The meromorphically convex hull of a compact set K in a complex manifold X collects the points z at which every meromorphic function on X that is holomorphic on K and z is bounded by its supremum norm on K; X is meromorphically convex when all such hulls are compact. This is what rational convexity in C^n becomes on an arbitrary complex manifold, and it comes with an inner hull that is the largest set to which those functions all extend holomorphically. Adding two conditions - that meromorphic functions separate points and supply local coordinates at every point - defines an M-manifold, the meromorphic counterpart of a Stein manifold: Stein manifolds, projective manifolds, every Riemann surface and the blow-up of a ball qualify, compact ones must be Moishezon, and Oka-Weil approximation by meromorphic functions holds on them. The striking lesson is that a convexity notion built from a chosen class of functions controls approximation independently of the holomorphic theory - there is a long C^2 that is an M-manifold yet carries no nonconstant holomorphic function at all.
Defines the meromorphically convex hull of a compact set K in a complex manifold X as the set of points z satisfying |f(z)| <= sup_K |f| for every meromorphic function on X that is holomorphic on K u…