Back to main page

Definition 2.1.1 (holomorphic atlas)

[definition] A holomorphic atlas on a differentiable manifold is an atlas $\{(U_i, \phi_i)\}$ of the form $\phi_i:U_i \simeq \phi_i(U_i) \subseteq \CC^n$, such that the transition functions

are holomorphic. The pair $(U_i, \phi_i)$ is called a holomorphic chart. Two holomorphic atlases $\{(U_i, \phi_i)\}$ and $\{(U_j’, \phi_j’)\}$ are called equivalent if all maps $\phi_i \circ \phi_j’^{-1}$ are holomorphic. [/definition]

Definition 2.1.2 (complex manifold)

[definition] A complex manifold $X$ of dimension $n$ is a real differentiable manifold of dimension $2n$ endowed with an equivalence class of holomorphic atlases. [/definition]

Topological convention

In addition to the above, we say that $X$ is connected, compact, simply-connected, etc. if the underlying topological manifold has said property.

Dimension naming convention

We also impose the convention that complex manifolds of dimension one (two, three, …) is called a complex curve (a complex surface, a complex threefold, …, respectively).

Definition 2.1.3 (holomorphic functions $X \to \CC$)

[definition] A holomorphic function on a complex manifold $X$ is a function $f: X \to \CC$ such that $f\circ \phi_{i}^{-1} : \phi_i(U_i) \to \CC$ is holomorphic for any chart $(U_i, \phi_i)$ of a holomorphic atlas in the equivalence class defining $X$. [/definition]

Definition 2.1.4 (sheaf of holomorphic functions)

[definition] Let $X$ be a complex manifold. By $\mathcal{O}_X$ we denote the sheaf of holomorphic functions on $X$, i.e., for any open subsets $U \subseteq X$ one has

[/definition]

Stalk isomorphism $\mathcal{O}_{X,x} \cong \mathcal{O}_{\CC^n, 0}$

There is an obvious isomorphism (as $\CC$-algebras) of stalks $\mathcal{O}_{X,x} \cong \mathcal{O}_{\CC^n, 0}$ by taking a holomorphic chart $(U, \phi)$ with $x \in U$ and $\phi(x) = 0$ by pulling back via $\phi$.

Quotient field of $\mathcal{O}_{X, x}$

Since $\mathcal{O}_{X, x} \cong \mathcal{O}_{\CC^n, 0}$, and the latter is an integral domain (in fact, its a local UFD by Proposition 1.1.15), we can take its quotient field which we denote by $Q(\mathcal{O}_{X, x})$. Thus, if $U \subseteq X$ is an open connected subset, then holomorphic functions $g, h : U\to \CC$, with $h \not\equiv 0$, define elements $g, h \in \mathcal{O}_{X, x}$ and $g/h \in Q(\mathcal{O}_{X, x})$ for all $x \in U$.

Proposition 2.1.5

[proposition] Let $X$ be a compact connected complex manifold. Then $\Gamma(X, \mathcal{O}_X) = \CC$, i.e. any global holomorphic function on $X$ is constant. [/proposition]

[proof] Let $f \in \mathcal{O}_X$ and $x \in X$ be a point where $f$ attains its maximum. There is a chart $(U, \phi)$ around $x$ in a holomorphic atlas in the equivalence class defining $X$. Then

is holomorphic. Taking $U_i$ to be sufficiently small, we can assume that $U_i$ is connected and so $f\circ \phi^{-1}$ is locally constant by the maximum principle on $\phi(U_i)$. But since $X$ is connected, it follows that $f$ is constant. [/proof]

Contrast to real differentiable manifolds

This result is obviously false compared to real differentiable manifolds. An counterexample is to consider $S^1$ as a real differentiable manifold. Then the map $f:S^1 \to \RR$ defined by $f(e^{i\theta}) = \sin\theta$ is smooth.

Indeed, we can cover $S^1$ by the sets

where $\phi_i(e^{i\theta}) = \theta$ are the chart maps. Now since

and

are smooth, $f$ is smooth and so the claim follows.

