Nonlinear CR automorphisms of Levi degenerate hypersurfaces and a new gap phenomenon

TL;DR

This paper classifies polynomial models of smooth real hypersurfaces in complex three-space that admit nonlinear CR automorphisms, establishing a new gap phenomenon and confirming a longstanding conjecture.

Contribution

It provides a complete classification of models with nonlinear automorphisms and describes the possible dimensions of their automorphism Lie algebras, revealing a new gap phenomenon.

Findings

Complete classification of polynomial models with nonlinear CR automorphisms

Sharp 1-jet determination result for such hypersurfaces

Identification of a new secondary gap in automorphism algebra dimensions

Abstract

We give a complete classification of polynomial models for smooth real hypersurfaces of finite Catlin multitype in $\mathbb C^3$, which admit nonlinear infinitesimal CR automorphisms. As a consequence, we obtain a sharp 1-jet determination result for any smooth hypersurface with such model. The results also prove a conjecture of the first author about the origin of such nonlinear automorphisms (AIM list of problems, 2010). As another consequence, we describe all possible dimensions of the Lie algebra of infinitesimal CR automorphisms, which leads to a new "secondary" gap phenomenon.

Equations

$$M_H:=\{\mathsf{Im}w=P(z,\bar{z})\},(z,w)\in {\mathbb{C}}^2\times \mathbb{C}.$$

$$\mathfrak{g}={\mathfrak{g}}_{-1}\oplus {\mathfrak{g}}_{-1/d}\oplus {\mathfrak{g}}_0\oplus {\mathfrak{g}}_{\tau /d}\oplus {\mathfrak{g}}_{1-1/d}\oplus {\mathfrak{g}}_1,$$

$$\dim {\mathfrak{g}}_{1-1/d}>0.$$

$$\mathsf{Im}w=\mathsf{Re}{z}_1{\bar{z}}_2^l.$$

$$M_H:=\{\mathsf{Im}w=P_C(z,\bar{z})\},(z,w)\in {\mathbb{C}}^2\times \mathbb{C},$$

$$\mathfrak{g}={\mathfrak{g}}_{-1}\oplus \underset{j=1}{\overset{2}{⨁}}{\mathfrak{g}}_{-{\mu }_j}\oplus {\mathfrak{g}}_0\oplus {\mathfrak{g}}_c\oplus {\mathfrak{g}}_n\oplus {\mathfrak{g}}_1,$$

$$M_H:=\{\mathsf{Im}w=P_C(z,\bar{z})\},(z,w)\in {\mathbb{C}}^2\times \mathbb{C},$$

$$\dim {\mathfrak{g}}_n>0.$$

$$\mathsf{Im}w=\mathsf{Re}{z}_1{\bar{z}}_2^l$$

$$\mathsf{Im}w=|z_1{|}^2\pm |z_2{|}^{2l}.$$

$$Y(U^n)=0,Y(U^j)=c_j{U}^{j+1},j=1,\dots ,n-1,$$

$$Y(V^n)=0,Y(V^j)=d_j{V}^{j+1},j=1,\dots ,n-1,$$

$$c_j=-{\bar{d}}_{n-j}.$$

$$P_C=\sum _{j=1}^M{T}_j,$$

$$T_j=\mathsf{Re}(\sum _{k=1}^{N_j}{U}_j^k\overline{V_j^{N_j-k+1}}),$$

$$P_C(z,\bar{z})=\sum _{|\alpha {|}_{\Lambda }=|\bar{\alpha }{|}_{\Lambda }=1}{A}_{\alpha ,\bar{\alpha }}{z}^{\alpha }{\bar{z}}^{\bar{\alpha }},$$

$$|\alpha {|}_{\Lambda }:={\lambda }_1{\alpha }_1+{\lambda }_2{\alpha }_2.$$

$$Y={\lambda }_1{z}_1{\partial }_{z_1}+{\lambda }_2{z}_2{\partial }_{z_2}$$

$$\dim \text{aut}(M_H,0)\le 10.$$

$$\{2,3,4,5,6,7,9,10\}.$$

$$v=F(z_1,z_2,{\bar{z}}_1,{\bar{z}}_2,u),$$

$$F(z_1,z_2,{\bar{z}}_1,{\bar{z}}_2,u)=P_m(z,\bar{z})+o(u,|z{|}^m),$$

$$q(z,\bar{z},u)=c_{\alpha \beta l}{z}^{\alpha }{\bar{z}}^{\beta }{u}^l,l\in \mathbb{N},$$

