Holomorphic line bundle

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.

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.