Estimates of Bergman kernels and Bergman metric on compact Picard surfaces

Abstract

Let Γ⊂SU((2,1),C) be a torsion-free cocompact subgroup. Let B2 denote the 2-dimensional complex ball endowed with the hyperbolic metric μhyp, and let XΓ:=Γ﹨B2 denote the quotient space, which is a compact complex manifold of dimension 2. Let Λ:=ΩXΓ2 denote the line bundle on XΓ, whose sections are holomorphic (2,0)-forms. For any k≥1, the hyperbolic metric induces a point-wise metric on H0(XΓ,Λ⊗k), which we denote by |⋅|hyp. For any k≥1, let BΛk denote the Bergman kernel of the complex vector space H0(XΓ,Λ⊗k). For any k≥3, and z,w∈XΓ, the first main result of the article is the following off-diagonal estimate of the Bergman kernel BΛk|BΛk(z,w)|hyp=OXΓ(k2cosh3k−8⁡(dhyp(z,w)/2)), where dhyp(z,w) denotes the geodesic distance between the points z and w on XΓ, and the implied constant depends only on the Picard surface XΓ.For any k≥1, let μberk(z):=−i2π∂z∂z‾log⁡|BΛk(z,z)|hyp denote the Bergman metric associated the line bundle Λ⊗k, and let μberk,vol denote the associated volume form. For k≫1 sufficiently large, and ϵ>0, the second main result of the article is the following estimatesupz∈XΓ⁡|μberk,vol(z)μhypvol(z)|=OXΓ,ϵ(k4+ϵ), where μhypvol denotes the volume form associated to the hyperbolic metric μhyp, and the implied constant depends on the Picard surface XΓ, and on the choice of ϵ>0.