Calabi-Yau

From Conservapedia

This article or section needs to be written in plain English, using plain English that most of our readers can understand. Articles that depend excessively on technical terms accessible only to specialists are useless for our purposes, so writers are admonished to avoid jargon

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

Views

Personal tools

Search

Popular Links

Help

World History

edit console