2019/04/04 by Wei Guo Foo, Foo, Wei Guo, Joël Merker +3
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.1904.02562
We study the local equivalence problem for real-analytic (Cω) hypersurfaces M5 ⊂ ℂ3 which, in coordinates (z1, z2, w) ∈ ℂ3 with w = u+i v, are rigid: u = F(z1,z2,z1,z2), with F independent of v. Specifically, we study the group \sf Hol\sf rigid(M) of rigid local biholomorphic transformations of the form: (z1,z2,w) \longmapsto ( f1(z1,z2), f2(z1,z2), a w + g(z1,z2) ), where a ∈ ℝ \backslash \0\ and (D(f1,f2))/(D(z1,z2)) ≠ 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 ( J5F ) ≡ V0 ( J5F ) then provides a necessary and sufficient condition for M to be locally rigidly-biholomorphic to the known model hypersurface: M\sf LC \colon u = \fracz1 z1 +(1)/(2) z12z2 +(1)/(2) z12z2 1-z2z2. We establish that dim \sf Hol\sf rigid (M) ≤ 7 = dim \sf Hol\sf rigid ( M\sf LC ) always. If one of these two primary invariants I0 \not≡ 0 or V0 \not≡ 0 does not vanish identically, we show that this rigid equivalence problem between rigid hypersurfaces reduces to an equivalence problem for a certain 5-dimensional \e\-structure on M.