Fiberwise Rossi extension domains for rank-one Matsuki orbits in complex Grassmannians

Résumé fourni par la source

Let G0 = SU(p, q) act on the complex Grassmannian Z = Grk(C p+q ), and let M be a Matsuki orbit, i.e., the intersection of a G0-orbit with its Matsuki-dual K-orbit. Such intersections are compact homogeneous real-analytic Cauchy–Riemann (CR) manifolds. This paper studies the holomorphic extension problem for rank-one Matsuki orbits with residual signature s ≥ 2. The fibers of the Matsuki fibration are isotropic CR spheres in projective spaces of signature (1, s) or (s, 1), whose one-sided Rossi envelope is the corresponding projective ball. We construct the resulting fiberwise Rossi extension domain Dℓ,m,1 in a naturally associated projective residual bundle Xℓ,m → Grℓ(E+) × Grm(E−), where the Matsuki orbit embeds as the boundary. A key clarification is that the evaluation image of Dℓ,m,1 in the ambient Grassmannian Z is generally not open, so extended holomorphic functions need not descend. Our main theorem establishes a precise fiberwise Rossi extension: every real-analytic CR function extends holomorphically to Dℓ,m,1, and this domain is maximal among those obtained by the fiberwise Rossi-envelope construction. The proof uses horizontal CR vector fields, joint holomorphicity, and fiberwise patching. We also relate the construction to rank-one cycle domains via the evaluation map and formulate the higher-rank problem.

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