This is a somewhat simple question: consider a complex manifold $M$ and its canonical bundle $\omega\_X$. It is clear that in $H^2(X,\mathbb{R})$, 
$$c\_1(\omega\_X) = - c\_1(T\_X)$$
(Obvious using Chern-Weil theory). 
Does this remain true in $H^2(X,\mathbb{Z})$ ? If not, is there a way to relate the two ? 



First Chern class of canonical bundle ? 
Yes. This is true for every vector bundle. By functorialuty, it is sufficient to check on just the infinite Grassmanian. But its integral cohomology is torsion-free, so Chern-Weil works.