Corollary 2.1.6

[corollary] Let $X$ be a complex manifold of dimension at least two and let $x \in X$. Then

If $X$ is, in addition, compact and connected, then

[/corollary]

[proof] Clearly any global section restricts to a section of $\mathcal{O}_X$ over $X \setminus x$ so we must have $\subseteq$. For $\supseteq$, we take a chart $(U, \phi)$ in a holomorphic atlas in the equivalence class defining $X$ so that $x \in U$ and $\phi(x) = 0$. Then $(U\setminus x, \phi)$ is a chart in a holomorphic atlas in the equivalence class defining $X \setminus x$. Furthermore, we can take $U$ to be sufficiently small so that $\phi(U) = B_\eps(0)$. So, then we have $\phi(U \setminus x) = B_\eps(0) \setminus 0$ and taking $\eps’$ to be sufficient small in each coordinate gives an extension of $f\circ \phi^{-1}$, where $f\in \Gamma(X \setminus x, \mathcal{O}_X)$, to $B_\eps(0)$ by Hartogs’ theorem.

The second claim follows immediate from Proposition 2.1.5 and the first claim. [/proof]

Partitions of unity

Notice that partitions of unity on a complex manifold are significantly more limited in use in complex geometry — the identity theorem in conjunction with Corollary 2.1.6 implies that (holomorphic) partitions of unity would be forced to be constant.

Definition 2.1.7 (holomorphic maps of manifolds)

[definition] Let $X$ and $Y$ be complex manifolds. A continuous map $f:X \to Y$ is a holomorphic map if, for any holomorphic charts $(U, \phi)$ and $(U’, \phi’)$ of $X$ and $Y$, respectively, the map

is holomorphic. $X$ and $Y$ are biholomorphic (isomorphic) if there exists a holomorphic homeomorphism $f:X \to Y$. [/definition]

[remark] Notice that we don’t actually have to specify the inverse map to be a holomorphism — that is taken care of by the inverse function theorem (Proposition 1.1.13) in the same way it is for real differentiable manifolds. [/remark]

Definition 2.1.8 (meromorphic function)

[definition] A meromorphic function on a complex manifold $X$ is a map

which associates to any $x\in X$, and element $f_x \in Q(\mathcal{O}_{X, x})$ such that for any $x_0 \in X$, there exists an open neighborhood $U \subseteq X$ of $x_0$ and two holomorphic functions $g, h:U \to \CC$ with $f_x = g/h$ for all $x \in U$.

The sheaf of meromorphic functions is denoted by $\mathcal{K}_X$ and its space of global sections is denoted $K(X) \coloneqq \Gamma(X, \mathcal{K}_X)$. [/definition]

Example ($X$ is dense open subset of connected open $U \subseteq \CC$)

While the definition of a meromorphic function looks pretty strange, it completely recovers the usual definition of a meromorphic function in single-variable complex analysis. Recall that a meromorphic function in that context can be written as the quotient of two holomorphic functions. And, so $f_x$ would have the same representation for any $x \in X$.

Notice that the same reasoning comes over verbatim if $U \subseteq \CC^n$.

Function field of $X$

If $U$ is connected, then $\mathcal{K}_X(U)$ is an integral domain and it is obvious that it is also a field. In the particular case where $X$ itself is connected, we call $K(X)$ the function field of $X$.

Proposition 2.1.9 (Transcendence degree of $K(X)$)

[proposition] Let $X$ be a compact connected complex manifold of dimension $n$. Then

[/proposition]

Definition 2.1.10 (Algebraic dimension)

[definition] The algebraic dimension of a compact connected complex manifold $X$ is

[/definition]

Examples of complex manifolds

Affine space $\CC^n$

By definition, affine $n$-space is $\CC^n$ (or any $\CC$-vector space isomorphic to $\CC^n$ with the induced Euclidean topology). Notice that the open subsets of $\CC^n$ serve as local models for arbitrary complex manifolds.

