Infinitesimal symmetries of weakly pseudoconvex manifolds

TL;DR

This paper classifies the infinitesimal CR automorphisms of weakly pseudoconvex hypersurfaces in complex space, revealing structural restrictions and jet determination properties crucial for understanding local equivalence.

Contribution

It provides a complete classification of Lie algebras of infinitesimal CR automorphisms for weakly pseudoconvex hypersurfaces, including nonexistence of certain automorphisms and jet determination results.

Findings

No nonlinear rigid automorphisms exist

CR automorphisms are determined by 1-jets for non-balanced hypersurfaces

Lie algebra of automorphisms is three graded for homogeneous polynomials

Abstract

We classify the Lie algebras of infinitesimal CR automorphisms of weakly pseudoconvex hypersurfaces of finite multitype in $\mathbb C^N$. In particular, we prove that such manifolds admit neither nonlinear rigid automorphisms, nor real or nilpotent rotations. As a consequence, this leads to a proof of a sharp 2-jet determination result for local automorphisms. Moreover, for hypersurfaces which are not balanced, CR automorphisms are uniquely determined by their 1-jets. The same classification is derived also for special models, given by sums of squares of polynomials. In particular, in the case of homogeneous polynomials the Lie algebra of infinitesimal CR automorphisms is always three graded. The results provide an important necessary step for solving the local equivalence problem on weakly pseudoconvex manifolds.

Equations

$$Imw=P(z,\bar{z})+o_w(1),$$

$$M_P:=\{\mathsf{Im}w=P(z,\bar{z})\}.$$

$$\mathfrak{g}={\mathfrak{g}}_{-1}\oplus \underset{j=1}{\overset{n}{⨁}}{\mathfrak{g}}_{-{\mu }_j}\oplus {\mathfrak{g}}_0\oplus \underset{\eta \in E}{⨁}{\mathfrak{g}}_{\eta }\oplus {\mathfrak{g}}_1,$$

$$\mathfrak{g}={\mathfrak{g}}_{-1}\oplus \underset{j=1}{\overset{n}{⨁}}{\mathfrak{g}}_{-{\mu }_j}\oplus {\mathfrak{g}}_0\oplus {\mathfrak{g}}_{\frac{1}{2}}\oplus {\mathfrak{g}}_1.$$

$$\mathsf{Im}(w)=\sum _{j=1}^k|P_j(z){|}^2.$$

$$\mathfrak{g}={\mathfrak{g}}_{-1}\oplus {\mathfrak{g}}_0\oplus {\mathfrak{g}}_1$$

$$\mathfrak{g}={\mathfrak{g}}_{-1}\oplus {\mathfrak{g}}_{-\frac{1}{2}}\oplus {\mathfrak{g}}_0\oplus {\mathfrak{g}}_{\frac{1}{2}}\oplus {\mathfrak{g}}_1,$$

$$\sum _j{\alpha }_j{\mu }_j=1.$$

$$|\alpha {|}_{\mu }=\sum _{j=1}^n{\alpha }_j{\mu }_j.$$

$$|(\alpha ,\hat{\alpha }){|}_{\mu }=\sum _{j=1}^n({\alpha }_j+{\hat{\alpha }}_j){\mu }_j.$$

$$f(z,\bar{z}){\partial }_{z_j}\text{ or }f(z,\bar{z}){\bar{\partial }}_{z_j},$$

$$f(z,w){\partial }_{z_j}\text{ and }g(z,w){\partial }_w,$$

$$P(z,\bar{z})=\sum _{|(\alpha ,\hat{\alpha }){|}_{\mu }=1}{A}_{\alpha ,\hat{\alpha }}{z}^{\alpha }{\bar{z}}^{\hat{\alpha }}.$$

$$v=P(z,\bar{z})+\text{higher order terms}$$

$$\mathsf{Im}(w)=P(z,\bar{z})$$

$$\mathsf{Im}(w)=\sum _{j=1}^k|P_j(z){|}^2.$$

$${\partial }_w={\partial }_u-i{\partial }_v,$$

$${\partial }_u(Im(w)-P(z,\bar{z}))=0$$

$$\mathfrak{g}={\mathfrak{g}}_{-1}\oplus \underset{j=1}{\overset{n}{⨁}}{\mathfrak{g}}_{-{\mu }_j}\oplus {\mathfrak{g}}_0\oplus {\mathfrak{g}}_c\oplus {\mathfrak{g}}_n\oplus {\mathfrak{g}}_1.$$

