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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.

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.

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

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  

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