8.2 Calculus on complex manifolds

8.2.1 Holomorphic maps

Let $f:M\to N$ , $M$ and $N$ being complex manifolds with ${\dim }_{\mathbb{C}}M=m$ and ${\dim }_{\mathbb{C}}N=n$ . Take a point $p$ in a chart $(U,\phi )$ of $M$ . Let $(V,\psi )$ be a chart of $N$ such that $f(p)\in V$ . If we write $\{z^{\mu }\}=\phi (p)$ and $\{w^{\nu }\}=\psi (f(p))$ , we have a map $\psi \circ f\circ {\phi }^{-1}:{\mathbb{C}}^m\to {\mathbb{C}}^n$ . If each function $w^{\nu }$ $(1\le \nu \le n)$ is a holomorphic

function of $z^{\mu }$ , $f$ is called a holomorphic map. This definition is independent of the special coordinates chosen. In fact, let $(U^{\prime },{\phi }^{\prime })$ be another chart such that $U\cap U^{\prime }\ne \varnothing$ and $z^{\prime \mu }=x^{\prime \lambda }+\mathrm{i}{y}^{\prime \lambda }$ be the coordinates. Take a point $p\in U\cap U^{\prime }$ . If $w^{\nu }=u^{\nu }+\mathrm{i}{v}^{\nu }$ is a holomorphic function with respect to $z$ , then

$$\frac{\partial u^{\nu }}{\partial x^{\prime \lambda }}=\frac{\partial u^{\nu }}{\partial x^{\mu }}\frac{\partial x^{\mu }}{\partial x^{\prime \lambda }}+\frac{\partial u^{\nu }}{\partial y^{\mu }}\frac{\partial y^{\mu }}{\partial y^{\prime \lambda }}=\frac{\partial v^{\nu }}{\partial y^{\mu }}\frac{\partial y^{\mu }}{\partial y^{\prime \lambda }}+\frac{\partial v^{\nu }}{\partial x^{\mu }}\frac{\partial x^{\mu }}{\partial y^{\prime \lambda }}=\frac{\partial v^{\nu }}{\partial y^{\prime \lambda }}.$$

We also find $\partial u^{\nu }/\partial y^{\prime \lambda }=-\partial v^{\nu }/\partial x^{\prime \lambda }$ . Thus, $w^{\nu }$ is holomorphic with respect to $z^{\prime }$ too. It can be shown that the holomorphic property is also independent of the choice of chart in $N$ .

Let $M$ and $N$ be complex manifolds. We say $M$ is biholomorphic to $N$ if there exists a diffeomorphism $f:M\to N$ which is also holomorphic (then $f^{-1}:N\to M$ is automatically holomorphic). The map $f$ is called a biholomorphism.

A holomorphic function is a holomorphic map $f:M\to \mathbb{C}$ . There is a striking theorem; any holomorphic function on a compact complex manifold is constant. This is a generalization of the maximum principle of elementary complex analysis, see Wells (1980). The set of holomorphic functions on $M$ is denoted by $\mathcal{O}(M)$ . Similarly, $\mathcal{O}(U)$ is the set of holomorphic functions on $U\subset M$ .

8.2.2 Complexifications

Let $M$ be a differentiable manifold with ${\dim }_{\mathbb{R}}M=m$ . If $f:M\to \mathbb{C}$ is decomposed as $f=g+\mathrm{i}h$ where $g,h\in \mathcal{F}(M)$ , then $f$ is a complex-valued smooth function. The set of complex-valued smooth functions on $M$ is called the complexification of $\mathcal{F}(M)$ , denoted by $\mathcal{F}(M{)}^{\mathbb{C}}$ . A complexified function does not satisfy the Cauchy-Riemann relation in general. For $f=g+\mathrm{i}h\in \mathcal{F}(M{)}^{\mathbb{C}}$ , the complex conjugate of $f$ is $\overline{f}\equiv g-\mathrm{i}h$ . $f$ is real if and only if $f=\overline{f}$ .

