Generalized CRF Geometry on Matsuki Orbits in Complex Grassmannians

Abstract

 From the ambient complex-type generalized complex (GC) structure on Z = Grk(Cn), we prove an Inheritance Theorem establishing that each Matsuki orbit M = (G0 · z) ∩ (K · z) inherits a canonical generalized CRF structure (in the sense of Vaisman) ΦM via eigenbun dle restriction; the construction is intrinsic and invariant under closed B fields. We establish a restricted Darboux normal form splitting (M,ΦM) locally into a complex block and a symplectic transverse factor, yielding the invariants type and transverse symplectic rank. Thus we obtain a natural generalized CRFK structure on each Matsuki orbit in the Grassmanian, together with explicit formulas for its type and transverse symplectic rank. Our results fit into the project of describing the complex geometry of lower-dimensional G0-orbits in complex flag manifolds. For G0 = SU(p,q) these are computed directly from the signature data (a,b,r) of h|W, with detailed calculations for SU(2,2)  Gr2(C4); in purespinor terms a generator is ϕM = ηtCR,0∧eiωT withωT thetransverseLeviform. These results connect classical CR geometry of real orbits with generalized complex/Dirac geometry and provide the first explicit homogeneous   examples of generalized CRFK manifolds from real group actions, opening avenues for deformation, cohomology, and quantization.