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Ved V. Datar

Publications and source records attributed to Ved V. Datar.

5 recordsLinked to original sources
arXiv2026

The top Yau--Yang conjecture for Kähler manifolds with positive sectional curvature

We prove that the top wedge power of the Ricci form of a complete non-compact Kähler manifold with positive sectional curvature has finite integral. Using a result of Chen-Zhu, an immediate consequence is the quasiprojectivity of such manifolds under the assumption of bounded sectional curvature. A key new idea to prove Bézout estimates along with a Lipschitz weight with finite Monge-Ampère mass is used in the proof of the main result.

math.DG↗
arXiv2021

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

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.

math.DG↗
arXiv2019

On coupled constant scalar curvature Kähler metrics

We provide a moment map interpretation for the coupled Kähler-Einstein equations introduced by Hultgren and Witt Nyström, and in the process introduce a more general system of equations, which we call coupled cscK equations. A differentio-geometric formulation of the corresponding Futaki invariant is obtained and a notion of K-polystability is defined for this new system. Finally, motivated by a result of Székelyhidi, we prove that if there is a solution to our equations, then small K-polystable perturbations of the underlying complex structure and polarizations also admit coupled cscK metrics.

math.DG↗
arXiv2014

On convexity of the regular set of conical Kahler-Einstein metrics

In this note we prove convexity, in the sense of Colding-Naber, of the regular set of solutions to some complex Monge-Ampere equations with conical singularities along simple normal crossing divisors. In particular, any two points in the regular set can be joined by a smooth minimal geodesic lying entirely in the regular set. We show that as a result, the classical theorems of Myers and Bishop-Gromov extend almost verbatim to this singular setting.

math.DG↗