On the algebraicization of certain Stein manifolds

Abstract

To every real analytic Riemannian manifold M there is associated a complex structure on a neighborhood of the zero section in the real tangent bundle of M. This structure can be uniquely specified in several ways, and is referred to as a Grauert tube. We say that a Grauert tube is entire if the complex structure can be extended to the entire tangent bundle. We prove here that the complex manifold given by an entire Grauert tube is, in a canonical way, an affine algebraic variety. In the special case M = the 2-sphere, we show that any entire Grauert tube associated to a metric (not necessarily round) on M must be algebraically biholomorphic to the Grauert tube of the round metric, that is, the non-singular quadric surface in complex affine 3-space. (This second result has been discovered independently by B. Totaro.)