Premium
Globalization of Holomorphic Actions on Principal Bundles
Author(s) -
Gilligan Bruce,
Heinzner Peter
Publication year - 1998
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19981890109
Subject(s) - holomorphic function , complexification , mathematics , pure mathematics , complex manifold , lie group , group (periodic table) , homogeneous , automorphism , action (physics) , algebra over a field , combinatorics , physics , quantum mechanics
Suppose G is a Lie group acting as a group of holomorphic automorphisms on a holomorphic principal bundle P → X. We show that if there is a holomorphic action of the complexification G C of G on. X , this lifts to a holomorphic action of G C on the bundle P → X. Two applications are presented. We prove that given any connected homogeneous complex manifold G/H with more than one end, the complexification G C 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 /Γ, where Γ is a Zariski dense discrete subgroup of a semisimple complex Lie group S with S and Γ being explicitly constructed in terms of G and H.