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.
Domains
Mathematics [math]
Origin : Files produced by the author(s)