The asymptotic behaviour of Bergman kernels

Let $(X,d,p)$ be the pointed Gromov-Hausdorff limit of a sequence of pointed complete polarized K"ahler manifolds $( M_l ,\omega_l ,$\mathcal{L}$_l ,h_l ,p_l ) $with$Ric ( h_l ) =2\pi \omega_l $,$Ric ( \omega_l ) \geq -\Lambda \omega_l $and$Vol \big( B_1 ( p_l ) \big) \geq v $,$\forall l\in$\mathbb{N}$ $,where$\Lambda ,v>0$areconstants\mathrm{.}Then$X$ is a normal complex space [Liu-Sz'ekelyhidi, 2022, GAFA]. In this paper, we discuss the convergence of the Hermitian line bundles $( $\mathcal{L}$_l ,h_l )$ and the Bergman kernels. In particular, we show that the K"ahler forms ${\omega }_l$ converge to a unique closed positive current ${\omega }_X$ on $X_{reg}$. By establishing a version of $L^2$ 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 $L^p$ asymptotic expansion of Bergman kernel on the collection of $n$-dimensional polarized K"ahler manifolds $(M,\omega ,\mathcal{L},h)$ with Ricci lower bound $-\mathrm{\Lambda }$ and non-collapsing condition $Vol \big( B_1 (x) \big) \geq v >0 $. Under the additional orthogonal bisectional curvature lower bound, we will also give a uniform $C^0$ asymptotic estimate of Bergman kernel for all sufficiently large $m$, which improves a theorem of Jiang [Jiang, 2016, crelle]. By calculating the Bergman kernels on orbifolds, we disprove a conjecture of Donaldson-Sun in [Donaldson-Sun, 2014, Acta].