A Sheaf-Theoretic CR Reduction for Levi-Flat Manifolds Beyond the Kähler Setting

This paper develops a sheaf-theoretic reduction formalism for real-analytic homogeneous CR manifolds realized as orbits in complex homogeneous manifolds. Let M=G0/H0 be such a CR manifold, let M↪X be a complex realization, and let Ï€:X⟶Y be the holomorphic reduction of the ambient complex manifold. For Z=Ï€M and Ï =Ï€M, it is proved that, under a global CR-extension hypothesis, the fibers of Ï are exactly the equivalence classes determined by global CR functions on M. Thus, the set-theoretic CR reduction of M is induced by the ambient holomorphic reduction. The local problem is then separated from the global one: Under a local descent hypothesis, the direct image Ï âˆ—OMCR is canonically identified with the restricted ambient quotient sheaf OZres≔i−1OY/IZ, where IZ is the sheaf of ambient holomorphic germs vanishing on Z. A practical descent criterion is given, showing that local CR extension, together with sheaf-level holomorphic descent along Ï€, implies the required local CR descent. The resulting morphism carries an ordinary Leray spectral sequence E2a,b=HaZ,RbÏ âˆ—OMCR⟹Ha+bM,OMCR, with no use of Kähler identities or Hodge-theoretic assumptions. As a genuinely Levi-flat application, compact Levi-flat CR manifolds with a dense bounded-Liouville Levi leaf are shown to have trivial global CR reduction; in particular, dense Cousin-type leaves force all global CR functions to be constant. A fully explicit Kronecker Levi-flat three-torus illustrates the dense-leaf mechanism. A second product example, built from a Stein homogeneous curve, an Iwasawa factor, and a Kronecker Levi-flat torus, gives a nonpoint CR reduction for which both global extension and local descent are verified directly. Finally, a separated holomorphic suspension model over compact complex curves is worked out: The higher direct images are sheaves of holomorphic sections of flat holomorphic vector bundles associated with monodromy on HqF,OF, giving corrected curve-base dimension formulas. The paper is a conditional reduction framework whose applicability depends on verifying extension and descent hypotheses in concrete homogeneous non-Kähler models.