Projective Hulls and the Projective Gelfand Transformation

Abstract

We introduce the notion of a ``projective hull'' for subsets of complex projective varieties, parallel to the idea of the polynomial hull in affine varieties. With this concept, a generalization of J. Wermer's classical theorem on the hull of a curve in $C^n$ is established in the projective setting. The projective hull is shown to have interesting properties and is related to various extremal functions and capacities in pluripotential theory. A main analytic result asserts that for any point x in the projective hull K^ of a compact set K in $P^n$, there exists a positive current T of bidimension (1,1) with support in the closure of K^, and a probability measure $μ$ on K with $d d^cT=μ- [x]$. This result generalizes to any Kahler manifold and has strong consequences for the structure of K^. We also introduce the notion of a projective spectrum for Banach graded algebras, parallel to the Gelfand spectrum of a Banach algebra. This projective spectrum has universal properties exactly like those in the Gelfand case. Moreover, the projective hull is shown to play a role (for graded algebras) completely analogous to that played by the polynomial hull in the study of finitely generated Banach algebras. This paper gives foundations for generalizing many of the results on boundaries of varieties in $C^n$ to general algebraic manifolds.