Beyond wrote:Well, then i am one of the few. I have never heard of a Calabi-Yau manifold.
I googled "Calabi-Yau manifold". Goodness! I've never read a wikipedia article that managed to confuse me so thoroughly! Or rather, I've never run into so many impossible words in an article that is supposedly in English. Or what about K3 surfaces, Kähler manifolds, canonical bundle, Calabi conjecture and Ricci flat metrics? Wanna know what a compact n-dimensional Kähler manifold M is? Easy as a pie!
http://en.wikipedia.org/wiki/Calabi–Yau_manifold wrote:
The canonical bundle of M is trivial.
M has a holomorphic n-form that vanishes nowhere.
The structure group of M can be reduced from U(n) to SU(n).
M has a Kähler metric with global holonomy contained in SU(n).
Still confused? Don't worry, here's a further explanation:
These conditions imply that the first integral Chern class c1(M) of M vanishes, but the converse is not true. The simplest examples where this happens are hyperelliptic surfaces, finite quotients of a complex torus of complex dimension 2, which have vanishing first integral Chern class but the canonical bundle is not trivial.
Riiight. Okay. I feel enlightened. Let's look at a picture for a change:
The picture is from the wikipedia article about Calabi-Yau manifold and shows a section of a quintic Calabi–Yau three-fold (3D projection).
I'm sure you'll be glad to know!
Ann
[quote="Beyond"]Well, then i am one of the few. I have never heard of a Calabi-Yau manifold.[/quote]
I googled "Calabi-Yau manifold". Goodness! I've never read a wikipedia article that managed to confuse me so thoroughly! Or rather, I've never run into so many impossible words in an article that is supposedly in English. Or what about K3 surfaces, Kähler manifolds, canonical bundle, Calabi conjecture and Ricci flat metrics? Wanna know what a compact n-dimensional Kähler manifold M is? Easy as a pie! http://en.wikipedia.org/wiki/Calabi–Yau_manifold wrote:
[quote]The canonical bundle of M is trivial.
M has a holomorphic n-form that vanishes nowhere.
The structure group of M can be reduced from U(n) to SU(n).
M has a Kähler metric with global holonomy contained in SU(n).[/quote]
Still confused? Don't worry, here's a further explanation:
[quote]These conditions imply that the first integral Chern class c1(M) of M vanishes, but the converse is not true. The simplest examples where this happens are hyperelliptic surfaces, finite quotients of a complex torus of complex dimension 2, which have vanishing first integral Chern class but the canonical bundle is not trivial.[/quote]
[float=right][img]http://upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Calabi_yau.jpg/220px-Calabi_yau.jpg[/img][/float]Riiight. Okay. I feel enlightened. Let's look at a picture for a change: :arrow:
The picture is from the wikipedia article about Calabi-Yau manifold and shows a section of a quintic Calabi–Yau three-fold (3D projection).
I'm sure you'll be glad to know! :D
Ann