Contents

# Contents

## Idea

A germ of a space is an equivalence class of pointed spaces, where two such spaces regarded as equivalent if they are isomorphic on small open neighbourhoods of the base points.

## Definition

The category of germs of spaces has as objects pointed spaces $(X,x_0)$, where space may depend on the context (topological space, complex manifold, …).

Morphisms $(X,x_0) \to (Y,y_0)$ are equivalence classes of basepoint-preserving morphisms $U \to Y$ defined on arbitrary open neighbourhoods $U$ of $x_0$. Two such morphisms are considered equal if they agree on an open neighbourhood of $x_0$ contained in the intersection of their domains.

## Examples

Let $f$ be a germ of a holomorphic function on a complex manifold $X$, defined on an open neighbourhood $U$ of some point $x_0$. Then the vanishing set of $f$, $V(f) \coloneqq \{ x \in U \,|\, f(x) = 0 \}$, is not well-defined as a space. However, it gives rise to a well-defined germ of a space.

The base space of a universal deformation? of a complex manifold is usually considered as a germ of a space.

## As a localization

The category of germs of space is the localization of the category of pointed spaces at the class of those morphisms which restrict to isomorphisms on open neighbourhoods of the basepoints.

Examples of sequences of local structures

geometrypointfirst order infinitesimal$\subset$formal = arbitrary order infinitesimal$\subset$local = stalkwise$\subset$finite
$\leftarrow$ differentiationintegration $\to$
smooth functionsderivativeTaylor seriesgermsmooth function
curve (path)tangent vectorjetgerm of curvecurve
smooth spaceinfinitesimal neighbourhoodformal neighbourhoodgerm of a spaceopen neighbourhood
function algebrasquare-0 ring extensionnilpotent ring extension/formal completionring extension
arithmetic geometry$\mathbb{F}_p$ finite field$\mathbb{Z}_p$ p-adic integers$\mathbb{Z}_{(p)}$ localization at (p)$\mathbb{Z}$ integers
Lie theoryLie algebraformal grouplocal Lie groupLie group
symplectic geometryPoisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization

A short exposition is contained in the textbook

• Daniel HuybrechtsComplex geometry - an introduction. Springer (2004). Universitext. 309 pages. (pdf)

For germs in deformation theory, see for instance

• Marco Manetti, Deformation theory via differential graded Lie algebras (arXiv:0507284).