Holomorphic vector bundles are quite essential objects in the study of the geometry of a complex manifold. On one hand, using holomorphic vector bundles, we can consider the analogy of Riemannian geometry in the complex case. On the other hand, the study of vector bundles leads to the concept of divisors, which are frequently considered in algebraic geometry as well.
Definitons and constructions
The definition of a holomorphic vector bundle is analogous to the definition of a smooth vector bundle. Suppose is a complex manifold. A holomorphic vector bundle (of rank ) on is a complex manifold together with a holomorphic map satisfiying that
each fiber where is an -dimenional complex vector space;
for each , there is an open neighborhood of in together with a holomorphic homeomorphism such that is mapped linearly isomorphically onto .
A holomorphic vector bundle of rank is usually called a holomorphic line bundle.
Suppose and are holomorphic vector bundles on . Then a homomorphism from to is a holomorphic map such that for each and that is linear with rank independent of . If is linear isomorphism for each , then we say is an isomorphism, and and are isomorphic.
Note that if and are open subsets in with noempty intersection and trivializations then we have a holomorphic map
given by
These are called transitions functions of . The transition functions of necessarily satisfy the identities
Conversely, given an open cover of and holomorphic maps
satisfying the above identities, we may consider the complex manifold which has the structure of a holomorphic vector bundle on such that are the transition functions. This holomorphic vector bundle is unique up to an isomorphism.
Suppose and are holomorphic vector bundles of rank and on with transition functions given by and . Using the description of a holomorphic vector bundle by transition functions, we have the following constructions.
The direct sum is given by transition functions
The fiber is canonically isomorphic to .
The tensor product is given by transition functions
The fiber is canonically isomorphic to .
The dual bundle is given by transition functions
The fiber is canonically isomorphic to the dual space of .
The exterior product is given by transition functions
We also have the canonical isomorphism of fibers .
In particular, is the line bundle given by transition functions called the determinant line bundle of .
Suppose for each the matrix can be written as
then is naturally a holomorphic subbundle of . The quotient bundle is then given by the transition functions .
Suppose is a vector bundle homomorpism, then there exist holomorphic subbundles and such that we have the canonical isomorphism of fibers These are called the kernel bundle and image bundle of . We can also define the cokernel bundle of , which is exactly the quotient bundle Using these definitions, we may consider the exact sequences of vector bundles in a natural way.
Suppose is a holomorphic map between complex manifolds and is a holomorphic vector bundle on with transition function . The pullback bundle of along is the holomorphic vector bundle on given by transition functions . For each there is a canonical isomorphism
If is a submanifold of and is the inclusion, then we call the restriction of on .
For a holomorphic vector bundle , we define a holomorphic section of on an open subset to be a holomorphic map such that is identity on , i.e., for each . The collection of all holomorphic sections of on is denoted by . The assignment gives a sheaf on , denoted by . We see that is naturally a -module.
Tangent bundles and related bundles
Suppose is an -dimensional complex manifold. From its complex manifold structure, we may construct some holomorphic vector bundles on which are not trivial in general.
Viewing as an -dimensional smooth manifold, we can consider its tangent bundle and its complexification . Note that the fiber of at is exactly
Then we have smooth subbundles of given by
and
The bundle is called the holomorphic tangent bundle of and is called the antiholomorphic tangent bundle of .
We claim that the holomorphic tangent bundle is actually a holomorphic vector bundle on . Consider a coordinate covering of . The transition functions of with respect to the trivializations induced by the coordinate maps are given by These are holomorphic, and hence is a holomorphic vector bundle. We may also write for the holomorphic tangent bundle of to specify that it is a holomorphic vector bundle.
Similarly we can define the holomorphic cotangent bundle and the antiholomorphic cotangent bundle on , and is a holomorphic vector bundle, also denoted by . Note that is the dual bundle of .
Using , we can defined the bundle of holomorphic -forms on . In particular, is called the canonical bundle of . We can see that the transition functions of is given by
Analogously to the construction of -forms, we may also construct the bundle of -forms on given as The smooth sections of are exactly the -forms on .
Now suppose is a complex submanifold of . Using the local expression of a complex submanifold, we see that the holomorphic tangent bundle of is naturally a subbundle of . Then we define the normal bundle of in by the following exact sequence which called the normal bundle sequence:
Proposition (Adjunction formula). Suppose is a complex submanifold of . Then the canonical bundle of is naturally isomorphic to the line bundle .
This is actually a corollary of the following result, which can be shown by considering the relations of the transition functions.
Proposition. Suppose the following sequence of vector bundles is exact: then there is a canonical isomorphism