Embedding of compact complex manifolds - 3

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For a positive line bundle on a compact complex manifold, the Kodaira vanishing thoerem tells us the higher cohomology groups can be controlled, and hence ensures that we can move from the local case to the global case. In this post we provide a proof of the Kodaira vanishing theorem and discuss some related results.

Note that a compact complex manifold with a positive line bundle admits a Kähler metric from the metric on the bundle. First we extend some results about operators on the space of forms to those on the space of bundle-valued forms. Suppose is a compact Kähler manifold with the Kähler form , and is a holomorphic vector bundle on with a hermitian metric . Recall that we have operators and on , which satisfy on that and on that .

On the space , the operators and are still well-defined. Meanwhile, gives an operator on , whose dual operator clearly exists and is denoted by . Since the metric connection on plays the role of the differential on , we can consider its holomorphic part as the analogy of . Then we have a well-defined Using the star operator, it is not hard to see that the operator provides the dual of . (Note that in the expression of , acts on , where is equipped with the induced metric from .)

Lemma. Suppose is a holomorphic vector bundle on a compact Kähler manifold . Then we have on each that

Proof: Suppose is a smooth local frame of , with respect to which the connection matrix of is , where consists of -forms and consists of -forms. For , we can express it as Then it can be seen that and Meanwhile, it is clear that Thus

For each , we can take a smooth frame such that vanishes at , which implies that By the arbitrarity of , we see that holds as operators on .

It is direct to see that holds for any .

Now we can prove the Kodaira vanishing theorem.

Theorem (Kodaira vanishing theorem). Suppose is an -dimensional compact complex manifold and is a positive line bundle on . Then

Proof: Suppose is a metric on such that the associated real -form is positive. Then provides a Kähler stucture on . Assume that the metric connection of on is . Comparing the type of forms in the equality we see that

Now by Hodge theorem, there is a canonical isomorphism . Thus it suffices to show that a harmonic form must be zero whenever . Take any with . Then implying that On the other hand, since we have Therefore Since , it must be true that , proving our assertion.

Using the Dolbeault theorem, it is direct that In particular, taking , we obtain If we consider the Kodaira-Serre duality, then

As an corollary, we see that vanishes for several cases, where is the hyperplane bundle on . Note that since is positive, any tensor power with is positive.

  • if , then is positive, implying that
  • if , then note that by computations of transition functions we have , suggesting that

Therefore

There is another interesting vanishing theorem which can be shown in a similar method to the Kodaira vanishing theorem.

Theorem. Suppose is an -dimensional compact complex manifold and is a positive line bundle on . Then for any holomorphic vector bundle on , there exists such that

Proof: Note that it suffices to show that for any holomorphic vector bundle and sufficiently large we have

Suppose is the metric on such that gives a Kähler metric on , and is a metric on with curvature form . Then provides a metric on , whose metric connection is denoted by . Then the curvature of is exactly By a similar argument in the proof of the Kodaira vanishing theorem, we see that Thus

For each , is an operator on the fiber such that the norm is independent of . Let then we have Therefore whenever , proving our assertion.