Finite-jet status and a complex-locus similarity sieve

Definitions and mathematical statement

D. Zaitsev's source question asks about $\operatorname{Aut}(M,p)$ at nonminimal points of a generically minimal, holomorphically nondegenerate real submanifold $M\subset{\mathbb{C}}^n$. A CR point is minimal if its CR orbit is locally open in $M$; generic minimality means this holds on a dense open subset. Holomorphic nondegeneracy excludes nonzero holomorphic $(1,0)$ vector fields complex-tangent to $M$, not all holomorphic vector fields whose real parts preserve it. The appended "CR embeddings" is the next section heading. The original does not specify real analyticity or whether automorphisms mean ambient biholomorphic germs or abstract smooth CR germs.

The aid uses real-analytic geometry and ambient biholomorphic stability germs fixing $p$. It reports Juhlin's finite-jet determination at every point for connected holomorphically nondegenerate real-analytic hypersurfaces: for each germ there is a finite order $k(M,p)$ whose jet determines the automorphism. No uniform order over all points or hypersurfaces is asserted. It also reports Baouendi--Ebenfelt--Rothschild finite-dimensionality of the infinitesimal CR automorphism algebra at every point under holomorphic nondegeneracy and generic minimality in the relevant arbitrary-codimension real-analytic setting. Full genericity/codimension hypotheses are not supplied. Finite-dimensionality of this algebra is not itself all-point finite-jet determination, which the aid leaves unresolved at nonminimal points in codimension at least two.

For an explicit hypersurface germ, write $(z,w)\in{\mathbb{C}}^d\times{\mathbb{C}}$, $d\ge1$, and for an integer $m\ge2$ put \[ M_m=\{\operatorname{Im}w=(\operatorname{Re}w)^m\|z\|^2\}. \] Its complex nonminimal locus is $\{w=0\}$. The aid states that every $H=(F,G)\in\operatorname{Aut}(M_m,0)$ satisfies \[ F(z,0)=|a|^{(1-m)/2}Uz, \] \[ G(z,w)=aw+c_2w^2+\cdots+c_mw^m+w^{m+1}H_1(z,w), \] where $U\in\mathrm U(d)$, $a\in{\mathbb{R}}\setminus\{0\}$, $a>0$ if $m$ is even, all $c_j\in{\mathbb{R}}$ are constants, and $H_1$ is holomorphic. This is a necessary restriction on the complex-locus action and transverse coefficients, not a converse realizing every such collection or a full automorphism classification.

The model contains the dilation-unitary subgroup \[ (z,w)\longmapsto(a^{(1-m)/2}Uz,aw),\qquad a>0,\ U\in\mathrm U(d), \] which is noncompact. Thus the qualitative hypotheses do not force finite, discrete, or compact isotropy. The example and cited real-analytic status do not answer the unspecified smooth-category question or the remaining higher-codimension all-point jet problem.

Short English statement

The aid distinguishes finite-dimensional infinitesimal symmetry from finite-jet determination and records known real-analytic hypersurface results. Its nonminimal model restricts the complex-locus action to a constant similarity but still has noncompact isotropy, so the source hypotheses do not imply a small compact automorphism group.

Sources

• Local UnsolvedMath registry, permanent ID 20002878, source code catalogue entry; created and updated 14 August 2026. Canonical aim-several-complex-variables-notes.json, zero-based notes[35], tag problem, item 7 attributed to D. Zaitsev.
• AIM "Emerging applications of complexity for CR mappings": https://aimath.org/WWN/crmappings/crmappings.pdf. Wording is marked exact. The supplied verification reports direct PDF inspection; no inspection is claimed here. Original remarks and literature field are empty.
• Research attempt 1: partial_result, problem status at run partially_solved, rigor label proved, novelty confidence low. It names Juhlin and Baouendi--Ebenfelt--Rothschild without full titles, dates, theorem statements, or authoritative bibliographic links.
• Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Machine-generated research aid, not peer reviewed; independently verify before citation.