We prove that generalised Monge-Ampere equations (a family of equations which includes the inverse Hessian equations like the J-equation, as well as the Monge-Ampere equation) on projective manifolds have smooth solutions if certain intersection numb
ers 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 Szekelyhidi in the projective case on the solvability of inverse Hessian equations. 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.
The aim of this article is to study expansions of solutions to an extremal metric type equation on the blow-up of constant scalar curvature Kahler surfaces. This is related to finding constant scalar curvature Kahler (cscK) metrics on K-stable blow-ups of extremal Kahler surfaces
In this note we prove convexity, in the sense of Colding-Naber, of the regular set of solutions to some complex Monge-Ampere equations with conical singularities along simple normal crossing divisors. In particular, any two points in the regular set
can be joined by a smooth minimal geodesic lying entirely in the regular set. We show that as a result, the classical theorems of Myers and Bishop-Gromov extend almost verbatim to this singular setting.
