Abstract
In the paper we study properties of symmetric powers of complex manifolds. We investigate a number of function theoretic properties [e. g. (quasi) c-finite compactness, existence of peak functions] that are preserved by taking the symmetric power. The case of symmetric products of planar domains is studied in a more detailed way. In particular, a complete description of the Carathéodory and Kobayashi hyperbolicity and Kobayashi completeness in that class of domains is presented.
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