Convergent normal forms for Levi-degenerate real hypersurfaces beyond the everywhere 2-nondegenerate case in $\mathbb{C}^3$

The Chern–Moser theorem (1974) solves the local biholomorphic equivalence problem for Levi-nondegenerate real hypersurfaces by a complete convergent normal form, from which one reads off a sphericity criterion and a description of the moduli space. Recent work established convergent normal forms in several Levi-degenerate classes: at a generic Levi-degeneracy point in any dimension (Kossovskiy–Zaitsev 2019), for finite-type hypersurfaces in $\mathbb{C}^2$ under a geometric condition (Kossovskiy–Zaitsev 2015), and — by Kolář–Kossovskiy (2022) — for everywhere 2-nondegenerate real-analytic hypersurfaces in $\mathbb{C}^3$, the first class of infinite Catlin multitype ever treated. The open problem studied here is the analogous complete convergent normal-form theory for the remaining degenerate classes: everywhere 2-nondegenerate hypersurfaces in $\mathbb{C}^N$ for $N\geq 4$; everywhere $k$-nondegenerate hypersurfaces for $k\geq 3$; and the remaining $\mathbb{C}^2$ regimes, namely convergence of formal normal forms at infinite-type points and the convergent normal form at an isolated weakly pseudoconvex point. In each settled case the normal form yielded a sphericity criterion in terms of its coefficients and an explicit moduli-space description; these are the expected deliverables for the unsettled everywhere $k$-nondegenerate classes, while for the remaining $\mathbb{C}^2$ regimes the objective is the normal-form construction itself together with resolution of the convergence issue; Kolář–Kossovskiy (2022) explicitly state that the moduli space of everywhere $k$-nondegenerate hypersurfaces in $\mathbb{C}^N$ remains undescribed for every $(k,N)$ other than $(2,3)$.

Background

Poincaré's problème local (1907) asks for a complete classification of germs of real hypersurfaces in complex space up to local biholomorphic equivalence. For a real-analytic hypersurface $M\subset\mathbb{C}^N$ with defining function $r$, the Levi form at a point $p\in M$ is the Hermitian form on the complex tangent space $T_pM\cap J(T_pM)$ ($J$ being the complex structure) obtained from the complex Hessian of $r$ restricted to that space; $M$ is Levi-nondegenerate at $p$ if this form is nondegenerate and Levi-degenerate otherwise. Chern and Moser (1974) solved the equivalence problem in the Levi-nondegenerate case: suitable coordinates $(z,w=u+iv)$ bring $M$ to a normal form $v=\langle z,\bar z\rangle+\Phi(z,\bar z,u)$ in which the lowest-order terms are those of the hyperquadric (the model hypersurface, i.e. the simplest local representative of the class) and the higher-order coefficients are normalized order by order, the normalizing transformation being convergent, i.e. an actual holomorphic change of coordinates rather than a formal power series. The normal form is complete (unique up to a finite-dimensional residual group, as opposed to partial normalizations), and from it one reads off: a sphericity criterion — $M$ is locally biholomorphic to the hyperquadric exactly when certain normal-form coefficients vanish; and a moduli space — the suitably residual part of the remaining coefficients parametrizes equivalence classes of germs. The construction relies on chains, distinguished curves on $M$ used to normalize the tangential variables, and on the homological method of solving the associated operator equation on the space of model perturbations. Beyond the Levi-nondegenerate case the central notions are: finite type (the order of contact of complex lines with $M$ is finite, in the sense of D'Angelo and of Bloom–Graham); finite nondegeneracy — $M$ is $k$-nondegenerate at $p$ (for the Levi form this means $k=1$) if the Levi kernel has constant dimension near $p$ and the successive fundamental forms of higher order recover nondegeneracy in the kernel directions, so that 2-nondegenerate hypersurfaces are exactly those whose Levi form has a one-dimensional kernel that is compensated at second order; and the Catlin multitype, the tuple of weights measuring the degeneracy, which is infinite for $k$-nondegenerate hypersurfaces with $k\geq 2$, so that no finite-weight polynomial model is available. Progress on the degenerate problem has been class-by-class. Kolář (2005) built formal (not convergent) normal forms for finite-type hypersurfaces in $\mathbb{C}^2$, developed by Ezhov–Kolář–Schmalz (2013), but Kolář (2012) also produced finite-type hypersurfaces whose normal-form power series diverge, showing that convergence is a genuine issue depending on the chosen normalization scheme (the divergence was later traced to an ill-posed cross-section, i.e. a badly ordered choice of normalized coordinates). Kossovskiy–Zaitsev (2015) constructed convergent normal forms, together with a canonical connection, for finite-type hypersurfaces in $\mathbb{C}^2$ satisfying a geometric condition, and (2019) a complete convergent normal form at points of generic Levi degeneracy in all $\mathbb{C}^N$, introducing degenerate chains; Ebenfelt (1998) had earlier given complete partial-to-complete normal forms in $\mathbb{C}^3$. The breakthrough of Kolář–Kossovskiy (2022) is a complete convergent normal form for everywhere 2-nondegenerate real-analytic hypersurfaces in $\mathbb{C}^3$ — the first construction for hypersurfaces of infinite Catlin multitype, based not on a polynomial model but on the rational Fels–Kaup realization of the tube over the light cone — with a Chern–Moser-style sphericity criterion ($\Phi_{3002}=\Phi_{5001}=0$ at every point) and an explicit description of the moduli space as $\mathcal{D}/G$ for a 5-dimensional group $G$. Foo–Merker–Ta (2022) treated the constant-Levi-rank-1 class $\mathfrak{C}_{2,1}$ in $\mathbb{C}^3$ by convergent Poincaré–Moser reduction, and Sabzevari (2026) completed the normal forms at 2-nondegenerate points of Levi non-uniform rank zero, the last finitely nondegenerate pointwise class in $\mathbb{C}^3$, by the equivariant moving frame method (a systematic, symbolically computable modern form of Cartan's method in which a cross-section to the pseudogroup orbits produces the normal form). For $\mathbb{C}^2$ at infinite-type points, Ebenfelt–Lamel–Zaitsev (2017) constructed complete formal normal forms in the least degenerate (1-infinite-type, nonresonant) case, with resonances governed by a universal polynomial in the 7-jet; Ebenfelt–Kossovskiy–Lamel (2022) developed the general infinite-type equivalence theory with smooth (not holomorphic) normal forms. Two general convergence mechanisms are known: Lamel–Stolovitch (2020), giving convergence of Chern–Moser–Beloshapka normalizing transformations for Levi-nondegenerate submanifolds under a condition on the formal normal form; and Olver–Sabzevari–Valiquette (2025), a general theorem guaranteeing convergence of moving-frame normal-form power series for suitably nonsingular analytic submanifolds under broad Lie pseudogroup actions, with Chern–Moser as a special case — but with no Levi-degenerate complex-geometric application yet worked out. What remains outside all of this is the subject of the present problem.

