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Potential theory on complex projective space: Application to characterization of pluripolar sets and growth of analytic varieties

A set ECPnC is said to be locally pluripolar if for each pointp E E there exists a neighborhood U of p and a plurisubharmonic function defined on U such that E C U Ix b(x) oo and is not identically oo on each component of U. A basic problem in function theory of several complex variables is to characterize those sets which are pluripolar. In his paper on projective capacity [1], Alexander gives a characterization of pluripolar sets in PnC in terms of a Tchebycheff constant z(E). His theorem says that E is locally pluripolar if and only if z(E) 0. The constant z(E) is defined in terms of normalized homogeneous polynomials on PnC. Another characterization of pluripolar sets was recently given by Bedford and Taylor [3]. Their characterization involves the Monge-Ampere equation and a "balayage" for a set ECC. In this paper I give a characterization of locally pluripolar sets in P’C in terms of a singular integral with respect to a probability measure, supported on E; the set in question. The kernel of this singular integral is defined on