$$w{\partial }_w+\sum _{j=1}^n{\mu }_j{z}_j{\partial }_{z_j}$$

$$\mathsf{Re}E(Im(w)-P(z,\bar{z}))=0$$

$$i\sum _{j=1}^n{\mu }_j{z}_j{\partial }_{z_j},$$

$$P^j(z_1,\dots ,z_j):=P(z_1,\dots ,z_j,0,0,\dots ,0)$$

$$\frac{\partial P^j}{\partial z_j}\ne 0.$$

$$\sum _{j=1}^n{\lambda }_j{z}_j{\partial }_{z_j},$$

$$X(z^{\alpha }{\bar{z}}^{\hat{\alpha }})=\sum _{j=1}^n{\lambda }_j{\alpha }_j{z}^{\alpha }{\bar{z}}^{\hat{\alpha }},\bar{X}(z^{\alpha }{\bar{z}}^{\hat{\alpha }})=\sum _{j=1}^n{\lambda }_j{\hat{\alpha }}_j{z}^{\alpha }{\bar{z}}^{\hat{\alpha }}$$

$$0=2ReX(z^{\alpha }{\bar{z}}^{\hat{\alpha }})=\sum _{j=1}^n{\lambda }_j({\alpha }_j+{\hat{\alpha }}_j)z^{\alpha }{\bar{z}}^{\hat{\alpha }}$$

$$\sum _{j=1}^n{\lambda }_j({\alpha }_j+{\hat{\alpha }}_j)=0.$$

$$P(z,\bar{z})=\sum _{\alpha ,\hat{\alpha }}{A}_{\alpha ,\hat{\alpha }}{z}^{\alpha }{\bar{z}}^{\hat{\alpha }}.$$

$$XP(z,\bar{z})$$

Full text

The authors were supported by the GACR grant GA17-19437S.

1. Introduction

The study of weakly pseudoconvex manifolds goes back to the pioneering work of J. J. Kohn [14], in which he defined type of a point $p\in M\subseteq\mathbb{C}^{2}$ as the lowest order (integer valued) local CR invariant.

More refined rational local invariants of pseudoconvex boundaries in $\mathbb{C}^{N}$ turned out crucial for characterizing subellipticity of the $\bar{\partial}$ - Neumann problem, as follows from the work of D. Catlin ([8, 9]).

Recent work of D. Zaitsev and Sung-Yeon Kim shows that qualitative understanding of higher order CR invariants is also essential to understand effective termination of the Kohn algorithm. In relation to the work [13, 28], Dmitri Zaitsev posed a question on possible symmetries of weakly psedoconvex manifolds of finite Catlin multitype. Can the structure results of [16] be improved under the pseudoconvexity assumption?

The main result of this paper shows that the structure of $\mathfrak{g}=\operatorname{aut}(M_{P},0)$ for weakly pseudoconvex hypersurfaces is indeed substantially simpler compared to the general case. In particular, there exist no nonlinear rigid vector fields on such manifolds. As one consequence, this leads to a sharp 2 - jet determination result. Moreover, 2-jets are needed precisely in the case of a balanced model. In the non balanced case we obtain determination of local automorphisms by their 1-jets.

We also consider the sum of squares models. In the case of homogeneous Levi degenerate sums of squares models the structure of $\operatorname{aut}(M_{P},0)$ is particularly simple, having precisely three nonvanishing components, where $\mathfrak{g}_{-1}$ and $\mathfrak{g}_{1}$ are one dimensional and $\mathfrak{g}_{0}$ is generated by the Euler field and a subalgebra of $\mathfrak{u}(n)$.

The ultimate motivation for our work comes from the Poincaré problem on local equivalence of real hypersurfaces in $\mathbb{C}^{N}$. Let us briefly recall some of the history and some recent developments, which motivated our work. In the classical case of Levi nondegenerate manifolds, the problem was solved in the works of Cartan, Tanaka, Chern and Moser ([6, 7, 10, 25, 27]. While Cartan, Tanaka and Chern applied differential geometric techniques, Moser developed a normal form approach, inspired by the normal form solution to the equivalence problem for analytic vector fields, originating also in the work of Poincaré.

The case of singular Levi form presents completely new challenges, which are often more of algebraic than of differential-geometric nature. A construction of a normal form for finite type Levi-degenerate hypersurfaces in $\mathbb{C}^{2}$ , and the description of their symmetries, was given by the second author in [20]. In combination with a convergence result of Baouendi-Ebenfelt-Rothschild [1], this normal form solves the biholomorphic equivalence problem for this class. The normal form has been shown to be convergent under some additional geometric conditions by Kossovskiy and Zaitsev, [23] but is divergent in general, as shown by Kolář [21].

In complex dimensions higher than two, local geometry of Levi degenerate hypersurfaces is far more complicated, even on the initial level. For pseudoconvex hypersurfaces, D. Catlin ([8]) introduced a notion of multitype. The entries of the Catlin multitype take rational values, but need not be integers, anymore. This approach provides a defining equation of the form

[TABLE]

where $P$ is a weighted homogeneous polynomial in the complex tangential variables $z=(z_{1},\dots,z_{n})$, with respect to the multitype weights $(\mu_{1},\dots,\mu_{n})$, $w$ is the normal variable and $o_{w}(1)$ denotes terms of weight bigger than one. Moreover, the multitype weights are the lexicographically smallest weights for which such a description is possible, hence providing a fundamental CR invariant.

