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… 

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References

SHOWING 1-10 OF 52 REFERENCES

A generalised Monge-Ampère equation

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

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

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

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

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

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*

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

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

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

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
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