# Equivalence between parabolic bundles and bundles on orbicurves

"Everyone knows" that there is an equivalence between vector bundles on a smooth complete orbicurve and parabolic bundles with rational weights on its coarse moduli space. Roughly, it proceeds by pulling back the bundle, then performing elementary modifications along the flags.

But is this thoroughly explained anywhere in the literature? I know the paper of Furuta-Steer, but that is really about a Mehta-Seshadri-type result in differential geometry, associating flat connections with poles to both, whereas I seek a purely algebraic statement.

Also, is anything similar true for principal bundles with reductive structure group? Why?

There is a paper of Niels Borne, "Sur les représentations du groupe fondamental d'une variété privée d'un diviseur à croisements normaux simples", http://arxiv.org/abs/0704.1236, in which this is carefully written and generalized to any smooth variety with a divisor with simple normal crossing.

I don't know about principal bundles.

• To Michael: enjoy you stay; I'd like to be there too. – Angelo Aug 13 '11 at 7:42

I got interested in this question too, and this is how I understood it. I do not know the exact definition of a bundle over an orbicurve, but what you are asking should be the definition, and here is why.

Let $$E$$ be a bundle on a curve $$X$$. A parabolic structure at $$x\in X$$ is a sequence of bundles and maps

$$E=E_0 \rightarrow E_1 \rightarrow \cdots \rightarrow E_{N-1} \rightarrow E_N=E(x)$$

such that the composition is the canonical map $$E\rightarrow E(x)$$. If we drop the assumption that these are bundles and allow any coherent sheaves, but impose further that for any $$i$$ we have $$E_i\to E_i(x)$$ is the canonical map, we obtain the notion of a parabolic coherent sheaf.

Now let us take the direct sum

$$\tilde{E}:=\bigoplus_{i=0}^{N-1} E_i$$

If $$t$$ was the local parameter at $$x$$, we have the action of $$t^{1/N}$$ on $$\tilde{E}$$ as follows:

$$t^{1/N}(s_1,\ldots,s_N) = (x s_N, s_1,\ldots,s_{N-1}).$$

The group of $$N$$-th roots of unity $$\mu_N$$ acts on the choices of $$t^{1/N}$$, and also on $$\tilde{E}$$:

$$\zeta(s_1,\ldots,s_N) = (s_1, \zeta s_2,\ldots,\zeta^{N-1} s_N).$$

The action of $$t^{1/N}$$ should be the lifting of our bundle to the $$N$$-fold cover of a small disk around $$x$$, and the action of $$\mu_N$$ gives the equivariant structure. You can go back, i.e. given $$\tilde{E}$$, use the $$\mu_N$$ to decompose it into summands, and then multiplication by $$t^{1/N}$$ will give the maps. So parabolic structure = our bundle lifts to an equivariant bundle on the $$N$$-fold cover of a small disk around $$x$$.