Complex projective structure

Definition

Let $X$ be a Riemann surface. A complex projective structure on $X$ is a maximal atlas of charts

\varphi_\alpha:U_\alpha\longrightarrow \varphi_\alpha(U_\alpha)\subseteq\mathbb P^1(\mathbb C)

such that every transition map

\varphi_\beta\circ\varphi_\alpha^{-1}

is locally the restriction of a Möbius transformation wherever it is defined.

Underlying complex structure

Möbius transformations are biholomorphic, so a complex projective atlas is in particular a complex atlas. Thus a projective structure refines the complex structure of $X$ ; it is extra data, not merely the assertion that $X$ is a Riemann surface.

Developing map and holonomy

Analytically continuing projective charts on the universal cover produces a locally biholomorphic developing map

\operatorname{dev}:\widetilde X\to\mathbb P^1(\mathbb C)

and a holonomy representation

\rho:\pi_1(X)\to PGL_2(\mathbb C)

for which $\operatorname{dev}(\gamma x)=\rho(\gamma)\operatorname{dev}(x)$ . The pair is defined only up to simultaneous postcomposition and conjugation by a Möbius transformation.

Associated projective connection

Taking the Schwarzian derivative of projective charts in an ordinary holomorphic coordinate produces a holomorphic projective connection.

References

R. C. Gunning, Lectures on Riemann Surfaces, Princeton University Press, 1966. Relevant: projective structures and projective connections.
William M. Goldman, “Projective structures with Fuchsian holonomy,” Journal of Differential Geometry 25 (1987), 297–326. Project Euclid record. Relevant: developing maps and holonomy.