On compact complex surfaces with finite homotopy rank-sum

TL;DR

This paper characterizes compact complex Kähler surfaces with finite homotopy rank-sum and explores the Steinness of their universal covers under certain conditions, contributing to the understanding of their topological and complex-analytic properties.

Contribution

It provides a characterization of Kähler surfaces with finite homotopy rank-sum and establishes conditions for the Steinness of their universal covers.

Findings

Characterization of Kähler surfaces with finite homotopy rank-sum

Proof of Steinness of universal covers under holomorphic convexity

Connection between homotopy rank-sum and complex geometric properties

Abstract

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.