complex manifolds are complex analytic spaces

Question

Are complex manifolds complex analytic spaces? https://en.wikipedia.org/wiki/Complex_analytic_space

What does it mean that there can be singularities in a complex analytic space but not on a complex manifold?

Answer 1

A complex manifold is by definition smooth. It looks locally like $\mathbb C^n$ with smoothly differentiable transition functions.

A complex analytic space however, is more general. It looks locally like the zero set of finitely many polynomials.

Example: The zero set $V$ of $f=x^3-y^2$ in $\mathbb C^2$ is a cusp. Topologically, it looks like a pinched sphere with one point removed.

One can also consider the closure of $V$ in $\mathbb C \mathbb P^2$, which is defined by the zero set of the homogeneous polynomial $\overline f = x^3-y^2z$. This looks topologically like a pinched sphere (without any points removed).

It is obtained by glueing the spaces defined by the dehomogenizations $f_1 = x^3-y^2$, $f_2=x^3-z$ and $f_3=1-y^2z$.

Answer 2

Complex manifolds are indeed complex analytic spaces. As in the wikipedia link you've attached, complex analytic spaces is the generalization of complex manifolds. Complex manifolds are locally isomorphic to $\mathbb{C}^n$, while complex analytic spaces are locally isomorphic to zero sets of holomorphic functions.