$ and $m$ .

Example 8.3

The complex projective space ${\mathbb{CP}}^n$ is defined similarly to ${\mathbb{RP}}^n$ ; see example 5.4. The tuple $z=(z^0,\dots ,z^n)\in {\mathbb{C}}^{n+1}$ determines a complex line through the origin provided that $z\ne 0$ . Define an equivalence relation

by $z\sim w$ if there exists a complex number $a\ne 0$ such that $w=az$ . Then ${\mathbb{CP}}^n\equiv ({\mathbb{C}}^{n+1}-\{0\})/\sim$ . The $(n+1)$ numbers $z^0,z^1,\dots ,z^n$ are called the homogeneous coordinates, which is denoted by $[z^0,z^1,\dots ,z^n]$ where $(z^0,\dots ,z^n)$ is identified with $(\lambda z^0,\dots ,\lambda z^n)(\lambda \ne 0)$ . A chart $U_{\mu }$ is a subset of ${\mathbb{C}}^{n+1}-\{0\}$ such that $z^{\mu }\ne 0$ . In a chart $U_{\mu }$ , the inhomogeneous coordinates are defined by ${\xi }_{(\mu )}^{\nu }=z^{\nu }/z^{\mu }$ ( $\nu \ne \mu$ ). In $U_{\mu }\cap U_{\nu }\ne \varnothing$ , the coordinate transformation ${\psi }_{\mu \nu }:{\mathbb{C}}^n\to {\mathbb{C}}^n$ is

$${\xi }_{(\nu )}^{\lambda }\mapsto {\xi }_{(\mu )}^{\lambda }=\frac{z^{\nu }}{z\mu }{\xi }_{(\nu )}^{\lambda }.\text{\hspace{1em}(8.6)}$$

Accordingly, ${\psi }_{\mu \nu }$ is a multiplication by $z^{\nu }/z^{\mu }$ , which is, of course, holomorphic.

Example 8.4

The complex Grassmann manifolds $G_{k,n}(\mathbb{C})$ are defined similarly to the real Grassmann manifolds; see example 5.5. $G_{k,n}(\mathbb{C})$ is the set of complex $k$ -dimensional subspaces of ${\mathbb{C}}^n$ . Note that $\mathbb{C}{P}^n=G_{1,n+1}(\mathbb{C})$ .

Let $M_{k,n}(\mathbb{C})$ be the set of $k\times n$ matrices of rank $k$ ( $k\le n$ ). Take $A,B\in M_{k,n}(\mathbb{C})$ and define an equivalence relation by $A\sim B$ if there exists $g\in \mathrm{GL}(k,\mathbb{C})$ such that $B=gA$ . We identify $G_{k,n}(\mathbb{C})$ with $M_{k,n}(\mathbb{C})/\mathrm{GL}(k,\mathbb{C})$ . Let $\{A_1,\dots ,A_l\}$ be the collection of all the $k\times k$ minors of $A\in M_{k,n}(\mathbb{C})$ . We define the chart $U_{\alpha }$ to be a subset of $G_{k,n}(\mathbb{C})$ such that $\det A_{\alpha }\ne 0$ . The $k(n-k)$ coordinates on $U_{\alpha }$ are given by the non-trivial entries of the matrix $A_{\alpha }^{-1}A$ . See example 5.5 for details.

Example 8.5

The common zeros of a set of homogeneous polynomials are a compact submanifold of $\mathbb{C}{P}^n$ called an algebraic variety. For example, let $P(z^0,\dots ,z^n)$ be a homogeneous polynomial of degree $d$ . If $a\ne 0$ is a complex number, $P$ satisfies

$$P(a{z}^0,\dots ,a{z}^n)=a^{d}P(z^0,\dots ,z^n).$$

This shows that the zeros of $P$ are defined on $\mathbb{C}{P}^n$ ; if $P(z^0,\dots ,z^n)=0$ then $P([z^0,\dots ,z^n])=0$ . For definiteness, consider

$$P(z^0,z^1,z^2)=(z^0{)}^2+(z^1{)}^2+(z^2{)}^2$$

and define $N$ by

$$N=\{[z^0,z^1,z^2]\in \mathbb{C}{P}^2|P(z^0,z^1,z^2)=0\}.\text{\hspace{1em}(8.7)}$$

We define $U_{\mu }$ as in example 8.3. In $N\cap U_0$ , we have

$$[{\xi }_{(0)}^1{]}^2+[{\xi }_{(0)}^2{]}^2+1=0$$

where ${\xi }_{(0)}^{\mu }=z^{\mu }/z^0$ (note that $z^0\ne 0$ ). Consider a holomorphic change of coordinates $({\xi }_{(0)}^1,{\xi }_{(0)}^2)\mapsto ({\eta }^1={\xi }_{(0)}^1,{\eta }^2=[{\xi }_{(0)}^1{]}^2+[{\xi }_{(0)}^2{]}^2+1)$ . Note that $\partial ({\eta }^1,{\eta }^2)/\partial ({\xi }_{(0)}^1,{\xi }_{(0)}^2)\ne 0$ unless ${\xi }_{(0)}^2=z^2=0$ . Then $N\cap U_0\cap U_2=\{({\eta }^1,{\eta }^2)\in {\mathbb{C}}^2|{\eta }^2=0\}$ is clearly a one-dimensional submanifold of ${\mathbb{C}}^2$ . If ${\xi }_{(0)}^2=z^2=0$ , we have $({\xi }_{(0)}^1,{\xi }_{(0)}^2)\mapsto ({\zeta }^1=[{\xi }_{(0)}^1{]}^2+[{\xi }_{(0)}^2{]}^2+1,{\zeta }^2={\xi }_{(0)}^2)$ for which the Jacobian does not vanish unless ${\xi }_{(0)}^1=z^1=0$ . Then $N\cap U_0\cap U_1=\{({\zeta }^1,{\zeta }^2)\in {\mathbb{C}}^2|{\zeta }^1=0\}$ is a one-dimensional submanifold of ${\mathbb{C}}^2$ . On $N\cap U_0\cap U_1\cap U_2$ , the coordinate change ${\eta }^1\mapsto {\zeta }^2$ is a multiplication by $z^2/z^1$ and is, hence, holomorphic. In this way, we may define a one-dimensional compact submanifold $N$ of ${\mathbb{CP}}^2$ .

A complex manifold is a differentiable manifold. For example, ${\mathbb{C}}^m$ is regarded as ${\mathbb{R}}^{2m}$ by the identification $z^{\mu }=x^{\mu }+\mathrm{i}{y}^{\mu }$ , $x^{\mu },y^{\mu }\in \mathbb{R}$ . Similarly, any chart $U$ of a complex manifold has coordinates $(z^1,\dots ,z^m)$ which may be understood as real coordinates $(x^1,y^1,\dots ,x^m,y^m)$ . The analytic property of the coordinate transformation functions ensures that they are differentiable when the manifold is regarded as a $2m$ -dimensional differentiable manifold.

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