Intereting Posts

Why are harmonic functions called harmonic functions?
Combinatorial argument for $1+\sum_{r=1}^{r=n} r\cdot r! = (n+1)!$
How to prove the formulation of mode-$n$ matricization and preclusive mode-$n$ product?
Confused about models of ZFC and passage of book
Limit: $\lim_{x\to 0}\frac{\tan3x}{\sin2x}$
Show that $k/(xz-y^2)$ is not a UFD.
Natural Logarithm and Integral Properties
log base 1 of 1
Quadratic with integer roots and coefficients in arithmetic progression
What's wrong with this “backwards” definition of limit?
Given $f, g \in k$ coprime, why can we find $u,v \in k$ such that $uf + vg \in k\setminus\{0\}$?
Partitioning $\Bbb{N}$
Help with a Probability Proof
Topological spaces as model-theoretic structures — definitions?
Determine in How many ways $N!$ can be expressed as sum of consecutive numbers

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?

- Cross Ratio is positive real if four points on a circle
- Geometry of the dual numbers
- the sum of $1-\frac{1}{5}+\frac{1}{9}-\frac{1}{13}+…$
- Do odd imaginary numbers exist?
- Roots of a equation on a complex plane
- Why is the complex number $z=a+bi$ equivalent to the matrix form $\left(\begin{smallmatrix}a &-b\\b&a\end{smallmatrix}\right)$

- Convergence of a sequence with assumption that exponential subsequences converge?
- What does order topology over Ordinal numbers look like, and how does it work?
- $E'$ is closed, where $E'$ is the set of limit points of $E$
- Complex Analysis Solution to the Basel Problem ($\sum_{k=1}^\infty \frac{1}{k^2}$)
- Limit of sequence of sets - Some paradoxical facts
- Bijective mapping from $(-1,1)$ to $\Bbb R$
- Form a space $X$ by identifying the boundary of $M$ with $C$ by a homeomorphism. Compute all the homology groups of $X$.
- Interior of arbitrary products
- Why is $1/i$ equal to $-i$?
- Is a closed subset of a compact set (which is a subset of a metric space $M$) compact?

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.

- When do Harmonic polynomials constitute the kernel of a differential operator?
- When is a cyclotomic polynomial over a finite field a minimal polynomial?
- Smooth classification of vector bundles
- Prove that the preimage of a prime ideal is also prime.
- Can the product of an $4\times 3$ matrix and a $3\times 4$ matrix be invertible?
- General Lebesgue Dominated Convergence Theorem
- Visual references for the Riemann-Stieltjes integral.
- Binary Strings of the form *111*
- How to show that disjoint closed sets have disjoint open supersets?
- Find new region after applying polar change of coordinates
- Factorization of $p\mathcal{O}_K$ in $\mathcal{O}_K$
- Radon-Nikodym Decomposition
- Integral $\int_0^\pi \theta^2 \ln^2\big(2\cos\frac{\theta}{2}\big)d \theta$.
- Optimization involving integrals with varying limits
- How many distinct functions can be defined from set A to B?