Rigid Biholomorphic Equivalences of Rigid C2,1 Hypersurfaces M5⊂C3

We study the local equivalence problem for real-analytic (Cω) hypersurfaces M5⊂C3 that, in some holomorphic coordinates (z1,z2,w)∈C3 with w=u+ −1v, are rigid sense that their graphing functions u=F(z1,z2,z‾1,z‾2) independent of v. Specifically, we group Holrigid(M) biholomorphic transformations form (z1,z2,w)⟼(f1(z1,z2),f2(z1,z2),aw+g(z1,z2)), where a∈R∖{0} and D(f1,f2)/D(z1,z2)≠0, which preserve rigidity hypersurfaces. After performing a Cartan-type reduction to an appropriate {e}-structure, find exactly two primary invariants I0 V0, express explicitly terms 5-jet function F M. The identical vanishing 0≡I0(J5F)≡V0(J5F) then provides necessary sufficient condition M be locally rigidly known model hypersurface MLC:u=z1z‾1+1/2z12z‾2+1/2z‾12z2 1−z2z‾2. establish always dimHolrigid(M)≤7=dimHolrigid(MLC). If one these I0≢0 or V0≢0 does not vanish identically, on either Zariski-open sets {p∈M:I0(p)≠0} {p∈M:V0(p)≠0}, show this between reduces certain five-dimensional {e}-structure M, is, get invariant absolute parallelism M5. Hence dimHolrigid(M) drops from 7 5, illustrating gap phenomenon.