Let $G$ be an algebraic group acting on a variety $V$. Which information can be obtained by looking the action of $G$, and subgroups of $G$ that fixes points of $V$?. In other words how we obtain $V$ from the group $G$?
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$\begingroup$ It's not clear to me exactly what you are after. Do you have a specific variety and group action in mind that you want to know more about? $\endgroup$– Daniel LoughranCommented Dec 22, 2012 at 16:08
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3$\begingroup$ The literature on algebraic group actions is enormous; you'd better ask for a more specific question. $\endgroup$– AngeloCommented Dec 22, 2012 at 16:57
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$\begingroup$ @Loughran, I consider the case where $V$ is an algebraic curve and $G$ is a semisimple algebraic group. What I need to know about this curve is its genus. $\endgroup$– albert cohenCommented Dec 22, 2012 at 18:13
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3 Answers
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A lot if $G$ is transitive. Then $V=G/H$ for a subgroup $H$ (if it has a point), or a $G$-torsor mod $H$ (if it doesn't). Then most questions about the geometry of the variety are best answered by studying the group action. For instance, we can study line bundles on a flag variety of a reductive group using the root lattice for that group.
Another case where you gain a lot of information is where $G$ acts almost transitively, i.e., there is a dense orbit, as in the case of toric varieties. Then it is not as simple to "obtain" our variety as just choosing a group $G$ and subgroup $H$. We must also include some information on how to glue on the other orbits. But usually, because of the extra symmetry the group structure provides, this description is not so complex as defining an entire algebraic variety, since if you know something about the geometry of a point, you can deduce the equivalent statement about all other points in its orbit.
answered Dec 22, 2012 at 17:00
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$\begingroup$ What about if stabilizer of a point is solvable?. Can we know someting about this variety?. For example its cohomology groups? $\endgroup$ Commented Dec 22, 2012 at 18:18
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$\begingroup$ I don't know any theorems with only that condition. What do you know about the group? $\endgroup$ Commented Dec 22, 2012 at 19:42
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1$\begingroup$ @Will: It seems that you mean that if an algebraic group $G$ defined over a field $k$ acts transitively on a $k$-variety $V$, then $V$ can be dominated by a principal homogeneous space (torsor) $P$ of $G$. However, this is not true in general. The action of $G$ on $V$ defines a certain second nonabelian cohomology class $\eta(V)$ of ${\rm Gal}(\bar k/k)$ with coefficients in a band (lien) related to the stabilizer $\overline{H}$ of a $\bar k$-point of $V$. Such a torsor $P$ exists if and only if this cohomology class is neutral. $\endgroup$ Commented Jan 1, 2013 at 16:27
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$\begingroup$ @Will: For details see Giraud's book, or Springer, "Non-abelian $H^2$ in Galois cohomology", or my paper "Abelianization of the second nonabelian Galois cohomology", Duke Math. J. 72 (1993), 217-239. $\endgroup$ Commented Jan 1, 2013 at 16:35
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As indicated by the other answers, you question is not specific enough.
In the particular case where $G$ is the multiplicative group, a theorem of Białynicki-Birula (On fixed point schemes of actions of multiplicative and additive groups. Topology 12 (1973), 99–103, MR:313261), furnishes a decomposition of $V$ into locally closed subsets $V_i$, each of them being stable under the action of $G$ and a trivial fibration over the fixed point set $V_i^G$.
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Not much if $G$ is trivial (or more generally if the action is trivial).
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1$\begingroup$ In my case $G$ is not trivial $\endgroup$ Commented Dec 22, 2012 at 18:09