non-algebraic variety definition

Question

In this MO question it was several times used the notion of a non-algebraic variety. and I ask myself what is non-algebraic variety by definition. I'm unsure, if it is really a standard notation, since google couldn't provide a satisfying answer. any idea what it might be?

Answer 1

As far as I can tell, the linked post uses "non-algebraic variety" to mean a complex-analytic space which is not isomorphic to the analytification of any algebraic variety. Complex-analytic spaces are locally ringed spaces which are locally modeled on the vanishing locus of a finite set of analytic functions. Here's a full formal definition:

Let $\mathcal{O}_{\Bbb C^n}$ be the sheaf of anayltic functions on $\Bbb C^n$. Let $U$ be an open connected subset of $\Bbb C^n$, and fix finitely many analytic functions $f_1,\cdots,f_m\in\mathcal{O}_{\Bbb C^n}(U)$. Let $V\subset U$ be the common vanishing locus of the functions. Define a sheaf of rings $\mathcal{O}_V$ on $V$ as the restriction of $\mathcal{O}_U/(f_1,\cdots,f_m)$, where $\mathcal{O}_U$ is the restriction of $\mathcal{O}_{\Bbb C^n}$ to $U$. We call the locally ringed space $(V,\mathcal{O}_V)$ a model. A complex-analytic space $X$ is then a locally ringed space $(X,\mathcal{O}_X)$ so that every point $x\in X$ has an open neighborhood $U\subset X$ isomorphic as a locally ringed space to a model as described previously.

Googling "complex-analytic space" should give you more to look at.