I will be please if someone can prove this to me. Let X be a compact complex surface and G a compact Lie group acting on X with a fixed point p in X. Then G has a 2 dimensional faithful representation i.e., there exists a faithful homomorphism from G to Gl_2(C) or more precisely from G to T_pX (the tangent space of X at p which is isomorph to C^2).
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$\begingroup$ $X$ is $2$-dimensional? $\endgroup$– Daniel LittFeb 1, 2011 at 3:02
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$\begingroup$ yes, sorry. It is a surface. $\endgroup$– ruhiFeb 1, 2011 at 3:04
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$\begingroup$ I don't get it - if your claim is true then for every such $G$ acting on $X$ one can define an action of $G\times S^1$ on $X$ by making the circle act trivially. Then the induced homomorphism $G\times S^1\to GL_2(\mathbb{C})$ is not faithful any more! $\endgroup$– Somnath BasuFeb 1, 2011 at 3:36
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$\begingroup$ It would help if you told us why you are trying to prove this statement (or whatever version of it that happens to be correct). Maybe you want the isotropy group to be trivial in a sphere around $p$? $\endgroup$– S. Carnahan ♦Feb 1, 2011 at 3:52
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$\begingroup$ Yeah this is clearly false as stated. It must be something like--$p$ is an isolated fixed point, $G$ acts holomorphically on $X$. $\endgroup$– Daniel LittFeb 1, 2011 at 4:14
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