Before we consider the complexification of $T_{p}M$ , we define the complexification $V^{\mathbb{C}}$ of a general vector space $V$ with ${\dim }_{\mathbb{R}}V=m$ . An element of $V^{\mathbb{C}}$ takes the form $X+\mathrm{i}Y$ where $X,Y\in V$ . The vector space $V^{\mathbb{C}}$ becomes a complex vector space of complex dimension $m$ if the addition and the scalar multiplication by a complex number $a+\mathrm{i}b$ are defined by

$$\begin{array}{rl}& \left(X_1+\mathrm{i}{Y}_1\right)+\left(X_2+\mathrm{i}{Y}_2\right)=\left(X_1+X_2\right)+\mathrm{i}\left(Y_1+Y_2\right)\\ & (a+\mathrm{i}b)(X+\mathrm{i}Y)=(aX-bY)+\mathrm{i}(bX+aY)\end{array}$$

$V$ is a vector subspace of $V^{\mathbb{C}}$ since $X\in V$ and $X+\mathrm{i}0\in V^{\mathbb{C}}$ may be identified. Vectors in $V$ are said to be real. The complex conjugate of $Z=X+\mathrm{i}Y$ is $\overline{Z}=X-\mathrm{i}Y$ . A vector $Z$ is real if $Z=\overline{Z}$ .

A linear operator $A$ on $V$ is extended to act on $V^{\mathbb{C}}$ as

$$A(X+\mathrm{i}Y)=A(X)+\mathrm{i}A(Y).\text{\hspace{1em}(8.8)}$$

If $A\to \mathbb{R}$ is a linear function $(A\in V^{\ast })$ , its extension is a complex-valued linear function on $V^{\mathbb{C}}$ , $A:V^{\mathbb{C}}\to \mathbb{C}$ . In general, any tensor defined on $V$ and $V^{\ast }$ is extended so that it is defined on $V^{\mathbb{C}}$ and $(V^{\ast }{)}^{\mathbb{C}}$ . An extended tensor is complexified as $t=t_1+\mathrm{i}{t}_2$ , where $t_1$ and $t_2$ are tensors of the same type. The conjugate of $t$ is $\overline{t}\equiv t_1-\mathrm{i}{t}_2$ . If $t=\overline{t}$ , the tensor is said to be real. For example $A:V^{\mathbb{C}}\to \mathbb{C}$ is real if $\overline{A(X+\mathrm{i}Y)}=A(X-\mathrm{i}Y)$ .

Let $\{e_k\}$ be a basis of $V$ . If the basis vectors are regarded as complex vectors, the same basis $\{e_k\}$ becomes a basis of $V^{\mathbb{C}}$ . To see this, let $X=X^k{e}_k$ , $Y=Y^k{e}_k\in V$ . Then $Z=X+\mathrm{i}Y$ is uniquely expressed as $(X^k+\mathrm{i}{Y}^k)e_k$ . We find ${\dim }_{\mathbb{R}}V={\dim }_{\mathbb{C}}{V}^{\mathbb{C}}$ .

Now we are ready to complexify the tangent space $T_{p}M$ . If $V$ is replaced by $T_{p}M$ , we have the complexification $T_p{M}^{\mathbb{C}}$ of $T_{p}M$ , whose element is expressed as $Z=X+\mathrm{i}Y$ ( $X,Y\in T_{p}M$ ). The vector $Z$ acts on a function $f=f_1+\mathrm{i}{f}_2\in \mathcal{F}(M{)}^{\mathbb{C}}$ as

$$\begin{array}{rl}Z[f]& =X[f_1+\mathrm{i}{f}_2]+\mathrm{i}Y[f_1+\mathrm{i}{f}_2]\\ & =X[f_1]-Y[f_2]+\mathrm{i}\{X[f_2]+Y[f_1]\}.\end{array}\text{\hspace{1em}(8.9)}$$

The dual vector space $T_p^{\ast }M$ is complexified if $\omega ,\eta \in T_p^{\ast }M$ are combined as $\zeta =\omega +\mathrm{i}\eta$ . The set of complexified dual vectors is denoted by $(T_p^{\ast }M{)}^{\mathbb{C}}$ . Any tensor $t$ is extended so that it is defined on $T_p{M}^{\mathbb{C}}$ and $(T_p^{\ast }M{)}^{\mathbb{C}}$ and then complexified.

Exercise 8.1

Show that $(T_p^{\ast }M{)}^{\mathbb{C}}=(T_p{M}^{\mathbb{C}}{)}^{\ast }$ . From now on, we denote the complexified dual vector space simply by $T_p^{\ast }{M}^{\mathbb{C}}$ .

