It seems worthwile to point out that it is not true that the vanishing of the integral first Chern class of a compact Kahler manifold implies that the canonical bundle is holomorphically trivial. It only implies that it is torsion, i.e. some positive multiple is holomorphically trivial, as indicated by Misha and Dmitri (an English version of Bogomolov's paper is here).

An example where $c_1(K_M)$ vanishes in integral cohomology while $K_M$ is not holomorphically trivial is any bielliptic surface (a finite unramified quotient of a complex $2$-torus), in which case $2K_M$ is trivial. This fact is remarked for example in this paper of Tian-Jun Li, Remark 6.4 (see also the paper of McDuff-Salamon cited there as [30]).

It seems worthwile to point out that it is not true that the vanishing of the integral first Chern class of a compact Kahler manifold implies that the canonical bundle is holomorphically trivial. It only implies that it is torsion, i.e. some positive multiple is holomorphically trivial, as indicated by Misha and Dmitri (an English version of Bogomolov's paper is here).

An example where $c_1(K_M)$ vanishes in integral cohomology while $K_M$ is not holomorphically trivial is any bielliptic surface (a finite unramified quotient of a complex $2$-torus), in which case $12K_M$ is trivial. This fact is remarked for example in this paper of Tian-Jun Li, Remark 6.4 (see also the paper of McDuff-Salamon cited there as [30]).