Why is it that the vanishing of the integral first Chern class of a compact Kahler manifold is equivalent to the canonical bundle being trivial? I can see that it implies that the canonical bundle must be topologically trivial, but not necessarily holomorphically trivial. Does proving the equivalence require Yau's theorem, in order to produce a flat connection on the canonical bundle, or is there a more elementary proof? 



Two definitions of Calabi-Yau manifolds
I have looked for a while for a proof
which does not use the Calabi-Yau theorem
and nobody seems to know it.


Also, there are plenty of non-Kaehler
manifolds with canonical bundle trivial
topologically and non-trivial as a holomorphic
bundle (the Hopf surface is an easiest
example).


The argument actually uses the Calabi-Yau
theorem, Bochner's vanishing, Berger's classification 
of holonomy and Bogomolov's decomposition theorem.


From Calabi-Yau theorem you infer that
there exists a Ricci-flat Kaehler metric.
Since the Ricci curvature is a curvature
of the canonical bundle, this implies
that the canonical bundle admits a flat
connection.


Of course, this does not mean that
it is trivial holomorphically; in fact,
the canonical bundle is flat on Hopf surface 
and on the Enriques surface, which are
*not* Calabi-Yau.


For Calabi-Yau manifolds, however,
it is known that the Albanese map
is a locally trivial fibration and 
and has Calabi-Yau fibers with trivial
first Betti number. This is shown using the 
Bochner's vanishing theorem which implies 
that all holomorphic 1-forms are parallel.


Now, by adjunction formula, you prove that
the canonical bundle of the total space is
trivial, if it is trivial for the base and the fiber. 
The base is a torus, and the fiber is a Calabi-Yau
with $H^1(M)=0$. For the later, triviality
of canonical bundle follows from Bogomolov's
decomposition theorem, because such a 
Calabi-Yau manifold is a finite quotient
of a product of simple Calabi-Yau manifolds
and hyperkaehler manifolds having holonomy
$SU(n)$ and $Sp(n)$. Bogomolov's decomposition
is itself a non-trivial result, and (in this generality)
I think it can be only deduced from the Berger's
classification. The original proof of Bogomolov was
elementary, but he assumed holomorphic triviality
of a canonical bundle, which we are trying to prove.


This argument is extremely complicated; also, 
it is manifestly useless in non-Kaehler situation 
(and in many other interesting situations).
I would be very interested in any attempt
to simplify it.


**Update:** Just as I was writing the reply, 
Dmitri has posted a link to Bogomolov's article, where
he proves that some power of a canonical bundle is 
always trivial, without using the Calabi-Yau theorem.


