Formal and biholomorphic CR equivalence diverge

Definitions and mathematical statement

Let $(M,p)$ and $(M',p')$ be germs of smooth embedded CR submanifolds of ${\mathbb{C}}^n$, with locally constant CR dimension. After translating $p,p'$ to the origin, let $I_M,I_{M'}\subset{\mathbb{C}}[[Z,\bar Z]]$ be the formal Taylor ideals of their local smooth defining equations. Formal equivalence means there exists $F\in{\mathbb{C}}[[Z]]^n$ with $F(0)=0$ and $\det F_Z(0)\ne0$ such that substitution by $(F,\overline F)$ identifies these ideals. Biholomorphic equivalence means that an actual locally invertible holomorphic germ $h:({\mathbb{C}}^n,p)\to({\mathbb{C}}^n,p')$ maps the germ of $M$ onto that of $M'$. The source asks whether formal equivalence always implies the existence of such an $h$. This is an existence question, not the stronger assertion that every specified formal equivalence converges.

The source-verification field reports the workshop convention that manifolds are smooth unless otherwise stated. Under that convention the supplied explicit example is as follows. For every ambient dimension $N\geq2$, use $(z,w)\in{\mathbb{C}}^{N-1}\times{\mathbb{C}}$, $w=u+iv$, and define germs at $0$ by \[ M_0=\{v=\|z\|^2\},\qquad M_{\mathrm{flat}}=\{v=\|z\|^2+\chi(u)\},\qquad \chi(u)=\begin{cases}0,&u\leq0,\\e^{-1/u^2},&u>0.\end{cases} \] These are smooth CR hypersurface germs of CR dimension $N-1$, strongly pseudoconvex after restricting to a sufficiently small neighborhood. Since every derivative of $\chi$ at zero vanishes, their formal Taylor defining ideals coincide and the identity formal map is an equivalence. The aid asserts that no biholomorphic equivalence exists: $M_0$ is real analytic, while $M_{\mathrm{flat}}$ is not real analytic at zero, and local biholomorphisms preserve real analyticity. Nonanalyticity here is that of the hypersurface germ itself; its graph over $(z,u)$ would be real analytic if the hypersurface were real analytic.

The aid separately reports that the intended real-analytic version is also false. It attributes to Kossovskiy and Shafikov the assertion that, for every pair of integers $d,c\geq1$, there are real-analytic generic holomorphically nondegenerate CR submanifold germs in ${\mathbb{C}}^{d+c}$, of CR dimension $d$ and real codimension $c$, that are formally equivalent but not biholomorphically equivalent. Generic means $T_xM+iT_xM=T_x{\mathbb{C}}^{d+c}$; holomorphic nondegeneracy means the absence of a nonzero holomorphic vector-field germ tangent to the real-analytic germ. The record supplies no full theorem citation or detailed hypotheses for this reported assertion. The smooth flat example does not itself establish the real-analytic assertion.

The original workshop remark says that non-CR submanifolds need not have the property, that many CR cases have affirmative answers, and that the general CR question was open in that source. This historical remark is retained separately from the later supplied counterexample claims; no assertion is made that positive finite-type or other restricted equivalence results fail.

Short English statement

Formal equivalence asks whether a power-series coordinate change matching the defining ideals guarantees some convergent holomorphic equivalence. Under the reported smooth convention, a flat nonanalytic perturbation has the same formal data as a real-analytic strongly pseudoconvex hypersurface but cannot be biholomorphically equivalent to it. The supplied aid also attributes real-analytic counterexamples to Kossovskiy and Shafikov, without giving a complete citation.

Sources

• Local registry: permanent catalogue entry, exact source code catalogue entry; aim-several-complex-variables-notes.json, notes[34].
• AIM workshop list Emerging applications of complexity for CR mappings, Problem 6 attributed to D. Zaitsev, https://aimath.org/WWN/crmappings/crmappings.pdf. Verification reports the smooth-manifold preamble and printed page 3, section "CR manifolds and mappings", but ends before reproducing the wording.
• Original remark: non-CR cases can fail, many CR cases are affirmative, and the general question was open at the source. Original literature field is empty; no references for these historical assertions are given.
• Supplied research attempt 1, registry updated 14 August 2026: counterexample, refuted-at-run label, low novelty confidence, machine-generated and not peer reviewed. It gives the smooth flat example and attributes the real-analytic counterexample theorem to Kossovskiy and Shafikov without a title, date, theorem number, or complete publication citation.