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

Datar, VV and Pingali, VP (2021) A numerical criterion for generalised Monge-Ampère equations on projective manifolds. In: Geometric and Functional Analysis .

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Official URL: https://doi.org/10.1007/s00039-021-00577-1


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. © 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

Item Type: Journal Article
Publication: Geometric and Functional Analysis
Publisher: Birkhauser
Additional Information: The copyright for this article belongs to Authors
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Date Deposited: 03 Dec 2021 06:50
Last Modified: 03 Dec 2021 06:50
URI: http://eprints.iisc.ac.in/id/eprint/70102

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