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

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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.

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Notes

  1. Indeed, if \(dim(Z)=1\), then for large N, \(Y \cap Z\) is a non-empty collection of points and is hence a divisor in Z. If \(dim(Z)\ge 2\), then since \(dim(Y)+dim(Z)>dim(M)\) the analytic set \(Y\cap Z\) is a union of connected subvarieties by the Fulton-Hansen connectedness theorem. Hence, if we ensure that there exists a Y that intersects every component of Z in at least one smooth point transversally, we are done by Bertini’s theorem.

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Acknowledgements

This is work partially supported by grant F.510/25/CAS-II/2018(SAP-I) from UGC (Govt. of India). The first author (V. Datar) is grateful to the Infosys Foundation for their support through the Young Investigator Award. The second author (V. Pingali) is partially supported by a MATRICS grant MTR/2020/000100 from SERB (Govt. of India) We are deeply indebted to Gao Chen for many clarifications regarding his paper, and for his comments on all drafts of this work. We thank Jian Song for his comments on the first draft of the paper, and for many useful discussions on the issue of relaxing uniform positivity in the Theorem 1.3. We also thank Indranil Biswas, Apoorva Khare, Kapil Paranjape, Venkatesh Rajendran for help with regard to some algebraic aspects of the work. Finally, we thank the anonymous reviewer for detailed and useful comments.

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  1. Department of Mathematics, Indian Institute of Science, Bangalore, 560012, India

    Ved V. Datar & Vamsi Pritham Pingali

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  1. Ved V. Datar
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Datar, V.V., Pingali, V.P. A numerical criterion for generalised Monge-Ampère equations on projective manifolds. Geom. Funct. Anal. 31, 767–814 (2021). https://doi.org/10.1007/s00039-021-00577-1

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