Difference between revisions of "Calabi-Yau"
(New page: 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...) |
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0\to TM \to T\mathbb{P}^n \to \mathcal{O}(n+1)|_M \to 0 | 0\to TM \to T\mathbb{P}^n \to \mathcal{O}(n+1)|_M \to 0 | ||
</math> | </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>. | ||
Revision as of 08:20, July 8, 2008
Let be a Kaehler manifold. Then 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} is Calabi-Yau if 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(K_M) = 0} . If 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} is a variety, we say 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 M} is a Calabi-Yau variety if it has vanishing canonical class.
For example, a degree 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 n+1} hypersurface 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} 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} .