Timeline for Do we have an equivariant Newlander-Nirenberg theorem for finite group action?
Current License: CC BY-SA 4.0
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| Jan 24, 2023 at 4:27 | comment | added | Michael Albanese | In that case, the answer to your question is yes. If $J$ is integrable, then $X$ is a complex manifold, and a map $g : X \to X$ is holomorphic if and only if $g_*J = Jg_*$. This condition on $g_*$ is equivalent to the Cauchy-Riemann equations for $g$ in smooth coordinates obtained as the real and imaginary parts of holomorphic coordinates, see here for the one-dimensional case. | |
| Jan 24, 2023 at 0:57 | comment | added | Zhaoting Wei | @MichaelAlbanese You are right. I made the edit according to your comments. | |
| Jan 24, 2023 at 0:56 | history | edited | Zhaoting Wei | CC BY-SA 4.0 |
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| Jan 23, 2023 at 23:16 | comment | added | Michael Albanese | Viewing $g$ as a smooth map $X \to X$, do you mean $g_*Jg_*^{-1} = J$? Also, an almost complex structure is a map $J : TX \to TX$, not $T_{\mathbb{C}}X \to T_{\mathbb{C}}X$ (of course, the former extends to the latter). | |
| Jan 23, 2023 at 15:56 | history | edited | YCor |
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| Jan 23, 2023 at 15:54 | history | asked | Zhaoting Wei | CC BY-SA 4.0 |