Function Theory and Holomorphic Maps on Symmetric Products of Planar Domains

Abstract

We show that the ∂-problem is globally regular on a domain in C n , which is the n-fold symmetric product of a smoothly bounded planar domain. Remmert-Stein type theorems are proved for proper holomorphic maps between equidimensional symmetric products and proper holomorphic maps from Cartesian products to symmetric products. It is shown that proper holomorphic maps between equidimensional symmetric products of smooth planar domains are smooth up to the boundary.

Certain non-homogeneous matricial domains and Pick–Nevanlinna interpolation problem

“…Given Ω, as in the statement of Theorem 1.1, Zwonek in [20, Theorem 16] proved that not only Σ n (Ω) is Kobayashi hyperbolic (see Section 2 for the definition), in fact, it is Kobayashi complete. Now, since Kobayashi complete domains are taut, the same proof as in [3] could be followed to establish that Σ n (Ω) has the property (P) above. So, in fact, under the condition on the domain Ω as in Theorem 1.1, Σ n (Ω) satisfies the condition (P) automatically.…”

“…It is a nontrivial result of Chakrabarty-Gorai [3, Corollary 1.3] -who generalized the analogous result of Edigarian-Zwonek [8, Theorem 1] for Σ n (D) -that Σ n (Ω) satisfies the property (P) above. The proof of this latter fact as given in [3] crucially depended on the ability to extract subsequences -given an auxiliary sequence constructed from the given proper map -that converge locally uniformly.…”

Function Theoretic Properties of Symmetric Powers of Complex Manifolds

“…The starting point for considerations in the paper was inspired by recent developments on the function theory in symmetric powers (see e. g. [2,3] and [4]).…”

Proper holomorphic self-maps of symmetric powers of balls

“…Then the image of D m Sym is a pseudoconvex domain Σ m D in C m , called the symmetrized polydisc. See [5] for more details. Consequently, the symmetric power D m Sym is biholomorphically identified with the domain Σ m D in C m , which shows that Theorem 2 is indeed an extension of Theorem 1.…”