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

@article{Datar2021ANC, title={A numerical criterion for generalised Monge-Amp{\`e}re equations on projective manifolds}, author={Ved V. Datar and Vamsi Pingali}, journal={Geometric and Functional Analysis}, year={2021} }

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 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 Szekelyhidi in the projective case on the solvability of inverse…

## 15 Citations

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## References

SHOWING 1-10 OF 52 REFERENCES

### A generalised Monge-Ampère equation

- Mathematics
- 2012

We consider a generalised complex Monge-Amp\`ere equation on a compact K\"ahler manifold and treat it using the method of continuity. For complex surfaces, we prove an easy existence result. We also…

### Estimates for the Complex Monge-Ampère Equation on Hermitian and Balanced Manifolds

- Mathematics
- 2010

We generalize Yau's estimates for the complex Monge-Ampere equation on compact manifolds in the case when the background metric is no longer Kahler. We prove $C^{\infty}$ a priori estimates for a…

### Fully non-linear elliptic equations on compact Hermitian manifolds

- MathematicsJournal of Differential Geometry
- 2018

We derive a priori estimates for solutions of a general class of fully non-linear equations on compact Hermitian manifolds. Our method is based on ideas that have been used for different specific…

### A note on the deformed Hermitian Yang-Mills PDE

- Mathematics
- 2015

ABSTRACT We prove a priori estimates for a generalised Monge–Ampère PDE with ‘non-constant coefficients’ thus improving a result of Sun in the Kähler case. We apply this result to the deformed…

### On a class of fully nonlinear flows in Kähler geometry

- Mathematics
- 2009

Abstract In this paper, we study a class of fully nonlinear metric flows on Kähler manifolds, which includes the J-flow as a special case. We provide a sufficient and necessary condition for the long…

### The Dirichlet problem for degenerate complex Monge-Ampere equations

- Mathematics
- 2009

The Dirichlet problem for a Monge-Ampere equation corresponding to a nonnegative, possible degenerate cohomology class on a Kaehler manifold with boundary is studied. C^{1,\alpha} estimates away from…

### On The Ricci Curvature of a Compact Kahler Manifold and the Complex Monge-Ampere Equation, I*

- Mathematics
- 1978

Therefore a necessary condition for a (1,l) form ( G I a ' r r ) I,,, Rlr dz' A d? to be the Ricci form of some Kahler metric is that it must be closed and its cohomology class must represent the…

### A special Lagrangian type equation for holomorphic line bundles

- Mathematics
- 2014

Let L be a holomorphic line bundle over a compact Kähler manifold X. Motivated by mirror symmetry, we study the deformed Hermitian–Yang–Mills equation on L, which is the line bundle analogue of the…

### Moment Maps, Nonlinear PDE and Stability in Mirror Symmetry, I: Geodesics

- Mathematics
- 2018

In this paper, the first in a series, we study the deformed Hermitian–Yang–Mills (dHYM) equation from the variational point of view as an infinite dimensional GIT problem. The dHYM equation is mirror…

### The deformed Hermitian Yang–Mills equation on three-folds

- MathematicsAnalysis & PDE
- 2022

We prove an existence result for the deformed Hermitian Yang-Mills equation for the full admissible range of the phase parameter, i.e., $\hat{\theta} \in (\frac{\pi}{2},\frac{3\pi}{2})$, on compact…