Calabi-Yau
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Let 
be a Kaehler manifold. Then is Calabi-Yau if 
. If is a variety, we say that is a Calabi-Yau variety if it has vanishing canonical class.
Examples
All elliptic curves are Calabi-Yau, since they are parallelizeable.
A degree 
hypersurface in 
is Calabi-Yau. For from the exact sequence
we get that 
. But this implies that
from which one easily concludes that 
.
Calabi Conjecture
The first Chern class of a Kaehler manifold is homologous to the Ricci curvature. For this reason, Calabi conjectured that when 
, there existed a metric on the manifold whose Ricci form identically vanished. This, the Calabi-Conjecture, was eventually proven by Shing-Tung Yau.