$$\kappa :=l+\sum _{i=1}^2({\alpha }_i+{\beta }_i){\lambda }_i.$$

$$|\alpha {|}_{\Lambda }:={\lambda }_1{\alpha }_1+{\lambda }_2{\alpha }_2.$$

$$|(\alpha ,\hat{\alpha }){|}_{\Lambda }:={\lambda }_1({\alpha }_1+{\hat{\alpha }}_1)+{\lambda }_2({\alpha }_2+{\hat{\alpha }}_2).$$

$$v=P(z,\bar{z})+o_{\Lambda }(1),$$

$$(m_1,m_2),$$

$$m_j=\{\begin{array}{l}\frac{1}{{\mu }_j}\text{ if}{\mu }_j\ne 0\\ \infty \text{ if}{\mu }_j=0.\end{array}$$

$$M_H=\{(z,w)\in {\mathbb{C}}^{n+1}|v=P_C(z,\bar{z})\}.$$

Full text

The first author was supported by the project CZ.1.07/2.3.00/20.0003 of the Operational Programme Education for Competitiveness of the Ministry of Education, Youth and Sports of the Czech Republic.

The second author was supported by Swiss NSF Grant 2100-063464.00/1

1. Introduction

This paper provides an important necessary step towards solving the local equivalence problem for hypersurfaces of finite Catlin multitype, by giving a full classification of their polynomial models with nonlinear infinitesimal CR automorphisms. Note that by the classical Chern-Moser theory, the only strongly pseudoconvex hypersurface which admits such automorphisms is the sphere (see [10], [27]).

The Levi degenerate case has recently attracted considerable attention and has led to the discovery of new types of nonlinear symmetries (see e.g. [13], [17], [19]). In contrast to the $\mathbb{C}^{2}$ case, nonlinear infinitesimal CR automorphisms with coefficients of arbitrarily high degree may arise in $\mathbb{C}^{n},n>2$ ([12], [23]).

Our aim in this paper is to study systematically nonlinear infinitesimal CR automorphisms of Levi degenerate hypersurfaces in complex dimension three. The results provide a description of hypersurfaces of finite Catlin multitype in $\mathbb{C}^{3}$, whose models admit such automorphisms. Since the Lie algebra of infinitesimal CR automorphisms of a polynomial model is in one-to-one correspondence with the kernel of the generalized Chern-Moser operator ([18]), we can prove a sharp 1-jet determination result for the biholomorphisms of any such hypersurface.

Moreover, we identify the common source of such automorphisms. In all cases they arise from suitable holomorphic mappings into a quadric in $\mathbb{C}^{K}$, $K\geq 3$, as a “pull-back” of an automorphism of the quadric.

As an application, we will determine all possible dimensions of the Lie algebra of infinitesimal CR automorphisms for such models, which reveals a ”secondary” gap phenomenon at dimension eight.

Our starting point are the sharp results of [18] which give an effective bound for the weighted degree of the coefficients of an infinitesimal CR automorphism, and put restrictions on the possible form of such vector fields.

Our first result deals with a holomorphically nondegenerate model hypersurface $M_{H}$ given by a homogeneous polynomial $P$ of degree $d>2$ without pluriharmonic terms,

[TABLE]

Notice that the case of $d=2$ corresponds to Levi nondegenerate models, i.e. hyperquadrics. We showed in [18] and in [19] that the Lie algebra $\mathfrak{g}=\text{\rm aut}(M_{H},0)$ of all germs of infinitesimal automorphisms of $M_{H}$ at [math] admits the weighted grading

[TABLE]

for some integer $\tau$, where $1\leq\tau\leq d-2.$ The following theorem shows that if $\dim\mathfrak{g}_{1-1/d}\neq 0,$ then there is a unique choice for $M_{H}.$