We will denote by $M_{P}$ the corresponding polynomial model,

[TABLE]

The paper [15] introduced an alternative approach to the Catlin multitype, which can be applied to a general smooth hypersurface in $\mathbb{C}^{n+1}$, not necessarily pseudoconvex. Using this approach, it proved biholomorphic equivalence of models, and gave an explicit description of biholomorphisms between different models. It also provided a constructive finite algorithm for computing the multitype.

Using this approach, Kolář, Meylan, Zaitsev showed in [KMZ14] that hypersurfaces of finite Catlin multitype provide the natural class of manifolds for which a generalization of the Chern-Moser operator is well defined.

Using this operator, Kolář, Meylan, Zaitsev proved that the Lie algebra of infinitesimal automorphisms $\mathfrak{g}=\operatorname{aut}(M_{P},0)$ of $M_{P}$ admits the weighted grading given by

[TABLE]

where $E$ is the set of integer combinations of the multitype weights, which lie between zero and one. They also obtained an explicit description of the graded components. As a consequence, they proved that the automorphisms of M at p are uniquely determined by their weighted 2-jets at p.

Since the kernel of the generalized Chern-Moser operator corresponds to the Lie algebra $\operatorname{aut}(M_{P},0)$ of infinitesimal CR automorphisms of the model, this result gives a necessary tool for addressing the equivalence problem. However, full classification of such Lie algebras seems still unattainable. One of the main difficulties is the presence of the component (denoted below as $\mathfrak{g}_{c}$), containg nonlinear rigid vector fileds, with arbitrarily high degree coefficients (only the weighted degree is controlled).

In this paper we show that in the most interesting case of pseudoconvex Levi degenerate manifolds, the structure of $\operatorname{aut}(M_{P},0)$, is in fact much simpler, thus opening the possibility for a complete solution of the Poincaré equivalence problem in this class.

Throughout the paper, $M$ will denote a hypersurface of finite Catlin multitype, described by (1.1).

We now formulate the main results of the paper.

Theorem 1.1**.**

Assume that $M$ is pseudoconvex in a neighbourhood of $p$ and the associated model hypersurface $M_{P}$, given by (1.2) is holomorphically nondegenerate.

Then the Lie algebra of infinitesimal automorphisms $\mathfrak{g}=\operatorname{aut}(M_{P},0)$ of $M_{P}$ admits the weighted grading given by

[TABLE]

As a consequence, we obtain the following sharp jet determination result.

Theorem 1.2**.**

Assume that $M$ is pseudoconvex in a neighbourhood of $p$ and the associated model hypersurface (1.2) is holomorphically nondegenerate. Then the automorphisms of $M$ at $p$ are uniquely determined by their jets of order $2$. Moreover, if $M_{P}$ is not balanced, the automorphisms of $M$ at $p$ are uniquely determined by their jets of order $1$.

Special domains have been used as a manageable test case in the context of the d-bar problem and effectivity of the Kohn algorithm (see e.g. [26], [8]).

The model coresponding to a special domain is a sum of squares models. Of course, it may happen that the model for a holomorphically nondegenerate hypersurface is holomorphically degenerate, a thus does not provide useful information concerning symmetries of the original manifold itself.

For special models, given by sums of squares of homogeneous polynomials, we prove the following results. By $M_{S}$, we will denote a model given by

[TABLE]

Theorem 1.3**.**

Let $M_{S}$ be the sum of squares-type homogeneous polynomial model of degree $k>2$. Then the Lie algebra of infinitesimal automorphisms $\mathfrak{g}=\operatorname{aut}(M_{S},0)$ of $M_{S}$ admits the weighted grading given by

[TABLE]

where $\mathfrak{g}_{-1}$ and $\mathfrak{g}_{1}$ are of real dimension one and $\mathfrak{g}_{0}$ is generated by the Euler field and a subalgebra of $\mathfrak{u}(n)$.

We obtain an analogous result in the case of a weighted homogeneous sum of squares model. Let us denote by $\kappa_{M}$ the number of multitype weights with $\mu_{j}=\frac{1}{2}$.

Theorem 1.4**.**

Let $M_{S}$ be the sum of squares-type weighted homogeneous polynomial model of degree $k>2$. Then the Lie algebra of infinitesimal automorphisms $\mathfrak{g}=\operatorname{aut}(M_{S},0)$ of $M_{S}$ admits the weighted grading given by

[TABLE]

where $\mathfrak{g}_{-1}$ and $\mathfrak{g}_{1}$ are of real dimension one, $\mathfrak{g}_{-\frac{1}{2}}$ and $\mathfrak{g}_{-\frac{1}{2}}$ are of real dimension $2\kappa_{M}$, and $\mathfrak{g}_{0}$ is generated by the Euler field and a subalgebra of $\mathfrak{u}(n)$.

The paper is organized as follows. In Section 2 we recall the needed definitions and notation, which is used in the sequel. In Section 3 we consider rotations and prove that psedoconvex models do not admit any real or nilpotent rotations. Section 4 considers generalized rotation. We show that such symmetries cannot occur on weakly psedoconvex manifolds. In Section 5 we consider sum of squares models. We prove Theorem 1.3 and 1.4. Theorems 1.1 and 1.2 are proved in Section 6.

