On the asymptotic behavior of Bergman kernels for positive line bundles
  Let L be a positive line bundle on a projective complex manifold. We study the asymptotic behavior of Bergman kernels associated with the tensor powers L^p of L as p tends to infinity. The emphasis is the dependence of the uniform estimates on the positivity of the Chern form of the metric on L. This situation appears naturally when we approximate a semi-positive singular metric by smooth positively curved metrics. 
Tien-Cuong Dinh, Xiaonan Ma and Viet-Anh Nguyen
math.CV