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

Abstract

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.

Hypercritical deformed Hermitian-Yang-Mills equation revisited

“…In [3,Conjecture 1.4], Collins-Jacob-Yau predicted that the existence of solution to the supercritical dHYM equation is equivalent to a stability condition in terms of holomorphic intersection numbers for any irreducible subvarieties V ⊂ X, modeled on the Nakai-Moishezon criterion, and confirmed it for complex surfaces. In [2], the authors and Takahashi confirmed the conjecture in the projective case building on the works of Chen [1] and Song [12], see also [7,9].…”

Special representatives of complexified Kähler classes

“…[11,32,36]. This equation, when solvable (see [6,9,15,38]), gives a unique way, compatible with mirror symmetry and expressing a vanishing moment map condition, to fix the B-field, for each choice of Kähler form ω.…”

$J$-equations on holomorphic submersions

“…The uniform version of the conjecture is proved by Gao Chen [Che21] and it is also proved that the uniform conditions are equivalent to the uniform J-stability. The original version of the conjecture is proved by Datar-Pingali [DP21] in a projective case and by Song [Son20] in general. Now we state the converse implication.…”