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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 ?

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$H^2(X, \mathbf{Z})$ is the group of isomorphism classes of complex line bundles; the first Chern class of a vector bundle is the isomorphism class of the determinant bundle. This gives $c_1(T_X) = c_1(det T_X)$. Since $\omega_X = (det T_X)^{-1}$, you also get $c_1(\omega_X) = -c_1(det T_X)$. This works in pretty much any theory of Chern classes ($H^\bullet(X, \mathbf{Z})$, Chow groups, etc). – Pavel Safronov May 24 '13 at 21:28
up vote 3 down vote accepted

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.

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