Non-Isolated Singularities in Complex Orbifolds

Question

From my understanding for each point in an orbifold there is a neighborhood structurally equivalent to $\mathbb{C}^{m}/G$ (homeomorphism?) for some finite group G which depends continuously on the point. If the singularity is isolated it can be said that this singularity is locally modeled on $\mathbb{C}^{m}/G$ but I am a little confused because for non-isolated singularities I have read that they can be modeled on $\mathbb{C}^{k}/G$ x $\mathbb{C}^{m-k}$. How is this consistent with the definition of an orbifold? In the case of a weighted complex projective space are all singularities either isolated or of the product form above and how can you distinguish the types of singularities in terms of the weights etc?

Answer 1

Joyce gave a wrong definition of a complex orbifold. What he defined is (more or less) the underlying space of a complex orbifold. I wish he had not done this, but alas...

You can find a general definition of a topological orbifold in this wikipedia article.

As it is mentioned in Definition 1.27 in the book

"Orbifolds and Stringy Topology" by A.Adem, J.Leida and Y.Ruan, Cambridge Univ. Press, 2007,

a complex orbifold is an orbifold where all the defining data is holomorphic. In terms of the wikipedia article, this means that groups $\Gamma_i$ are acting complex-linearly on ${\mathbb C}^n$ (instead of ${\mathbb R}^n$) and the gluing maps $\phi_{ij}$ are biholomorphic.