Talk:Chern class

message: I reverted this anon edit:
:The field strength F may be viewed as a matrix of 4-forms (or 2 2-forms), and thus the eigenvalues of F are 4-forms as well. Each Chern class is then a 4k (2x2k)form. 
back to the original
:The field strength F may be viewed as a matrix of 2-forms, and thus the eigenvalues of F are 2-forms as well. Each Chern class is then a 2k form.
I'm not sure what the anonymous editor was thinking ... perhaps they were counting the number of spinor indecies instead of vector indecies? Or were they counting two real components instead of one complex component? linas 02:26, 18 October 2005 (UTC)
thread_title:  Revert anonymous edit 

message: I'm not sure that it is correct to attribute the splitting principal to Grothendieck.  I thought it was due to Atiyah, although I could be wrong.  Can anyone provide a reference?
151.204.6.171
*The basic Grothendieck paper is here: . 1958 - I'd thought it was a little earlier. If I recall, he has a principle about reduction to Borel subgroups; and his theory is axiomatic. It is possible Atiyah has an earlier published version. Charles Matthews 19:13, 2 November 2005 (UTC)
thread_title:  Grothendieck or Atiyah? 

message: There are a lot of different ways to define Chern Classes. Some of them differ by signs. I think the article should point this out. A good reference is Borel/Hirzebruch, Characteristic Classes and Homogeneous Spaces II, Amer. J. Math 81, 1959. They really figure out the different definitions in the Appendix. Spaetzle (talk) 11:39, 27 October 2011 (UTC)
thread_title:  Different ways to define Chern Classes 

message: Substituted at 01:52, 5 May 2016 (UTC)
thread_title: Assessment comment

message: A TeX oddity: Of course \dots between commas gets rendered as \ldots and \dots between plus signs and some other binary operation symbols and binary relation symbols gets rendered as \cdots, but \dots between two instances of \oplus gets rendered as \ldots, not as \cdots. (Whether it works that way in genuine TeX and LaTeX as opposed to the thing we use here I haven't checked yet. 19:10, 5 May 2016 (UTC)
thread_title:  various dots 

message: This page should discuss chern classes of box products of vector bundles over a product space.  — Preceding unsigned comment added by 71.212.185.82 (talk) 04:30, 24 August 2017 (UTC)
thread_title:  Chern classes box product 

message: The material in "Properties of Chern classes" and "classical axiomatic definition" seems like it ought to be much higher up. I don't even think the article says that the kth Chern class lives in H^{2k} before section 6!
Cyrapas (talk) 15:42, 21 September 2017 (UTC)
thread_title:  Buried basics 

message: === Chern Classes in Algebraic Geometry ===
This page should have a section for chern classes in algebraic geometry. This should start with basic computations but also give non-trivial examples from enumerative geometry. Some places to look are
* http://www.numdam.org/article/BSMF_1958__86__137_0.pdf
* https://math.boisestate.edu/~zteitler/math/expository/chern.pdf
* https://www2.bc.edu/dawei-chen/Lecture-2014.pdf
* https://www.math.stonybrook.edu/~azinger/mat620/EGnotes.pdf
* http://www.math.harvard.edu/theses/senior/patrick/patrick.pdf
* Calabi-Yau Manifolds and Related Geometries
Todo:
* dualize euler sequence and analyze
* https://mathoverflow.net/questions/41970/how-to-resolve-a-wedge-product-of-vector-bundles to compute chern classes of k-forms.
* give examples based on this - https://math.stackexchange.com/questions/989147/quick-question-chern-classes-of-sym-wedge-hom-and-tensor
* explain how to use this with the grothendieck group
* construct chern classes for coherent sheaves using resolutions by vector bundles
* use this to compute the characteristic classes for virtual vector bundles. For example, try the cotangent complex for a smooth local complete intersection
* give toy examples using the kernel and cokernel of morphisms of line bundles
* https://www.math.fsu.edu/~aluffi/archive/paper348.pdf  — Preceding unsigned comment added by Wundzer (talk • contribs) 22:44, 7 February 2020 (UTC)
replies: text: Great. Please keep in mind that on Wikipedia it is even more important that material be reliably sourced than that it be correct. So please work on the citations as you add new material. Mgnbar (talk) 02:13, 30 September 2017 (UTC)
thread_title:  Todo 

