Asymptotics for Bergman-Hodge kernels for high powers of complex line   bundles
  In this paper we obtain the full asymptotic expansion of the Bergman-Hodge kernel associated to a high power of a holomorphic line bundle with non-degenerate curvature. We also explore some relations with asymptotic holomorphic sections on symplectic manifolds 
Robert Berman, Johannes Sjoestrand