Arc K-semistability is claimed to be a very general algebraic property in flat families via openness of Paul-stability of pairs, but the proof of the pair-stability lemma contains a false assertion about orbit-closure fibers.
A Hilbert--Mumford criterion for generalised Monge--Amp\`ere equations
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abstract
We give a new numerical criterion in the spirit of GIT for existence of solutions to inverse Hessian equations, including in particular the J-equation. Our criterion is formulated in terms of stability of pairs in the sense of Paul. To that end, we build on previous work of the author with Dervan, and generalise a result of Zhang, proving isometry between generalised Chow line bundles and mixed Deligne pairings.
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The constant scalar curvature K\"ahler condition is very general
Arc K-semistability is claimed to be a very general algebraic property in flat families via openness of Paul-stability of pairs, but the proof of the pair-stability lemma contains a false assertion about orbit-closure fibers.