Theorem 1.1**.**

Let $M_{H}$ be the holomorphically nondegenerate model hypersurface given by (1.1) with $d>2$, and let $\mathfrak{g}_{1-1/d}$ in (1.2) satisfy

[TABLE]

Then $M_{H}$ is biholomorphically equivalent to

[TABLE]

Further we consider the more general case of a holomorphically nondegenerate weighted homogeneous model of finite Catlin multitype. Let

[TABLE]

where $P_{C}$ is a weighted homogeneous polynomial of degree one with respect to the multitype weights $\mu_{1},\mu_{2}$ (see Section 2 for the needed definitions).

As proved in [18], the Lie algebra of infinitesimal automorphisms $\mathfrak{g}=\text{\rm aut}(M_{H},0)$ of $M_{H}$ admits the weighted decomposition given by

[TABLE]

where $\mathfrak{g}_{c}$ contains vector fields commuting with $W=\partial_{w}$ and $\mathfrak{g}_{n}$ contains vector fields not commuting with $W$, whose weights in both cases lie in the interval $(0,1)$. Note that by a result of [18], vector fields in $\mathfrak{g}_{j}$ with $j<0$ are regular and vector fields in $\mathfrak{g}_{0}$ are linear.

Theorem 1.2**.**

Let $P_{C}(z,\bar{z})$ be a weighted homogeneous polynomial of degree 1 with respect to the multitype weights, such that the hypersurface

[TABLE]

is holomorphically nondegenerate. Let $\mathfrak{g}_{n}$ in (1.6) satisfy

[TABLE]

Then $M_{H}$ is biholomorphically equivalent to

[TABLE]

*or *

[TABLE]

Note that the Levi nondegenerate case, corresponding to $l=1$, is covered by Theorem 1.2. The following result, which deals with the component $\mathfrak{g}_{c}$ was obtained in [19].

Definition 1.3**.**

Let $Y$ be a weighted homogeneous vector field. A pair of finite sequences of holomorphic weighted homogeneous polynomials $\{U^{1},\dots,U^{n}\}$ and $\{V^{1},\dots,V^{n}\}$ is called a symmetric pair of $Y-$chains if

[TABLE]

[TABLE]

where $c_{j}$, $d_{j}$ are non zero complex constants, which satisfy

[TABLE]

If the two sequences are identical we say that $\{U^{1},\dots,U^{n}\}$ is a symmetric $Y$ - chain.

The following theorem shows that in general the elements of $\mathfrak{g}_{c}$ arise from symmetric pairs of chains.

Theorem 1.4**.**

Let $M_{H}$ be a holomorphically nondegenerate hypersurface given by (1.7), which admits a nontrivial $Y\in\mathfrak{g}_{c}$. Then $P_{C}$ can be decomposed in the following way

[TABLE]

where each $T_{j}$ is given by

[TABLE]

where $\{{{{U_{j}^{1}},\dots,{U_{j}^{N_{j}}}}}\}$ and $\{{{{V_{j}^{1}},\dots,{V_{j}^{N_{j}}}}}\}$ are a symmetric pair of $Y-$ chains.

*Conversely, if $Y$ and $P_{C}$ satisfy (1.11) – (1.15), then $Y\in\mathfrak{g}_{c}$. *

Note that $Y$ is uniquely and explicitely determined by $P$ (see [19]). Hence for a given hypersurface this result also provides a constructive tool to determine $\mathfrak{g}_{c}$.

Definition 1.5**.**

If $P_{C}$ satisfies (1.11) – (1.15), the associated hypersurface $M_{H}$ is called a chain hypersurface.

The description of the remaining component $\mathfrak{g}_{1}$ is a consequence of Theorem 4.7 in [18] (see section 2 for the notation).

Definition 1.6**.**

We say that $P_{C}$ given by (1.7) is balanced if it can be written as

[TABLE]

for some nonzero pair of real numbers $\Lambda=(\lambda_{1},\lambda_{2})$, where

[TABLE]

The associated hypersurface $M_{H}$ is called a balanced hypersurface.

