Toward a classification of compact complex homogeneous spaces.
The author considers: \(M\) a compact complex homogeneous manifold, \(G\) a connected complex Lie group acting almost effectively, holomorphically and transitively on \(M,\Gamma\) a discrete cocompact subgroup of \(G\) such that \(M=G/ \Gamma\), \(R\) the radical of \(G\). The following beautiful and important theorems which give a classification of compact complex homogeneous spaces are proved. Theorem A. If \(S\) is a Levi subgroup of \(G\), then any simple factor of \(S\) which acts nontrivially on \(R\) is of type \(A_\ell\). Theorem B. Let \(G,\Gamma, S\), and \(R\) be as in Theorem A. \(S_1\) (respectively \(S_2)\) be the normal subgroup of \(S\) such that each simple factor of \(S_1\) (respectively of \(S_2)\) acts trivially (respectively nontrivially) on \(R\). Then, up to a finite covering, \(G/ \Gamma\) is a torus fiber bundle over a product \(S_1/ \Gamma_1\times S_2R/ \Gamma_2C^0\), where \(C_0\) is the identity component of the center \(C\) of \(G\) and \(\Gamma_1\) (respectively \(\Gamma_2)\) is a discrete subgroup of \(S_1\) (respectively \(S_2 R/C^0)\). In particular, \(S_2\) has only factors of type \(A_\ell\). Theorem C. Let \(M\) be a compact complex homogeneous space. Then \(M\), up to a finite covering, is a torus bundle over \(S_1/H_1\times S_2 R/H_2\) with \(S_1\), \(S_2\) semisimple and \(R\) being the radical of \(S_2 R\) such that each factor of \(S_2\) acts nontrivially on \(R\). If \(J_1= N_{S_1} (H_1^0)\), \(J_2=N_{S_2R} (H_2^0)\), then \(J_2/H_1^0\) is semisimple and \(H_2^0\cap S_2\) is nilpotent, \(J_2\) has only simple factors of type \(A_\ell\), which are not in \(H_2\). Moreover, each simple factor of \(S_2\) is a classical Lie group and each simple factor of \(J_2\) acts nontrivially on \(R/R\cap H\). A classification of the data, for each type of classical Lie algebra, is given in the lists from Theorem D. Theorem E. The reduced primitive compact complex homogeneous spaces, up to actions of some torus, are isogenous on the Lie algebra level to one of following 12 cases: \[ \begin{alignedat}{3} & (1)\;k_1A_\ell^{a,1}\times k_2 a^a_{1, \ell}, \;& (7)\quad & k_1A^{b,1} \times k_2a_1^b,\\ & (2)\;k_1 A_\ell^{a, 2} \times k_2a_{2,\ell}^a, \;& (8)\quad & k_1 A^{b,2}\times k_2 D^4 \times k_3a_2^b,\\ &(3)\;k_1A_\ell^{a,3}\times k_2a^a_{3, \ell}, \;& (9)\quad & k_1A^{b,3} \times k_2D^1 \times k_3D^3\times k_4 d_1,\\ &(4)\;k_1 A_{\ell}^{a,4} \times k_2a^a_{4,\ell}, \;& (10)\quad & k_1 B^2\times k_2b,\\ &(5)\;k_1 A_\ell^{a,5} \times k_2B^3\times k_3a_5^a \times (\prod_i A_{\ell_i}^{a,6}), \;& \;(11)\quad & k_1C\times k_2c,\\ &(6)\;\left( \prod^*_i A^{a,7}_{n_i}\right) \times\left( \prod^*_jB^1_{n_j} \right) \times \left( \prod^*_kA^{b,4}_{n_\ell} \right)\times\left( \prod^*_\ell a^a_{7, n_\ell} \right), \;& (12)\quad & k_1D^2 \times k_2d_2,\end{alignedat} \] where \(k_i\), \(i=1,2,3,4\), are nonnegative integers and \(kB\) means \(k\) copies of \(B\). Theorem F. Any reduced space of 1-step is isogenous to \(T\times L\) with \(T\) a torus and \(L\) is, up to an action of a torus, isogenous (i.e., \(\leq)\) to (up to a finite covering, this induces an embedding) a product of a parallelizable manifold and several reduced primitive compact complex homogeneous spaces such that each projection of the image is onto. Theorem G. Any reduced space \(M\) of 1-step is in an isogeny class \(\geq\) a product \(M_0\) of a parallelizable manifold and some complete reduced primitive homogeneous spaces. In particular, \(M\) is a homogeneous \(M_0\) bundle over a torus. Theorem H. Any reduced space \(M\) is in an isogeny class \(\geq\) a product \(M_0\) of a parallelizable manifold and some primary spaces. In particular, \(M\) is a homogeneous \(M_0\) bundle over a torus.
scientific article; zbMATH DE number 2055657
scientific article