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.

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We consider the following question: Let S1 and S2 be two smooth, totally-real surfaces in C 2 that contain the origin. If the union of their tangent planes is locally polynomially convex at the origin, then is S1 ∪ S2 locally polynomially convex at the origin? If T0S1 ∩ T0S2 = {0}, then it is a folk result that the answer is yes. We discuss an obstruction to the presumed proof, and provide a different approach. When dim R (T0S1 ∩T0S2) = 1, we present a geometric condition under which no consistent answer to the above question exists. We then discuss conditions under which we can expect local polynomial convexity.2000 Mathematics Subject Classification. Primary: 32E20, 46J10.

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Let Γ n , n ≥ 2, denote the symmetrized polydisc in C n , and Γ 1 be the closed unit disc in C. We provide some characterizations of elements in Γ n . In particular, an element (s 1 , . . . , s n−1 , p) ∈ C n is in Γ n if and only if s