Talk:Calabi–Yau manifold

message: "These conditions imply that the first integral Chern class c_1(M) of M vanishes. Nevertheless, 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 non-trivial canonical bundle."
Should it really be written "integral Chern class" and not "real Chern class"? It seems to me that the conditions above the sentence should be equivalent to the vanishing of the INTEGRAL first Chern class since in that case c_1(canonical bundle)=-c_1(M)=0 and since line bundles are classified by INTEGRAL c_1 the canonical line bundle would be trivial. Do hyperelliptic surfaces really have vanishing integral first Chern class?
:I think it is correct. Integral first Chern classes classify smooth line bundles. A holomorphic line bundle can have vanishing first Chern class (i.e. it is smoothly trivial) without being holomorphically trivial. For example on an elliptic curve, every line bundle of degree zero has vanishing first Chern class.JonasEhrhard (talk) 08:45, 25 April 2024 (UTC)
thread_title: Confused with the sentence...

message: It's really starting to bug me how even a serious page about a difficult mathematical topic has a "... in popular culture" section. These sections add nothing. I don't care if Calabi-Yau manifolds were mentioned in a particular episode of care bears, or in a particular track off of an obscure icelandic death metal band. What could this possibly add to my understanding of Calabi-Yau manifolds? They are, at best, arbitrary factoids introduced by followers of particular fandoms, with no business being discussed in a serious scientific topic.  —Preceding unsigned comment added by 131.142.152.101 (talk) 17:34, 23 July 2008 (UTC)
replies: text: I see some cause to challenge most of these on notability grounds since most seem merely to use the term as the buzzword of the day. But I see the "in popular culture" section fulfilling a vital role in the discussion of a topic. Namely it is important to illuminate any significant influences that a concept has on the rest of society. For example, relativity and quantum mechanics both have a profound effect on popular culture. -- KarlHallowell (talk) 16:17, 25 July 2008 (UTC)
replies: text: The section has been removed now... I'm not a great wikipedia expert, so maybe this information is out there but unfindable to me, but was there some reason given? Otherwise I agree with KarlHallowell on the indicative nature of a "In Popular Culture" section. For equality's sake, was the "In Popular Culture"-like section also removed from the for instance Wormhole page? Hmm... no it turns out. In fact the section has its *own* page. Just wondering here. -- Zsejk (talk) 15:07, 11 February 2009 (UTC)
thread_title: Pop culture

message: Here's my rather superficial understanding or a CY manifold. I am sorry if any of this "simplistic" definition is wrong.
«A Calabi-Yau manifold might look like a folded lasagna pasta in higher dimensions. It is a complicated surface and, when projected in 3D or in 2D, it would appear to twist and swirl and even cross itself.  It has a smooth surface (without angles, singularities or holes) but it may or may not have an inside volume (compare a balloon to a coffee cup, for instance).» 207.134.187.165 (talk) 13:50, 16 April 2008 (UTC)
thread_title: In simple words

message: A Calabi-Yau (C-Y) manifold isn't required to be compact (it's probably better to say that compactness is an optional part of the definition). Otherwise, it makes little sense to discuss compactification of C-Y manifolds as is done in the "Applications in String Theory" section. If the manifold is compact, then it is its compactification. -- KarlHallowell 20:04, 15 February 2006 (UTC)
replies: text: Look at this preprint, Topological Strings and Their Physical Applications , by Andrew Neitzke and Cumrun Vafa (see pp. 3-4). They replace the conditions of compactness and the vanishing of the first Chern class with the condition that the manifold admits a global, nonvanishing, holomorphic n-form. It coincides with with our definition when the manifold is compact. -- KarlHallowell 20:21, 15 February 2006 (UTC)
replies: text: Still seems to be a problem, but perhaps not a significant one? In string theory, Calabi-Yau manifolds aren't necessarily compact (I see the compactification comment still is there), but I gather they are in mathematics proper. -- KarlHallowell 02:22, 10 March 2007 (UTC)
thread_title:  The compactness requirement 

