The conjugate points of ℂℙ∞ and the zeroes of bergman kernel

Abstract

Two points of the infinite dimensional complex projective space ℂℙ∞ with homogeneous coordinates a = (a0,a1,a2,…) and b = (b0,b1,b2,…), respectively, are conjugate if and only if they are complex orthogonal, i.e., ab†=∑j=0∞ajbj¯=0. For a complete ortho-normal system φ(t)=(φ(t), φ1(t), φ2(t), …) of L2H(D) the space of the holomorphic and absolutely square integrable functions in the bounded domain D of ℂn, φ(t),t ∈ D is considered as the homogeneous coordinate of a point in ℂℙ∞. The correspondence t ↦ φ(t) induces a holomorphic imbedding ιφ:D → ℂℙ∞. It is proved that the Bergman kernel ▪ of D equals to zero for the two points t and v in D if and only if their image points under ιφ are conjugate points of ℂℙ∞.