On the existence of unitary flat connections over the punctured sphere with given local monodromy around the punctures
A parabolic structure at a point \(x\) on a vector bundle \(E\) on a smooth curve \(X\) consists of a flag on the fiber \(E_x\) together with real numbers called weights attached to it. A parabolic vector bundle is a vector bundle with such a parabolic data at finitely many points of \(X\). Let \(S\) be the set of parabolic points. In case \(X = \mathbb{P}^1\), the author gives sufficient and necessary conditions for the existence of (semi)stable parabolic bundles with given parabolic data. These conditions are inequalities involving parabolic data. This generalises the author's results in the rank two case [\textit{I. Biswas}, Int. J. Math. 9, 523-533 (1998; Zbl 0939.14015)]. There is a one to one correspondence between the parabolic bundles and representations of the fundamental group \(\pi ({\mathbb{P}}^1 - S)\) with the parabolic data at a point \(s\in S\) corresponding to the conjugacy class of the local monodromy at \(s\). Then the above conditions can be interpreted as sufficient and necessary conditions for realising conjugacy classes \(C_s, s\in S\), as the local monodromies of a unitary local system, the conditions being in terms of eigenvalues of \(C_s\) and their multiplicities.
representations of the fundamental group
scientific article; zbMATH DE number 1563747