Geometry and Topology of Complex Manifolds - Lecture 1

This is the first post in a series on the content from the course taught Winter 2025 at Stanford University on Math 270: Geometry and Topology of Complex Manifolds by Professor Eleny Ionel. The assumed background is the content of Math 215ABC, i.e. the basics of algebraic topology, differential topology, and Riemannian geometry.

The topics we plan to cover include the following.

Complex and Kähler Manifolds
Complex vector bundles, holomorphic vector bundles (connections and curvature)
Dolbeault cohomology, Hodge theorem, Lefschetz theorem
Several vanishing theorems (e.g. Kodaira vanishing)
Kähler-Einstein metrics on manifolds and Hermitian-Einstein metrics on holomorphic vector bundles.

Second Perspective on Complex Manifolds

One might be able to guess the usual way of defining complex manifolds, which is the usual manifold axioms along with local charts in $\bbc^n$ such that the transition maps are holomorphic, blah blah blah. We will introduce a second perspective on complex manifolds that will prove useful.

Linear Algebra Motivation

We make the following observation. A complex vector space $V$ is equivalent to a real vector space $V$ along with some $J \in \End_{\bbr}(V)$ such that $J^2 = - \id_V$ , called a complex structure. Given a complex vector space $V$ , we can let $J$ be multiplication by $i$ . Given $(V,J)$ as above, we can get a natural $\bbc$ -structure on $V$ by letting

(a+bi)v = av + bJv

for all $a,b \in \bbr$ and $v \in V$ .

Example Let $M = \bbc^n$ , $p \in M$ and $V = T_p M$ with local coordinates $z = (z_1, \ldots, z_n)$ . Then each $z_k = x_k + i y_k$ where the $x,y$ are real-valued coordinates, and so $T_p M = \bbr^n \times \bbr^n$ with the local coordinates

\left\{\frac{\del}{\del x_1}, \ldots, \frac{\del}{\del x_n}, \frac{\del}{\del y_1}, \ldots, \frac{\del}{\del y_n}\right\}

Define the following complex structure on $T_pM$

J = \begin{pmatrix}0 & I_n \\ -I_n & 0 \end{pmatrix} \in \End(V)

Unimportant Remark Note that $J$ satisfies the following equations which resemble a PDE.

J \frac{\del}{\del x_k} = \frac{\del}{\del y_k} \qquad J \frac{\del}{\del y_k} = -\frac{\del}{\del x_k}

We will see later that on manifolds whose tangent bundle has a complex structure $J$ , we can always find coordinates around a point such that $J$ takes the above canonical form and satisfies these equations. This might not be true in a neighborhood of $p$ , but if it is then $J$ is said to be integrable.

Ignore this for now, but note that this is the reason for the terminology almost complex structure below. If a manifold has an almost complex structure and is integrable, then it has a complex structure (i.e. it is a complex manifold).

Important Remark Note that this extends naturally to families, i.e. to vector bundles, by defining everything fiberwise. That is, a complex vector bundle $\cale$ over a manifold $M$ is a real vector bundle $\cale$ over $M$ together with a $J \in \End(\cale)$ such that for every $p \in M$ , the map $J_p \colon \cale_p \to \cale_p$ on the fibers satisfies $J_p^2 = -\id$ . Because the notation will get annoying, for $v \in \cale_p$ an element of the fiber over $p \in M$ we write $Jv$ instead of $J_p v$ .

Almost Complex Structure

Definition An almost complex manifold (ACM) is a manifold $M$ with an almost complex structure (ACS), i.e. a complex structure $J \in \End(TM)$ on the tangent bundle $TM$ .

For example, if $M$ is a complex manifold then multiplying by $i$ in each local holomorphic chart induces a well-defined almost complex structure on $M$ . The details are easily obtained from the $M = \bbc^n$ example in the previous subsection.

Remark Suppose $M$ is a complex manifold, $J$ an ACS on $M$ , and $f \colon M \to \bbc$ a function. Then the following are equivalent.

