On compact complex surfaces with finite homotopy rank-sum

A topological space (not necessarily simply connected) is said to have finite homotopy rank-sum if the sum of the ranks of all higher homotopy groups (from the second homotopy group onward) is finite. In this article, we characterize the smooth compact complex Kaehler surfaces having finite homotopy rank-sum. We also prove the Steinness of the universal cover of these surfaces assuming holomorphic convexity of the universal cover.
Comment: Final version; Proc. Amer. Math. Soc. (to appear)