[remark=(IMPORTANT)] A massive difference between real (differentiable) manifolds and complex manifolds is that real manifolds can be covered by open subsets that are diffeomorphic to $\RR^n$. In contrast, a general complex manifold cannot be covered by open subsets biholomorphic to $\CC^n$. [/remark]

Projective space $\PP^n$

Complex projective space

is the most important example of a compact complex manifold. As usual, we denote the points of $\PP^n$ by

The usual choice of charts gives a holomorphic atlas. In particular,

and

The transition maps

are defined by

Clearly these maps are bijective and holomorphic by Osgood’s lemma.

Complex tori

Let $X$ be the quotient $\CC^n/\ZZ^{2n}$ where we identify $\ZZ^{2n}$ with the obvious subset of $\CC^n$ via

Endowing $X$ with the quotient topology, we can construct an open covering by letting $U = B_\eps(z)$ with

Then $U$ bijectively maps onto its image in the quotient.

More generally, let $V$ be a complex vector space and $\Gamma \subseteq V$ a free abelian discrete subgroup of order $2n$. Then $X = V/\Gamma$ is a complex manifold.

What’s particularly interesting here is that if we pick two lattices $\Gamma_1, \Gamma_2 \subseteq \CC^n$, then $\CC^n/\Gamma_1$ and $\CC^n/\Gamma_2$ are diffeomorphic (they are both diffeomorphic to $(S^1)^{2n}$). However, in general, these two tori are not necessarily biholomorphic.

Affine hypersurfaces

Let $f:\CC^n \to \CC$ be holomorphic with $0 \in \CC$ a regular value (recall this means that for all $x \in f^{-1}(0)$, the differential $df(x)$ is surjective.) Let

By the implicit function theorem, $X$ is a complex manifold of dimension $n - 1$.

Projective hypersurfaces

Let $f \in \CC[z_0, \ldots, z_n]$ be homogeneous. Also assume that $0 \in \CC$ is a regular value for $f:\CC^{n+1} \setminus 0 \to \CC$. By the previous example, $f^{-1}(0) = Z(f)$ is a complex manifold. Note that

is also a complex manifold of dimension $n - 1$ using the usual charts of $\PP^n$ intersected with $X$.

Complete affine intersections

One can consider $k$ holomorphic functions $f_1, \ldots, f_k$ so that

has $0$ as a regular value. Then

is a complex manifold of dimension $n - k$.

Complete projective intersections

One can consider $k$ homogeneous polynomials $f_1, \ldots, f_k$ so that

has $0$ as a regular value. Then

is a complex manifold of dimension $n - k$.

Complex lie groups

Let $G$ be a group and complex manifolds. Then $G$ is a complex Lie group if $G\times G \to G$, $(x, y) \mapsto xy^{-1}$ is a holomorphic map. Examples of complex Lie groups are:

• $\GL(n, \CC)$ not abelian for $n > 1$
• $\SL(n, \CC)$ not abelian for $n > 1$
• $\Sp(n, \CC)$ not abelian for $n > 1$
• the complex torus $\CC^n/\Gamma$ – abelian and compact
• $\CC^n$ – abelian but not compact

Notice that $U(n)$ or $O(n)$ are often not complex lie groups but just real Lie groups. For instance, $U(1) \cong S^1$ which has odd real-dimension.

Grassmannian manifolds

Let $V$ be a complex vector space of dimension $n + 1$. Then the Grassmannian $\Gr_k(V)$ is the collection of $k$-dimsnional subspaces of $V$. Showing it is a complex manifold typically is done by representing the subspaces by the row span of matrices.

We begin by picking a basis so that we may assume $V = \CC^{n+1}$. Then $W \in \Gr_k(V)$ is the row space of a $k\times(n+1)$-matrix $A$ of full rank. We denote these matrices by $M_{k,n+1}$ (notice that this is an open subset of the set of all $k\times(n+1)$ — perturbing such a matrix by a small amount is usually still full rank). Thet set of all $k\times(n+1)$ matrices is a complex manifold canonically isomorphic to $\CC^{k(n+1)}$ so there is a natural surjection $\pi:M_{k,n+1} \to \Gr_k(\CC^{n+1})$ given by quotienting by the usual action of $\GL(k,\CC)$ on $M_{k,n+1}$.

