Timeline for Principal Bundles over Complex Projective Varieties
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
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| Jun 5, 2012 at 21:06 | comment | added | Fran Burstall | The "compact" way of saying "Levi" is: centraliser of a torus. At the complex level these are the generalised flag manifolds $H/Q$ for $H$ complex semisimple and $Q$ parabolic. These are the only projective homogeneous spaces with $H$ semisimple (theorem of Wang from the 1950's). | |
| Feb 6, 2010 at 21:50 | comment | added | Ben Webster♦ | Good points all. This is why I should never try to phrase things in terms of compact groups. Complex ones are so much nicer. | |
| Feb 6, 2010 at 21:49 | history | edited | Ben Webster♦ | CC BY-SA 2.5 |
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| Feb 6, 2010 at 21:40 | comment | added | Pavel Etingof | Also I think "G containing the maximal torus of P" is not sufficient: G has to be a Levi subgroup of P (i.e. $P/G$ is a coadjoint orbit of $P$). E.g. $SO(5)/SO(4)=S^4$ is not a complex manifold. | |
| Feb 6, 2010 at 21:36 | comment | added | Pavel Etingof | I think the statement that this is the only way to get a projective quotient is not quite right (although easy to correct). Take P=2-torus, G=1. Perhaps the irreducible projective varieties admitting the representation P/G are products of an abelian variety and a coadjoint orbit of a compact group as explained above. | |
| Feb 6, 2010 at 17:34 | comment | added | Jason DeVito - on hiatus | Using biquotient constructions (which are natural generalizations of homogenous actions), one can find an actions of $T^2$ on $S^3\times S^3$ whose quotient spaces are$\mathbb{C}P^2$#$\pm \mathbb{C}P^2$ . One still gets a principal fiber bundle $H\rightarrow G\rightarrow G/H$ just as in the homogeneous action case. That said, I don't know enough algebraic geometry to know whether either of these two spaces are projective varieties. | |
| Feb 6, 2010 at 16:00 | history | answered | Ben Webster♦ | CC BY-SA 2.5 |