A Hilbert--Mumford criterion for generalised Monge--Amp\`ere equations

R\'emi Reboulet

A Hilbert--Mumford criterion for generalised Monge--Amp\`ere equations

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arxiv 2504.06612 v1 pith:K6R24IUS submitted 2025-04-09 math.DG math.AG

Rémi Reboulet This is my paper · ORCID

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keywords criterionequationsgeneralisedauthorbuildbundleschowdeligne

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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
math.AG 2025-04 reject novelty 6.0 of 10

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.

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