2. Preliminaries

In this section we recall the definitions and notation needed in the rest of the paper (for more details, see e.g. [16]).

Definition 2.1**.**

Let $n\in\mathbb{N}$ be an integer. A weight is an $n$-tuple of nonnegative rational numbers $\mu=(\mu_{1},\cdots,\mu_{n})$, where $0\leq\mu_{j}\leq\frac{1}{2}$ and $\mu_{j}\geq\mu_{j+1}$ such that there exist an $n$-tuple of non-negative integers $(\alpha_{1},\cdots,\alpha_{n})$ satisfying $\alpha_{j}\neq 0$ if $\mu_{j}\neq 0$ for each $j$ and

[TABLE]

If $\alpha=(\alpha_{1},\cdots,\alpha_{n})$ is a multiindex, for the given weight $\mu=(\mu_{1},\cdots,\mu_{n})$, the weighted length of $\alpha$ is

[TABLE]

Similarly, if $\alpha=(\alpha_{1},\cdots,\alpha_{n})$ and $\hat{\alpha}=(\hat{\alpha}_{1},\cdots,\hat{\alpha}_{n})$ are two multiindices, for the given weight $\mu=(\mu_{1},\cdots,\mu_{n})$, the weighted length of $(\alpha,\hat{\alpha})$ is

[TABLE]

Definition 2.2**.**

For the given weight $\mu=(\mu_{1},\cdots,\mu_{n})$, the weighted degree of a monomial $A_{\alpha,\hat{\alpha}}z^{\alpha}\bar{z}^{\hat{\alpha}}$ is $l$, where $A_{\alpha,\hat{\alpha}}\in\mathbb{C}\setminus\{0\}$ and $z=(z_{1},\cdots,z_{n})\in\mathbb{C}^{n}$, if $|(\alpha,\hat{\alpha})|_{\mu}=l$. A homogeneous polynomial $P$ is called $\mu$-homogeneous of weighted degree $l$, or simply weighted homogeneous polynomial of degree $l$ , if it is a sum of monomials with weighted degree $l$.

On the other hand, a vector field $Y$ is called $\mu$-homogeneous of weighted degree $l$, or simply weighted vector field of degree $l$, if it is a sum of vector fields of the form

[TABLE]

where $f(z,\bar{z})$ is a weighted homogeneous polynomial of degree $l+\mu_{j}$. Moreover, with respect to coordinates $(z,w)=(z_{1},\cdots,z_{n},w)$ of $\mathbb{C}^{n+1}$ with the weight $(\mu,1)=(\mu_{1},\cdots,\mu_{n},1)$, a holomorphic vector field $Y$ is called $\mu$-homogeneous of weighted degree $l$, or simply weighted holomorphic field of degree $l$, if it is sum of holomorphic vector fields with polynomial coefficients, which are of the form

[TABLE]

where $f(z,w)$ is a $(\mu,1)$-homogeneous of weighted degree $l+\mu_{j}$ and $g(z,w)$ is a $(\mu,1)$-homogeneous of weighted degree $l+1$.

Remark 2.3*.*

If $P$ is a $\mu$-homogeneous polynomial of weighted degree 1, we can write $P$ as

[TABLE]

Remark 2.4*.*

The weighted degree of the vector fields $\partial_{z_{j}}$ and $\bar{\partial}_{z_{j}}$ are $-\mu_{j}$. The weighted degree of the vector field $\partial_{w}$ is $-1$.

Definition 2.1**.**

Let $M$ be a hypersurface of $\mathbb{C}^{n+1}$ and $p\in M$ be a point. The weight $\mu$ is called distinguished if there exist local holomorphic coordinates $(z,w)$ centered at $p\in M$, where $z=(z_{1},\cdots,z_{n})\in\mathbb{C}^{n}$ and $w=u+iv$, such that $M$ is described as

[TABLE]

at $p$, where $P(z,\bar{z})$ is a weighted homogeneous polynomial of degree 1 without pluriharmonic terms. For the distingushed weight $\mu$, the local holomorphic coordinates are called $\mu$-adapted. Let $\mu_{M}=(\mu_{1},\cdots,\mu_{n})$ be the infimum of distingushed weights with respect to lexicographic ordering. The mulitype of $M$ at $p$ is the $n$-tuple $(m_{1},m_{2},\cdots,m_{n})$, where $m_{j}=\frac{1}{\mu_{j}}$ if $\mu_{j}\neq 0$ or $m_{j}=\infty$ if $\mu_{j}=0$. If $\mu_{j}\neq 0$ for all $j$, we say that $M$ is finite mulitype at $p$. The weight $\mu_{M}$ is called the mulitype weight and $\mu_{M}$-adapted coordinate are called mulitype coordinate.

