How do we prove that a compact Kahler manifold whose 1st Chern class vanishes admits a globally defined nowhere vanishing volume form? Thanks.
One can use Yau's theorem here. It states that $M$ admits a Kahler Ricci flat metric. With this metric, the holomony group lie inside $SU(n)$, thus the standard holomorphic $(n,0)$ form $$dz^1 \wedge \cdots \wedge dz^n$$ can be extended to the whole $M$. This can be found in the book "Riemannian holomony groups and Calibrated Geometry" by D. Joyce.
Edit: If the first Chern class is defined by the image of boundary homomorphism $$H^1 (M, \mathbb C^*) \to H^2(M, \mathbb Z)$$ of the anitcanonical line bundle, then your statement is almost trivial as $L \mapsto c_1(L)$ is a one-one correspondence. (I am thinking of $c_1(M)$ as defined by curvature, so it is the torsion free part of the image of the boundary homomorphism)