Holomorphic invariant forms of a bounded domain

Abstract

Given a complete ortho-normal system ϕ = (ϕ₀,ϕ₁,ϕ₂,…) of L ² H($\mathcal{D}$), the space of holomorphic and absolutely square integrable functions in the bounded domain $\mathcal{D}$ of ℂⁿ, we construct a holomorphic imbedding $\iota _\phi :\mathcal{D} \to \mathfrak{F}(n,\infty )$, the complex infinite dimensional Grassmann manifold of rank n. It is known that in $\mathfrak{F}(n,\infty )$ there are n closed (μ, μ)-forms τ_μ (μ = 1,…,n) which are invariant under the holomorphic isometric automorphism of $\mathfrak{F}(n,\infty )$ and generate algebraically all the harmonic differential forms of $\mathfrak{F}(n,\infty )$. So we obtain in $\mathcal{D}$ a set of (μ, μ)-forms ι ^*_ϕ τ_μ (μ = 1,…, n), which are independent of the system ϕ chosen and are invariant under the bi-holomorphic transformations of $\mathcal{D}$. Especially the differential metric ds ¹₂ associated to the Kähler form ι ^*_ϕ τ₁ is a Kähler metric which differs from the Bergman metric ds ² of $\mathcal{D}$ in general, but in case that the Bergman metric is an Einstein metric, ds ²₁ differs from ds ² only by a positive constant factor.