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Rigid Biholomorphic Equivalences of Rigid C_2,1 Hypersurfaces M^5 in C^3

Abstract : We study the local equivalence problem for real-analytic (C^omega) hypersurfaces M^5 in C^3 which, in some holomorphic coordinates (z1, z2 , w) in C^3 with w = u + i v, are rigid in the sense that their graphing functions: u = F (z1 , z2, z1bar, z2bar) are independent of v. Specifically, we study the group Hol rigid (M) of rigid local biholomorphic transformations of the form: (z1, z2, w) --> f1 (z1, z2), f2 (z1, z2), a w + g(z1, z2), where a ∈ R-0 and D(f1,f2) / D(z1,z2) \neq 0, which preserve rigidity of hypersurfaces. After performing a Cartan-type reduction to an appropriate {e}-structure, we find exactly two primary invariants I0 and V0, which we express explicitly in terms of the 5-jet of the graphing function F of M. The identical vanishing 0 = I0 (J^5 F) = V0 (J^5 F) then provides a necessary and sufficient condition for M to be locally rigidly-biholomorphic to the known model hypersurface: M_LC: u = (z1 z1bar + (1/2) z1^2 z2bar + (1/2) z1bar z2) / (1 - z2 z2bar). We establish that dim Hol rigid (M) <= 7 = dim Hol rigid (M_LC) always. If one of these two primary invariants I0 \neq 0 or V0 \neq 0 does not vanish identically, then on either of the two Zariski-open sets {p in M: I0(p) \neq 0} or {p in M: V0(p) \neq 0}, we show that this rigid equivalence problem between rigid hypersurfaces reduces to an equivalence problem for a certain 5-dimensional {e}-structure on M, that is, we get an invariant absolute parallelism on M^5. Hence dim Hol rigid (M ) drops from 7 to 5, illustrating the gap phenomenon.
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https://hal-universite-paris-saclay.archives-ouvertes.fr/hal-03286292
Contributor : Joël Merker <>
Submitted on : Wednesday, July 14, 2021 - 10:13:28 AM
Last modification on : Wednesday, July 21, 2021 - 3:47:17 AM

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Wei-Guo Foo, Joël Merker, The-Anh Ta. Rigid Biholomorphic Equivalences of Rigid C_2,1 Hypersurfaces M^5 in C^3. Michigan Mathematical Journal, University of Michigan, In press. ⟨hal-03286292⟩

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