Holomorphic projective connection

Definition

Let $X$ be a Riemann surface with local coordinates $z$ . In the convention of this knowl, a holomorphic projective connection is a collection of holomorphic functions $R_z$ , one in each coordinate, such that under a change of coordinate $w=w(z)$ ,

R_z(z)=R_w(w(z))\bigl(w'(z)\bigr)^2+S(w)(z),

where $S(w)$ is the Schwarzian derivative of the coordinate change.

Why this transformation law

If $f$ is a locally univalent projective coordinate, set $R_z=S(f)(z)$ . The Schwarzian chain rule gives exactly the displayed change-of-coordinate formula. Postcomposing $f$ with a Möbius transformation leaves $R_z$ unchanged.

Relation to projective structures

A complex projective structure determines a projective connection by taking the Schwarzians of its projective charts in ordinary holomorphic coordinates. Conversely, the local equation $S(f)=R_z$ produces projective charts. The affine-space theorem for projective connections describes how any two such connections differ.

Convention warning

Some sources put a factor of $1/2$ , or the opposite sign, in the Schwarzian term. The displayed transformation law fixes the normalization used here. In higher-dimensional differential geometry, “projective connection” can instead mean a projective equivalence class of affine connections; that is related but not this one-dimensional holomorphic definition.

References

R. C. Gunning, Lectures on Riemann Surfaces, Princeton University Press, 1966. Relevant: projective structures and projective connections.
Otto Forster, Lectures on Riemann Surfaces, Springer, 1981. Publisher record. Relevant: analytic continuation and meromorphic functions.