Weak Kähler hyperbolicity is birational

Abstract

We show that a compact Kähler manifold bimeromorphic to a weakly Kähler hyperbolic manifold is weakly Kähler hyperbolic, providing an answer to a problem raised by J. Kollár in his 1995 book "Shafarevic maps and automorphic forms"

16 pages, no figures, comments are very welcome! v2: fixed some typos (including in the title), Corollary 1.3 and Theorem 5.1 are improved, and Corollary 1.4 is new