The asymptotic behavior of Bergman kernels

Let (X,d,p) be the pointed Gromov-Hausdorff limit of a sequence of pointed complete polarized Kähler manifolds (Ml,ωl,Ll,hl,pl) with Ric(hl)=2πωl, Ric(ωl)≥−Λωl and Vol(B1(pl))≥v, ∀l∈N, where Λ,v>0 are constants. Then X is a normal complex space [33]. In this paper, we discuss the convergence of the Hermitian line bundles (Ll,hl) and the Bergman kernels. In particular, we show that the Kähler forms ωl converge to a unique closed positive current ωX on Xreg. By establishing a version of L2 estimate on the limit line bundle on X, we give a convergence result of Fubini-Study currents on X. Then we prove that the convergence of Bergman kernels implies a uniform Lp asymptotic expansion of Bergman kernel on the collection of n-dimensional polarized Kähler manifolds (M,ω,L,h) with Ricci lower bound −Λ and non-collapsing condition Vol(B1(x))≥v>0. Under the additional orthogonal bisectional curvature lower bound, we will also give a uniform C0 asymptotic estimate of Bergman kernel for all sufficiently large m, which improves a theorem of Jiang [28]. By calculating the Bergman kernels on orbifolds, we disprove a conjecture of Donaldson-Sun in [20].