Note that $P_{C}$ is balanced if and only if the linear vector field

[TABLE]

is a complex reproducing field in the terminology of [18], i.e., $Y(P_{C})=P_{C}$.

Theorem 1.7**.**

The component $\mathfrak{g}_{1}$ satisfies $\dim\mathfrak{g}_{1}>0$ if and only if in suitable multitype coordinates $M_{H}$ is a balanced hypersurface.

As a consequence, we obtain the following result for a general hypersurface of finite Catlin multitype.

Theorem 1.8**.**

Let $M$ be a smooth hypersurface and $p\in M$ be a point of finite Catlin multitype with holomorphically nondegenerate model. If its model at $p$ is neither a balanced hypersurface nor a chain hypersurface, then its automorphisms are determined by the 1-jets at $p$.

Our results also confirm a conjecture about the origin of nonlinear automorphisms of Levi degenerate hypersurfaces formulated by the first author (see [19], [20]). Recall that two vector fields $X_{1}$ and $X_{2}$ are $f$-related if $f_{*}(X_{1})=X_{2}.$

Theorem 1.9**.**

*Let $M_{H}$ be a holomorphically nondegenerate hypersurface given by (1.7) and $Y$ be a vector field of strictly positive weight. Then $Y\in\text{\rm aut}(M_{H},0)$, if and only if there exists an integer $K\geq 3$ and a holomorphic mapping $f$ from a neighbourhood of the origin in $\mathbb{C}^{3}$ into $\mathbb{C}^{K}$ which maps $M_{H}$ into a Levi nondegenerate hyperquadric $H\subseteq\mathbb{C}^{K}$ such that $Y$ is $f$-related with an infinitesimal CR automorphism of $H$. *

Let us remark that mappings of CR manifolds into hyperquadrics have been studied intensively in recent years (see e.g. [2], [11]). Here we ask in addition that the mapping be compatible with a symmetry of the hyperquadric.

As an application of our results, we will determine all possible dimensions of $\text{\rm aut}(M_{H},0)$. Recall that the dimension of the symmetry group of a Levi nondegenerate hyperquadric in $\mathbb{C}^{3}$ is equal to 15. In complex dimension two, the possible dimensions of the symmetry group of a general hypersurface are known to be $1,2,3,4,5,8$ ([24]).

The problem of possible dimensions of the symmetry groups of a general hypersurface in $\mathbb{C}^{3}$ was considered by Beloshapka in [5]. He proved that for a germ of (an arbitrary) smooth real hypersurface, which is not equivalent to a hyperquadric, the dimension is at most 11. The following result shows that for polynomial models of finite Catlin multitype the largest possible dimension is 10. Further, it demonstrates the existence of a secondary gap phenomenon in dimension 8.

Theorem 1.10**.**

Let $M_{H}$ be a holomorphically nondegenerate hypersurface given by (1.7), such that [math] is a Levi degenerate point. Then

[TABLE]

If the dimension is equal to 10, then $M_{H}$ is equivalent to (1.4). If the dimension is equal to 9, then $M_{H}$ is equivalent to (1.10). Further, the dimension of $\text{\rm aut}(M_{H},0)$ can attain any value from the set

[TABLE]

In particular, there is no $M_{H}$ with $\dim\text{\rm aut}(M_{H},0)=8$.

The paper is organized as follows. Section 2 contains the necessary definitions used in the rest of the paper. Section 3 deals with the $\mathfrak{g}_{n}$ component of the Lie algebra $\mathfrak{g}$. Section 4 contains the proofs of the results, up to Theorem 1.9. Section 5 deals with possible dimensions of $\text{\rm aut}(M_{H},0)$ and contains the proof of Theorem 1.10.

2. Preliminaries

In this section we introduce notation and recall briefly some needed definitions (for more details, see e.g. [21]).

Consider a smooth hypersurface $M\subseteq\mathbb{C}^{3}$ and let $p\in M$ be a point of finite type $m\geq 2$ (in the sense of Kohn and Bloom-Graham, [6], [16]).

