Holomorphic section - Imported Knowlpedia Content

Definition

Let $\pi:E\to X$ be a holomorphic vector bundle. A holomorphic section over an open set $U\subseteq X$ is a holomorphic map $s:U\to E$ satisfying $\pi\circ s=\operatorname{id}_U$ ; in particular, it is a section of the underlying smooth bundle. In a holomorphic trivialization $E|_V\cong V\times\mathbb C^r$ , the section has the form $x\mapsto(x,s^1(x),\ldots,s^r(x))$ , where all coefficient functions $s^j$ are holomorphic. This local condition is independent of the chosen holomorphic trivialization.

Sheaf of sections

Holomorphic sections restrict to smaller open sets and uniquely glue when they agree on overlaps. Hence $U\mapsto\Gamma(U,E)$ is a sheaf and a module over the sheaf of holomorphic functions. In a local holomorphic frame it is a free module of rank $r$ .

Zeros and trivializations

A rank- $r$ bundle is holomorphically trivial over $U$ exactly when it has $r$ holomorphic sections that form a basis in every fiber. In particular, a holomorphic line bundle is trivial exactly when it admits a nowhere-vanishing global holomorphic section. A nonzero holomorphic section of a line bundle may vanish, and its zero set carries analytic information.

Comparison with smooth sections

Every holomorphic section is smooth, but a smooth section need not be holomorphic. In a holomorphic frame, the distinction is precisely whether its coefficient functions are holomorphic. This is stronger than pointwise complex-linearity, which is automatic for the selected vectors and imposes no differential condition Huybrechts, §2.2.

References

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