Timeline for Chern classes generating cohomology
Current License: CC BY-SA 2.5
7 events
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| Nov 16, 2010 at 18:57 | history | edited | roy smith | CC BY-SA 2.5 |
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| Nov 16, 2010 at 15:48 | comment | added | roy smith | The argument that the dth homogeneous part of the chern character of the structure sheaf of a subvariety of codimension d, corresponds to the cohomology class of the subvariety, is sketched on page 187 of Griffiths and Adams. I.e. as you realized better than I, it is not tautological. | |
| Nov 16, 2010 at 12:11 | comment | added | Donu Arapura | Vamsi: The ref that Roy gives is a good one and it probably answers all your questions. Another, is Grothendieck, Theorie de classes de Chern. On p 151 eq. (16) you'll see a formula for the cycle class of a subvariety in terms of the Chern classes of its structure sheaf. | |
| Nov 16, 2010 at 6:36 | comment | added | roy smith | I'm a little over my head here, but apparently all algebraic subvarieties define coherent algebraic sheaves. Then the distinction between algebraic classes and classes of vector bundles is that those algebraic subvarieties occurring as zeroes of sections of vector bundles apparently define locally free coherent sheaves. So showing that all algebraic classes are generated by classes of vector bundles, means showing that the sheaves associated to arbitrary subvarieties can be expressed in terms of the locally free sheaves associated to subvarieties cut out by sections of vector bundles. | |
| Nov 16, 2010 at 5:11 | comment | added | Vamsi | Thanks for the reply. But, why is it that the group of algebraic cohomology classes is the image of the Chern character map of coherent sheaves? Anyway, I will try to look it up. Thanks for the reference. | |
| Nov 16, 2010 at 4:31 | history | edited | roy smith | CC BY-SA 2.5 |
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| Nov 16, 2010 at 4:24 | history | answered | roy smith | CC BY-SA 2.5 |