The asymptotic behaviour of Bergman kernels

Abstract

Let $(X,d,p)$ be the pointed Gromov-Hausdorff limit of a sequence of pointed complete polarized Kähler manifolds $(M_l,{\omega }_l,{\mathcal{L}}_l,h_l,p_l)$ with $Ric(h_l)=2\pi {\omega }_l$, $Ric({\omega }_l)\ge -\Lambda {\omega }_l$ and $Vol(B_1(p_l))\ge v$, $\forall l\in \mathbb{N}$, where $\Lambda ,v>0$ are constants. Then $X$ is a normal complex space [Liu-Székelyhidi, 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 Kähler 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 Kähler manifolds $(M,\omega ,\mathcal{L},h)$ with Ricci lower bound $-\Lambda$ and non-collapsing condition $Vol(B_1(x))\ge v>0$. 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].

29 pages, all comments are welcome