Function theoretic properties of symmetric powers of complex manifolds

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 Carath'eodory and Kobayashi hyperbolicity and Kobayashi completeness in that class of domains is presented.