$f$ is holomorphic
$df \colon TM \to T\bbc$ is complex linear, i.e. $df \circ J = J_0 \circ df$ where $J_0$ is the canonical ACS on $T \bbc$
$f = u + iv$ are the real and imaginary components, $du \circ J_0 = dv$
The $u,v$ above satisfy the Cauchy-Riemann equations in each local chart, i.e.

\frac{\del u}{\del x_k} = \frac{\del v}{\del y_k} \qquad \frac{\del u}{\del y_k} = -\frac{\del v}{\del x_k}

Differential Calculus on Almost Complex Manifolds

Note that $J \in \End(TM)$ being such that $J^2 = -\id$ on the fibers implies that $J$ has eigenvalues $\pm i$ . Considering the eignspaces will allow us to decompose almost complex manifolds into direct sums, which will prove useful. I'll expand on this with some motivation from linear algebra first.

Linear Algebra Motivation

Let $V$ be a real vector space. We can construct the complexification of $V$ by taking $V^{\bbc} = V \otimes_{\bbr} \bbc$ . We denote elements of $V^{\bbc}$ by $v + iw$ , which is shorthand for $v \otimes 1 + w \otimes i$ . This is naturally a complex vector space.

Let $(V,J)$ be a complex vector space, i.e. $V$ is a real vector space and $J$ is a complex structure on $V$ . Note then that $J$ extends to $V^{\bbc}$ by $J(v + iw) = Jv + i Jw$ . As an aside, we denote the conjugate $(V,-J)$ by $\bar{V}$ .

Since $J$ has two eigenvalues on $V^{\bbc}$ , namely $\pm i$ , there are two eigenspaces

V^{1,0} = \{v \mid Jv = iv\} = \{v - i Jv \mid v \in V\}

V^{0,1} = \{v \mid Jv = -iv\} = \{v + i J v \mid v \in V\}

corresponding to the eigenvalues $i$ and $-i$ of $J$ , respectively. Note that $V^{1,0} = \bar{V^{0,1}}$ are conjugate.

Important There is a natural decomposition of $V^{\bbc}$ into a direct sum of its eigenspaces.

V^{\bbc} = V^{1,0} \oplus V^{1,0}

This again extends in families, i.e. to vector bundles by applying the definitions fiberwise. That is, if $(\cale, J)$ is a complex vector bundle then we can extend $\cale^{\bbc} = \cale \otimes_{\bbr} \bbc$ and $J$ extends on the fibers $\cale^{\bbc}_p$ as before. We then have a decomposition

\cale^{\bbc} = \cale^{1,0} \oplus \cale^{0,1}

where

\cale^{1,0} = \{(p, v - i J v) \mid p \in M, v \in \cale_p\}

\cale^{0,1} = \{(p, v + i J v) \mid p \in M, v \in \cale_p\}

are the eigenbundles corresponding to $i$ and $-i$ , respectively.

Holomorphic Tangent Bundle

We can apply the above motivation to the setting of a manifold $M$ with an almost complex structure $J$ . Note $(TM, J)$ is a complex vector bundle, so we can complexify it to $TM^{\bbc} = TM \otimes_{\bbr} \bbc$ . As above, we have a decomposition

TM^{\bbc} = T^{1,0} M \oplus T^{0,1} M

Since the notation will get particularly bad otherwise, let us denote the cotangent bundle $T^*M$ by $\Omega M$ .

Given a complex structure $J$ on $TM$ , we can obtain a complex structure $J^*$ on $\Omega M$ by $J^* \omega = -\omega \circ J$ for $\omega \in \Omega_p M = T^*_p M$ . The negative sign is not necessary but its a useful convention.

