Complex manifolds are basic objects in complex geometry, just like smooth manifolds are basic objects in differential geometry. We introduce complex manifolds as an analogy of smooth manifolds. Yet it is worth noting some differences between the complex case and the real case.
Definition and examples
A complex manifold (of dimension ) is a -dimensional smooth manifold with an open cover and coordinate maps , such that is holomorphic on .
Like the real case, we can define holomorphic functions on a complex manifold and holomorphic maps between complex manifolds. A holomorphic map which has a holomorphic inverse is called a holomorphic homeomorphism.
The holomorphic condition is actually a quite strong restriction to functions. This can be seen from the following proposition.
Proposition. Any holomorphic function on a compact connected complex manifold is constant.
Proof: Suppose is holomorphic. Since is compact, the continuous function attains its maximum at some point . Consider a coordinate chart with . Then the maximal module principal on implies that is constant. As is connected, must be constant on the whole .
The above proposition suggests that any compact connected complex manifold cannot be holomorphicly embedded in any . However, we still want to put a compact complex manifold into some good space. That is why we introduce the projective spaces.
Use to specify a point in . Define an equivalence relation on by that if and only if for some . The collection of equivalence classes is called the (-dimensional) projective space, denoted by . The equivalence class containing is denoted by , with called the homogeneous coordinates of .
Similar to the real case, we can show that is an -dimensional complex manifold. Consider the unit sphere . The restriction of the natural projection to gives a continuous surjection onto . Since is compact, is also compact.
The projective space can be viewed as a compactification of . Consider the inclusion given by This inclusion is a holomorphic homeomorphism from to an open subset of , and the complement of its image is naturally identified with . When , we see that is exactly the Riemann sphere.
Another example of a compact complex manifold is a complex torus. Suppose are linearly independent as vectors in a real vector space. Let be the lattice generated by . Then is a compact smooth manifold together with a complex manifold structure induced from the covering map . This is called the (-dimensional) complex torus. We see that when , is diffeomorphic to an ordinary torus .
We try to generalize the construction of the complex torus. Suppose is a topological covering map and is a complex manifold. A desk transformation is a homeomorphism from to itself such that . If each desk transformation on is holomorphic, then we can equip with a well-defined complex manifold structure induced from that of . This is similar to the process that defines a quotient smooth manifold.
Consider the group of self-homeomorphisms on generated by the map . Let be the quotient smooth manifold of this group action. Then the natural projection gives a complex manifold structure on , making into a -dimensional complex manifold, called the Hopf surface. The Hopf surface is a compact complex manifold that cannot be embedded in any projective space .
Tangent spaces and cotangent spaces
Next we introduce the tangent space of a complex manifold. Suppose is a -dimensional complex manifold and . Since is a -dimensional smooth manifold, we can consider the real tangent space of at . Let be the complexified tangent space. An element in , called a complex tangent vector of at , is identified with a -linear derivation on the ring of germs of complex-valued smooth functions on a neighborhood of .
Let be the ring of germs of holomorphic functions around , and be that of antiholomorphic functions, i.e., the functions whose conjugates are holomorphic.
If a complex tangent vector vanishes on , then we call a holomorphic tangent vector, and if vanishes on , then we call an antiholomorphic tangent vector. Then collection of all holomorphic (resp. antiholomorphic) tangent vectors is called the holomorphic tangent space (resp.antiholomorphic tangent space) of at , denoted by (resp.).
We try to show that and are both -dimensional complex linear spaces with
Suppose is a holomorphic coordinate chart of around , with . Then form a (-linear) basis of . Let Then Cauchy-Riemann equations imply that form a basis of , while form a basis of . The direct sum decomposition follows.
We can also construct the corresponding cotangent spaces of at . Let be the dual space of , whose elements are called complex cotangent vectors of at . A complex cotangent vector vanishing on (resp.) is called a holomorphic cotangent vector (resp.antiholomorphic cotangent vector), and then we can similarly define the holomorphic cotangent space and the antiholomorphic cotangent space. Note that can be naturally identified with the dual of , while is identified with the dual of .
If we use the local coordinates around , then form a basis of . The corresponding bases of and are then given by and which are exactly the dual basis of and , respectively.
Tangent maps and Jacobians
Since we have tangent spaces of a complex manifold, we can consider the corresponding tangent maps. Suppose and are complex manifolds of dimensional and , respectively. Then any smooth map with induces a tangent map and then a map However, in general, this does not induce a holomorphic tangent map from to , as well as a antiholomorphic tangent map from to .
Proposition. Suppose and are complex manifolds of dimensional and , respectively, and is a smooth map. Consider the tangent map . Then is holomorphic if and only if .
Proof: Note that is equivalent to that is antiholomorphic whenever is antiholomorphic. The latter is directly equivalent to that is holomorphic.
Now suppose is a holomorphic map. Consider the local coordinates around and the local coordinates around . The holomophic tangent map can be expressed by the holomorphic Jacobian Suppose for and for . With respect to the bases and , the real tangent map is given by the real Jacobian If we pass through the tensor product with to consider the complex tangent map, and change our bases into and , we obtain the complex Jacobian
Since and only differ by a change of basis, we have Moreover, when , we have As an corollary, any complex manifold is orientable, as we can fix a natural orientation on by the -form and pull this back through the holomorphic coordinate maps. The pullbacks agree with each other on the intersections of coordinate neighborhoods and hence give a well-defined orientation on the -dimensional complex manifold .