A numerical criterion for generalised Monge-Ampère equations on projective manifolds

By V. Datar, Vamsi Pingali. Geometric and Functional Analysis, 2020

52 citations recorded by the source corpus.

We prove that generalised Monge-Ampère equations (a family of equations which includes the inverse Hessian equations like the J-equation, as well as the Monge-Ampère equation) on projective manifolds have smooth solutions if certain intersection numbers are positive. As corollaries of our work, we improve a result of Chen (albeit in the projective case) on the existence of solutions to the J-equation, and prove a conjecture of Székelyhidi in the projective case on the solvability of certain inverse Hessian equations. The key new ingredient in improving Chen’s result is a degenerate concentration of mass result. We also prove an equivariant version of our results, albeit under the assumption of uniform positivity. In particular, we can recover existing results on manifolds with large symmetry such as projective toric manifolds.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/386d20c47758b8a764c60200c59a868a5d89fc35

Citations in the corpus

• Nakai-Moishezon criterions for complex Hessian equations (2020, unknown): “toward this conjecture by Chen [6], proving Conjecture 1.1 under a slightly stronger topological condition (uniform J-positivity, see Definition 1.1 (3)). Chen’s result is generalized by Datar-Pingali [8] to a more general family of complex Hessian equations. Conjecture 1.1 is also proved in [8] for ”