Zero varieties for the Nevanlinna class in convex domains of finite type in \(\mathbb{C}^n\). (English) Zbl 0912.32001

Bruna, Joaquim; Charpentier, Philippe; Dupain, Yves

Ann. Math. (2) 147, No. 2, 391-415 (1998).

Let \(\Omega\) be a smoothly bounded domain in \(\mathbb{C}^n\), \(\Omega= \{\rho<0\}\), \(\partial\Omega= \{\rho=0\}\) and \(\nabla\rho\neq 0\) on \(\partial \Omega\) and \(N(\Omega)\) denotes the Nevanlinna class of the functions analytic in \(\Omega\). For any \(f\in N(\Omega)\), the zero-variety \(X= \{f=0\}\) satisfies the Blaschke condition, i.e. if \(X_k\) are the irreducible components of \(X\) and \(n_k\) the corresponding multiplicities of \(f\), then \[ \sum_k n_k\int_{X_k} \bigl| \rho(z) \bigr| d\mu/X_k (z)<+ \infty, \tag{B} \] where \(d\mu_{X_k}\) is the Euclidean measure on the regular part of \(X_k\).
Now, let \(\Omega\) be a convex domain in \(\mathbb{C}^n\).
The authors solve the problem whether in dimension \(n>1\) the Blaschke condition (B) is sufficient for the divisor \(\widehat X=\{X_k,n_k\}\) to be defined by a function \(f\in N(\Omega)\). The main result is: If \(\Omega\) is a convex domain of finite strict type, every divisor \(\widehat X\) in \(\Omega\) satisfying the Blaschke condition can be defined by a function \(f\in N(\Omega)\). The proof consists of the following steps: To each divisor \(\widehat X\) is associated a (1,1) current \(\Theta\), and the nonisotropic estimates for the current \(\Theta\) are obtained, and the corresponding \(\partial\)-equation are solved.

Reviewer: A.Klíč (Praha)

MSC:

32A10	Holomorphic functions of several complex variables
32A17	Special families of functions of several complex variables
32C25	Analytic subsets and submanifolds
32C30	Integration on analytic sets and spaces, currents

Keywords:

Poincaré homotopy formula; \(\overline\partial\)-equation; Nevanlinna class; Blaschke condition; divisor