A moment counterexample and a Levi wall for sphere links

Definitions and mathematical statement

Let $n\geq2$ and $M\subset{\mathbb{C}}^n$ be a compact smooth embedded CR manifold of real dimension $2m-1$ and CR dimension $m-1$, where $1\leq m\leq n$. This is the source's maximally complex condition: its CR codimension is one, which is not generally its real codimension in ${\mathbb{C}}^n$. The source asks whether being homotopic, within the class of maximally complex CR manifolds, to the standard sphere $S^{2m-1}$ in a linear complex $m$-plane forces $M$ to bound a smooth complex analytic submanifold. The precise deformation notion and the boundedness and boundary regularity of the filling are not specified. For the supplied counterexample interpret a filling as a bounded complex $m$-manifold with smooth compact closure and oriented boundary $M$; the aid also discusses finite-mass holomorphic chains.

The literal dimensions allow $m=1$, hence CR dimension zero. Fix $\epsilon>0$ and, in ${\mathbb{C}}^2\subset{\mathbb{C}}^n$, define the embedded oriented circles \[ \gamma_t(\theta)=(e^{i\theta},t\epsilon e^{-i\theta}),\qquad 0\leq t\leq1,\quad\theta\in{\mathbb{R}}/(2\pi{\mathbb{Z}}). \] Their complex tangent spaces have dimension zero, so this is a smooth isotopy through maximally complex CR manifolds, starting at the linear unit circle. The supplied moment obstruction is \[ \int_{\gamma_t}z_2\,{\,\mathrm{d}} z_1=2\pi i\,t\epsilon. \] For an oriented boundary of a bounded smooth complex curve this integral must vanish: Stokes' formula applies and ${\,\mathrm{d}}(z_2\,{\,\mathrm{d}} z_1)={\,\mathrm{d}} z_2\wedge{\,\mathrm{d}} z_1$ restricts to zero on a complex curve. The same moment condition holds for the bounded finite-mass holomorphic one-chain fillings under discussion. Thus $\gamma_1$ is the supplied counterexample to the literal bounded-filling assertion, even with an embedded CR isotopy.

For positive CR dimension, the aid separately reports the Brieskorn link \[ \Sigma(2,3,5,7)=\{z\in{\mathbb{C}}^4:z_1^2+z_2^3+z_3^5+z_4^7=0\}\cap S^7_\delta \] for sufficiently small $\delta>0$. It has real dimension five and CR dimension two. According to the aid it is homeomorphic to $S^5$, while its unique bounded Harvey--Lawson holomorphic filling is singular. This concerns mere sphere topology or ordinary homotopy and does not establish the existence of an isotopy through embedded maximally complex CR manifolds to a linear sphere.

The aid further reports that any such embedded CR isotopy from this link to the linear $S^5$ must contain a noncontact, hence Levi-degenerate, member. For a cooriented CR hypersurface-type structure with $H=TM\cap iTM=\ker\theta$, contact means $\theta\wedge({\,\mathrm{d}}\theta)^{m-1}\ne0$ everywhere. Along a smooth compact isotopy that stays contact, the endpoint contact structures must be contactomorphic. Thus the reported Levi-wall assertion requires a verified obstruction to contactomorphism of the endpoint structures; the record supplies neither that invariant nor a full reference. It is retained as a qualified aid assertion. The stronger positive-CR-dimension problem with a genuine embedded CR deformation remains reported as unresolved.

Short English statement

The literal question includes circles, and the supplied explicit circle isotopy ends at a curve with a nonzero holomorphic moment, preventing a bounded complex filling. In higher CR dimension, a sphere's topological type alone is also reported insufficient. That topological obstruction does not settle the stronger question requiring an actual deformation through embedded maximally complex CR manifolds; the reported Levi-degeneracy obstruction needs its contact-topology source checked.

Sources

• Local registry: permanent catalogue entry, exact source code catalogue entry; aim-several-complex-variables-notes.json, notes[30].
• AIM workshop list Emerging applications of complexity for CR mappings, Problem 2 attributed to D. Zaitsev, https://aimath.org/WWN/crmappings/crmappings.pdf. The original remarks and literature field are empty. No full authoritative publication citation is supplied.
• Supplied research attempt 1, registry updated 14 August 2026: counterexample, refuted-at-run label, low novelty confidence, machine-generated and not peer reviewed. It supplies the circle moment example and reports the Brieskorn-link filling and contact-wall assertions, with Harvey--Lawson named but no complete theorem citation.
• The verification field consists only of "The AIM PDF states:"; it provides no fuller check or exact intended deformation convention.