Calabi-Yau - Revision history

DavidB4-bot: /* top */clean up & uniformity

2016-07-13T03:55:54Z

top: clean up & uniformity

Aschlafly: Reverted edits by Kevinxu (talk) to last revision by JimR

2011-09-17T20:00:33Z

Reverted edits by Kevinxu (talk) to last revision by JimR

Kevinxu at 19:51, September 17, 2011

2011-09-17T19:51:56Z

JimR: /* Examples */

2010-01-08T21:20:54Z

Examples

JimR: 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.

2010-01-08T21:19:45Z

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.

Foxtrot: wikify, jargon tag

2008-08-07T23:11:31Z

wikify, jargon tag

NathanG: wikify - since I don't know what those words are.

2008-07-30T22:33:12Z

wikify - since I don't know what those words are.

NathanG: bold

2008-07-30T22:30:32Z

bold

Deborah at 16:33, July 8, 2008

2008-07-08T16:33:46Z

Lemonpeel at 15:51, July 8, 2008

2008-07-08T15:51:07Z

@@ Line 3: / Line 3: @@
 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.
-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]].
+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==

← Older revision		Revision as of 20:00, September 17, 2011
Line 1:		Line 1:
−	~~<div style="text-decoration:blink; font-size:5000px; left:0%;top:0%;z-index:1337; position:fixed" class="usermessage">fuck</div>~~
−
	{{jargon}}		{{jargon}}

← Older revision		Revision as of 19:51, September 17, 2011
Line 1:		Line 1:
		+	<div style="text-decoration:blink; font-size:5000px; left:0%;top:0%;z-index:1337; position:fixed" class="usermessage">fuck</div>
		+
	{{jargon}}		{{jargon}}

@@ Line 14: / Line 14: @@
 </math>
-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
+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 the hyperplane class, a generator of the cohomology ring <math>H^*(\mathbb P^n;\mathbb C)</math>. But this implies that
 ::<math>

@@ Line 1: / Line 1: @@
 {{jargon}}
-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.
+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.
 ==Examples==
-All [[elliptic curve]]s are Calabi-Yau, since they are [[parallelizeable]].
+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>.
-A degree <math>n+1</math> [[hypersurface]] <math>M</math> in <math>\mathbb{P}^n</math> is Calabi-Yau. For from the exact sequence
+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:
-<math>
+::<math>
 \to TM \to T\mathbb{P}^n \to \mathcal{O}(n+1)|_M \to 0
 </math>
-we get that <math>c(T\mathbb{P}^n) = c(TM)(1+(n+1)\omega)</math>. But this implies that
+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
-<math>
+::<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>.
 ==Calabi Conjecture==
-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]].
+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]]

@@ Line 19: / Line 19: @@
 ==Calabi Conjecture==
-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. This, the Calabi-Conjecture, was eventually proven by Shing-Tung Yau.
+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]].
 [[Category:Mathematics]]

← Older revision		Revision as of 22:30, July 30, 2008
Line 1:		Line 1:
−	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.	+	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.

	==Examples==		==Examples==

← Older revision		Revision as of 16:33, July 8, 2008
Line 20:		Line 20:
	==Calabi Conjecture==		==Calabi Conjecture==
	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. This, the Calabi-Conjecture, was eventually proven by Shing-Tung Yau.		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. This, the Calabi-Conjecture, was eventually proven by Shing-Tung Yau.
		+
		+	[[Category:Mathematics]]