We will define a model hypersurface associated to $M$ at $p$ of the form (2.5). $M_{P}$ is the hypersurface of $\mathbb{C}^{n+1}$ given by

[TABLE]

in the coordinates $(w,z)$, where $P(z,\bar{z})$ is the weighted homogeneous polynomial of degree 1 with respect to the weight $\mu_{M}$, and we will assume the hypersurface is pseudoconvex.

As an important example, let $P_{j}$ be the weighted homogeneous polynomials in $z=(z_{1},\cdots,z_{n})\in\mathbb{C}^{n}$ of weight $\frac{1}{2}\mu$, where $\mu$ is a distinguished weight, and let $M_{P}$ be the hypersurface of $\mathbb{C}^{n+1}$ given by

[TABLE]

If we assume that $deg\;P_{j}\geq 1$, then $M_{P}$ is automatically pseudoconvex and we call it a sum of squares-type model.

Let $w=u+iv$ and let $W$ be the vector field of degree $-1$ given by

[TABLE]

then we have

[TABLE]

which gives us the first symmetry. As was proved in [16], the Lie algebra $\mathfrak{g}$ of the infinitesimal automorphism $\operatorname{aut}(M,0)$ at $0\in M\subset\mathbb{C}^{n+1}$ of $M$ admits a weighted decomposition:

[TABLE]

where the local vector field in $\mathfrak{g}_{c}$ commute with $W$, the non-zero local vector field in $\mathfrak{g}_{n}$ do not commute with $W$ and their weights $\mu_{j}$ are between 0 and 1. In particular, $W=\partial_{w}$ is contained in $\mathfrak{g}_{-1}$, which has real dimension one.

Let $E$ be the Euler field given by

[TABLE]

then it is immediate that

[TABLE]

which implies $E\in\mathfrak{g}_{0}$ and hence, $dim(\mathfrak{g}_{0})\geq 1$. Thus the Euler field $E$ gives the second symmetry in $\operatorname{aut}(M_{P},0)$, for an arbitrary model $M_{P}$.

3. Rotations

Let $X$ be a weight zero infinitesimal CR automorphism in $\mathfrak{g}_{0}$. By the result of [16], $X$ is a linear vector field. It’s Jordan normal form could be decomposed into $X^{Re}+X^{Im}+X^{Nil}$, where $X^{Re}$ is the real diagonal of the Jordan normal form, $X^{Im}$ is the imaginary diagonal of the Jordan normal form and $X^{Nil}$ is nilpotent part. In particular, if we let $\tilde{E}$ be the vector field given by

[TABLE]

then $\tilde{E}=\tilde{E}^{Im}$.

Lemma 3.1**.**

Let the model hypersurface (2.6) be pseudoconvex and consider the expanded polynomial of Remark 2.3. There exist multitype coordinates in which for each $j=1,\dots,n$ there exists a multiindex $\alpha$ such that $A_{\alpha,\alpha}\neq 0$, $\alpha_{j}\neq 0$ and $\alpha_{l}=0$ for all $l>j$.

Proof.

By Lemma 3.1 of [15], there exist multitype coordinates such that for each $j$ the j-th partial derivative of the polynomial

[TABLE]

is not identically zero, i.e.

[TABLE]

For any choice of the variables $z_{1},\dots,z_{j-1}$, consider the restriction of $P_{j}$ to the complex line $z_{1}=c_{1},\dots,z_{j-1}=c_{j-1}$ and denote this one variable polynomial by $Q$. Since $\Delta Q$ is nonnegative, it must contain a leading term of the form $|z_{j}|^{2m}$, of the lowest degree. We will consider the minimal such $m$ over the choice of the complex lines $z_{1}=c_{1},\dots,z_{j-1}=c_{j-1}$. Then $P^{j}$ contains the summand $S(z_{1},\dots,z_{j-1})|z_{j}|^{2m}$. $S$ has to be a nonnegative polynomial, hence it contains a nonzero term with $\alpha=\hat{\alpha}$ in $z_{1},\dots,z_{j-1}$. That finishes the proof. ∎

Let us remark that by Lemma 4.6. in [16], if $X$ lies in $\operatorname{aut}(M_{P},0)$, then in Jordan normal form both its diagonal and nilpotent part lie in $\operatorname{aut}(M_{P},0)$. Moreover, both the real and imaginary parts of the diagonal component also lie in $\operatorname{aut}(M_{P},0)$.

Lemma 3.2**.**

Let $X$ be a weight zero rigid infinitesimal CR automorphism in $\mathfrak{g}_{0}$ and $X^{Re}$ be the real diagonal of the Jordan normal form of $X$. Then $X^{Re}=0$

Proof.

Let $X=X^{Re}$ be the vector field

[TABLE]

where $\lambda_{j}\in\mathbb{R}$, and let $z^{\alpha}\bar{z}^{\hat{\alpha}}$ be the monomial, where $z=(z_{1},\cdots,z_{n})$, $\alpha=(\alpha_{1},\cdots\alpha_{n})$ and $\hat{\alpha}=(\hat{\alpha}_{1},\cdots\hat{\alpha}_{n})$. We apply $X$ and $\bar{X}$ to $z^{\alpha}\bar{z}^{\hat{\alpha}}$, then

[TABLE]

and hence,

[TABLE]

if and only if

[TABLE]

We denote this $\lambda\perp(\alpha+\hat{\alpha})$.

