In this post we focus on holomorphic line bundles and discuss some relations between divisors and line bundles.
Definition of divisors
Suppose is an -dimensional complex manifold and is an analytic subvariety of dimension . For each , there is a neighborhood of such that is defined as the zero set of a holomorphic function on this neighborhood. This functions is called a local defining function for near , and is unique up to multiplication by a function nonzero at .
Proposition. Suppose is a complex manifold and is an analytic hypersurface on . Then for any connected component of , the closure is an analytic variety.
Corollary. An analytic hypersurface is irreducible if and only if is connected.
We see from this proposition that each analytic hypersurface can expressed uniquely as the union of some irreducible analytic hypersurfaces.
Now we define a divisor on is a locally finite formal linear combination of irreducible analytic hypersurfaces of . The local finiteness here means that for each there is a neighborhood of intersecting with only finitely many which appear in . The divisors on form an additive group . If holds for all , then we call an effective divisor, written as . Note that if is an analytic hypersurface, then we can identified it with the divisor
Suppose is an irreducible analytic hypersurface, and is a defining function for near . For any meromorphic function defined in a neighborhood of , we can define the order of along at to be the unique integer such that
This is independent of the choice of the defining function and the point .
We say has a zero of order along if , and that has a pole of order along if . We can see that for any two meromorphic functions , it holds that
Now consider a nonzero meromorphic function on . The divisor associated to is given by This divisor can be written as the difference of two effective divisor, namely the zero divisor
and the pole divisor
The divisors can be constructed in sheaf-theoretic terms. We claim that a divisor on is equivalent to a global section of the quotient sheaf of with respect to , and then we have the natural isomorphism
On the one hand, a global section of the quotine sheaf is given by an open cover of and meromorphic functions on that are not identically zero with
Thus for any analytic hypersurface we have
if they are defined. The corresponding divisor is then given by
where is chosen such that .
On the other hand, given a divisor thee is an open cover of such that every has a local defining function in each . We then let
to obtain a global section of .
We see from the above constructions that the identification is actually a homomorphism and hence an isomorphism.
Suppose is a holomorphic map between complex manifolds. Then for each divisor on such that the image of is not contained in the support of , we may define the pullback of along . Suppose under the identification of a divisor with a section, is given by , then is the divisor on given by . For holomorphic functions such that is dense in , we then obtain a homomorphism
Relations between divisors and line bundles
Recall that a line bundle on a complex manifold can be given by an open cover of and transition functions of . Using this description, it is not hard to see that the set of line bundles on can be identified with .
The set of line bundles on can be given the structure of group with multiplication given by tensor product and inverses given by dual bundles. This group is called the Picard group of , denoted by . Note that the group structure of and the group structure of is actually the same, we have the natural isomorphism
Now we attempt to associate a line bundle to each divisor on . Let be a divisor on with local defining functions over an open cover of . Then the transition functions determine a line bundle on , called the associated line bundle of . This is independent of and hence well-defined.
We see the following properties of :
if and are divisors on , then implying that the corresponding map is homomorphism;
the line bundle is trivial if and only if there is a nonzero meromorphic function on such that .
Thus we say that two divisors and are linearly equivalent if for some nonzero meromorphic on , or equivalently , written as .
The above discussion can be interpreted in the sheaf-theoretic opinion. Consider the exact sequence of sheaves on This induces the exact sequence of cohomology groups that Identifying the corresponding cohomology groups with and , respectively, we see that the homomorphism maps each meromorphic function to the divisor , and the homomorphism maps each divisor to the line bundle . Hence the latter property of the associated line bundle is nothing else than the exactness of the sequence.
Holomorphic and meromorphic sections of line bundles
Suppose is a holomorphic line bundle on a complex manifold with trivializations and corresponding transition functions . These trivializations induce ismorphisms and then give a correspondence Thus a section of on is equivalent to a collection of functions satisfying on each
According to this point of view, we define a meromorphic section of on to be given by a collection of meromorphic functions satisfying on each . This is equivalent to the section on of the sheaf Note that the quotient of two nonzero meromorphic sections of is a well-defined meromorphic function on .
For a non-trivial global meromorphic section of , we have implying that for any irreducible hypersurface , we have Thus we can define the order of along by The divisor associated to is then given by It is clear that is holomorphic is and only if is effective.
Proposition. Suppose is a complex manifold. Then the image of the natural map consists of those line bundles admitting non-trivial meromorphic sections.
Proof: If is given by functions , then clearly these functions give a meromorphic section of with . Conversely, suppose is given by transition functions and is a non-trivial global meromorphic section of , then and hence .
The holomorphic sections of the line bundle associated to a divisor on may be constructed in the following way. Consider the set of meromorphic functions on such that Fixed a global meromorphic section of with . On the one hand, any holomorphic section of induces a meromorphic function on such that On the other hand, any meromorphic function gives a holomorphic section of . Thus we obtain a bijection
We end this post with some discussion of the relations between holomorphic sections of line bundles and holomorphic maps into projective spaces.
Suppose is a holomorphic line bundle on a complex manifold and are global holomorphic sections of having no common zero. Suppose is an open cover of with trivializations on each and transition functions . On each , the sections can be expressed by holomorphic functions . Then we have a map given by On , we have suggesting that we may define a well-defined function by for . This map is clearly holomorphic.
Conversely, assume we have a holomorphic map . Then we may take an open cover of such that can be expressed as with having no common zero. By the definition of homogeneous coordinate, there are nonvanishing holomorphic functions such that Using these as transition functions, we obtain a holomorphic line bundle on . Meanwhile, the holomorphic functions define a holomorphic section of for each , and these sections have no common zero.
We conclude that a holomorphic map from to is equivalent to holomorphic sections of a line bundle on with no common zero.