Fix an ordering $\{B_1, \ldots, B_m\}$ of all $k\times k$ minors (so $m = {n+1\choose k}$) of matrices $A \in M_{k,n+1}$. Define an open covering $\Gr_k(\CC^{n+1}) = \bigcup_{i=1}^m U_i$ where $U_i$ is the open subset

(Equivalently, $U_i$ is the complement of the vanishing locus of the Plucker coordinate corresponding to $B_i$.) Notice that these sets are well-defined because the vanishing of the determinant is invariant under the action of $\GL(k,\CC)$. After permuting the columns of $A \in \pi^{-1}(U_i)$, we may assume that $A$ is of the form $A = [B_i \mid C_i]$. Then the map $\phi_i:U_i \to \CC^{k(n+1-k)}$ given by

is well-defined. It turns out that this defines a holomorphic atlas of $\Gr_k(\CC^{n+1})$. Notice that this also shows that $\dim_\CC \Gr_k(V) = k(n + 1 - k)$.

Flag manifolds

Also of much interest are the flags. Let $V$ be a complex vector space and fix

Then $\Flag(V, k_1, \ldots, k_\ell)$ is the manifold of all flags

with $\dim(W_i) = k_i$. So, in particular, $\Flag(V, k) = \Gr_k(V)$.

Definition (Orbit space)

[definition] Let $X$ be a topological space and $G$ a group that acts on $X$ continuously. The quotient space (or orbit space) $X/G$ is the topological space $X/G$ endowed with the quotient topology via the projection map $\pi:X \to X/G$. [/definition]

In general, $X/G$ is not a topological manifold (let alone a complex manifold). An easy example is $\CC/\CC^*$ which only has two points. There are only two open subsets of this space: $\emptyset$, $\CC^*$, and $\CC$. So this space is pretty pathological (especially not Hausdorff).

Definition 2.1.11 (Free and proper actions)

[definition] The action of $G$ on $X$ is free if for all $1 \neq g\in G$ and $x \in \CC$, we have $gx \neq x$.

The action is proper if the map $G \times X \to X \times X$, $(g, x) \mapsto (gx, x)$ is proper (preimage of compact is compact). [/definition]

Proposition 2.1.13 (Free and proper implies quotient is complex manifold)

[proposition] Let $G \times X \to X$ be the proper and free action of a complex Lie group $G$ on a complex manifold $X$. Then the quotiant $X/G$ is a complex manifold in a natural way and the quotient map $\pi:X \to X/G$ is holomorphic. [/proposition]

As a matter of convention, if $X$ is a complex manifold and $G$ a complex Lie group acting on $X$, then we will assume that the action map is holomorphic.

Examples of quotient complex manifolds

Example 2.1.14i (Discrete lattice)

Let $\Gamma \subseteq \CC^n$ be a discrete lattice. As a lattice, $\Gamma$ acts discretely by translations. Obviously, this action is free. For properness, we claim that if $K \subseteq \CC^n\times \CC^n$ is compact, then its preimage under $\Gamma \times \CC^n$, $(\tau, z) \mapsto (\tau + z, z)$ is compact since the preimage is

Take an open cover $\{U_i\}$ of $S$ in $\Gamma \times \CC^n$. We claim that this open cover admits a finite open subcover. Since $K$ itself is bounded, $\tau$ is bounded by the diameter of $K$. So the projection $P$ of $S$ onto the first factor $\Gamma$ of $\Gamma \times \CC^n$ is finite. So it follows that

Note that $\{z \in \CC^n \mid (z, z) \in (-\tau, 0) + K\}$ is bounded by $K$ being compact in $\CC^n \times \CC^n$. It is also closed since this set is the intersection of the diagonal of $\CC^n\times \CC^n$ (which is closed) and the closed subset $(-\tau,0) + K$. So this set is compact in $\CC^n$. Therefore, we only need finitely many subsets in $\{U_i\}$ to cover this set. Since $P$ is finite, we only need finitely many sets overall to cover $S$.

