Average growth estimates for hyperplane sections of entire analytic sets (Q792515)

Let X be an analytic subvariety of \({\mathbb{C}}^ m\), and let \(P\subset {\mathbb{C}}^ m\) be a complex linear subspace. Although it is possible for X and \(X\cap P\) to have different rates of volume growth, it has been shown by Carlson, Gruman, and Stoll that X and \(X\cap P\) have the same rates of growth for ''almost all'' P. In this paper, the authors define a projective logarithmic capacity C on \({\mathbb{P}}^{m-1}\). They show that for a family of hyperplanes \(H\in {\mathbb{P}}^{m-1}\), the ''average'' of the rate of growth of \(H\cap X\) is bounded above and below by the rate of growth of X. This ''average'' is taken with respect to H as \(\int \mu(dH)\), where \(\mu\) is a measure on \({\mathbb{P}}^{m-1}\) arising in the construction of C. Several consequences are derived from this estimate. In particular, if \(C\{\) H:\(H\cap X\) is \(algebraic\}=0\) then X is algebraic.