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 ChernWeil theory). Does this remain true in $H^2(X,\mathbb{Z})$ ? If not, is there a way to relate the two ?
Take the 2minute tour
×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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 torsionfree, so ChernWeil works. 

