Automorphism algebras and jet determination for weighted-homogeneous polynomial models in CR geometry beyond Catlin multitype weights and holomorphic nondegeneracy

The generalized Chern–Moser theory of Kolář–Meylan–Zaitsev (2014) describes the graded Lie algebra of infinitesimal CR automorphisms of the polynomial model of a real hypersurface of finite Catlin multitype, and derives sharp finite jet-determination statements for local automorphisms, under two standing hypotheses: the weights of the model are the Catlin multitype weights, and the model is holomorphically nondegenerate. The open problem is to complete the theory by removing these hypotheses: determine the structure of the infinitesimal automorphism algebra $\operatorname{aut}(M_P,0)$ and the corresponding jet-determination statements for weighted-homogeneous models whose weights are not Catlin multitype weights (including infinite multitype), for which models the reduction of the hypersurface's stability group to the model's infinitesimal automorphisms remains valid, and for holomorphically degenerate models. Later work (2016–2026) resolves important subclasses — the complete $\mathbb C^3$ classification under nondegeneracy, the pseudoconvex and sums-of-squares classes, and general $\mathbb C^N$ nonlinearizable symmetries — but the two general regimes remain open, and the newest paper in the line (2026) still assumes holomorphic nondegeneracy and states that the full classification of kernels of the Chern–Moser operator does not seem tractable in full generality.

Background

A real hypersurface $M\subset\mathbb C^{n+1}$ carries a Cauchy–Riemann (CR) structure: the complex tangent spaces $T^{1,0}M$ and $T^{0,1}M$, together with the real normal direction. The Levi form is the Hermitian form measuring the extent to which the commutators of CR tangent directions leave the complex tangent space; $M$ is Levi nondegenerate when the Levi form has no null vectors, and Levi degenerate otherwise. The classical theory of Chern and Moser (Acta Math. 133 (1974)) solves the local equivalence and automorphism problem in the Levi-nondegenerate case: the model is the hyperquadric $\{\operatorname{Im} w=|z|^2\}$ (a quadric in complex space replacing the tangent plane), the algebra of infinitesimal CR automorphisms (holomorphic vector fields tangent to the hypersurface, i.e. one-parameter families of local biholomorphic self-maps) is graded with at most five components, and automorphisms are determined by their $2$-jets at the point.

For Levi-degenerate hypersurfaces the analogue of type is the finite type condition of Bloom–Graham (no complex analytic hypersurface has infinite-order contact), refined by Catlin's multitype: in suitable multitype coordinates $(z,w)\in\mathbb C^n\times\mathbb C$, $M$ is given by $\operatorname{Im} w=P(z,\bar z)+F$ where $P$ is a real weighted-homogeneous polynomial of weighted degree $1$ with respect to the multitype weights $\mu_1,\dots,\mu_n\in(0,1/2]$ (weight of $w$ equal to $1$) and $F$ has weighted order $>1$. The polynomial model is $M_P=\{\operatorname{Im} w=P(z,\bar z)\}$; it is an invariant up to biholomorphism (Kolář, IMRN 2010, building on Catlin's distinguished weights). A model is holomorphically nondegenerate if no nonzero holomorphic vector field is tangent to it; by a theorem of Stanton (Amer. J. Math. 118 (1996)), for finite-type real-analytic hypersurfaces holomorphic nondegeneracy is equivalent to finite dimensionality of the infinitesimal automorphism algebra, so the degenerate case is where the finite graded picture may genuinely break down.

