# Topology ad Geometry of $\mathbb{C}^n/\mathbb{Z}_k$

I would like to know topologically what the space $(\mathbb{C}^n – \{0\})/\mathbb{Z}_k$ may be thought of as.

The paper I am reading says that we let $\mathbb{Z}_k$ act on $\mathbb{C}^n – \{0\}$ via $z \rightarrow e^{\frac{2\pi i}{k}}z$.

Perhaps I am just not thinking of this in the right way but I cannot see what the quotient space will look like. Moreover this space is then endowed with the quotient flat metric and later referred to as a metric cone. So is this a topological cone space, if so is the group action incorrect for this?

#### Solutions Collecting From Web of "Topology ad Geometry of $\mathbb{C}^n/\mathbb{Z}_k$"

Topologically it’s not so interesting, but geometrically it’s a cone. The action is rotation by $2\pi / k$. It’s what you would obtain by taking a $2\pi / k$-sector of the complex plane and gluing the ends together via rotation. This gives a nice flat metric everywhere besides for the origin, because the angle around $z=0$ is just $2 \pi / k$, which is less than the requisite $2\pi$ for $k>1$. In other words, this is just a cone with the origin being the cone point, with the cone point then being removed.