Acta mathematica scientia,Series B ›› 2009, Vol. 29 ›› Issue (3): 480-492.doi: 10.1016/S0252-9602(09)60048-5

• Articles • Previous Articles     Next Articles

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)

Abstract:

Two points of the infinite dimensional complex projective space CP∞ with homogeneous coordinates α=(α0, α1, α2, … ) and b=b0, b1, b2, … ), respectively, are conjugate if and only if they are complex orthogonal, i.e., αb = Σ∞j=0
αj bj =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 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 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

CLC Number: 

  • 32A36
Trendmd