Cycle Spaces of Flag Domains: A Complex Geometric Viewpoint

This research monograph is a systematic exposition of the background, methods, and recent results in the theory of cycle spaces of ?ag domains. Some of the methods are now standard, but many are new. The exposition is carried out from the viewpoint of complex algebraic and differential geometry. Except for certain foundational material,whichisreadilyavailablefromstandardtexts,itisessentiallyself-contained; at points where this is not the case we give extensive references. After developing the background material on complex ?ag manifolds and rep- sentationtheory, wegiveanexposition(withanumberofnewresults)ofthecomplex geometric methods that lead to our characterizations of (group theoretically de?ned) cyclespacesandtoanumberofconsequences. Thenwegiveabriefindicationofjust how those results are related to the representation theory of semisimple Lie groups through, for example, the theory of double ?bration transforms, and we indicate the connection to the variation of Hodge structure. Finally, we work out detailed local descriptions of the relevant full Barlet cycle spaces. Cycle space theory is a basic chapter in complex analysis. Since the 1960s its importance has been underlined by its role in the geometry of ?ag domains, and by applications in the representation theory of semisimple Lie groups. This developed veryslowlyuntilafewofyearsagowhenmethodsofcomplexgeometry,inparticular those involving Schubert slices, Schubert domains, Iwasawa domains and suppo- ing hypersurfaces, were introduced. In the late 1990s, and continuing through early 2002, we developed those methods and used them to give a precise description of cycle spaces for ?ag domains. This effectively enabled the use of double ?bration transforms in all ?ag domain situations.