Scalar-parabolic rank obstructions for equivariant hyperquadric embeddings

Definitions and mathematical statement

The source concerns a germ $(M,p)\subset{\mathbb{C}}^2$ admitting a nonidentity local self-map fixing $p$ and "tangent to the identity of order $2$", and asks whether it embeds into a hyperquadric as in the preceding problem. The preceding problem is not supplied, so the exact target and equivariance requirement cannot be recovered. In coordinates $p=0$, two materially different jet conditions are $j^1_0h=j^1_0\operatorname{id}$ with a possibly nonzero quadratic term, and the stronger $j^2_0h=j^2_0\operatorname{id}$, namely $h(Z)=Z+O(|Z|^3)$. The source also does not specify here whether $M$ is a real-analytic hypersurface or which category of self-maps and embeddings is intended.

A nondegenerate hyperquadric model is \[ Q_{r,s}=\{(Z,W)\in{\mathbb{C}}^{r+s}\times{\mathbb{C}}:\operatorname{Im}W=\sum_{j=1}^r|Z_j|^2-\sum_{j=r+1}^{r+s}|Z_j|^2\}. \] A holomorphic germ $F:({\mathbb{C}}^2,0)\to({\mathbb{C}}^{r+s+1},0)$ is an embedding near zero if it has complex differential of rank two there and is locally injective; it maps $M$ into the quadric if $F(M)\subset Q_{r,s}$. CR transversality additionally requires the normal derivative of a target defining function pulled back by $F$ to be nonzero. For a specified source self-map $h$ and target automorphism $P$, equivariance means $F\circ h=P\circ F$, a restriction stronger than embedding alone.

The aid's finite-type discussion, in its intended real-analytic hypersurface setting, invokes Kolar's classification to reduce a nontrivial identity-first-jet symmetry to the circular model $\operatorname{Im}w=|z|^{2m}$. In that notation the model symmetry and scalar parabolic may be written \[ H_{m,t}(z,w)=\left(\frac{z}{(1-tw)^{1/m}},\frac{w}{1-tw}\right),\qquad P_t(Z,W)=\left(\frac{Z}{1-tW},\frac{W}{1-tW}\right), \] with integer $m\geq1$, fixed real $t\ne0$, and the holomorphic root branch equal to one at zero. The target $P_t$ preserves every displayed $Q_{r,s}$. The reported obstruction is \[ m>1,\quad F(0)=0,\quad F\circ H_{m,t}=P_t\circ F \quad\Longrightarrow\quad\operatorname{rank}_{{\mathbb{C}}}{\,\mathrm{d}} F_0\leq1. \] It uses equivariance for just one fixed nonzero $t$, and the aid reports the corresponding statement after pointed target conjugation. Thus it excludes a holomorphic immersion for this specified scalar-parabolic action, not all hyperquadric embeddings. The aid reports both a full-rank CR-transversal embedding of the circular model into an indefinite quadric and a branched equivariant map to the Heisenberg quadric. The latter has the form $(z,w)\mapsto(z^m,w)$ and is not an immersion at zero when $m>1$.

A second aid assertion concerns the Zaitsev--Kowalski higher-contact family and symmetries denoted $T_{k,t}$, $k\geq2$. It reports a full-rank nontransversal neutral-pair embedding and a branched equivariant map, but claims every germ intertwining $T_{k,t}$ with the scalar parabolic has ${\,\mathrm{d}} F_0=0$, again for one fixed nonzero parameter and pointed target conjugates. No defining hypersurface or formula for $T_{k,t}$ is supplied, so this family-specific claim cannot be made a standalone quantified theorem from this record.

Under the stronger identity-second-jet interpretation, the aid invokes finite-type two-jet determination to force infinite type for a nontrivial automorphism in its intended category, and reports that any nondegenerate hyperquadric target of an embedding must then be indefinite. These assertions need their real-analytic, nondegeneracy, and mapping hypotheses checked. The supplied progress does not settle the unspecified general equivariant embedding question.

Short English statement

The question asks whether a CR germ with a nontrivial symmetry tangent to the identity admits a hyperquadric embedding of the kind specified in an omitted preceding problem. The supplied progress distinguishes ordinary embeddings, branched equivariant maps, and embeddings intertwining a particular scalar parabolic. Its rank obstruction excludes the last of these for the circular models, while leaving the broader question and the meaning of the order-two jet condition qualified.

Sources

• Local registry: permanent catalogue entry, exact source code catalogue entry; aim-several-complex-variables-notes.json, notes[37].
• AIM workshop list Emerging applications of complexity for CR mappings, section "CR embeddings", Problem 2 attributed to D. Zaitsev, https://aimath.org/WWN/crmappings/crmappings.pdf. The preceding problem is referenced but absent. Original remarks and literature field are empty.
• Supplied research attempt 1, registry updated 14 August 2026: partial result, open-at-run label, low novelty confidence, machine-generated and not peer reviewed. It names Kolar's classification, the Zaitsev--Kowalski higher-contact family, and two-jet determination without full authoritative citations.
• The verification field says the section has two consecutive questions but ends before reproducing either; the aid's literature-checked open claim has no dated search details or references here.