  The main goal of the paper is to connect matrix polynomial biorthogonality on
a contour in the plane with a suitable notion of scalar, multi-point Pad\'e
approximation on an arbitrary Riemann surface endowed with a rational map to
the Riemann sphere. To this end we introduce an appropriate notion of (scalar)
multi-point Pad\'e\ approximation on a Riemann surface and corresponding notion
of biorthogonality of sections of the semi-canonical bundle
(half-differentials). Several examples are offered in illustration of the new
notions.
