[Math] First Chern class of canonical bundle

Question

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 ?

Answer 1

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.