On the existence of Kobayashi and Bergman metrics for Model domains

We prove that for a pseudoconvex domain of the form $${\mathfrak {A}} = \{(z, w) \in {\mathbb {C}}^2 : v > F(z, u)\}$$ A = { ( z , w ) ∈ C 2 : v > F ( z , u ) } , where $$w = u + iv$$ w = u + i v and F is a continuous function on $${\mathbb {C}}_z \times {\mathbb {R}}_u$$ C z × R u , the following conditions are equivalent: The domain $$\mathfrak {A}$$ A is Kobayashi hyperbolic. The domain $$\mathfrak {A}$$ A is Brody hyperbolic. The domain $$\mathfrak {A}$$ A possesses a Bergman metric. The domain $$\mathfrak {A}$$ A possesses a bounded smooth strictly plurisubharmonic function, i.e. the core $$\mathfrak {c}(\mathfrak {A})$$ c ( A ) of $$\mathfrak {A}$$ A is empty. The graph $$\Gamma (F)$$ Γ ( F ) of F can not be represented as a foliation by holomorphic curves of a very special form, namely, as a foliation by translations of the graph $$\Gamma ({\mathcal H})$$ Γ ( H ) of just one entire function $${\mathcal {H}} : {\mathbb {C}}_z \rightarrow {\mathbb {C}}_w$$ H : C z → C w .

Bergman metrics and geodesics in the space of Kähler metrics on principally polarized abelian varieties

It is well known in Kähler geometry that the infinite-dimensional symmetric space $\mathcal{H}$ of smooth Kähler metrics in a fixed Kähler class on a polarized Kähler manifold is well approximated by finite-dimensional submanifolds $\mathcal{B}_k\subset\mathcal{H}$ of Bergman metrics of height k. Then it is natural to ask whether geodesics in $\mathcal{H}$ can be approximated by Bergman geodesics in $\mathcal{B}_k$. For any polarized Kähler manifold, the approximation is in the C0 topology. For some special varieties, one expects better convergence: Song and Zelditch proved the C2 convergence for the torus-invariant metrics over toric varieties. In this article, we show that some C∞ approximation exists as well as a complete asymptotic expansion for principally polarized abelian varieties.