The asymptotic behaviour of Bergman kernels

2021/12/16 by Zhou, Shengxuan
#Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2112.08893

Abstract

Let ( X ,d ,p ) be the pointed Gromov-Hausdorff limit of a sequence of pointed complete polarized Kähler manifolds ( M_l ,ω_l ,L_l ,h_l ,p_l ) with Ric ( h_l ) =2πω_l , Ric ( ω_l ) ≥ -Λω_l and Vol ( B₁ ( p_l ) ) ≥ v , ∀ l∈ℕ , where Λ,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 ( L_l ,h_l ) and the Bergman kernels. In particular, we show that the Kähler forms ω_l converge to a unique closed positive current ω_X on X_reg. 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 L^p 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 [Jiang, 2016, crelle]. By calculating the Bergman kernels on orbifolds, we disprove a conjecture of Donaldson-Sun in [Donaldson-Sun, 2014, Acta].

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