The Dolbeault-cohomology ring of a compact, even-dimensional Lie group (Q749933)

A left invariant, integrable, complex structure (LICS) on a compact, connected, even-dimensional (real) Lie group G is an R-linear mapping J of the Lie algebra \({\mathfrak G}\) of left invariant vector fields on G into itself such that (1) \(J^ 2=-Id_{{\mathfrak G}};\) (2) \([JX,JY]=[X,Y]+J([JX,Y]+[X,JY])\) for all X,Y\(\in {\mathfrak G}\). The automorphism group Aut(G) acts on the closed subspace of \(End_ R({\mathfrak G})\) of these mappings by conjugation and the quotient of this subspace by Aut(G) is equipped with the quotient topology. The paper begins with the investigation of this quotient space. Afterwards, using Bott's Lie algebra formula [\textit{R. Bott}, Ann. Math., II. Ser. 66, 203-248 (1957; Zbl 0094.357)], the author studies the Dolbeault cohomology ring \(H_{{\bar \partial}}(G)\) for G semisimple endowed with an LICS. He points out the dependence of \(H_{{\bar \partial}}(G)\) on the LICS; using particular LICS's on SO(9), he shows how the conjecture that, in the Hodge-de Rham spectral sequence, \(E_ 2=E_{\infty}\) for every compact complex manifold is false.