message: It think this entry would be enriched if the reason that led Calabi to define Calabi-Yau manifolds from his study of Kahler structures is added.
:I know little of the historical motivation for these manifolds. Do you have any sources you can recommend? -- KarlHallowell 19:29, 3 August 2006 (UTC)
replies: text: Shing-Tung Yau and Steve Nadis; The Shape of Inner Space, Basic Books, 2010. The book is as much about the motivations and historical development as the manifolds themselves. — Cheers, Steelpillow (Talk) 14:55, 18 November 2015 (UTC)

text: Calabi conjectured, as a matter of pure geometry, that certain manifolds manifesting a Ricci curvature could be given a metric having zero curvature, i.e. their geometry would be Ricci-flat. Yau published a refutation, which was widely accepted at the time, but Calabi raised questions to Yau, who subsequently came to realise that the conjecture was after all true. His proof of the Calabi conjecture was therefore also the proof that Calabi-Yau manifolds existed. Ricci flatness is therefore a foundational defining cgharacteristic of these manifolds. I'll have to revisit the book to check exactly which class of surfaces the proof starts from, certainly something to do with Kahler manifolds. Calabi and Yau were intuitively aware of the implications to physics, but it took some time for physicists to discover their work. — Cheers, Steelpillow (Talk) 15:12, 18 November 2015 (UTC)
thread_title:  Historical Motivation 

message: As currently stated, the last sentence in the paragraph defining Calabi-Yau manifold directly contradicts the supposedly equivalent definition by the existence of a holomorphic n-form in the second paragraph. Arcfrk 05:35, 26 March 2007 (UTC)
thread_title:  Formal definition 

message: The following statement in the leading paragraph: "Physical insights about Calabi-Yau manifolds, especially mirror symmetry, led to tremendous progress in pure mathematics." isn't developed in the rest of the article (or at least I cannot extract it from the jungle). What were those insights? Why did they have such an impact? I find it very interesting and would be pleased if it is explained. --euyyn 17:23, 22 May 2007 (UTC)
thread_title:  Free statement 

message: Does anyone know what the proper way to say Calabi-Yau is. Could a phonetic spelling be included? 74.12.128.198 20:24, 23 July 2007 (UTC)
replies: text: Eugenio Calabi's name is pronounced with the vowel patter of "wasabi", with the stress on the second syllable; Shing-Tung Yau's name is pronounced to rhyme with "how". --Tristan 03:54, 5 July 2008 (UTC)
thread_title:  Phonetic 

message: This article was automatically assessed because at least one WikiProject had rated the article as start, and the rating on other projects was brought up to start class. BetacommandBot 09:45, 10 November 2007 (UTC)
thread_title:  WikiProject class rating

message: Could someone point to a reference that shows that the reducibility of the structure group is equivalent to the other statements in the definition (in particular to the one that defines a calabi yau as having a trivial canoninical bundle). Or should this be obvious?  —Preceding unsigned comment added by 145.97.232.199 (talk) 19:35, 16 December 2008 (UTC)
thread_title:  SU(m) structure group 

message: I agree with merging Calabi–Yau four-fold with calabi-yau manifold. Jakob.scholbach (talk) 12:33, 18 December 2008 (UTC)
thread_title:  merger proposal 

message: According to Shing-Tung Yau himself, along with Steve Nadis, in The Shape of Inner Space, Basic Books, 2010, Page 176, the humble 2-torus is the simplest Calabi-Yau manifold and the K3 surfaces are 4 (real) dimensional examples. I think this needs adding to the article and the lead modifying accordingly. But I'd like to clarify one thing first. The article section on different definitions is interminable and I am sure that many of these are simply relative to the topic of the paper concerned and are not meant to be global definitions. But do any global definitions refute the idea that Calabi-Yau manifolds exist in all even numbers of dimensions? — Cheers, Steelpillow (Talk) 14:51, 18 November 2015 (UTC)
thread_title:  Dimensions 

message: Substituted at 10:41, 29 April 2016 (UTC)
thread_title: Assessment comment

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