Difference between revisions of "Calabi-Yau"

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m (wikify, jargon tag)
(fluff up the beginning a bit, clarify proofs (but the jargon is unavoidable here), add more examples (and more jargon). i'll see if i can't find some motivation from physics to add to the intro.)
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{{jargon}}
 
{{jargon}}
  
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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.
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Calabi-Yau manifolds are a certain class of smooth [[complex analysis|complex]] [[manifolds]] that have proved to play an important role in several areas of mathematics and physics.  Calabi-Yau manifolds have proved to occupy a central place in [[string theory]] and are the subject of the important mirror symmetry conjecture.  Research here has in turn led to surprising results in the study of [[algebraic geometry]] and related fields.
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To be Calabi-Yau, a manifold must first have the structure of [[Kähler manifold]] -- this means that it is a manifold with some notion of distance provided by a [[metric|Riemannian manifold]] (called the Kähler metric), satisfying certain additional conditions. Then a Kähler manifold <math>M</math> is said to be '''Calabi-Yau''' if it carries a non-vanishing [[complex analysis|holomorphic]] [[Calc3.10|differential form]] of maximal dimension (in more sophisticated language, one says that "the canonical bundle is trivial", and this is turn is equivalent to the vanishing of the first Chern class). In the case that the manifold in question is [[compact space|compact]], this has many several equivalent statements, which may be interpreted directly as conditions imposed on the Kähler metric: <math>M</math> is Calabi-Yau if and only if the Kähler metric has zero [[curvature|Ricci curvature]].
  
 
==Examples==
 
==Examples==
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All [[elliptic curve]]s are Calabi-Yau, since they are [[parallelizeable]].
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All [[torus|elliptic curves]] are Calabi-Yau, since when regarded as quotients of the complex plane <math>\mathbb C</math> by a lattice, they carry a non-vanishing holomorphic 1-form <math>dz</math>.
  
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A degree <math>n+1</math> [[hypersurface]] <math>M</math> in <math>\mathbb{P}^n</math> is Calabi-Yau. For from the exact sequence  
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A degree <math>n+1</math> hypersurface <math>M</math> in <math>\mathbb{P}^n</math> is Calabi-Yau. This is a simple object, nothing more than the vanishing set of a polynomial of degree <math>n+1</math> in <math>n</math> dimensional space.  The proof is easy, but requires the development of some machinery: one computes the first Chern class <math>c_1(TM)</math> by using the normal bundle exact sequence and the adjunction formula, which gives:
  
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<math>
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::<math>
 
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>
  
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we get that <math>c(T\mathbb{P}^n) = c(TM)(1+(n+1)\omega)</math>. But this implies that  
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by additivity of the total Chern class this yields <math>c(T\mathbb{P}^n) = c(TM)(1+(n+1)\omega)</math>, where <math>\omega</math> is a generator of the cohomology ring <math>H^*(\mathbb P^n;\mathbb C)</math>. But this implies that  
  
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<math>
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::<math>
 
c(TM) = 1+c_1(TM)+\ldots = \frac{(1+\omega)^{n+1}}{1+(n+1)\omega}
 
c(TM) = 1+c_1(TM)+\ldots = \frac{(1+\omega)^{n+1}}{1+(n+1)\omega}
 
</math>
 
</math>
  
 
from which one easily concludes that <math>c_1(TM) = c_1(K_M) = 0</math>.
 
from which one easily concludes that <math>c_1(TM) = c_1(K_M) = 0</math>.
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There are many other ways to construct Calabi-Yau manifolds.  In any example similar to that above, one may start with a hypersurface that is not smooth (it might intersect itself, or have something like a corner: these are called singularities).  By smoothing out these manifolds (more technically, taking a resolution of singularities), one then obtains a smooth Calabi-Yau.  A vast array of other examples, useful for computations, arisse as so-called [[torus|toric]] [[algebraic geometry|varieties]].
  
 
==Calabi Conjecture==
 
==Calabi Conjecture==
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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]].
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The first Chern class of a Kähler manifold is homologous to the [[curvature|Ricci curvature]]. For this reason, Calabi conjectured that when <math>c_1(TM) = 0</math>, there exists a [[metric]] on the manifold whose Ricci form vanishes identically. This was the Calabi Conjecture, eventually proven by Shing-Tung Yau.
  
 
[[Category:Mathematics]]
 
[[Category:Mathematics]]

Revision as of 21:19, January 8, 2010

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

Calabi-Yau manifolds are a certain class of smooth complex manifolds that have proved to play an important role in several areas of mathematics and physics. Calabi-Yau manifolds have proved to occupy a central place in string theory and are the subject of the important mirror symmetry conjecture. Research here has in turn led to surprising results in the study of algebraic geometry and related fields.

To be Calabi-Yau, a manifold must first have the structure of Kähler manifold -- this means that it is a manifold with some notion of distance provided by a Riemannian manifold (called the Kähler metric), satisfying certain additional conditions. Then a Kähler manifold is said to be Calabi-Yau if it carries a non-vanishing holomorphic differential form of maximal dimension (in more sophisticated language, one says that "the canonical bundle is trivial", and this is turn is equivalent to the vanishing of the first Chern class). In the case that the manifold in question is compact, this has many several equivalent statements, which may be interpreted directly as conditions imposed on the Kähler metric: is Calabi-Yau if and only if the Kähler metric has zero Ricci curvature.

Examples

All elliptic curves are Calabi-Yau, since when regarded as quotients of the complex plane by a lattice, they carry a non-vanishing holomorphic 1-form .

A degree hypersurface in is Calabi-Yau. This is a simple object, nothing more than the vanishing set of a polynomial of degree in dimensional space. The proof is easy, but requires the development of some machinery: one computes the first Chern class by using the normal bundle exact sequence and the adjunction formula, which gives:

by additivity of the total Chern class this yields , where is a generator of the cohomology ring . But this implies that

from which one easily concludes that .

There are many other ways to construct Calabi-Yau manifolds. In any example similar to that above, one may start with a hypersurface that is not smooth (it might intersect itself, or have something like a corner: these are called singularities). By smoothing out these manifolds (more technically, taking a resolution of singularities), one then obtains a smooth Calabi-Yau. A vast array of other examples, useful for computations, arisse as so-called toric varieties.

Calabi Conjecture

The first Chern class of a Kähler manifold is homologous to the Ricci curvature. For this reason, Calabi conjectured that when , there exists a metric on the manifold whose Ricci form vanishes identically. This was the Calabi Conjecture, eventually proven by Shing-Tung Yau.

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