Given smooth vector fields $X,Y\in \mathcal{X}(M)$ , we define a complex vector field $Z=X+\mathrm{i}Y$ . Clearly $Z{|}_p\in T_p{M}^{\mathbb{C}}$ . The set of complex vector fields is the complexification of $\mathcal{X}(M)$ and is denoted by $\mathcal{X}(M{)}^{\mathbb{C}}$ . The conjugate vector field of $Z=X+\mathrm{i}Y$ is $\overline{Z}=X-\mathrm{i}Y$ . $Z=\overline{Z}$ if $Z\in \mathcal{X}(M)$ , hence $\mathcal{X}(M{)}^{\mathbb{C}}\supset \mathcal{X}(M)$ . The Lie bracket of $Z=X+\mathrm{i}Y$ , $W=U+\mathrm{i}V\in \mathcal{X}(M{)}^{\mathbb{C}}$ is

$$[X+\mathrm{i}Y,U+\mathrm{i}V]=\{[X,U]-[Y,V]\}+\mathrm{i}\{[X,V]+[Y,U]\}.\text{\hspace{1em}(8.10)}$$

The complexification of a tensor field of type $(p,q)$ is defined in an obvious manner. If $\omega ,\eta \in {\Omega }^1(M),\xi \equiv \omega +\mathrm{i}\eta \in {\Omega }^1(M{)}^{\mathbb{C}}$ is a complexified one-form.

8.2.3 Almost complex structure

Since a complex manifold is also a differentiable manifold, we may use the framework developed in chapter 5. We then put appropriate constraints on the results. Let us look at the tangent space of a complex manifold $M$ with ${\dim }_{\mathbb{C}}M=m$ . The tangent space $T_{p}M$ is spanned by $2m$ vectors

$$\left\{\frac{\partial }{\partial x^1},\dots ,\frac{\partial }{\partial x^m};\frac{\partial }{\partial y^1},\dots ,\frac{\partial }{\partial y^m}\right\}\text{\hspace{1em}(8.11)}$$

where $z^{\mu }=x^{\mu }+\mathrm{i}{y}^{\mu }$ are the coordinates of $p$ in a chart $(U,\phi )$ . With the same coordinates, $T_p^{\ast }M$ is spanned by

$$\left\{\mathrm{d}{x}^1,\dots ,\mathrm{d}{x}^m;\mathrm{d}{y}^1,\dots ,\mathrm{d}{y}^m\right\}.\text{\hspace{1em}(8.12)}$$

Let us define $2m$ vectors

$$\frac{\partial }{\partial z^{\mu }}\equiv \frac{1}{2}\left\{\frac{\partial }{\partial x^{\mu }}-\mathrm{i}\frac{\partial }{\partial y^{\mu }}\right\}\text{\hspace{1em}(8.13a)}$$

$$\frac{\partial }{\partial {\bar{z}}^{\mu }}\equiv \frac{1}{2}\left\{\frac{\partial }{\partial x^{\mu }}+\mathrm{i}\frac{\partial }{\partial y^{\mu }}\right\}\text{\hspace{1em}(8.13b)}$$

where $1\le \mu \le m$ . Clearly they form a basis of the $2m$ -dimensional (complex) vector space $T_p{M}^{\mathbb{C}}$ . Note that $\overline{\partial /\partial z^{\mu }}=\partial /\partial {\overline{z}}^{\mu }$ . Correspondingly, $2m$ one-forms

$$\mathrm{d}{z}^{\mu }\equiv \mathrm{d}{x}^{\mu }+\mathrm{id}{y}^{\mu }\mathrm{d}{\bar{z}}^{\mu }\equiv \mathrm{d}{x}^{\mu }-\mathrm{id}{y}^{\mu }\text{\hspace{1em}(8.14)}$$

form the basis of $T_p^{\ast }{M}^{\mathbb{C}}$ . They are dual to (8.13),

$$⟨\mathrm{d}{z}^{\mu },\partial /\partial {\bar{z}}^{\nu }⟩=⟨\mathrm{d}{\bar{z}}^{\mu },\partial /\partial z^{\nu }⟩=0\text{\hspace{1em}(8.15a)}$$

$$⟨\mathrm{d}{z}^{\mu },\partial /\partial z^{\nu }⟩=⟨\mathrm{d}{\bar{z}}^{\mu },\partial /\partial {\bar{z}}^{\nu }⟩={\delta }_{\nu }.\text{\hspace{1em}(8.15b)}$$

