✦ LIBER ✦

Globalization of Holomorphic Actions on Principal Bundles

✍ Scribed by Bruce Gilligan; Peter Heinzner

Publisher: John Wiley and Sons
Year: 1998
Tongue: English
Weight: 707 KB
Volume: 189
Category: Article
ISSN: 0025-584X
DOI: 10.1002/mana.19981890109

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✦ Synopsis

Suppose G is a Lie group acting as a group of holomorphic automorphisms on a holomorphic principal bundle P --t X . We show that if there is a holomorphic action of the complexification GC of G on X , this lifts to a holomorphic action of GC on the bundle P --t X . Two applications are presented. We prove that given any connected homogeneous complex manifold G / H with more than one end, the complexification GC of G acts holomorphically and transitively on G / H . We also show that the ends of a homogeneous complex manifold G / H with more than two ends essentially come from a space of the form S/r, where r is a Zariski dense discrete subgroup of a semisimple complex Lie group S with S and r being explicitly constructed in terms of G and H .

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