Holomorphic vector bundle

From Wikipedia, the free encyclopedia
  (Redirected from Holomorphic line bundle)
Jump to: navigation, search

In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold X such that the total space E is a complex manifold and the projection map π : EX is holomorphic. Fundamental examples are the holomorphic tangent bundle of a complex manifold, and its dual, the holomorphic cotangent bundle. A holomorphic line bundle is a rank one holomorphic vector bundle.

Definition through trivialization[edit]

Specifically, one requires that the trivialization maps

\phi_U :  \pi^{-1}(U) \to U \times \mathbf{C}^k

are biholomorphic maps. This is equivalent to requiring that the transition functions

t_{UV} : U\cap V \to \mathrm{GL}_k(\mathbf{C})

are holomorphic maps. The holomorphic structure on the tangent bundle of a complex manifold is guaranteed by the remark that the derivative (in the appropriate sense) of a vector-valued holomorphic function is itself holomorphic.

The sheaf of holomorphic sections[edit]

Let E be a holomorphic vector bundle. A local section s : UE|U is said to be holomorphic if, in a neighborhood of each point of U, it is holomorphic in some (equivalently any) trivialization.

This condition is local, meaning that holomorphic sections form a sheaf on X. This sheaf is sometimes denoted \mathcal O(E). Such a sheaf is always locally free of the same rank as the rank of the vector bundle. If E is the trivial line bundle \underline{\mathbf{C}}, then this sheaf coincides with the structure sheaf \mathcal O_X of the complex manifold X.

The sheaves of forms with values in a holomorphic vector bundle[edit]

If \mathcal E_X^{p, q} denotes the sheaf of C differential forms of type (p, q), then the sheaf of type (p, q) forms with values in E can be defined as the tensor product

\mathcal{E}^{p, q}(E) \triangleq \mathcal E_X^{p, q}\otimes E.

These sheaves are fine, meaning that they have partitions of the unity.

A fundamental distinction between smooth and holomorphic vector bundles is that in the latter, there is a canonical differential operator called the Dolbeault operator:

\overline{\partial} : \mathcal{E}^{p, q}(E) \to \mathcal{E}^{p, q+1}(E).

It is obtained by taking antiholomorphic derivatives in local coordinates.

Cohomology of holomorphic vector bundles[edit]

If E is a holomorphic vector bundle, the cohomology of E is defined to be the sheaf cohomology of \mathcal O(E). In particular, we have

H^0(X, \mathcal O(E)) = \Gamma (X, \mathcal O(E)),

the space of global holomorphic sections of E. We also have that H^1(X, \mathcal O(E)) parametrizes the group of extensions of the trivial line bundle of X by E, that is, exact sequences of holomorphic vector bundles 0 → EFX × C → 0. For the group structure, see also Baer sum.

The Picard group[edit]

In the context of complex differential geometry, the Picard group Pic(X) of the complex manifold X is the group of isomorphism classes of holomorphic line bundles with group law given by tensor product and inversion given by dualization. It can be equivalently defined as the first cohomology group H^1(X, \mathcal O_X^*) of the sheaf of non-vanishing holomorphic functions.