We focus on the Riemann-Roch theorem in the study of compact Riemann surface in this post. This will be a comprehensive application of the tools we have developed.
Suppose is a compact Riemann surface, i.e., a compact -dimensional complex manifold. Recall our discussions about divisors and line bundles. We have associated to each nontrivial meromorphic function on a divisor , and to each divisor a line bundle . The divisor group on is naturally identified with , while the Picard group of line bundles on is identified with . The exact seqeunce of sheaves induces the exct sequence
For a divisor on , the holomorphic sections of the associated line bundle can be constructed by meromorphic functions on satifying some properties. Precisely, there is an isomorphism between and . In particular, Here we consider meromorphic functions on , however, we can also consider meromorphic differential forms on . A meromorphic differential form on is a global meromorphic section of the differential form bundle on . Since is -dimensional, is a line bundle, and each nontrivial global meromorphic section of induces a divisor just as meromorphic functions. Consider In analogy to the way that each induces a holomorphic section of , each induces a holomorphic -valued differential form on , and hence we obtain an isomorphism Recall that by the generalized Dolbeault theorem and the Kodaira-Serre duality, we have implying that
Note that as is -dimensional, the hypersurfaces on are exactly points. Thus each divisors on can be expressed as where are distinct points on and are positive integers. We then define the degree of the divisor to be
Lemma. Suppose is a divisor defined by a nontrivial meromorphic function on . Then .
Proof: Consider a finite open cover of such that each is identified with an open subset of . We can take for each such that these are disjoint, cover , and there is no zero or pole of on any . By the argument principle for meromorphic functions, we see that the difference between the number of zeros of on and the number of poles of on , both counted with multiplicities, is the integral of along . Thus where the last equality holds by the definition of contour integrals.
Another important concept appeared in the statement of the Riemann-Roch theorem is the genus of a Riemann surface . This is just the genus of a compact surface of real dimension , and by the orientability of , we have the expression of as where the first is the de Rham cohomology with complex coefficients, and the second one is the sheaf cohomology of the locally constant sheaf .
Consider any hermitian metric on . As soon as is a hermitian Riemann surface, becomes a Kähler manifold. Then by the Hodge decomposition theorem for compact Kähler manifolds, we see that Using the Dolbeault theorem, it follows that
The following lemma will be repeatedly used in our discussion.
Lemma. Suppose is a compact Riemann surface, is a line bundle on and is an effective divisor. Then
Proof: Suppose is given as By the identification of with , we can take a nontrivial meromorphic section such that . Since is effective, is indeed holomorphic, and has zeros at each with order .
For each local holomophic section of , gives a holomorphic section of . If , then is zero on a dense subset of , implying that is the trivial section of on . Thus we obtain an injective sheaf morphism Put this into an exact sequence of sheaves:
For , is nonzero and hence is an isomorphism, implying that . For for some , the image of in consists of those local sections such that is a zero of with multiplicity no less than . Thus Therefore it is not hard to see that
We have the long exact sequence of cohomology groups The Hodge theorem and the generalized Dolbeault theorem have shown the finiteness of the dimensional of these cohomology groups. Thus implying the desired formula.
Proposition. Suppose is a compact Riemann surface and is a line bundle on . Then there is a divisor on such that .
Proof: Let and . By the lemma, we have For a sufficiently large , we must have i.e., has a nontrivial global holomorphic (and hence meromorphic) section. As we have shown before, is contained in the image of . Suppose . Then , proving our assertion.
Now we can define the degree of a line bundle on . Suppose is a line bundle on . Take a divisor on such that . We simply define the degree of to be the degree of . However, the well-definedness needs considering. Suppose is another divisor on such that . Then there exists a meromorphic function on such that . Hence showing that is actually well-defined.
Theorem. (Riemann-Roch theorem). Suppose is a compact Riemann surface with genus and is a line bundle on . Then
Proof: Take a divisor on such that . Clearly there exist effective divisors and on such that . By the lemma we have from that and from that It follows that Since is a compact complex manifold, we have Note that and that we conclude that
Using and , we can express the Riemann-Roch theorem as follows:
Theorem (Riemann-Roch theorem). Suppose is a compact Riemann surface with genus and is a divisor on . Then