Unitary representations of the fundamental group of orbifolds

Abstract

Abstract. There is a well known bijective correspondence between the isomorphism classes of polystable vector bundles E with c i (E) = 0 for i ≥ 1 on a smooth complex projective variety and the equivalence classes of unitary representations of the fundamental group of the variety. We show that this bijective correspondence extends to smooth orbifolds.

Topological invariants of parabolic G-Higgs bundles

“…whereas Formula (6) implies that deg(E) = par deg(F ). In conclusion, the equality of the degree provides the following proposition: Proposition 6.10 (Proposition 5.9 in [9]).…”

“…We have more invariants if we would like to classify the holomorphic bundle instead of the topological bundle. From the classical Narasinhan-Seshadri correspondence, there is a correspondence between unitary representations of fundamental group and rank n polystable bundles E with trivial first Chern class and it holds that c 2 (E) • c 1 (L) n−1 for an ample line bundle L. I. Biswas and A. Hogadi in [6] generalized this correspondence for a compact orbifold of any dimension and any rank: Theorem 7.7 (Theorem 1.2 in [6]). Let M be a complex projective orbifold of dimension n and E a vector bundle over X with L an ample line bundle.…”

“…Theorem 1.2 in[6]). Let M be a complex projective orbifold of dimension n and E a vector bundle over X with L an ample line bundle.…”

Vector bundles and modular forms for Fuchsian groups of genus zero

“…These results have been extended considerably in the direction of nonabelian Hodge theory, due initially to Hitchin, Donaldson and Simpson -see [Sim91] and the references therein. In particular, the Narasimhan-Seshadri result has been generalized [BH16], [Sim11] to the setting of compact orbifolds discussed in Proposition 2.2. For line bundles on a compact orbifold of genus zero, where all representations of rank one have finite image, one does not need the full strength of these results, as shown in the proof of Proposition 2.2.…”