(Inverse)-Hessian type equations and positivity in complex algebraic geometry

Abstract: In the early 2000's Demailly and Paun proved that a (1,1) cohomology class on a K\"ahler manifold is positive if and only if certain intersection numbers are positive. This is a generalization of the classical Nakai criteria for ampleness of line bundles on projective manifolds. The proof, somewhat surprisingly, relies on Yau's work on the complex Monge-Ampere equation, and his solution to the Calabi conjecture. In 2019, Gao Chen extended the method of Demailly-Paun to prove that another important PDE in Kahler geometry, namely the J-equation, is equivalent to the positivity of certain (twisted) intersection numbers, thereby settling a conjecture of Lejmi and Szekelyhidi. In my talk, I will describe this circle of ideas, concluding with a recent result obtained in collaboration with Vamsi Pingali extending the work of Gao Chen to more general inverse Hessian type equations, thereby settling a conjecture of Szekelyhidi for projective manifolds. In the process we obtain an equivariant version of Gao Chen's result, and in particular recover some results of Collins and Szekelyhidi on the J-equation on toric manifolds.

algebraic geometrycomplex variablesdifferential geometry

Audience: researchers in the topic

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