Let $P$ be the homogeneous polynomial, then we have decomposition based on the monomials

[TABLE]

Then $XP(z,\bar{z})$ and $\bar{X}P(z,\bar{z})$ are

[TABLE]

Hence,

[TABLE]

if and only if $\lambda\perp(\alpha+\hat{\alpha})$ for all $A_{\alpha,\hat{\alpha}}\neq 0$.

By Lemma 3.1, the subset $\{2\alpha\in\mathbb{C}^{n}|A_{\alpha,\alpha}\neq 0\}$ of $\{\alpha+\hat{\alpha}\in\mathbb{C}^{n}|A_{\alpha,\hat{\alpha}}\neq 0\}$ spans $\mathbb{C}^{n}$. Hence, $\lambda=0$. ∎

Lemma 3.3**.**

Let $X$ be a weight zero rigid infinitesimal CR automorphism in $\mathfrak{g}_{0}$ and $X^{Nil}$ be the nilpotent part of the Jordan normal form of $X$. Then $X^{Nil}=0$

Proof.

Let $Y=X^{Nil}$ be the nilpotent part of $X$,

[TABLE]

where $\lambda_{i}\in\{0,1\}$.

We expand $P$ based on the monomials;

[TABLE]

From the equation $YP(z,\bar{z})=-\overline{Y}P(z,\bar{z})$, we have

[TABLE]

where $\alpha(i)=(\alpha_{1},\cdots,\alpha_{i}+1,\alpha_{i+1}-1,\cdots,\alpha_{n})$ and $\hat{\alpha}(i)=(\hat{\alpha}_{1},\cdots,\hat{\alpha}_{i}+1,\hat{\alpha}_{i+1}-1,\cdots,\hat{\alpha}_{n})$.

For the set $\{\alpha\}=\{\hat{\alpha}\}$, we give a partial ordering by $\alpha\succ_{1}\beta$ if $\alpha(i)=\beta$ for some $i\in\{1,\cdots,n-1\}$: we denote $\alpha\succeq\beta$ if $\alpha=\beta$, $\alpha\succ_{1}\beta$ or there is a subset $\{\gamma_{i}|i=1,\cdots t,\gamma_{1}\succ_{1},\cdots,\succ_{1}\gamma_{t}\}$ of $\{\alpha\}$ such that $\alpha\succ_{1}\gamma_{1}$ and $\gamma_{t}\succ_{1}\beta$. Then $\{\alpha,\succeq\}$ is a partially ordered set.

Assume $Y\neq 0$. Let $i$ be the largest integer with $\lambda_{i}\neq 0$. By Lemma 3.1, there exists $\alpha$ such that $\alpha_{i+1}\neq 0$ and $A_{\alpha,\alpha}\neq 0$. In the equation (3.6), non-vanishing $\lambda_{i}\alpha_{i+1}A_{\alpha,\alpha}$ of the left-side implies there are $\beta$ and $\hat{\beta}$ such that $\lambda_{k}\hat{\beta}_{k+1}A_{\beta,\hat{\beta}}\neq 0$ for some $k$ and

[TABLE]

which gives us $\hat{\beta}\succ_{1}\alpha$. Since $\lambda_{k}\hat{\beta}_{k+1}\neq 0$, the equation (3.6) implies there are $\gamma$ and $\hat{\gamma}$ such that $\lambda_{k}\hat{\gamma}_{l+1}A_{\beta,\hat{\beta}}\neq 0$ for some $l$ and

[TABLE]

implies $\hat{\gamma}\succ_{1}\hat{\beta}\succ_{1}\alpha$. If we continue this, we get infinite-length chain, which is a contradiction, since the set $\{\alpha\}$ is finite. Hence, $Y=0$. ∎

4. Generalized rotations

In this section we prove the following result.

Theorem 4.1**.**

If $M_{P}$ given by (2.6) is pseudoconvex, then $\mathfrak{g}_{c}=0$.

Proof.

Since $M_{P}$ is pseudoconvex near [math], any neighbourhood of [math] contains strongly pseudoconvex points, by the finite multitype assumption. It follows that $\operatorname{aut}(M_{P},0)$ is a subalgebra of $\operatorname{aut}(Q,0)$, where $Q$ is the strongly pseudoconvex hyperquadric. We know that $\operatorname{aut}(Q,0)$ admits the weighted grading

[TABLE]

Here both $\mathfrak{g}_{-1}$ and $\mathfrak{g}_{1}$ are one dimensional. Moreover, $\mathfrak{g}_{0}$ can be split further as $\mathfrak{g}_{0}=\mathfrak{g}_{0e}\oplus\mathfrak{g}_{0r}$, where $\mathfrak{g}_{0e}$ is the one dimensional subspace contaning the Euler field and $\mathfrak{g}_{0r}$ contains rigid vector fields, i.e., rotations. We have the following commutation relations

[TABLE]

Moreover, $\mathfrak{g}_{0r}$ commutes both with $\mathfrak{g}_{-1}$ and $\mathfrak{g}_{1}$.

