Equivariant hyperquadric embeddings and a standard-parabolic obstruction

Definitions and mathematical statement

In a local fixed-point interpretation of M. Kolar's question, let $(M,0)\subset({\mathbb{C}}^2,0)$ be a real-analytic hypersurface germ, and let $H:({\mathbb{C}}^2,0)\to({\mathbb{C}}^2,0)$ be a biholomorphic germ with $H(M)=M$. Say that $H$ is holomorphically nonlinearizable if it is not conjugate, by a local holomorphic coordinate change fixing zero, to its linear part. Does there exist a finite $n$, a nondegenerate hyperquadric germ \[ Q=\{(Z,W)\in{\mathbb{C}}^{n-1}\times{\mathbb{C}}:\operatorname{Im}W=\sum_{j=1}^{n-1}\epsilon_j|Z_j|^2\},\qquad \epsilon_j\in\{1,-1\}, \] a nontrivial holomorphic embedding germ $G:({\mathbb{C}}^2,0)\to({\mathbb{C}}^n,0)$ with $G(M)\subset Q$, and an automorphism germ $A$ of $Q$, such that \[ G\circ H=A\circ G? \] Can one bound $n$, with the permitted dependence of that bound specified? An embedding here has complex differential of rank two. The source does not define "nontrivially," specify transversality or signature, distinguish local from global automorphisms, or specify analytic versus formal linearization. The displayed version is the local holomorphic interpretation used by the supplied aid, not a verified resolution of these choices.

The aid reports a restricted obstruction on the circular finite-type hypersurfaces \[ M_m=\{(z,w):\operatorname{Im}w=|z|^{2m}\},\quad m\ge2,\qquad H_{m,t}(z,w)=\left(\frac{z}{(1-tw)^{1/m}},\frac{w}{1-tw}\right),\quad t\in{\mathbb{R}}\setminus\{0\}, \] where the holomorphic root is chosen to equal one at zero. The branched map $(z,w)\mapsto(z^m,w)$ intertwines $H_{m,t}$ with the standard parabolic action on the Heisenberg hyperquadric. More generally, for every finite target dimension, if \[ G=(f_1,\ldots,f_{N-1},w),\qquad G\circ H_{m,t}=P_t\circ G,\qquad P_t(Z,W)=\left(\frac{Z}{1-tW},\frac{W}{1-tW}\right), \] the aid reports ${\,\mathrm{d}} G_0(\partial/\partial z)=0$. Thus this particular transverse coordinate and standard parabolic action cannot give an embedding, even after adding more such coordinates. This is not an obstruction to all target actions or all choices of transverse coordinate.

The aid separately reports ordinary, non-equivariant embeddings of $M_m$ into an indefinite hyperquadric in ${\mathbb{C}}^4$. For rigid polynomial models $\operatorname{Im}w=P(z,\bar z)$, a finite Hermitian coefficient rank $R$ yields an ordinary embedding bound $n\le R+3$ in its stated construction. These ordinary embeddings do not establish the required equivariance. The general question remains open according to the supplied report.

Short English statement

Ask whether a nonlinearizable hypersurface automorphism can be realized by an automorphism of a finite-dimensional hyperquadric containing an embedded copy of the hypersurface. The reported model obstruction applies only to a prescribed transverse coordinate and standard parabolic target action; ordinary embeddings alone do not answer the equivariant question.

Sources

• M. Kolar, Problem 1 in the CR embeddings section of the AIM workshop Emerging applications of complexity for CR mappings, https://aimath.org/WWN/crmappings/crmappings.pdf. No separate bibliographic references are supplied.
• Local UnsolvedMath registry, permanent ID 20002879, exact code catalogue entry; canonical location aim-several-complex-variables-notes.json, notes[36], dated 14 August 2026. The statement remains labeled reconstructed and unverified; the verification field reports recovery from the original PDF, without an independent verification in this draft.
• Research attempt 1: partial result, problem open at the run, rigor label proved, novelty confidence low; registry status partially solved. The obstruction and embedding constructions are supplied machine-generated research aids, not peer reviewed; independent verification is required before citation.