Calabi-Yau

Let <math>M</math> be a Kaehler manifold. Then <math>M</math> is Calabi-Yau if <math>c_1(K_M) = 0</math>. If <math>M</math> is a variety, we say that <math>M</math> is a Calabi-Yau variety if it has vanishing canonical class.

Examples

All elliptic curves are Calabi-Yau, since they are parallelizeable.

A degree <math>n+1</math> hypersurface <math>M</math> in <math>\mathbb{P}^n</math> is Calabi-Yau. For from the exact sequence

<math> 0\to TM \to T\mathbb{P}^n \to \mathcal{O}(n+1)|_M \to 0 </math>

we get that <math>c(T\mathbb{P}^n) = c(TM)(1+(n+1)\omega)</math>. But this implies that

<math> c(TM) = 1+c_1(TM)+\ldots = \frac{(1+\omega)^{n+1}}{1+(n+1)\omega} </math>

from which one easily concludes that <math>c_1(TM) = c_1(K_M) = 0</math>.

Calabi Conjecture

The first Chern class of a Kaehler manifold is homologous to the Ricci curvature. For this reason, Calabi conjectured that when <math>c_1 = 0</math>, there existed a metric on the manifold whose Ricci form identically vanished. This, the Calabi-Conjecture, was eventually proven by Shing-Tung Yau.