message: This page should have a section for chern classes in algebraic geometry. This should start with basic computations but also give non-trivial examples from enumerative geometry. Some places to look are
* http://www.numdam.org/article/BSMF_1958__86__137_0.pdf
* https://math.boisestate.edu/~zteitler/math/expository/chern.pdf
* https://www2.bc.edu/dawei-chen/Lecture-2014.pdf
* https://www.math.stonybrook.edu/~azinger/mat620/EGnotes.pdf
* http://www.math.harvard.edu/theses/senior/patrick/patrick.pdf
* Calabi-Yau Manifolds and Related Geometries
Todo:
* dualize euler sequence and analyze
* https://mathoverflow.net/questions/41970/how-to-resolve-a-wedge-product-of-vector-bundles to compute chern classes of k-forms.
* give examples based on this - https://math.stackexchange.com/questions/989147/quick-question-chern-classes-of-sym-wedge-hom-and-tensor
* explain how to use this with the grothendieck group
* construct chern classes for coherent sheaves using resolutions by vector bundles
* use this to compute the characteristic classes for virtual vector bundles. For example, try the cotangent complex for a smooth local complete intersection
* give toy examples using the kernel and cokernel of morphisms of line bundles
* https://www.math.fsu.edu/~aluffi/archive/paper348.pdf  — Preceding unsigned comment added by Wundzer (talk • contribs) 22:44, 7 February 2020 (UTC)
replies: text: Great. Please keep in mind that on Wikipedia it is even more important that material be reliably sourced than that it be correct. So please work on the citations as you add new material. Mgnbar (talk) 02:13, 30 September 2017 (UTC)
thread_title:  Chern Classes in Algebraic Geometry 

message: At one point the article reads:
"The observation that a Chern class is essentially an elementary symmetric polynomial can be used to "define" Chern classes. Let Gn be the infinite Grassmannian of n-dimensional complex vector spaces. It is a classifying space in the sense that, given a complex vector bundle E of rank n over X, there is a continuous map 
:f_E: X \to G_n
unique up to homotopy. "
This is ridiculous. It is necessary to say that the map pulls back the universal bundle to get the given complex vector bundle (or some equivalent qualifying phrase) for this map to be "unique up to homotopy".
Later the article states: 
"For example, consider the quintic threefold in \mathbb{P}^4."
The article does not bother to link this to any definition of "quintic threefold" — as though every reader knows exactly what this phrase means. 
Yet there is no such thing as "the quintic threefold" because there are many different quintic threefolds.
These types of problems are the result of sloppy writing. That does not help Wikipedia.50.205.142.50 (talk) 22:42, 3 May 2020 (UTC)
thread_title:  Some serious problems 

message: "The Chern classes of V can therefore be defined as the pullback of the Chern classes of the universal bundle." Such recursive statements are unhelpful to the uninitiated. 24.177.184.68
:One just has to read the next sentence, to see how the "recursion" is terminated. But I'll think about the wording. Mgnbar (talk) 11:56, 5 April 2021 (UTC)
thread_title:  The Chern classes of V can therefore be defined as the pullback of the Chern classes of the universal bundle 

message: There are a number of problems with the "In algebraic geometry" section. None of it is sourced. Axiom 5 of the axiomatic description is wrong (the total Chern class is not a ring homomorphism). There are also many grammar issues "In particular, we can find X is a spin 4-manifold if 4 - d is even..."
I'm not an expert, so there may be other mathematical errors I didn't notice. 
Hasire (talk) 16:20, 23 May 2021 (UTC)
thread_title:  Serious problems in the "In algebraic geometry" section