Timeline for Riemann Existence Theorem for Real Curve
Current License: CC BY-SA 2.5
7 events
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| Dec 4, 2010 at 18:15 | comment | added | rita | @Francesco: thanks for clarifying my comment. Since the monodromy rep. is defined (I think) only up to inner automorphisms of $S_n$, my impression is that it might be enough to ask that $p$ and $p\circ \sigma_*$ differ by an inner automorphism of $S_n$. I would also pick a real base point for $\pi_1(X-S)$. | |
| Dec 4, 2010 at 16:29 | comment | added | Francesco Polizzi | @Vivek: If $S$ is invariant for $\sigma$, then $\sigma$ restricts to an anti-holomorphic involution $\sigma \colon X-S \to X-S$, which induces an isomorphism $\sigma_* \colon \pi_1(X-S) \to \pi_1(X-S)$. Therefore a necessary condition for lifting $\sigma$ to $Y$ is that the monodromy representation $p \colon \pi_1(X-S) \to S_n$ satisfies $p \circ \sigma_*=p$, which is the $\sigma$-equivariance condition you are looking for. I suspect that this condition is also sufficient; at any rate, it is always satisfied for double covers. | |
| Dec 4, 2010 at 15:59 | comment | added | Vivek Shende | @rita: it seems to me that you would at least have to require that the monodromy can be made $\sigma$-equivariant. Is the point that this can always be done for double covers? Also, what does it precisely mean to make the monodromy $\sigma$-equivariant, anyway? | |
| Dec 3, 2010 at 21:26 | history | edited | Francesco Polizzi | CC BY-SA 2.5 |
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| Dec 3, 2010 at 14:30 | comment | added | rita | In general you can lift the involution to a double cover of $P^1$ if $S$ is invariant for $\sigma$. I think one can work out a similar condition for more general covers of $P^1$, but you have to give not only the branching order but the monodromy at each point of S, as one actually does in the Riemann construction. | |
| Dec 3, 2010 at 14:15 | history | edited | Francesco Polizzi | CC BY-SA 2.5 |
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| Dec 3, 2010 at 13:12 | history | answered | Francesco Polizzi | CC BY-SA 2.5 |