Bergman-Hartogs型域的全纯自同构群

The holomorphic automorphism group of the domains of the Bergman-Hartogs type

我们考虑一类以有界对称域D为底的Bergman-Hartogs型域Ω={(wm(1),...,w(r),z)∈C1×···×Cmr×D:∥w(1)∥2p1+···+∥w(r)∥2pr<KD(z,z)-q},其中KD(z,z)是D上的Bergman核函数,r 1且为正整数,参数p1,...,pr>1和q>0为实数.我们给出它的全纯自同构群,并且证明当r=1时此自同构群为最大全纯自同构群;当r>1时,若Ω的全纯自同构变换F将(0,z)∈{0}×D映到(0,z*)∈{0}×D,则F在我们给出的全纯自同构群中.

We consider the domain Ω of the Bergman-Hartogs type which bases on any bounded symmetric domain D,Ω = {（w（1）,..., w（r）, z） ∈ Cm1× · · · × Cmr× D ： ∥w（1）∥2p1＋ · · · ＋ ∥w（r）∥2pr〈 KD（z, z）-q},where KD（z, z） denotes the Bergman kernel on D, r is a positive integer, p1,..., pr 〉 1 and q 〉 0 are real parameters. We give the holomorphic automorphism group of Ω, and prove that the given holomorphic automorphism group is the full holomorphic automorphism group of Ω for r = 1. In addition, when r 〉 1, if the holomorphic automorphism mapping F on Ω maps（0, z） ∈ {0} × D to（0, z＊） ∈ {0} × D, then F belongs to the given holomorphic automorphism group.