It is well-known that the Chern class of a line bundle in $H^2(M,\mathbb Z)$ fully determines the bundle up to isomorphism. However, in this wikipedia entry on Calabi-Yau manifolds it is stated that there are manifolds that have a non-trivial canonical bundle, but a vanishing integral Chern class.
What is happening here? The only thing I can think of is that a differentiably trivial line bundle doesn't have to be holomorphically trivial. Is that the case? How should one think of such bundles that are topologically but not analytically trivial?