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Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
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First Chern class of canonical bundle ?
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-W …