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Pré-Publication, Document De Travail Année : 2022

Monge-Amp\`ere equations on compact Hessian manifolds

Résumé

We consider degenerate Monge-Amp\`ere equations on compact Hessian manifolds. We establish compactness properties of the set of normalized quasi-convex functions and show local and global comparison principles for twisted Monge-Amp\`ere operators. We then use the Perron method to solve Monge-Amp\`ere equations whose RHS involves an arbitrary probability measure, generalizing works of Cheng-Yau, Delano\"e, Caffarelli-Viaclovsky and Hultgren-\"Onnheim. The intrinsic approach we develop should be useful in deriving similar results on mildly singular Hessian varieties, in line with the Strominger-Yau-Zaslow conjecture.

Dates et versions

hal-03868775 , version 1 (23-11-2022)

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Vincent Guedj, Tat Dat Tô. Monge-Amp\`ere equations on compact Hessian manifolds. 2022. ⟨hal-03868775⟩
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