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References to convex analysis

I am a little confused by the references to convex analysis. I agree that the lemma transforms the Kahler–Einstein problem to a complex Monge–Ampère equation, but Yau's solution did not use any variational techniques, and what I understand of Chen–Donaldson–Sun/Tian's work also has nothing to do with it. Moreover I've often heard that the very difficulty of Kähler geometry is that convexity arguments just don't apply to plurisubharmonic functions. (Maybe it's meant to refer to constant scalar curvature on toric varieties, where Legendre transform puts the problem in convex-analytic terms?) Gumshoe2 (talk) 08:27, 27 October 2022 (UTC)

Yes it's a bit esoteric now that you mention it. What I was referring to was the study of Kahler metrics using convex analysis in the latest variational approach to Kahler geometry (a la Berman--Boucksom--Jonsson). I associate general pluripotential theory with convex geometry in my mind due to associations with that and toric geometry. I will remove the reference. Tazerenix (talk) 08:56, 27 October 2022 (UTC)

Gap in proof of the local ${\displaystyle \partial {\bar {\partial }}}$-lemma

The proof assumes that ${\displaystyle \gamma }$ is concentrated in bidegrees ${\displaystyle (p-1,q)}$ and ${\displaystyle (p,q-1)}$, but this is not true in general. The ${\displaystyle \gamma }$ produced by the ${\displaystyle d}$-Poincaire lemma can vary by a ${\displaystyle d}$-exact form, which can potentially have any bidegree. To fix the proof, you must inductively subtract off the most holomorphic and most anti-holomorphic bidegrees by observing they are ${\displaystyle \partial }$- and ${\displaystyle {\bar {\partial }}}$-closed. 135.180.50.23 (talk) 07:10, 5 April 2023 (UTC)

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