Problem Statement

Extend the Chern–Moser convergent normal-form program to the Levi-degenerate classes of real-analytic hypersurfaces for which it is still unavailable. Concretely, for the unsettled classes — (i) everywhere 2-nondegenerate real-analytic hypersurfaces in $\mathbb{C}^N$ for $N\geq 4$; (ii) everywhere $k$-nondegenerate real-analytic hypersurfaces for $k\geq 3$ in any dimension; and (iii) the remaining $\mathbb{C}^2$ degenerate regimes, namely convergence of the formal normal forms at infinite-type points and the construction of a convergent normal form at an isolated weakly pseudoconvex point (a point at which the Levi form is positive semidefinite and degenerate) — construct a complete convergent normal form: a convergent biholomorphic change of coordinates bringing an arbitrary germ of the class to a canonical defining equation whose normalizing transformations are unique up to a finite-dimensional group, with the normal-form coefficients convergent power series. For the classes (i) and (ii), derive from the normal form the Chern–Moser-type consequences that are the objective of the program for these classes: (a) a sphericity criterion, characterizing local biholomorphic equivalence to the appropriate model hypersurface by the vanishing of suitable normal-form coefficients at every point; and (b) an explicit description of the moduli space of germs of each such class up to local biholomorphic equivalence. These deliverables are jointly the solution of Poincaré's local equivalence problem for the classes (i)–(ii) via the normal-form method, in exact analogy with Chern–Moser (Levi-nondegenerate case) and Kolář–Kossovskiy (everywhere 2-nondegenerate $\mathbb{C}^3$). For the class (iii), the objective is the normal-form construction itself: resolving the convergence issue for the formal normal forms at infinite-type points, and constructing a convergent normal form at an isolated weakly pseudoconvex point (Kolář–Kossovskiy–Zaitsev survey, Problems 5.2–5.3). The classes (i)–(iii) are distinct open problems in the cited literature.

References

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