Potential theory on complex projective space: Application to characterization of pluripolar sets and growth of analytic varieties
The purpose of this paper is to characterize the non-locally pluripolar subsets of the projective space \(P({\mathbb{C}}^ n)\) in terms of a capacity on the homogeneous polynomials. If f is a homogeneous polynomial of degree d on \(P({\mathbb{C}}^ n)\) and \(f^*\) is the dual polynomial, the author sets \(K_ d(z,f)=\log \| z\|^ d| f^*(z)|^{- 1}.\) Then basically, the non-locally pluripolar subsets E are those such that, if \[ U_ d(E)=\inf_{\mu \geq 0, \int d\mu =1}\sup_{z\in P({\mathbb{C}}^ n)}\int_{f}K_ d(z,f) d\mu (f), \] then \(\limsup_{d\to \infty}U_ d(E)>0.\) This is closely related to the projective capacity of \textit{H. Alexander} [Recent developments in several complex variables, Proc. Conf., Princeton Univ. 1979, Ann. Math. Stud. 100, 3-27 (1981; Zbl 0494.32001)]. This capacity can be used to obtain average Bezout type theorems for algebraic hypersurfaces.
locally pluripolar subsets of the projective space
scientific article; zbMATH DE number 3933467