Note that $\mathfrak{g}_{0e}$ is spanned by the Euler field $E=E_{Q}$, which is the grading element. Let us emphasize the crucial fact that on the strongly psedoconvex hyperquadric the Euler field is determined uniquely, since there exist no real rotations.

Assume $\dim\mathfrak{g}_{c}>0$ and let $Z\in\mathfrak{g}_{c}$. Denote by $\mathfrak{h}$ the subalgebra of $\mathfrak{g}$ generated by the vector fields $E,W,Z$. $\mathfrak{h}$ is three dimensional with the commutation relations

[TABLE]

[TABLE]

where $0<c<1$, and

[TABLE]

We will prove that $\operatorname{aut}(Q,0)$ contains no subalgebra isomorphic to $\mathfrak{h}$. By contadiction, asuume that $\operatorname{aut}(Q,0)$ contains such an algebra and denote the corresponding vector fileds in $\operatorname{aut}(Q,0)$ by primes. Hence $E^{\prime},W^{\prime},Z^{\prime}$ generate a three dimensional subalgebra with the same relations as in (4.2) - (4.4).

We will decompose $E^{\prime},W^{\prime},Z^{\prime}$ into the graded components of $\operatorname{aut}(Q,0)$ and then express the commutators of those three vector fileds according to this grading. More precisely, we will be only interested in the $\mathfrak{g}_{-1}$ $\mathfrak{g}_{0e}$ and $\mathfrak{g}_{1}$ components of these commutators.

Note that for this purpose we may ignore the $\mathfrak{g}_{-\frac{1}{2}}$ and the $\mathfrak{g}_{\frac{1}{2}}$ components of $\mathfrak{g}$. Indeed, $[\mathfrak{g}_{-\frac{1}{2}},\mathfrak{g}_{\frac{1}{2}}]=0$. and the nonvanishing commutators of $\mathfrak{g}_{-\frac{1}{2}}$ and $\mathfrak{g}_{\frac{1}{2}}$ with the components $\mathfrak{g}_{-1}$, $\mathfrak{g}_{0}$ and $\mathfrak{g}_{1}$ lie in $\mathfrak{g}_{-\frac{1}{2}}$ or $\mathfrak{g}_{\frac{1}{2}}$.

Hence we can write

[TABLE]

[TABLE]

[TABLE]

where $W_{Q}\in\mathfrak{g}_{-1}$ $E_{Q}\in\mathfrak{g}_{0e}$ and $G_{1}\in\mathfrak{g}_{1}$ satisfying equation (4.8)-(4.10) and $\epsilon_{E}$, $\epsilon_{W},\epsilon_{Z}$, represent terms which will not contribute to the commutator components on the level of $\mathfrak{g}_{-1}$, $\mathfrak{g}_{0e}$ and $\mathfrak{g}_{1}$, in other words they belong to $\mathfrak{g}_{-\frac{1}{2}}\oplus\mathfrak{g}_{0r}\oplus\mathfrak{g}_{\frac{1}{2}}$.

We have

[TABLE]

[TABLE]

and

[TABLE]

Hence we obtain for $[E^{\prime},W^{\prime}]$

[TABLE]

For $[E^{\prime},Z^{\prime}]$ we have

[TABLE]

and for $[Z^{\prime},W^{\prime}]$ we obtain

[TABLE]

By the third set of equations (4.11), the vector product of $z$ and $w$ is zero, hence $z$ and $w$ are linearly dependent. From the first two sets we obtain immediate contradiction.

∎

5. Sum of squares models

In this section we consider sum of squares models and give a description of their symmetry algebra.

Theorem 5.1**.**

Let $M_{S}$ be the sum of squares homogeneous polynomial model of degree $k>2$. Then the subspace $\mathfrak{g}_{-\mu_{j}}$ $j=1,..n$ of the symmetry algebra vanishes.

Proof.

Since the local vector field $\partial_{z_{i}}$ has the weight $-\mu_{i}$, it is equivalent to show that

[TABLE]

Since $\partial_{z_{i}}Im(w)=0$, it is enough to prove $Re\partial_{z_{i}}\sum_{j=1}^{k}|P_{j}(z)|^{2}\neq 0$. Since $M_{S}$ is holomorphically non-degenerate, i.e., $\{\bigtriangledown P_{j}\}$ spans $\mathbb{C}^{n}$ at a generic point, we have $max_{j}\{deg_{z_{i}}P_{j}(z)\}>max_{j}\{deg_{z_{i}}\partial_{z_{i}}P_{j}(z)\}\geq 0$. It follows that

[TABLE]