Then $(\Omega M, J^*)$ is a complex vector bundle, so we can complexify it just like $(TM, J)$ . Let

\Omega M^{\bbc} = \text{ space of } \bbc \text{-valued functions } = (T^* M)^{\bbc}

We really want to think of this as the space of complex-valued differential 1-forms on $M$ . As before,

\Omega M^{\bbc} = \Omega^{1,0} M \oplus \Omega^{0,1}M

Let's unravel the definitions here real quick.

\begin{align*} \Omega^{1,0} M &= \{(p, \omega - i J^* \omega) \mid p \in M, \omega \in \Omega_p M\} \\ &= \{(p, \omega + i \omega \circ J ) \mid p \in M, \omega \in \Omega_p M\} \end{align*}

\begin{align*} \Omega^{0,1} M &= \{(p, \omega + i J^* \omega) \mid p \in M, \omega \in \Omega_p M\} \\ &= \{(p, \omega - i \omega \circ J ) \mid p \in M, \omega \in \Omega_p M\} \end{align*}

Let us extend this to the space of $\bbc$ -valued differential $k$ -forms $\Omega^k M^{\bbc}$ . Recall $\Omega^k M = \bigwedge^k \Omega M$ and naturally we define $\Omega^k M^{\bbc} = \Omega^k M \otimes_{\bbr} \bbc$ . Since complexifying commutes with the wedge product, we have

\begin{align*} \Omega^k M^{\bbc} &= \left(\wedge^k \Omega M\right)^{\bbc} \\ &= \wedge^k \Omega M^{\bbc} \\ &= \wedge^k \left(\Omega^{1,0} M \oplus \Omega^{0,1} M\right) \\ &= \oplus_{p + q = k} \left(\wedge^p\Omega^{1,0} M \otimes \wedge^q\Omega^{0,1} M \right) \\ &= \oplus_{p+q = k} \Omega^{p,q} M \end{align*}

where $\Omega^{p,q} M$ is called the space of complex-valued differential forms of type $(p,q)$ on $M$ .

\Omega^{p,q} M = \underbrace{\Omega^{1,0} M \wedge \ldots \wedge \Omega^{1,0} M}_{p \text{ times}} \otimes \underbrace{\Omega^{0,1} M \wedge \ldots \wedge \Omega^{0,1} M}_{q \text{ times}}

Let's denote $A^k M = \Gamma(\Omega^k M^{\bbc})$ by the space of smooth sections of complex-valued differential $k$ -forms on $M$ , and similarly $A^{p,q} = \Gamma(\Omega^{p,q} M)$ for $(p,q)$ -forms.

Definition If $M$ is a complex manifold, we have the following names for the spaces defined above.

$T^{1,0}M$ is called the holomorphic tangent bundle,
$T^{0,1}M$ is the anti-holomorphic tangent bundle,
$\Omega^{1,0} M$ is the holomorphic cotangent bundle, and, lastly
$\Omega^{0,1} M$ is the anti-holomorphic cotangent bundle.

We want to be able to do calculations, so it helps to know what these spaces look like locally. Let $M$ be a complex manifold with local coordinate $z = (z_1, \ldots, z_n)$ . A complex-valued $1$ -form can be written locally uniquely as

\eta = \underbrace{\sum_{i} \eta_i dz_i}_{\text{holomorphic part}} + \underbrace{\sum_j \eta_j d\bar{z}_j}_{\text{anti-holomorphic part}}

where $\eta_i, \eta_j \in C^{\infty}(M, \bbc)$ . Similarly, a complex-valued $(p,q)$ form can be written locally uniquely as

\eta = \sum_{I, J} \eta_{I, J} dz_I \wedge d\bar{z}_J

where $I$ is the multiindex $i_1 < \ldots < i_p$ (and $J$ is $j_1 < \ldots < j_q$ ), and where $dz_I = dz_{i_1} \wedge \ldots \wedge dz_{i_p}$ (and $d\bar{z}_J = d\bar{z}_{j_1} \wedge \ldots \wedge d\bar{z}_{j_q}$ ). As before, $\eta_{I,J} \in C^{\infty}(M, \bbc)$ .

In the coming lectures we will expand on this. Namely, we can consider what happens for example when $\eta_{I,J}$ take values in spaces other than $\bbc$ , or are holomorphic rather than smooth. We will also define the Dolbeault complex consisting of sections $A^{p,q}$ which leads us naturally into the topic of Dolbeault cohomology.