Let $M$ be a complex manifold and define a linear map $J_p:T_{p}M\to T_{p}M$ by

$$J_p\left(\frac{\partial }{\partial x^{\mu }}\right)=\frac{\partial }{\partial y^{\mu }}{J}_p\left(\frac{\partial }{\partial y^{\mu }}\right)=-\frac{\partial }{\partial x^{\mu }}\text{\hspace{1em}(8.16)}$$

$J_p$ is a real tensor of type (1, 1). Note that

$$J_p^2=-{\mathrm{id}}_{T_{p}M}.\text{\hspace{1em}(8.17)}$$

Roughly speaking, $J_p$ corresponds to the multiplication by $\pm \mathrm{i}$ . The action of $J_p$ is independent of the chart. In fact, let $(U,\phi )$ and $(V,\psi )$ be overlapping charts with $\phi (p)=z^{\mu }=x^{\mu }+\mathrm{i}{y}^{\mu }$ and $\psi (p)=w^{\mu }=u^{\mu }+\mathrm{i}{v}^{\mu }$ . On $U\cap V$ , the functions $z^{\mu }=z^{\mu }(w)$ satisfy the Cauchy-Riemann relations. Then we find

$$J_p\left(\frac{\partial }{\partial u^{\mu }}\right)=J_p\left(\frac{\partial x^{\nu }}{\partial u^{\mu }}\frac{\partial }{\partial x^{\nu }}+\frac{\partial y^{\nu }}{\partial u^{\mu }}\frac{\partial }{\partial y^{\nu }}\right)=\frac{\partial y^{\nu }}{\partial v^{\mu }}\frac{\partial }{\partial y^{\nu }}+\frac{\partial x^{\nu }}{\partial v^{\mu }}\frac{\partial }{\partial x^{\nu }}=\frac{\partial }{\partial v^{\mu }}.$$

We also find that $J_p\partial /\partial v^{\mu }=-\partial /\partial u^{\mu }$ . Accordingly, $J_p$ takes the form

$$J_p=\left(\begin{array}{cc}0& -I_m\\ I_m& 0\end{array}\right)\text{\hspace{1em}(8.18)}$$

with respect to the basis (8.11), where $I_m$ is the $m\times m$ unit matrix. Since all the components of $J_p$ are constant at any point, we may define a smooth tensor field $J$ whose components at $p$ are (8.18). The tensor field $J$ is called the almost

complex structure of a complex manifold $M$ . Note that any $2m$ -dimensional manifold locally admits a tensor field $J$ which squares to $-I_{2m}$ . However, $J$ may be patched across charts and defined globally only on a complex manifold. The tensor $J$ completely specifies the complex structure.

The almost complex structure $J_p$ is extended so that it may be defined on $T_p{M}^{\mathbb{C}}$ ,

$$J_p(X+\mathrm{i}Y)\equiv J_{p}X+\mathrm{i}{J}_{p}Y.\text{\hspace{1em}(8.19)}$$

It follows from (8.16) that

$$J_p\partial /\partial z^{\mu }=\mathrm{i}\partial /\partial z^{\mu }{J}_p\partial /\partial {\bar{z}}^{\mu }=-\mathrm{i}\partial /\partial {\bar{z}}^{\mu }.\text{\hspace{1em}(8.20)}$$

Thus, we have an expression for $J_p$ in (anti-)holomorphic bases,

$$J_p=\mathrm{id}{z}^{\mu }\otimes \frac{\partial }{\partial z^{\mu }}-\mathrm{id}{\bar{z}}^{\mu }\otimes \frac{\partial }{\partial {\bar{z}}^{\mu }}\text{\hspace{1em}(8.21)}$$

whose components are given by

$$J_p=\left(\begin{array}{cc}\mathrm{i}{I}_m& 0\\ 0& -\mathrm{i}{I}_m\end{array}\right).\text{\hspace{1em}(8.22)}$$

Let $Z\in T_p{M}^{\mathbb{C}}$ be a vector of the form $Z=Z^{\mu }\partial /\partial z^{\mu }$ . Then $Z$ is an eigenvector of $J_p$ ; $J_{p}Z=\mathrm{i}Z$ . Similarly, $Z=Z^{\mu }\partial /\partial {\overline{z}}^{\mu }$ satisfies $J_{p}Z=-\mathrm{i}Z$ . In this way $T_p{M}^{\mathbb{C}}$ of a complex manifold is separated into two disjoint vector spaces,

$$T_p{M}^{\mathbb{C}}=T_p{M}^{+}\oplus T_p{M}^{-}\text{\hspace{1em}(8.23)}$$

where

$$T_p{M}^{\pm }=\{Z\in T_p{M}^{\mathbb{C}}|J_{p}Z=\pm \mathrm{i}Z\}.\text{\hspace{1em}(8.24)}$$

