Let $S$ be a compact R.S of genus $\geq 2$. In the paper "Stable and unitary vector bundles on compact Riemann surfaces" (by Narasimhan and Seshadri), they claim that there is a branched covering map from the upper half plane to $S$ which is ramified at exactly one point (with index $N$) (i.e. $S$ is the quotient of $\mathbb{H}$ by the group $\langle A_i, B_i, C \vert \Pi [A_i, B_i] = C, [A_i,C] = [B_i, C] = I, C^N=I \rangle$). I am not able to access the reference (Grothendieck) pointed out in that paper. Is there any other reference (or any easy proof of this?) ? Also, does a similar fact hold for non-compact Riemann surfaces (compact minus a finite collection of points).
[Math] Branched covers of compact Riemann surfaces
ag.algebraic-geometrycomplex-geometry
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I think what you want probably follows from Theoreme 2.5 of the paper: Biquard, Olivier: Fibrés paraboliques stables et connexions singulières plates, Bull. Soc. Math. France 119 (1991), no. 2, 231–257.
Ciao Francesco!
The answer to your question is yes, and the work of Bogomolov you are looking for about semistability of the tangent space (for minimal surfaces of general type indeed, no need of ampleness of the canonical bundle) is
"Holomorphic tensors and vector bundles on projective varieties", Math. USSR Izvestija 13/3 (1979) 499-555.
But let me give you a more general framework.
Given a Hermitian holomorphic vector bundle $(E,h)$ over a compact Kähler manifold $(X,\omega)$, if $(E,h)\to (X,\omega)$ is Hermite-Einstein then its dual as well as its symmetric powers (with the induced Hermitian structures) also are Hermite-Einstein.
Next, if $(E,h)\to (X,\omega)$ is Hermite-Einstein, then $E\to X$ is $[\omega]$-semistable (polystable, indeeed). This is the "easy" direction of the Kobayashi-Hitchin correspondence.
If a projective manifold $X$ has ample canonical bundle, then it admits a Kähler-Einstein metric $\omega$ of negative Einstein constant, i.e. a Kähler metric such that $\operatorname{Ric}(\omega)=-\omega$ (by the Aubin-Yau Theorem).
In particular $(T_X,\omega)\to (X,\omega)$ is Hermite-Einstein, and then so does any symmetric power of the cotangent bundle (with the induced Hermitian structure). This makes these vector bundles $[\omega]$-semistable, as we saw above.
Finally, in this case, begin $[\omega]$-semistable means that these vector bundle are $K_X$-semistable, since $[\omega]=[-\operatorname{Ric}(\omega)]=c_1(K_X)$.
You can see all this and much more on S. Kobayashi "Differential geometry of complex vector bundles".
P.S. For quite recent advances about semistability of the tangent sheaf of possibly singular varieties (even in the logarithmic setting), you might want to take a look to this paper by H. Guenancia.
Best Answer
From your group-theoretic description, it seems to me that you are asking for a covering $S' \to S$ which is only ramified over one point of $S$, and the ramification index of each point of the inverse image divides $N$. So, Felipe's construction should work.
Here is a more direct construction. Suppose that $S$ is compact. Pick a point $p\in S$; the fundamental group of $S\smallsetminus \{p\}$ is the free group on $2N$ generators $A_1, \dots, A_g$, $B_1, \dots, B_g$; the product $C = \prod_{i=1}^g[A_i, B_i]$ represents a small loop around $p$. Consider the homomorphism $\pi_1(S\smallsetminus \{p\})\to \mathrm{S}_{2N}$ that sends $A_1$ into $(1 \dots N)$, $B_1$ into $(1 \dots 2N)$, and all the other $A_i$ and $B_i$ into the identity. It is easy to see that $C$ goes into $(1 \dots N)(N+1 \dots 2N)$, and that this homomorphism has transitive image. Hence the corresponding ramified cover $S' \to S$ has two ramification points over $p$, each of index $N$.
When $S$ is non-compact, then the small loop $C$ is part of a free set of generators of $\pi_1(S\smallsetminus \{p\}$, and the construction is even easier (for example, you can easily construct a cyclic cover $S' \to S$ of degree $N$ which is totally ramified over $p$, and nowhere else; this is impossible in the compact case).