Kodaira’s embedding theorem shows how to use potential theory on a compact complex manifold X to characterize when it admits a holomorphic embedding into the complex projective space, i.e., is projective algebraic. Cornalba and Griffiths, in the early ‘70’s, began trying to use complex analytic growth techniques to understand affine algebraic (in particular, non-compact) manifolds. There are several ways to approach this, using in various proportions complex differential geometry and potential theory on such open manifolds. The problem is to characterize the “slowest growing” entire functions on X and showing, eventually, these are the polynomial functions on an affine variety. There are several approaches to this problem, no one of which is, to this date, entirely satisfactory. Several such approaches will be described, each corresponding to a different characteristic picture of a closed algebraic variety in affine space, and various solutions of the homogeneous complex Monge-Ampère equation used to measure intrinsically the growth of X at infinity.
This represents work partly joint with Raul Aguilar and Zhou Zhang.