Projective connections form an affine space

Statement

Let $X$ be a Riemann surface. If the set of holomorphic projective connections on $X$ is nonempty, it is an affine space modeled on the vector space

H^0(X,K_X^{\otimes2})

of holomorphic quadratic differentials.

Difference of two connections

If $R$ and $\widetilde R$ are projective connections, their Schwarzian terms cancel under a coordinate change $w=w(z)$ :

(R_z-\widetilde R_z)(z) = (R_w-\widetilde R_w)(w(z))\bigl(w'(z)\bigr)^2.

This is exactly the transformation law for a holomorphic quadratic differential.

Translation by a quadratic differential

Conversely, if $q$ is a holomorphic quadratic differential and $R$ is a projective connection, then the local coefficients $R_z+q_z$ satisfy the projective-connection transformation law. Therefore quadratic differentials act freely and transitively on the set of projective connections.

References

R. C. Gunning, Lectures on Riemann Surfaces, Princeton University Press, 1966. Relevant: projective structures and projective connections.