Thus, by Proposition 2.1.13, it follows that the torus $\CC^n/\Gamma$ is a complex manifold with the natural choice of topological and complex structure.

Example 2.1.14ii ($\PP^n$ as a quotient)

By similar argumentation, $\PP^n$ realized as the quotient $\CC^{n+1}\setminus 0 /\CC^*$ is a complex manifold without having to go through the usual from scratch way.

Ball quotients

The (open) unit disct

can also be viewed as an open subset of the standard open subset $U_0 \subseteq \PP^n$ where

To see how, we recall that there is a hermition form $\langle -,-\rangle$ on $\CC^{n+1}$ defined by

Then $D^n$ is the open subset of $\PP^n$ of points $z$ with $\langle z, z\rangle > 0$. Notice that this actually makes sense since for this to be true, we’d have

which is invariant under the action of $\CC^*$ and, of course, this necessitates $z \in U_0$. Since we’ve normalized $U_0$ so that $|z_0| = 1$, $D^n$ is actually precisely the set of $z$ with $\langle z, z\rangle > 0$.

Of course, notice that $D^n$ is invariant under the action of $\SU(\langle -, -\rangle) = \SU(1, n)$

Ball quotients are quotients of $D^n$ by a discrete group $\Gamma \subseteq \SU(1, n)$ acting freely on $D$. Typically we also want $D^n/\Gamma$ to be compact too so we might just assume we’re working in that case.

Finite quotients of products of curves

Let $C$ be a complex curve and $E = \CC/\Gamma$ be an elliptic curve. If $G$ is a finite subgroup of $E$ (thus it acts by translation on $E$) that acts on $C$, then there is a natural free $G$-action on $C\times E$. Thus, the quotient $X = (C \times E)/G$ is a complex surface and the is a surjective holomorphic map $f:X \to C/G$. If $G$ acts freely in $z\in C$, then the fiber $f^{-1}(z)$ is isomorphic to the elliptic curve $E$. In general, $f^{-1}(z)$ is isomorphic to $E/\Stab(z)$.

Hopf manifolds

Let $\ZZ$ act on $\CC^n\setminus 0$ by

for $k \in \ZZ$. For $0 < \lambda < 1$, this action is free and discrete. The complex manifold $X = (\CC^n \setminus 0) /\ZZ$ is (real) diffeomorphic to $S^1 \times S^{2n - 1}$.

Iwasawa manifold

Let $G$ be the complex Lie group

Then $G$ is biholomorphic to $\CC^3$ (as a complex manifold). The group $G$ is called the Heisenberg group. Let $\Gamma$ be the subgroup $G \cap \GL(3, \ZZ + i\ZZ)$. Then $(w_1, w_2, w_3) \in \Gamma$ acts on $G$ by translation (which, of course, is properly discontinuous). So the quotient $X = G/\Gamma$ is a complex manifold of dimension three.

Definition 2.1.16 (Complex submanifolds)

[definition] Let $X$ be a complex manifold of complex dimension $n$ and $Y \subseteq X$ be a differentiable submanifold of real dimension $2k$. We say that $Y$ is a complex submanifold of (complex) dimension $k$ in $X$ if there is a holomorphic atlas $\{(U_i, \phi_i)\}$ of $X$ such that $\phi_i:U_i \cap Y \cong \phi(U_i) \cap \CC^k$.

Here $\CC^k$ is embedded into $\CC^n$ via the first $k$ coordinates. The codimension of $Y$ in $X$ is by definition $\dim(X) = \dim(Y) = n - k$. [/definition]

Definition 2.1.17 (Projective complex manifolds)

[definition] A complex manifold $X$ is projective if $X$ is biholomorphic to a closed complex submanifold of some projective space $\PP^N$. [/definition]