In differential geometry, we know that any smooth manifold can be embedded in a Euclidean space . In contrast, as any holomorphic function on a compact complex manifold is constant, we cannot put a compact complex manifold into any affine space . Therefore people turn to the embedding of compact complex manifolds in projective spaces, which will be dicussed in this post and the followings.
Before considering the sufficient conditions for embedding, the necessary conditions are also essential. Recall that a submanifold of is a Kähler manifold with the hermitian metric induced from the Fubini-Study metric on . Another necessary condition comes from the study of line bundles on .
Hyperplane bundle on
Consider the line bundle on given by together with the natural projection . is called the universal bundle on . Using the coordinate covering of , we can give the trivializations of . Let The local trivializations of are then given by , It follows that the transition functions of are
The hyperplane bundle on is then defined to be the dual bundle of , i.e., . In terms of transition functions, we see that corresponds to , where
In order to see why the line bundle is called the hyperplane bundle, we need to turn to an alternative definition of . Fix a nonzero . The zero set of the homogeneous function is a well-defined hyperplane on , and hence gives a divisor on . We claim that is exactly the line bundle . Let by the holomorphic function given by It is direct to see that Thus the divisor corresponds to . It follows that the line bundle given in terms of transition functions is , where As agrees with for any and , it is clear that .
Using the transition functions of , a hermitian metric on may be express by a collection of locally defined positive smooth functions satisfying some properties. Suppose is the natural projection and is the local trivialization. Let be the holomorphic section of on such that It is clear that is determined by the positive smooth functions and that these satisfy the following transition relations: Conversely, given positive smooth functions on each satisfying the above condition, we obtain a hermitian metric on up to a constant scalar.
In particular, we define It is direct to verify that satisfy the transition relations. The metric connection of is then given by the -form and hence the curvature form is It follows that the Fubini-Study metric on is expressed locally on each as
Positive line bundles
The above discussion about inspires the following definition of positive line bundles. Suppose is a complex manifold and is a holomorphic line bundle on . Then we say that is a positive line bundle on , if there is a hermitian metric on such that the associated curvature form satisfies that the real -form is positive definite everywhere on .
Note that if is a positive line bundle on with a curvature form satisfying the above condition, then as is a closed real -form positive definite everywhere, it defines a Kähler metric on . Thus a complex manifold admitting a positive line bundle must be a Kähler manifold. Moreover, if is a complex submanifold on , then it is not hard to see that is a positive line bundle on . Therefore, if a compact complex manifold can be embedded into , then admits a positive line bundle. This gives a necessary condition for the embedding, and we will show in the following posts that this is actually a sufficient condition.
The positivity of a line bundle can be described by its Chern class, as shown in the following propositions.
Lemma. Suppose is a compact Kähler manifold and is a -form on . If is -exact, then there exits a -form on such that . Moreover, if and is real, then we may take such that is real.
Proof: Since is a compact Kähler manifold, the harmonic forms with respect to and are all the same. Let be the Green operator with respect to , which is a real operator and also the Green operator with respect to . As is -exact, it is orthogonal to the space of harmonic forms, and . Hence the Hodge decompostion of with respect to is Note that we see that . Consider the Hodge decompostion of with respect to : which implies that It follows that . Recall that in the proof of the relation we have shown that , implying that Therefore .
Note that the above construction of can be written explicitly as In case is real, we have showing that is real.
Proposition. Suppose is a compact Kähler manifold and is a line bundle on . If is a closed -form on such that is real and then there exists a metric on with the associated curvature form .
Proof: Take any metric on with the associated curvature form . Then by the definition of the Chern class, Hence is a real -exact -form. By the above lemma, there exists a real smooth function on such that . Let , then is also a positive function on , and the curvature form oh is
Corollary. Suppose is a compact Kähler manifold and is a line bundle on . Then is a positive line bundle if and only if the first Chern class can be represented by a closed real -form that is positive definite everywhere.