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

TL;DR

This paper establishes a numerical criterion for the existence of smooth solutions to generalized Monge-Ampère equations on projective manifolds, improving known results and confirming conjectures in specific cases.

Contribution

It introduces a new intersection number criterion ensuring solutions, extends results to inverse Hessian equations, and verifies a conjecture in the projective setting.

Findings

Proved existence of solutions under positivity of intersection numbers.

Improved Chen's result on the J-equation in the projective case.

Confirmed Székelyhidi's conjecture for inverse Hessian equations in projective manifolds.

Abstract

We prove that generalised Monge-Amp\`ere equations (a family of equations which includes the inverse Hessian equations like the $J$-equation, as well as the Monge-Amp\`ere 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 Sz\'ekelyhidi 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.