Datar, VV and Pingali, VP (2021) A numerical criterion for generalised Monge-Ampère equations on projective manifolds. In: Geometric and Functional Analysis, 31 (4). pp. 767-814.
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Abstract
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.
| Item Type: | Journal Article |
|---|---|
| Publication: | Geometric and Functional Analysis |
| Publisher: | Birkhauser |
| Additional Information: | The copyright for this article belongs to the Authors. |
| Department/Centre: | Division of Physical & Mathematical Sciences > Mathematics |
| Date Deposited: | 15 May 2023 10:06 |
| Last Modified: | 15 May 2023 10:06 |
| URI: | https://eprints.iisc.ac.in/id/eprint/81654 |
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