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Abstract: Generalised Monge-Ampere equations are a family of PDE that contain inverse Hessian equations like the J-equation, and special cases of the deformed Hermitian-Yang-Mills equation. I shall describe a couple of results about them. Firstly, these equations can be solved on projective manifolds if and only if Nakai-Moizeshon / Demailly-Paun-style intersection numbers are positive. This result improves a recent theorem of Gao Chen (who assumed uniform strict-positivity), and settles a conjecture of Lejmi-Szkelyhidi in the projective case. Secondly, assuming uniform strict-positivity, an equivariant version of the same result holds.
Thus, earlier results due to Collins-Szekelyhidi are recovered. This work is joint with Ved Datar.
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