Timeline for Chern Classes: two approaches
Current License: CC BY-SA 3.0
11 events
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| Oct 25 at 19:50 | comment | added | Z. M | @JasonStarr Maybe I am confused, but the definition in terms of Atiyah class seems to only give us Chern classes in Hodge cohomology, not in de Rham cohomology? | |
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Jan 2, 2017 at 4:57 | comment | added | naf | Yes, applying the cycle class map to the Grothendieck Chern class gives the analytic Chern class; the fact that this map is neither injective nor surjective is irrelevant. Using the formal properties of Chern classes you only need to check this for line bundles and this is not hard. | |
| Jan 1, 2017 at 17:03 | comment | added | Dubious | Ok it is clear that there many constructions XD. But coming back to the two construction listed above: suppose that we have the Grothendieck Chern class $c_k$, then is it true that $\text{cycl}(c_k)$ is the analytic Chern class? I understand that the main problem is that $\text{cycl}$ is not injective nor surjective. | |
| Jan 1, 2017 at 16:59 | comment | added | Jason Starr | There are sooo many constructions of Chern classes. In addition to the two above, you can first define the Atiyah extension, and then use trace maps on self-products of this extension to define Chern classes. Also, you can define the gamma filtration on K-theory, and then you can define Chern classes as elements in the associated graded ring of this filtration as in Borel-Serre and Manin (i.e., Grothendieck's original proof of his version of Riemann-Roch). There are other constructions as well. The beauty of the axiomatic approach is that you can compare these many different constructions. | |
| Jan 1, 2017 at 16:04 | comment | added | მამუკა ჯიბლაძე | Where both make sense they give the same result. In fact there are at least two other definitions - through going to $H^*({\mathbb P}(\mathscr E))$ (the bundle of lines in $\mathscr E$) and through evaluating the universal Chern classes in $H^*(\mathrm{Grassmanian})$; these also make sense in different contexts and also give the same result where these contexts intersect. | |
| Jan 1, 2017 at 15:34 | comment | added | Dubious | So, are you saying that the two definitions are intrinsically different? | |
| Jan 1, 2017 at 15:30 | comment | added | Allen Knutson | (Nor is it surjective.) For a first place to inspect the difference, consider line bundles on an elliptic curve. | |
| Jan 1, 2017 at 15:09 | comment | added | user25309 | That's right: one is differential geometric and the other is algebraic. For example, the algebraic definition makes sense for algebraic varieties defined over any field. To go from the algebraic definition over the complex numbers to a differential/topological definition, one needs indeed the cycle class map from Chow groups to usual cohomology. The algebraic Chern classes valued in Chow groups usually contain algebraic information which is much finer than the topological Chern classes valued in usual cohomology: the cycle class map is very far of being injective in general. | |
| Jan 1, 2017 at 14:51 | history | edited | Dubious | CC BY-SA 3.0 |
added 4 characters in body
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| Jan 1, 2017 at 14:45 | history | asked | Dubious | CC BY-SA 3.0 |