Torsten Ekedahl
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Member for 14 years, 10 months
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Connected extensions of finite by connected algebraic groups
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Connected extensions of finite by connected algebraic groups
I am not sure what you mean by vector group.
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composition of covering map and bundle projection
It depends on your definition of fibration, I have used the one which says that it locally is a product. (You did say fibre bundle though which is more likely to mean one where all the fibres are the same.) It is then a fact that you have the same fibre if the base is connected. Without that condition I am not totally sure the statement is true. In any case that is something I think is most easily shown a posteriori.
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composition of covering map and bundle projection
Two things. One doesn't need a connected base to have a covering space theory and what I said about a homotopy equivalence inducing a category equivalence is true in general. Also that $Z$ is connected is part of the condition that $Z$ be contractible (or just simply connected).
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Cohomology of Theta divisor on Jacobian?
In characteristic zero yes, it turns out to be true irregardless (even though the Kodaira vanishing theorem is false in general).
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Cohomology of Theta divisor on Jacobian?
All cohomology groups but the zeroth vanish so the R-R formula gives what you want (see for instance David Mumford: Abelian varieties, Ch 16).
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composition of covering map and bundle projection
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Classification of the Kähler Structures on the Sphere
Except for the $2$-sphere there are none, the cohomology class of the Kähler form is non-zero so for a Kähler manifold the second Betti number must be non-zero.
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Central extensions of group schemes
@Kevin: Sorry for the sloppy first editing. @Michael: I have added some comments which hopefully clarifies.
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Central extensions of group schemes
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Central extensions of group schemes
It certainly should (fixed).
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Central extensions of group schemes
Fixed missing transposition
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Invariant Polynomes under group action - given the invariants looking for the group. algorithmic solution?
Sorry but now your example is still wrong, we have that $x_1x_2x_3$ is also invariant under your scalar matrices (you have to replace the scalar matrices with all diagonal matrices whose non-zero entries are third roots of unity).
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Simple sheaves, definition and characterization
To begin with this is mostly used for proper varieties. In nay case as a definition it seems OK (though I guess some people might make some restrictions on $F$ such as being torsion free). It is definitely not equivalent to having no non-trivial subsheaves. Consider for instance a proper geometrically integral scheme over a field and the structure sheaf, it is simple in this sense but there are lots of coherent ideals (unless the scheme is zero-dimensional).