∎

Let $X=X^{Im}$ be the vector field

[TABLE]

where $\lambda_{j}\in\mathbb{R}$, and let $z^{\alpha}\bar{z}^{\hat{\alpha}}$ be the monomial, where $z=(z_{1},\cdots,z_{n})$, $\alpha=(\alpha_{1},\cdots\alpha_{n})$ and $\hat{\alpha}=(\hat{\alpha}_{1},\cdots\hat{\alpha}_{n})$. We apply $X$ and $\bar{X}$ to $z^{\alpha}\bar{z}^{\hat{\alpha}}$, then

[TABLE]

and hence,

[TABLE]

if and only if

[TABLE]

Lemma 5.2**.**

If $P(z,\bar{z})=\sum_{j=1}^{k}|P_{j}(z)|^{2}$ is a sum of squares polynomial of weighted degree 1, then $\tilde{E}\in\mathfrak{g}_{0}$.

Proof.

If $P(z,\bar{z})=\sum_{j=1}^{k}|P_{j}(z)|^{2}$ is a sum of squares-type, then

[TABLE]

which implies the result. ∎

In this case: if $P(z,\bar{z})=\sum_{j=1}^{k}|P_{j}(z)|^{2}$ is a sum of square-type of degree $l$, by Theorem 4.7 of [16] and by Lemma 5.2, the vector field

[TABLE]

is contained in $\mathfrak{g}_{1}$. It provides the third symmetry.

As a consequence, we obtain the following precise description of the Lie algebra of infinitesimal automorphisms of $M_{S}$.

Theorem 5.3**.**

Let $M_{S}$ be the sum of square-type homogeneous polynomial model $M_{S}$ of degree $k>2$. Then the Lie algebra of infinitesimal automorphisms $\mathfrak{g}=\operatorname{aut}(M_{S},0)$ of $M_{S}$ admits the weighted grading given by

[TABLE]

where $\mathfrak{g}_{-1}$ and $\mathfrak{g}_{1}$ are of real dimension one and $\mathfrak{g}_{0}$ is generated by the Euler field and a subalgebrs of $\mathfrak{u}(n)$.

Now we consider the general case of a weighted homogeneous polynomial model. We will detone by $\kappa_{M}$ the number of multitype weights with $\mu_{j}=\frac{1}{2}$.

Theorem 5.4**.**

Let $M_{S}$ be the sum of square-type weighted homogeneous polynomial model $M_{S}$ of degree $k>2$. Then the Lie algebra of infinitesimal automorphisms $\mathfrak{g}=\operatorname{aut}(M_{S},0)$ of $M_{S}$ admits the weighted grading given by

[TABLE]

where $\mathfrak{g}_{-1}$ and $\mathfrak{g}_{1}$ are of real dimension one, $\mathfrak{g}_{-\frac{1}{2}}$ and $\mathfrak{g}_{-\frac{1}{2}}$ are of real dimension $2\kappa_{M}$, and $\mathfrak{g}_{0}$ is generated by the Euler field and a subalgebra of $\mathfrak{u}(n)$.

Proof.

It follows from the assumptions that in suitable multitype coordinates we can write

[TABLE]

where $Q$ is weighted homogeneous and balanced. We verify that the vector fields

[TABLE]

for each $j=1,\dots,\kappa_{M}$ and $a\in\mathbb{C}$ lie in $\mathfrak{g}_{-\frac{1}{2}}$, and can be integrated to vector fields

[TABLE]

which lie in $\mathfrak{g}_{\frac{1}{2}}$. By the above Theorem 5.1, there are no other elements of $\mathfrak{g}_{-\frac{1}{2}}$, hence no other elements in $\mathfrak{g}_{\frac{1}{2}}$, which gives the claim. ∎

6. Proof of the main results

We give now the proof of Theorem 1.1.

Proof.

By Theorem 1.3 of [16] and Theorem 4.1, we obtain that $\mathfrak{g}=\operatorname{aut}(M_{P},0)$ admits the weighted grading given by

[TABLE]

It remains to prove that the weight of $g_{n}$ is $\frac{1}{2}$. We will use the following characterization of manifolds with nonvanishing $\mathfrak{g}_{n}$.

Let $M_{P}$ have nontrivial $\mathfrak{g}_{n}$ and

[TABLE]

for some $l$ be an infinitesimal symmetry of $M_{P}$, which can be integrated. Let us write $P$ as

[TABLE]

for some homogeneous polynomials ${P_{j}}$ in the variables $z^{\prime}=(z_{1},\dots,\hat{z}_{l},\dots,z_{n})$, with $P_{m}\neq 0.$ Then $M_{P}$ has one of the following two forms. Either

[TABLE]

or

[TABLE]

This characterization was proved by Kolář and Meylan in [19] in the $\mathbb{C}^{3}$ case, and in [18] in general. It is immediate to verify that in the second case, the manifold is not pseudoconvex. On the other hand, in the first case we have $\mu_{l}=\frac{1}{2}$, which leads to the claim of Theorem 1.1.

∎

Theorem 1.2 is an immediate consequence of Theorem 1.1