Timeline for Unipotent vector bundles
Current License: CC BY-SA 3.0
11 events
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| Aug 10, 2012 at 21:04 | history | edited | Veen | CC BY-SA 3.0 |
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| Jan 15, 2012 at 0:04 | answer | added | Thomas | timeline score: 1 | |
| Dec 7, 2011 at 18:59 | comment | added | Hacon | For abelian varieties you may want to look at Mukai's paper in Nagoya (1981) (especially example 2.9), also see 1.5 and 1.6 of 0802.1060. | |
| Dec 5, 2011 at 16:36 | vote | accept | Veen | ||
| Dec 3, 2011 at 10:48 | answer | added | Niels | timeline score: 9 | |
| Dec 3, 2011 at 8:33 | comment | added | Veen | @Keerthi: thanks, that's a definition which makes good sense in my case. Two things: first, is the stability under extensions really clear? I don't immediately see how you get your filtration on the middle term. Second, where exactly do you see the problem of the category to be abelian? For example, if you denote by $Conn(X)$ the category of vector bundles (of finite rank) on a smooth variety over say $\mathbb C$, then one knows that this is abelian. Now in our case one would have to endow kernels and cokernels of unipotent bundles with a unipotent structure. But this is not clear, isn't it? | |
| Dec 3, 2011 at 7:47 | comment | added | Keerthi Madapusi | In general, you might want to define a vector bundle with flat connection to be unipotent if it admits a filtration by sub-bundles, the associated gradeds of which are trivial bundles with connection. It is clear that the category of such objects is closed under extensions in the category of all vector bundles with flat connection. It is not immediately clear to me that this category will be abelian, though that does not seem unlikely. | |
| Dec 3, 2011 at 7:44 | comment | added | Keerthi Madapusi | Over $\mathbb{C}$, a vector bundle with flat connection can be viewed as a local system and hence as a representation of $\pi_1$. I think a vector bundle here is 'unipotent' if the corresponding representation of $\pi_1$ is unipotent. This seems to be an abelian category to me: it is the category of representations of the unipotent algebraic envelop of $\pi_1$. | |
| Dec 2, 2011 at 23:42 | history | edited | Veen | CC BY-SA 3.0 |
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| Dec 2, 2011 at 23:07 | comment | added | Moosbrugger | There's a general notion of G-bundle with flat connection for an algebraic group G, in particular for G unipotent. However, they form a groupoid and not an abelian category, so perhaps it's not what you want. Can you clarify what you're looking for? Does some author use this phrase? Or else can you give examples of what you want to axiomatize? | |
| Dec 2, 2011 at 22:16 | history | asked | Veen | CC BY-SA 3.0 |