arXiv subjects
Publications and source records attributed to Kartick Ghosh.
Xiuxiong Chen proved the existence of $C^{1, \bar 1}$ geodesics in the space of K\"{a}hler potentials and the convexity of the $J$-functional along $C^{1, \bar 1}$ geodesics. Analogous results were proved by Collins and Yau for the hypercritical Leung--Yau--Zaslow equation. In this paper, we study a similar picture for generalised Monge--Amp\`{e}re equations associated with two right-Noetherian polynomials in proper position.
In this paper, we study the ellipticity of the vector bundle versions of the Monge-Amp\`ere, $J$, dHYM and $\sigma_{k}$-equations at a point. These are nonlinear geometric partial differential equations defined on a holomorphic vector bundle over a compact K\"ahler manifold. We show that when both the dimension of the manifold and the rank of the bundle are greater than or equal to three, these equations do not preserve ellipticity along continuity paths in the connected component of the trivial solution. However, the $\sigma_{2}$-equation does preserve ellipticity along continuity paths.
Alvarez-Consul--Garcia-Fernandez--Garcia-Prada introduced the K\"ahler-Yang-Mills equations. They also introduced the $\alpha$-Futaki character, an analog of the Futaki invariant, as an obstruction to the existence of the K\"ahler-Yang-Mills equations. The equations depend on a coupling constant $\alpha$. Solutions of these equations with coupling constant $\alpha>0$ are of utmost importance. In this paper, we provide a formula for the $\alpha$-Futaki character on certain ample line bundles over toric manifolds. We then show that there are no solutions with $\alpha>0$ on certain ample line bundles over certain toric manifolds and compute the value of $\alpha$ if a solution exists. We also relate our result to the existence result of Keller-Friedman in dimension-two.
We introduce a new system of equations coupling Kähler-Einstein and Hermitian-Yang-Mills equations. We provide a moment map interpretation of these equations. We identify a Futaki type invariant as an obstruction to the existence of solutions to these equations. We also prove a Matsushima-Lichnerowicz type theorem. We prove a deformation result that produces nontrivial solutions of these equations under some conditions. We produce examples on some projective bundles using Calabi ansatz.
In this paper, we prove \emph{a priori} estimates for some vortex-type equations on compact Riemann surfaces. As applications, we recover existing estimates for the vortex bundle Monge-Ampère equation, prove an existence and uniqueness theorem for the Calabi-Yang-Mills equations on vortex bundles, and get estimates for $J-$vortex equation. We prove an existence and uniqueness result relating Gieseker stability and the existence of almost Hermitian Einstein metrics, i.e., a Kobayashi-Hitchin type correspondence. We also prove Kählerness of the negative of the symplectic form which arises in the moment map interpretation of the Calabi-Yang-Mills equations in \cite{Vamsi3}