Holomorphic germ - Knowlpedia

Definition

Let $X$ be a complex manifold and $a\in X$ . Two holomorphic functions $f:U\to\mathbb C$ and $g:V\to\mathbb C$ , defined on neighborhoods of $a$ , determine the same holomorphic germ at $a$ if they agree on some neighborhood $W\subseteq U\cap V$ of $a$ . The equivalence class is denoted $[f]_a$ or $f_a$ .

Local ring

Holomorphic germs at $a$ form a local ring $\mathcal O_{X,a}$ under pointwise addition and multiplication of representatives. Its unique maximal ideal consists of germs vanishing at $a$ ; a germ is a unit exactly when its value at $a$ is nonzero.

One complex variable

For a plane domain, a germ is completely determined by its convergent Taylor series at $a$ . The order of vanishing and the local inverse of a nonvanishing germ depend only on the germ, not on a representative.

Continuation

Analytic continuation transports a germ through chains of overlapping neighborhoods. The monodromy theorem gives conditions under which the transported germ is independent of the chosen path.

References

Otto Forster, Lectures on Riemann Surfaces, Springer, 1981. Publisher record. Relevant: Chapter 1, §§5–7.