Abstract
The paper presents a classification of all homogeneous (integrable) complex structures on compact, connected lie groups of even dimension. Thereafter, using lie algebraic methods it proves theorems about the Dolbeault cohomology rings of these complex manifolds in the semisimple case and exhibits the dramatic variation of ring structure of the Dolbeault rings of groups of rank 2. Using some specific computations forSO(9), it gives a counter-example to a long-standing conjecture about the Hodge-deRham (Frohlicher) spectral sequence.
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References
Atiyah M I, Complex analytic connections in fibre bundles,Trans. AMS 85 (1957) 181–207
Borel A, A spectral sequence for complex analytic bundles,Appendix II in Hirzebruch’s book pp. 202–217
Bott R, Homogeneous vector bundles,Ann. Math. 65 (1957) 203–248
Helgason S,Differential geometry and symmetric spaces (Academic Press) (1962)
Hirzebruch F,Topological methods in algebraic geometry (Springer Verlag) Grundlagen der Mathematischen Wissenschaften Band 131, 3rd edn. (1966)
Hochschild G and Serre J-P, Cohomology of Lie algebras,Ann. Math. 57 (1953) 591–603
Griffiths P A, Some geometric and analytic properties of homogeneous complex manifolds: Part I,Acta. Math. 110 (1963) 115–155
Kostant B, Lie algebra cohomology and generalized Schubert cells,Ann. Math. 77 (1963) 77–144
Tate J, Homology of local rings and Noetherian rings,Illinois J. Math. 1 (1957) 14–27
Wang H C, Closed manifolds with homogeneous complex structures,Am. J. Math. 76 (1954) 1–32
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Pittie, H.V. The Dolbeault-cohomology ring of a compact, even-dimensional lie group. Proc. Indian Acad. Sci. (Math. Sci.) 98, 117–152 (1988). https://doi.org/10.1007/BF02863632
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DOI: https://doi.org/10.1007/BF02863632