Holomorphic vector bundle - Imported Knowlpedia Content

Definition

Let $X$ be a complex manifold. A holomorphic vector bundle of rank $r$ over $X$ is a complex vector bundle $\pi:E\to X$ whose total space is a complex manifold and which admits local trivializations

\Phi_i:\pi^{-1}(U_i)\longrightarrow U_i\times\mathbb{C}^r

that are fiberwise complex-linear holomorphic maps with holomorphic inverses and satisfy $\operatorname{pr}_1\Phi_i=\pi$ . Equivalently, the transition maps have the form $(x,v)\mapsto(x,g_{ij}(x)v)$ , where each $g_{ij}:U_i\cap U_j\to\operatorname{GL}_r(\mathbb{C})$ is holomorphic and the cocycle identities hold.

Holomorphic sections and morphisms

A holomorphic section is a section $s:X\to E$ that is holomorphic as a map of complex manifolds. In a holomorphic trivialization it is represented by a holomorphic map $U_i\to\mathbb{C}^r$ , and the representatives transform by $g_{ij}$ . A morphism of holomorphic vector bundles is a bundle map whose local matrix entries are holomorphic; an isomorphism is such a morphism with a holomorphic inverse.

Standard constructions

Direct sums, tensor products, duals, exterior powers, and pullbacks along holomorphic maps inherit holomorphic transition functions by applying the corresponding matrix operations. The tangent bundle of a complex manifold is holomorphic because changes of holomorphic coordinates have holomorphic Jacobian matrices. A holomorphic line bundle is the rank-one case, with transition functions valued in $\mathbb{C}^{\times}$ .

Comparison with smooth bundles

Forgetting the complex structures on the total space and transition maps gives an underlying smooth complex vector bundle. The converse is not automatic: smooth transition functions need not be holomorphic, and a fixed smooth complex bundle may support inequivalent holomorphic structures. Holomorphic local triviality is therefore extra analytic structure, not merely complex-linear fiber data.

References

D. Huybrechts, Complex Geometry: An Introduction, Springer, 2005. DOI record. Relevant: Definition 2.2.1 and the ensuing discussion of holomorphic bundles.