The asymptotic behaviour of Bergman kernels

Abstract

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 $\Lambda ,v>0$ are constants. 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 $-\Lambda$ 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].