Complex manifolds - 3

In analogy to the de Rham cohomology, we define the Dolbeault cohomology for complex manifolds. To compare this new cohomology with the sigular cohomology, the sheaf theory is applied. It is worth reminding that the sheaf theory is of great importance in the further study of complex geometry.

Dolbeault cohomology

Suppose is a complex manifold. Let be the kernel of the linear map The forms in are said to be -closed. Since , we have The Dolbeault cohomology group (of order ) of is then defined to be the quotient space

We may consider the functorial property of the Dolbeault cohomology. Suppose is a holomorphic map between compex manifolds. The pullback of differential forms gives maps As is holomorphic, it can be verified that

Noting that commutes with , we obtain the induced homomorphisms

The Poincaré lemma for impies that the de Rham cohomology groups are locally trivial. In analogy, we have the following -Poincaré lemma which suggets that the positive-order Dolbeault cohomology groups of a polydisc are trivial.

Theorem (-Poincaré lemma). Suppose is a polydisc which can be unbounded and . If is -closed, then there exists such that .

We begin with the one-dimenional case.

Lemma. Suppose is an open neighborhood of the closure of a bounded disc . Consider . Then the function is well-defined on and satisfies .

Proof: Consider any . By the existence of the partition of unity, we may take the decomposition with supported on and supported outside . Let It is clear that is well-defined for . Consider the change of variable , we may write which is certainly well-defined. Thus is well-defined for .

Since is holomorphic in for in the support of , we see for that Meanwhile, using the above expression of , we can compute that For a fixed and any sufficiently small , we obtain from the Stokes’ formula that Letting , this yields Since is arbitrary, we see that on the whole .

Lemma. Suppose is an open neighborhood of the closure of a bounded polydisc and . If is -closed, then there exists such that on .

Proof: It suffices to prove for . Suppose is given as We claim that if the decomposition of does not involve any for , then there exists such that does not involve any for . This is sufficient since the conclusion can be shown by induction after this assertion.

Let Then Comparing the terms of and , we have for any and such that .

Consider the functions Let Then the preceding lemma implies that this is the desired up to a sign.

Now we can prove the complete -Poincaré lemma.

Proof of -Poincaré lemma: Consider a monotone increasing sequence tending to and let . The previous lemma yields for each some such that on It remains to show that these can be chosen such that they converge uniformly on each compact subset of .

We procedure by induction on . Suppose has been constructed. Take any such that holds on . Then holds on . When , there is some such that on . The can then be given by

Now consider the case . We see that is holomorphic on . Consider the power series expansion of this function and truncate it to get a polynomial such that Then we can set

In both cases we obtain a series such that exists and satisfies on .

Sheaf theory on complex manifolds

The definitons and basic results in sheaf theory can be found in this PDF file.

There are many examples of sheaves on a complex manifold :

• the locally constant sheaves ;
• the additive sheaf of holomorphic functions;
• the multiplicative sheaf of nonvanishing holomorphic functions;
• the sheaf of holomorphic -forms, which can be expressed as with holomorphic;
• the sheaf of -forms;
• the sheaf of -closed -forms;
• the sheaf of holomorphic functions vanishing on a fixed analytic subvariety .

Suppose is an open subset of . A meromorphic function on is given locally as the quotient of two holomorphic functions. Precisely, there is a covering of such that the restriction of on is given by for each , where and are relatively prime in and in . We can then consider the sheaf of meromorphic functions and the multiplicative sheaf of nonzero meromorphic functions.

The exact sequences of sheaves on are widely used in the study of complex manifolds.

• The exponential morphism yields an exact sequence

  This is called the exponential sheaf sequence.

• Suppose is a submanifold. The sheaf can be viewed as a sheaf on . Then the sequence is exact, where the morphisms are given by inclusion and restriction.

• By the -Poincaré lemma, the sequence is exact.

For a sheaf on a topological space we can consider its cohomology groups . Basic results concerning the sheaf cohomology can also be found in this PDF file.

Applying the results of sheaf theory to a complex manifold , we obtain the following theorem.

Theorem (Dolbeault theorem). Suppose is a complex manifold. Then we have the canonical isomorphism for each nonnegative and .

Proof: Note that the exact sequence gives an acyclic resolution of the sheaf , we have the canonical isomorphism

Another application of the sheaf theory is the answer to the Cousin problem.

By -Poincaré lemma, we have Meanwhile, by the singular cohomology, Then the exponential exact sequence yields a long exact sequence of cohomoogy groups which implies that

Proposition. Any analytic hypersurface in is the zero set of an entire function.

Proof: Suppose is an analytic hypersurface. For each , there is a neighborhood of such that is given by a holomorphic in this neighborhood, and can be chosen to be of no square factor uniquely up to a unit.

Thus there is a cover of and functions such that for each , and that

for each and . Since

after a refinement of covering if necessary, there exists for each such that . Therefore we obtain an entire function whose zero set is exactly .