Calabi-Yau

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Let $M$ be a Kaehler manifold. Then $M$ is Calabi-Yau if $c 1 (K M) = 0$ . If $M$ is a variety, we say that $M$ is a Calabi-Yau variety if it has vanishing canonical class.

Examples

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

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

$0\to TM \to T\mathbb{P}^n \to \mathcal{O}(n+1)|_M \to 0$

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

$c(TM) = 1+c_1(TM)+\ldots = \frac{(1+\omega)^{n+1}}{1+(n+1)\omega}$

from which one easily concludes that $c 1 (T M) = c 1 (K M) = 0$ .

Calabi Conjecture

The first Chern class of a Kaehler manifold is homologous to the Ricci curvature. For this reason, Calabi conjectured that when $c 1 = 0$ , there existed a metric on the manifold whose Ricci form identically vanished. Thus, the Calabi-Conjecture, was eventually proven by Shing-Tung Yau.

Calabi-Yau

From Conservapedia

Examples

Calabi Conjecture

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