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Let $M$ be a connected closed complex manifold with an antiholomorphic involution.

Must there be an atlas and a choice of a reference point in each chart such that the transition functions are ratios of holomorphic functions each having Taylor series with coefficients in $\mathbb{R}$?

Todd Trimble
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  • Closely related to your previous question: https://mathoverflow.net/questions/373095/complex-projective-manifold-with-an-antiholomorphic-involution, which if I understand correctly covers the projective case. – YCor Oct 02 '20 at 09:51
  • Not really a counterexample, but still. Take a conic in $\mathbb{P}^2_{\mathbb{R}}$ without real points, by base change to $\mathbb{C}$ you get a non-standard antiholomorphic involution on $\mathbb{P}^{1}{\mathbb{C}}$ without fixed points. Obviously $\mathbb{P}^{1}{\mathbb{C}}$ does descend to a real manifold, but this has nothing to do with the given involution. – Giulio Bresciani Oct 05 '20 at 12:07
  • @GiulioBresciani I am not sure I understand you. Both of those curves are real models of $\mathbb{P}^1_{\mathbb{C}}$ and provide an atlas as in the question. The presence of real points is not essential. –  Oct 05 '20 at 20:00
  • @JoeT Sorry, my fault! – Giulio Bresciani Oct 06 '20 at 06:51
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    @JoeT: I noticed that you have now changed the question completely. You should at least admit that I gave a correct solution to the original question. – Robert Bryant Oct 06 '20 at 12:11
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    @JoeT Indeed, I would recommend reverting this back to the original question, accepting Robert's answer, and posing this new question as a distinct question. – Steven Gubkin Oct 07 '20 at 18:26

1 Answers1

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There is a trivial construction that shows that the answer is 'yes' for all complex manifolds, not just those that admit an anti-holomorphic involution.

Let $(M,J)$ be a (finite-dimensional) complex $n$-manifold and let $\mathscr{U}$ be an open cover of $M$ with the properties that (i) for each $U\in\mathscr{U}$, there is a $J$-holomorphic chart $\zeta:U\to\mathbb{C}^n$, and (ii) For each $U\in\mathscr{U}$ there is a point $p\in U$ that does not lie in any $V\in\mathscr{U}$ other than $U$. (Using paracompactness, it is not difficult to construct such a chart.) Then by choosing one such 'reference point' $p_U\in U$ with $p_U\not\in V\in\mathscr{U}$ for $V\not=U$ and one $J$-holomorphic chart $\zeta_U:U\to\mathbb{C}^n$ so that $\zeta_U(p_U) = 0\in\mathbb{C}^n$, we arrive at a 'pointed atlas' $$ \widehat{\mathscr{U}} = \{ (U,\zeta_U,p_U)\ |\ U\in \mathscr{U}\ \} $$ with all the stated properties. The reason is that the only time the point $p_U$ is in the domain of a transition function for the pointed atlas $ \widehat{\mathscr{U}}$ is when one is 'transitioning' from $U$ to $V=U$, and, in that case, the only transition function is the identity mapping on $\zeta_U(U)\subset\mathbb{C}^n$, whose Taylor series at $\zeta_U(p_U) = 0\in\mathbb{C}^n$ clearly has all coefficients in $\mathbb{R}$ (in fact, all the coefficients are in $\mathbb{Z}$).

Robert Bryant
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    Note that this answer is a correct answer to the original question, which, I admit, was badly posed. – Robert Bryant Oct 06 '20 at 12:12
  • Moderators have turned back to the original question, which is the one answer here. – YCor Oct 10 '20 at 21:59
  • As currently worded, the question asks about "transition functions". These are functions $\zeta_U\circ\zeta_V^{-1}:\mathbb{C}^n\to\mathbb{C}^n$ defined on the "transition" $U\cap V$ that move from one chart to another. This answer makes no mention of the transition functions, only the charts. But perhaps the original wording of the question failed to say "transition" !? – Linas Oct 11 '20 at 00:35
  • @Linas: I'm not sure what point you are trying to make. Also, I'm not sure that you have read the answer since, contrary to what you claim, I explicitly talk about the transition functions in the last sentence. – Robert Bryant Oct 11 '20 at 09:35
  • I read the question as asking "for all transitions $U\cap V$ ..." and I don't understand how your answer implies that $\zeta_U \circ\zeta_V^{-1}$ is meromorphic with coefficients in $\mathbb {R}$ (for all pairs $U$, $V$). – Linas Oct 12 '20 at 17:14
  • @Linas: You may have missed the point of the question. The OP originally wanted the transition functions of the atlas to have the property that their Taylor series at every point had coefficients in $\mathbb{R}$. When it was pointed out that this is impossible, the OP modified the question to require only that the Taylor series at a specified 'reference point' in each chart have coefficients in $\mathbb{R}$. I showed, in my answer, that there is such a 'real pointed atlas' for any complex manifold. In other words, it wasn't a useful notion. The OP then changed the question completely. – Robert Bryant Oct 12 '20 at 17:43
  • @RobertBryant OK, thank you! Clearly, I've layered my own muddled thinking on top of a muddled situation. I'm now embarrassed by my own earlier questions, which I'm sorely tempted to delete because they're just ... wrong. Blame it on late-night cross-eyed bleariness. – Linas Oct 14 '20 at 19:23