Hodge theory on compact hermitian manifolds - 3

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The proof of Theorem A and Theorem C appeared in the previous post will be given in this one. Note that the standard Rellich lemma in functional analysis will be applied without a proof.

We always suppose that is a compact hermitian manifold and is a holomorphic vector bundle on endowed with a hermitian metric .

Theorem A

Recall that the hermitian structures on and induce an inner product on the space , and hence the norms on given by where is the metric connection on with respect to . Considering the completion of with respect to and its dual space, we obtain the Sobolev spaces and . The operators and on extends to differential operators of order one from to , while becomes differential operators of order two from to .

The statement of Theorem A is the following.

Theorem A (Gårding’s inequality). There exist positive constants such that for each , we have

To prove Theorem A, we need to estimate the lower bound of . The compactness of suggests that we can process the estimation locally. First introduce some notations. Suppose form a local orthonormal frame of with dual frame . Let be a local holomorphic frame of , whose dual frame of is given by . Then an -valued form can be expressed by where the summation is taken over subsets and of with and . For a local smooth function , denote that We can see that However, in general we do not have Indeed, it should be where we use to denote a term containing only instead of any derivative of them. The specific expression of is of no importance as we just need to know that it can be bounded by some multiple of in our estimation. Similarly we can denote by the terms only containing the derivatives of of order no more than and hence dominated by . An example is that which permits us to ignore the ordering of partial differentiations up to lower-order terms.

Lemma (Weitzenböck formula). Suppose is given locally as Then

Proof: Clearly we can reduce the assertion to . We claim that this can further reduce to . Let for . Consider the following computations:

Thus for we have This proves our assertion.

Now it remains to show for that By linearility, we may drop the summation for and , and as and have nothing to do with , we can assume that . The symmetry in the indices further implies that there is no loss of generality assuming that . So we only have to deal with

Direct computation yields that and that As for each and , we obtain which completes our proof.

Now we can write down the proof of Theorem A.

Proof of Theorem A: As are orthogonal to each other, it suffices to show the assertion for where and are fixed. By the definition of the action of on , we see that is decomposed into with Moreover, using the local expression of as we see that and that

Consider the -form which can be verified to be well-defined (up to some ) globally on the whole . Since is of type , we have , and then the Stokes’ formula implies that Direct computation yields that and then by the Weitzenböck formula and the local expression of we see that Note the inequality we obtain

Analogously, consider we have Thus Taking sufficiently small and , we obtain which is exactly the Gårding’s inequality.

Rellich lemma and Theorem C

Recall our statement of Theorem C:

Theorem C. Suppose is a sequence in such that is bounded. Then there exists a subsequence of which is a Cauchy sequence with respect to , i.e., converges in .

Let us first consider the local case, which relies on some discussion of Sobolev spaces , where is an open subsets. Recall that with is the completion of with respect to , where Clearly we have for any , and hence we can consider the closure of in , which is exactly the completion of with respect to and usually denoted by . As we have whenever , there is a natural inclusion .

We have the following result in functional analysis.

Theorem (Rellich lemma). Suppose is a bounded open subset and . Then the natural inclusion is a compact operator.

Suppose is an open cover of such that on each we have a holomorphic coordinate map and a holomorphic frame of . Suppose is a smooth partition of unity subordinate to the open cover . Write for each that We have seen that the norm given by

is equivalent to .

Proof of Theorem C: Consider the above open cover and . Then each is compactly supported on , and the sequence is bounded in . As , , and has only finitely many choices, we obtain from the Rellich lemma that there is a subsequence of such that is Cauchy in . This directly implies that is Cauchy in after the equivalence of and .