Calabi-Yau

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{P}^n} is Calabi-Yau. For from the exact sequence

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0\to TM \to T\mathbb{P}^n \to \mathcal{O}(n+1)|_M \to 0 }

we get that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c(T\mathbb{P}^n) = c(TM)(1+(n+1)\omega)} . But this implies that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c(TM) = 1+c_1(TM)+\ldots = \frac{(1+\omega)^{n+1}}{1+(n+1)\omega} }

from which one easily concludes that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_1(TM) = 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_1 = 0} , there existed a metric on the manifold whose Ricci form identically vanished. This, the Calabi-Conjecture, was eventually proven by Shing-Tung Yau.