Kolář–Meylan–Zaitsev (Adv. Math. 263 (2014)) extended the Chern–Moser machinery to this setting. Their generalized Chern–Moser operator $L(f,g)=\operatorname{Re}\left\{i\,g\big(z,u+iP(z,\bar z)\big)+2\sum_{j=1}^n P_{z_j}(z,\bar z)\,f^j\big(z,u+iP(z,\bar z)\big)\right\}$ linearizes the tangency condition for an infinitesimal coordinate change $(f,g)$ of $M_P$. Its kernel corresponds to the infinitesimal CR automorphisms. A transformation-law expansion (equation (2.16) of the paper) shows that, for a hypersurface of finite Catlin multitype, the first nonzero weighted component of any local automorphism of $M$ lies in $\operatorname{aut}(M_P,0)$, reducing the stability group of the hypersurface to the model's algebra; whether an analogous reduction remains valid beyond the finite-multitype weights is left open and is the question posed in part (a). Under the two standing hypotheses — Catlin multitype weights and holomorphic nondegeneracy — they prove the weighted grading $\mathfrak g=\mathfrak g_{-1}\oplus\bigoplus_j\mathfrak g_{-\mu_j}\oplus\mathfrak g_0\oplus\bigoplus_{\eta\in E}\mathfrak g_\eta\oplus\mathfrak g_1$ (with $E$ an explicit finite set of intermediate weights) and sharp finite jet-determination statements of Chern–Moser type, generalizing the classical five-component decomposition; in the homogeneous case of degree $d$ the grading refines to the classical form $\mathfrak g_{-1}\oplus\mathfrak g_{-1/d}\oplus\mathfrak g_0\oplus\bigoplus_{\tau=1}^{d-2}\mathfrak g_{\tau/d}\oplus\mathfrak g_{1-1/d}\oplus\mathfrak g_1$. A fundamentally new phenomenon appears with no Levi-nondegenerate analogue: the intermediate components $\mathfrak g_\eta$ can contain nonlinear rigid vector fields (exotic symmetries) of arbitrarily high polynomial degree, and the jet-determination order is not uniformly bounded. Subsequent work by the same school classified these phenomena: complete classification of models with nontrivial exotic component $\mathfrak g_c$ in $\mathbb C^3$ via symmetric $Y$-chains with the hyperquadric-pullback origin of nonlinear symmetries (Kolář–Meylan, Proc. Amer. Math. Soc. 144 (2016)); the rarity of the noncommuting component $\mathfrak g_n$ in $\mathbb C^3$ (Kolář–Meylan, Ann. Sc. Norm. Super. Pisa 19 (2019)); general $\mathbb C^N$ classification of models with nonlinearizable infinitesimal automorphisms and optimal $1$-jet determination (Kolář–Meylan, J. Geom. Anal. 33 (2023)); the pseudoconvex case, where the algebra simplifies to $\mathfrak g_{-1}\oplus\bigoplus_j\mathfrak g_{-\mu_j}\oplus\mathfrak g_0\oplus\mathfrak g_{1/2}\oplus\mathfrak g_1$ with sharp $2$-jet ($1$-jet for unbalanced models) determination, including graded descriptions of sums-of-squares models $\operatorname{Im} w=\sum_j|P_j(z)|^2$, which in the paper's own analysis are holomorphically nondegenerate (Kim–Kolář, Math. Z. 2022); and the classification of all $\mathbb C^3$ models without $2$-jet determination (Kolář–Liczman–Meylan, J. Geom. Anal. 2026). All of these results retain the two standing hypotheses in some form, and the 2026 classification paper states that classifying all possible kernels of the Chern–Moser operator does not seem tractable in full generality.

Problem Statement

Let $M_P=\{\operatorname{Im} w=P(z,\bar z)\}\subset\mathbb C^{n+1}$ be a polynomial model with $P$ a real weighted-homogeneous polynomial without pluriharmonic terms (pluriharmonic meaning locally the real part of a holomorphic function, so that such terms can be removed from the model equation). Throughout, following the conventions of Kolář–Meylan–Zaitsev, hypersurfaces $M$ are smooth, their local automorphisms are germs of local biholomorphisms mapping $M$ into itself and fixing the point, and $\operatorname{aut}(M_P,0)$ consists of germs of holomorphic vector fields whose real part is tangent to $M_P$. The problem is to extend the generalized Chern–Moser theory of Kolář–Meylan–Zaitsev beyond its two standing hypotheses, namely that the weights are Catlin multitype weights of finite multitype and that $M_P$ is holomorphically nondegenerate:

(a) For weight systems that are not the Catlin multitype weights of any finite-multitype hypersurface (including systems with vanishing weights, i.e. infinite multitype), two sub-claims are asked independently: (a1) determine for which weighted-homogeneous models $M_P$ the reduction of the stability group remains valid — that is, the first nonzero weighted component of every local automorphism of a hypersurface $M$ whose leading weighted term is $P$ belongs to $\operatorname{aut}(M_P,0)$; and (a2) establish the structure of $\operatorname{aut}(M_P,0)$ (graded decomposition, dimensions and normal forms of the graded components) together with the corresponding sharp finite jet-determination statements for automorphisms of $M$. Each sub-claim may be resolved either positively or by an explicit negative example: the reduction sub-claim (a1) by a weighted-homogeneous model in this regime together with a local automorphism of a hypersurface whose leading weighted term is $P$ and whose first nonzero weighted component is not in $\operatorname{aut}(M_P,0)$; the jet-determination part of (a2) by an explicit model in this regime for which finite jet determination fails. A negative resolution settles only the sub-claim it addresses and leaves the remaining sub-claims mandatory: unless refuted, (a1) requires the characterization of the models admitting the reduction, and (a2) requires the graded decomposition, dimensions and normal forms of $\operatorname{aut}(M_P,0)$ together with the sharp jet-determination order.

(b) In the finite-multitype setting without assuming holomorphic nondegeneracy, two sub-claims are asked independently: (b1) describe the structure of $\operatorname{aut}(M_P,0)$ for holomorphically degenerate models — noting that by Stanton's theorem the algebra of a finite-multitype holomorphically degenerate model is infinite-dimensional, so a finite graded decomposition as in the 2014 theory cannot exist and the structural description must cover the holomorphically degenerate finite-multitype regime rather than a single selected subclass; and (b2) derive the corresponding jet-determination conclusions for smooth hypersurfaces with such models. The jet-determination sub-claim (b2) may be resolved positively, or negatively by a proof that finite jet determination fails, namely that for every finite order $k$ there exist two distinct local automorphisms of such a hypersurface whose $k$-jets at the point coincide. A negative resolution settles (b2) only and leaves the structural description (b1) mandatory.

The common objective is a complete picture of the infinitesimal automorphism algebras of weighted-homogeneous polynomial models and of the finite jet determination of local automorphisms governed by them, with the two regimes (a) and (b) covering all weighted-homogeneous models to which the 2014 theory does not apply.

References

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