Dolbeault cohomology of compact complex homogeneous manifolds
If a complex transformation group \(G\) acts on a complex manifold \(M\), there is an induced action of \(G\) on the Dolbeault cohomology of \(M\). This action must be trivial if \(M\) is compact Kähler, but may be non-trivial in general (as was shown by examples of Kodaira and Lescure). The authors prove the following result: Let \(M\) be a compact complex homogeneous manifold which can be realized as a fiber bundle with a flag variety as base and a torus \(T\) as fiber. Then the connected component \(\Aut_1(M)\) of the automorphism group of \(M\) acts trivially on the Dolbeault cohomology of \(M\). For certain classes of these manifolds they compute the Dolbeault cohomology and the Picard group. In remark 2.3 the authors raise the question whether \(\Aut_1(M)\) is algebraic if \(T\) is an abelian variety. The reviewer would like to point out that the answer is positive: There is a short exact sequence \(1 \to T \to \Aut_1(M)\to S\to 1\) where \(S\) is a semisimple Lie group. The universal covering \(\widetilde S\to S\) is finite and the pull-back \(\widetilde G\to\widetilde S\) of the bundle \(G=\Aut_1(M)\to S\) to \(\widetilde S\) yields a short exact sequence \(1\to T\to \widetilde G\to \widetilde S\to 1\) which is split, because \(\widetilde S\) is semi-simple and simply-connected. Therefore \(\widetilde G\simeq\widetilde S\times T\) is an algebraic group as well as \(G=\Aut_1(M)\) which is a quotient of \(\widetilde G\) by a finite subgroup.
scientific article; zbMATH DE number 1340031
scientific article