Timeline for Sheaves and bundles in differential geometry
Current License: CC BY-SA 2.5
16 events
| when toggle format | what | by | license | comment | |
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| Jan 7, 2017 at 7:43 | comment | added | Chill2Macht | An exception which proves the rule: Chapter 5 of Warner's Differentiable Manifolds and Lie Groups discusses sheaves on manifolds extensively (while as far as I am aware the book makes no mention of fiber bundles), but this all preparation for a discussion later in the chapter of the equivalence of various cohomology theories on manifolds. | |
| Oct 1, 2015 at 21:08 | comment | added | Samantha Y | Not to nitpick, but the above link is actually to part 2 of the article. Zariski's article is located here: ams.org/journals/bull/1956-62-02/S0002-9904-1956-10018-9 | |
| May 22, 2010 at 2:22 | history | edited | Emerton | CC BY-SA 2.5 |
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| May 22, 2010 at 0:51 | comment | added | Anweshi | The article page at AMS: ams.org/journals/bull/1956-62-02/S0002-9904-1956-10015-3 | |
| Mar 9, 2010 at 9:11 | comment | added | Joel Fine | @fpqc, Emerton, I would say that most definitely one uses sheaves when studying complex manifolds. Indeed, on an arbitrary complex manifold, cohomology of the "obvious" sheaves is just about all we have. Slowly, however, people are turning to more differential geometric ways to try and understand complex non-Kähler manifolds. In brief, one looks for a Hermitian metric whose 2-form satisfies some PDE analogous to being closed. Eg. work of Streets-Tian on pluriclosed metrics and work of Fu-Li-Yau on balanced metrics. | |
| Mar 9, 2010 at 7:03 | comment | added | Cam McLeman | Aha, thanks. For anyone else who's looking for the link, it's part 3 (pages 117-141) of: Martin, W. T.; Chern, S. S.; Zariski, Oscar. Scientific report on the Second Summer Institute, several complex variables. Bull. Amer. Math. Soc. 62 (1956), 79--141. | |
| Mar 9, 2010 at 6:48 | comment | added | Emerton | No; that is Zariski's report on his own work, I think. The article I am referreing to is in the Bulletin of the AMS, and is part of a series of reports on a conference on sheaves and cohomology in algebraic geometry. Zariski reports on Serre's FAC. It is one of the best introductions to sheaves and cohomology in algebraic geometry, and how they relate to concrete questions of projective geometry of the kind that Zariski (and before him the Italians) studied. | |
| Mar 9, 2010 at 6:42 | comment | added | Cam McLeman | Is this the Zariski paper you're referencing? Sounds fantastic. The fundamental ideas of abstract algebraic geometry. Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, vol. 2, 77--89. Amer. Math. Soc., Providence, R. I., 1952. | |
| Mar 9, 2010 at 4:04 | comment | added | Emerton | cont'd: ... they could be much worse when the functions are less rigid. | |
| Mar 9, 2010 at 4:03 | comment | added | Emerton | Typically, one considers maps of bundles which are locally of the form $\mathbb R^m \hookrightarrow \mathbb R^n$, so that the quotient is again a bundle. In fact, even more geometrically inclined algebraic geometers tend to distinguish between a map of bundles (in the above sense), and a map of sheaves (such as the map $\mathcal O \hookrightarrow \mathcal O(1)$) given by choosing a hyperplane in projective space.) One point to consider is that the latter kinds of maps work well in algebraic geometry in part because the singularities of maps are so tame (zeroes or poles) ... | |
| Mar 9, 2010 at 3:44 | comment | added | Mike Skirvin | Regarding the statement that studying bundles allows one to stay in the realm of manifolds. I'm curious, do differential geometers not usually care that bundles do not form an Abelian category, i.e., that kernels and cokernels of maps between bundles are not necessarily bundles? When working with locally free sheaves in an algebro-geometric setting, I tend to find it necessary to view them in the larger category of coherent sheaves for this reason. | |
| Mar 9, 2010 at 3:42 | comment | added | Emerton | Dear fpqc, In complex geometry sheaves and cohomology certainly play a role, although I'm not close enough to the field to know whether they are as dominant as they were in the heyday of Oka and Cartan. I would guess that the closer the investigations are to algebraic geometry, the more likely these methods are to play an important role. | |
| Mar 9, 2010 at 3:42 | comment | added | Sam Derbyshire | That is a great answer! | |
| Mar 9, 2010 at 3:36 | comment | added | Harry Gindi | If we're looking at complex or analytic manifolds, do sheaves again come to the fore? | |
| Mar 9, 2010 at 3:35 | vote | accept | Harry Gindi | ||
| Mar 9, 2010 at 3:29 | history | answered | Emerton | CC BY-SA 2.5 |