Holomorphic line bundle - Imported Knowlpedia Content

Definition

Let $X$ be a complex manifold. A holomorphic line bundle on $X$ is a holomorphic vector bundle $L\to X$ of complex rank one. Equivalently, $X$ has an open cover $\{U_i\}$ and holomorphic trivializations $L|_{U_i}\cong U_i\times\mathbb C$ whose transition functions $g_{ij}:U_i\cap U_j\to\mathbb C^\times$ are nowhere-vanishing holomorphic functions satisfying $g_{ij}g_{jk}=g_{ik}$ on triple overlaps. Isomorphisms are invertible holomorphic vector-bundle morphisms over $X$ . Its fibers are one-dimensional complex vector spaces varying holomorphically over the base.

Tensor operations

The tensor product $L\otimes L'$ , dual $L^\vee$ , and pullback $f^*L$ along a holomorphic map are holomorphic line bundles. Their transition functions are respectively $g_{ij}g'_{ij}$ , $g_{ij}^{-1}$ , and $g_{ij}\circ f$ . Isomorphism classes form an abelian group under tensor product, with the trivial bundle as identity and the dual as inverse.

Sections and local data

A holomorphic section is represented in each trivialization by a holomorphic function $s_i$ satisfying $s_i=g_{ij}s_j$ under the stated transition convention. A nowhere-vanishing global holomorphic section gives a holomorphic trivialization. A smooth nowhere-vanishing section proves only smooth triviality and need not trivialize the holomorphic structure.

Standard examples

The trivial bundle $X\times\mathbb C$ is holomorphic. The top exterior power of the holomorphic cotangent bundle is the canonical line bundle. Holomorphic line bundles also encode divisors and the Picard group; this relationship depends on the analytic setting and is developed in Griffiths–Harris, Chapter 1, §1.

References

P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley, 1978. DOI record. Relevant: Chapter 1, §1, holomorphic line bundles and divisors.
D. Huybrechts, Complex Geometry: An Introduction, Springer, 2005. DOI record. Relevant: §2.2, holomorphic vector bundles and line bundles.