Calabi-Yau

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.

For example, 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.