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Explicit construction of moduli space of bounded complete Reinhardt domains in \(\mathbb C^n\). (English) Zbl 1229.32004

The authors construct an explicit and complete infinite set of biholomorphic invariants for bounded, complete, pseudoconvex Reinhardt domains in \(\mathbb{C}^{n}\) with class \(C^{1}\) boundary. The idea is that, in view of T. Sunada’s theorem about linear equivalence of biholomorphic Reinhardt domains [Math. Ann. 235, 111–128 (1978; Zbl 0357.32001)], suitably normalized \(L^{2}\) norms of monomials should serve as biholomorphic invariants after accounting for permutations of the variables.
The authors implement this idea to obtain invariants that initially live in the Cartesian product of \(n!\) copies of the group ring of the symmetric group, modulo a natural equivalence relation. By applying a result of M. Göbel [J. Symb. Comput. 19, No. 4, 285–291 (1995; Zbl 0832.13006)] about generators of rings of permutation-invariant polynomials, the authors realize the invariants as numbers. Previously, the first and the third author used a different method to address the problem of moduli of Reinhardt domains in dimension two [J. Differ. Geom. 82, No. 3, 567–610 (2009; Zbl 1181.32003)].

MSC:

32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
32G13 Complex-analytic moduli problems
32H99 Holomorphic mappings and correspondences
32Q57 Classification theorems for complex manifolds
32T99 Pseudoconvex domains
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