On the Bergman representative coordinates

We study the set where the so-called Bergman representative coordinates (or Bergman functions) form an immersion. We provide an estimate of the size of a maximal geodesic ball with respect to the Bergman metric, contained in this set. By concrete examples we show that these estimates are the best possible.Comment: 20 page

Mixed Hessian inequalities and uniqueness in the class E(X,ω,m)

We prove a general inequality for mixed Hessian measures by global arguments. Our method also yields a simplification for the case of complex Monge–Ampère equation. Exploiting this and using Kołodziej’s mass concentration technique we also prove the uniqueness of the solutions to the complex Hessian equation on compact Kähler manifolds in the case of probability measures vanishing on (Formula presented.)-polar sets