We define the projection operators ${\mathcal{P}}^{\pm }:T_p{M}^{\mathbb{C}}\to T_p{M}^{\pm }$ by

$${\mathcal{P}}^{\pm }\equiv \frac{1}{2}\left(I_{2m}\mp \mathrm{i}{J}_p\right).\text{\hspace{1em}(8.25)}$$

In fact, $J_p{\mathcal{P}}^{\pm }Z=\frac{1}{2}(J_p\mp \mathrm{i}{J}_p^2)Z=\pm \mathrm{i}{\mathcal{P}}^{\pm }Z$ for any $Z\in T_p{M}^{\mathbb{C}}$ . Hence,

$$Z^{\pm }\equiv {\mathcal{P}}^{\pm }Z\in T_p{M}^{\pm }.\text{\hspace{1em}(8.26)}$$

Now $Z\in T_p{M}^{\mathbb{C}}$ is uniquely decomposed as $Z=Z^{+}+Z^{-}$ ( $Z^{\pm }\in T_p{M}^{\pm }$ ). $T_p{M}^{+}$ is spanned by $\{\partial /\partial z^{\mu }\}$ and $T_p{M}^{-}$ by $\{\partial /\partial {\overline{z}}^{\mu }\}$ . $Z\in T_p{M}^{+}$ is called a holomorphic vector while $Z\in T_p{M}^{-}$ is called an anti-holomorphic vector. We readily verify that

$$T_p{M}^{-}=\overline{T_p{M}^{+}}=\{\bar{Z}|Z\in T_p{M}^{+}\}.\text{\hspace{1em}(8.27)}$$

Note that

$${\dim }_{\mathbb{C}}{T}_p{M}^{+}={\dim }_{\mathbb{C}}{T}_p{M}^{-}=\frac{1}{2}{\dim }_{\mathbb{C}}{T}_p{M}^{\mathbb{C}}=\frac{1}{2}{\dim }_{\mathbb{C}}M.$$

Exercise 8.2

Let $(U,\phi )$ and $(V,\psi )$ be overlapping charts on a complex manifold $M$ and let $z^{\mu }=\phi (p)$ and $w^{\mu }=\psi (p)$ . Verify that $X=X^{\mu }\partial /\partial z^{\mu }$ , expressed in the coordinates $w^{\mu }$ , contains a holomorphic basis $\{\partial /\partial w^{\mu }\}$ only. Thus, the separation of $T_p{M}^{\mathbb{C}}$ into $T_p{M}^{\pm }$ is independent of charts (note that $J$ is defined independently of charts).

Given a complexified vector field $Z\in \mathcal{X}(M{)}^{\mathbb{C}}$ , we obtain a new vector field $JZ\in \mathcal{X}(M{)}^{\mathbb{C}}$ defined at each point of $M$ by $JZ{|}_p=J_p\cdot Z{|}_p$ . The vector field $Z$ is naturally separated as

$$Z=Z^{+}+Z^{-}{Z}^{\pm }={\mathcal{P}}^{\pm }Z\text{\hspace{1em}(8.28)}$$

where $Z^{\pm }={\mathcal{P}}^{\pm }Z$ . The vector field $Z^{+}\left(Z^{-}\right)$ is called a holomorphic (antiholomorphic) vector field. Accordingly, once $J$ is given, $\mathfrak{X}(M{)}^{\mathbb{C}}$ is decomposed uniquely as

$$\mathcal{X}(M{)}^{\mathbb{C}}=\mathcal{X}(M{)}^{+}\oplus \mathcal{X}(M{)}^{-}.\text{\hspace{1em}(8.29)}$$

$Z=Z^{+}+Z^{-}\in \mathfrak{X}(M{)}^{\mathbb{C}}$ is real if and only if $Z^{+}=\overline{Z^{-}}$

Exercise 8.3

Let $X,Y\in \mathcal{X}(M{)}^{+}$ . Show that $[X,Y]\in \mathcal{X}(M{)}^{+}$ . [If $X,Y\in \mathcal{X}(M{)}^{-}$ , then $[X,Y]\in \mathcal{X}(M{)}^{-}.]$

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