The monodromy representations of projective structures (Q1109921)

Let M be a complex manifold of complex dimension n, identified with \(\tilde M/\pi_ 1(M)\), where \(\tilde M\) is the universal covering of M. Thena (flat) projective structure on M is a locally biholomorphic map \(f: \tilde M\to P^ n({\mathbb{C}})\) for which there is a homomorphism \(h: \pi_ 1(M)\to PGL(n+1,{\mathbb{C}})\) such that \(f\circ \gamma =h(\gamma)\circ f\) for all \(\gamma\) in \(\pi_ 1(M)\). Here, h (if it exists) is uniquely determined by f, and is called the monodromy representation of the projective structure. In this note, it is shown that the obstruction to lifting the monodromy representation to one of \(\pi_ 1(M)\) into \(GL(n+1,{\mathbb{C}})\) is the same for all projective structures (if any) on M, namely the image of the first Chern class of M in \(H^ 2(M,{\mathbb{Z}}/(n+1))\). Corollary: this obstruction vanishes in the case of a Riemann surface (a classical but often-proved result). The proof involves only linear-algebraic considerations on projective space.