A numerical criterion for generalised MongeAmpere equations on projective manifolds
Abstract
We prove that generalised MongeAmpère equations (a family of equations which includes the inverse Hessian equations like the $J$equation, as well as the MongeAmpère 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2020
 arXiv:
 arXiv:2006.01530
 Bibcode:
 2020arXiv200601530D
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Algebraic Geometry;
 Mathematics  Complex Variables
 EPrint:
 Final version. To appear in Geometric and Functional Analysis