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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.

==Examples==

==Examples==

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

All [[elliptic curve]]s 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  

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>

<math>

==Calabi Conjecture==

==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. Thus, the Calabi-Conjecture, was eventually proven by [[Shing-Tung Yau]].

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. Thus, the Calabi-Conjecture, was eventually proven by [[Shing-Tung Yau]].

[[Category:Mathematics]]

[[Category:Mathematics]]