数学物理学报(英文版) ›› 2009, Vol. 29 ›› Issue (3): 480-492.doi: 10.1016/S0252-9602(09)60048-5

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THE CONJUGATE POINTS OF CP AND THE ZEROES OF BERGMAN KERNEL

陆启铿   

  1. Institute of Mathematics, Academy of Mathematics &|System Science, Chinese Academy of Sciences, |Beijing 100190, China
  • 收稿日期:2008-06-16 修回日期:2009-01-13 出版日期:2009-05-20 发布日期:2009-05-20
  • 基金资助:

    Partially support by NSF of China (A01010501 and 10731080)

THE CONJUGATE POINTS OF CP AND THE ZEROES OF BERGMAN KERNEL

Lu Qikeng   

  1. Institute of Mathematics, Academy of Mathematics &|System Science, Chinese Academy of Sciences, |Beijing 100190, China
  • Received:2008-06-16 Revised:2009-01-13 Online:2009-05-20 Published:2009-05-20
  • Supported by:

    Partially support by NSF of China (A01010501 and 10731080)

摘要:

Two points of the infinite dimensional complex projective space CP∞ with homogeneous coordinates α=(α0, α1, α2) and b=b0, b1b2), respectively, are conjugate if and only if they are complex orthogonal, i.e., αb = Σj=0
αbj =0. For a complete ortho-normal system φ(t)=(φ0(t), φ1(t), φ2(t), ) of L2H(D), the space of the holomorphic and absolutely square integrable functions in the bounded domain D of Cnφ(t), t ∈ D, is considered as the homogeneous coordinate of a point in CP. The correspondence t → φ(t) induces a holomorphic imbedding tφ D → CP. It is proved that the Bergman kernel K(t, v) of  D equals to zero for the two points t and v in D if and only if their image points under tφ are conjugate points of CP.

关键词: conjugate points, Bergman kernel

Abstract:

Two points of the infinite dimensional complex projective space CP∞ with homogeneous coordinates α=(α0, α1, α2) and b=b0, b1b2), respectively, are conjugate if and only if they are complex orthogonal, i.e., αb = Σj=0
αbj =0. For a complete ortho-normal system φ(t)=(φ0(t), φ1(t), φ2(t), ) of L2H(D), the space of the holomorphic and absolutely square integrable functions in the bounded domain D of Cnφ(t), t ∈ D, is considered as the homogeneous coordinate of a point in CP. The correspondence t → φ(t) induces a holomorphic imbedding tφ D → CP. It is proved that the Bergman kernel K(t, v) of  D equals to zero for the two points t and v in D if and only if their image points under tφ are conjugate points of CP.

Key words: conjugate points, Bergman kernel

中图分类号: 

  • 32A36