第二类超Cartan域上的消没定理

Vanishing Theorem on Super-Cartan Domain of the Second Type

第二类超Cartan域(也称为第二类Cartan-Hartogs域)为:YⅡ(N,p;k)={w∈CN,Z∈RⅡ(p):‖w‖2k<det(I-ZZT)}(k>0),其中RⅡ(p)为华罗庚意义下的第二类Cartan域;ZT表示Z的共轭和转置;det表示行列式;N,p,k都是自然数.证明在第二类超Cartan域上,对于Bergman度量下平方可积调和(r,s)形式空间,有Hr2,s(YⅡ(N,p;k))=0,r+s≠N+p(p+1)2.

The super-Cartan domain of the second type is： YⅡ（N,p;k）={w∈C^N,Z∈RⅡ（P）：‖w‖^2k〈det（I-ZZ^T）}（k〉O）, where RⅡ（p） denotes the Cartan domains of the second type in the sence of Hua, respectively. Z^T denotes the conjugate and transposed matrix of Z, ＂det＂ denotes ＂determinant＂ and N,p,k are positive integers. In this paper, it is proved that the space of square integrable harmonic （r,s）-form is vanished relative to the Bergman metric for r＋s≠N＋p（p＋1）/2 on the super-Cartan domain of the second type in C^N＋p（p＋1）/2 .