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Projective hulls and the projective Gelfand transform. (English) Zbl 1178.32008
Summary: We introduce the notion of a projective hull for subsets of complex projective varieties parallel to the idea of a polynomial hull in affine varieties. With this concept, a generalization of J. Wermer’s classical theorem on the hull of a curve in \(\mathbb C^n\) 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 \(\widehat{K}\) of a compact set \(K\subset\mathbb P^n\) there exists a positive current \(T\) of bidimension \((1,1)\) with support in the closure of \(\widehat {K}\) and a probability measure \(\mu\) on \(K\) with \(dd^cT= \mu-\delta_x\). This result generalizes any Kähler manifold and has strong consequences for the structure of \(\widehat{K}\).
We also introduce the notion of a projective spectrum for Banach graded algebras parallel to the Gelfand spectrum of a Banach algebra. This projective 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 \(\mathbb C^n\) to general algebraic manifolds.

32E99 Holomorphic convexity
46J99 Commutative Banach algebras and commutative topological algebras
14C99 Cycles and subschemes
32U15 General pluripotential theory
46J10 Banach algebras of continuous functions, function algebras
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