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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..."
709:
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
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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
625:
I know that the Chern classes are obstructions to finding non-vanishing vector fields. What I don't know, and hence i think it would be nice to add to this entry, is what the converse implies. I.e. if I know that the first Chern class vanishes, then what can I conclude? for exemple, if the first
501:
what the article should say. It is 100% irrelevant if that seems "pedantic" to a professional who is well familiar with the subject matter. Many people reading this article will be students just learning the material, and if the addition of only a few words makes the article clearer, then
626:
Stiefel-Whitney class vanishes then the bundle is orientable - but it doesn't imply that the bundle has a nowhere vanishing section (eg.: the tangent space of the 2-sphere. All of its SW classes vanishes, as it does for any n-sphere, but it doesn't have a nowhere zero section).
162:
The hairy ball theorem is a very bad example to give not so much because it concerns the real differentiable case (after all, the tangent bundle to the
Riemann sphere is indeed a complex bundle) but because the bundle is stably trivial (in particular, its
744:
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!
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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?
983:
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".
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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
140:
1059:"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
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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?
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use this to compute the characteristic classes for virtual vector bundles. For example, try the cotangent complex for a smooth local complete intersection
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The article does not bother to link this to any definition of "quintic threefold" — as though every reader knows exactly what this phrase means.
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The observation that a Chern class is essentially an elementary symmetric polynomial can be used to "define" Chern classes. Let
664:, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.
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One just has to read the next sentence, to see how the "recursion" is terminated. But I'll think about the wording.
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The Chern classes of V can therefore be defined as the pullback of the Chern classes of the universal bundle
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in genuine TeX and LaTeX as opposed to the thing we use here I haven't checked yet. 19:10, 5 May 2016 (UTC)
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Can this be made more accessible? A longer lead would help. Maybe more motivation and some applications?
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https://math.stackexchange.com/questions/989147/quick-question-chern-classes-of-sym-wedge-hom-and-tensor
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vanish) and stably trivial bundles are precisely the sort of thing Chern classes cannot detect. --
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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606:. 1958 - I'd thought it was a little earlier. If I recall, he has a principle about reduction to
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This page should discuss chern classes of box products of vector bundles over a product space.
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The lines: For example, the first few terms are ch(V) = dim(V) + c1(V) + c1(V)2/2 − c2(V) +...
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would benefit by extension to a further term by clarifying the general form of the terms.
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https://mathoverflow.net/questions/41970/how-to-resolve-a-wedge-product-of-vector-bundles
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Great. Please keep in mind that on
Knowledge it is even more important that material be
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is the hyperplane. Is your complaint simply that the text does not say "... dual to the
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These types of problems are the result of sloppy writing. That does not help
Knowledge.
610:; and his theory is axiomatic. It is possible Atiyah has an earlier published version.
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than that it be correct. So please work on the citations as you add new material.
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may be viewed as a matrix of 4-forms (or 2 2-forms), and thus the eigenvalues of
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construct chern classes for coherent sheaves using resolutions by vector bundles
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I'm not an expert, so there may be other mathematical errors I didn't notice.
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give toy examples using the kernel and cokernel of morphisms of line bundles
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the hyperplane..." (which is technically correct but pedantic, I think)?
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quintic threefold" because there are many different quintic threefolds.
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may be viewed as a matrix of 2-forms, and thus the eigenvalues of
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This ought to be completely obvious to anyone editing
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https://math.boisestate.edu/~zteitler/math/expository/chern.pdf
226:{\displaystyle \mathbb {CP} ^{n-1}\subseteq \mathbb {CP} ^{n}}
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http://www.math.harvard.edu/theses/senior/patrick/patrick.pdf
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https://www.math.stonybrook.edu/~azinger/mat620/EGnotes.pdf
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Serious problems in the "In algebraic geometry" section
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http://www.numdam.org/article/BSMF_1958__86__137_0.pdf
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264:{\displaystyle \mathbb {CP} ^{n-1}}
38:It is of interest to the following
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806:dualize euler sequence and analyze
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115:Knowledge:WikiProject Mathematics
1014:{\displaystyle \mathbb {P} ^{4}}
967:{\displaystyle f_{E}:X\to G_{n}}
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304:{\displaystyle H_{2n-2}}
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