We prove that certain possibly non-smooth Hermitian metrics are Griffiths-semipositively curved if and only if they satisfy an asymptotic extension property. This result answers a question of Deng--Ning--Wang--Zhou in the affirmative.
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.
We give a complete solution to the existence problem for gravitating vortices with non-negative topological constant $c geqslant 0$. Our first main result builds on previous results by Yang and establishes the existence of solutions to the Einstein-B
ogomolnyi equations, corresponding to $c=0$, in all admissible Kahler classes. Our second main result completely solves the existence problem for $c>0$. Both results are proved by the continuity method and require that a GIT stability condition for an effective divisor on the Riemann sphere is satisfied. For the former, the continuity path starts from a given solution with $c = 0$ and deforms the Kahler class. For the latter result we start from the established solution in any fixed admissible Kahler class and deform the coupling constant $alpha$ towards $0$. A salient feature of our argument is a new bound $S_g geqslant c$ for the curvature of gravitating vortices, which we apply to construct a limiting solution along the path via Cheeger-Gromov theory.
We prove an existence result for the deformed Hermitian Yang-Mills equation for the full admissible range of the phase parameter, i.e., $hat{theta} in (frac{pi}{2},frac{3pi}{2})$, on compact complex three-folds conditioned on a necessary subsolution
condition. Our proof hinges on a delicate analysis of a new continuity path obtained by rewriting the equation as a generalised Monge-Amp`ere equation with mixed sign coefficients.
We come up with infinite-dimensional prequantum line bundles and moment map interpretations of three different sets of equations - the generalised Monge-Amp`ere equation, the almost Hitchin system, and the Calabi-Yang-Mills equations. These are all p
erturbations of already existing equations. Our construction for the generalised Monge-Amp`ere equation is conditioned on a conjecture from algebraic geometry. In addition, we prove that for small values of the perturbation parameters, some of these equations have solutions.