Let $(z,w)$ be local holomorphic coordinates centered at $p$, where $z=(z_{1},z_{2})$ and $z_{j}=x_{j}+iy_{j}$, $j=1,2$, and $w=u+iv$. We assume that the hyperplane $\{v=0\}$ is tangent to $M$ at $p$, so $M$ is described near $p$ as the graph of a uniquely determined real valued function

[TABLE]

where $\ dF(0)=0.$ We can assume that (see e.g. [6])

[TABLE]

where $P_{m}(z,\bar{z})$ is a nonzero homogeneous polynomial of degree $m$ without pluriharmonic terms.

The definition of Catlin multitype involves rational weights associated to the variables $w,z_{1},z_{2}$. The variables $w$, $u$ and $v$ are given weight one, reflecting our choice of tangential and normal variables. The complex tangential variables $(z_{1},z_{2})$ are treated as follows.

By a weight we understand a pair of nonnegative rational numbers $\Lambda=(\lambda_{1},\lambda_{2})$, where $0\leq\lambda_{j}\leq\frac{1}{2}$, and $\lambda_{1}\geq\lambda_{2}$. Let $\Lambda=(\lambda_{1},\lambda_{2})$ be a weight, and \alpha=(\alpha_{1},\alpha_{2}),\ $\ \beta=(\beta_{1},\beta_{2})$ be multiindices. The weighted degree $\kappa$ of a monomial

[TABLE]

is then

[TABLE]

A polynomial $Q(z,\bar{z},u)$ is weighted homogeneous of weighted degree $\kappa$ if it is a sum of monomials of weighted degree $\kappa$.

Definition 2.1**.**

For a weight $\Lambda$, the weighted length of a multiindex $\alpha=(\alpha_{1},\alpha_{2})$ is defined by

[TABLE]

Similarly, if $\alpha=(\alpha_{1},\alpha_{2})$ and $\hat{\alpha}=(\hat{\alpha}_{1},\hat{\alpha}_{2})$ are two multiindices, the weighted length of the pair $(\alpha,\hat{\alpha})$ is

[TABLE]

Definition 2.2**.**

A weight $\Lambda$ will be called distinguished for $M$ if there exist local holomorphic coordinates $(z,w)$ in which the defining equation of $M$ takes form

[TABLE]

where $P(z,\bar{z})$ is a nonzero $\Lambda$ - homogeneous polynomial of weighted degree $1$ without pluriharmonic terms, and $o_{\Lambda}(1)$ denotes a smooth function whose derivatives of weighted order less than or equal to one vanish.

Distinguished weights do always exist, as follows from (2.2). For these coordinates $(z,w),$ we have $\Lambda=(\dfrac{1}{m},\dfrac{1}{m}).$

In the following we shall consider the standard lexicographic order on the set of pairs. We recall the following definition (see [8]).

Definition 2.3**.**

Let $\Lambda_{M}=(\mu_{1},\mu_{2})$ be the infimum of all possible distinguished weights $\Lambda$ with respect to the lexicographic order. The multitype of $M$ at $p$ is defined to be the pair

[TABLE]

where

[TABLE]

If none of the $m_{j}$ is infinity, we say that $M$ is of finite multitype at $p$. Clearly, since the definition of multitype includes all distinguished weights, the infimum is a biholomorphic invariant.

Coordinates corresponding to the multitype weight $\Lambda_{M}$, in which the local description of $M$ has form (2.3), with $P$ being $\Lambda_{M}$-homogeneous, are called multitype coordinates.

Definition 2.4**.**

Let $M$ be given by (2.3). We define a model hypersurface $M_{H}$ associated to $M$ at $p$ by

[TABLE]

Next let us recall the following definitions.

Definition 2.5**.**

Let $X$ be a holomorphic vector field of the form

[TABLE]

We say that $X$ is rigid if $f^{1},f^{2},g$ are all independent of the variable $w$.

We can divide homogeneous rigid vector fields into three types, and introduce the following terminology.

Definition 2.6**.**

Let $X\in\text{\rm aut}(M_{H},0)$ be a rigid weighted homogeneous vector field. $X$ is called

1. (1)

a shift if the weighted degree of $X$ is less than zero; 2. (2)

a rotation if the weighted degree of $X$ is equal to zero; 3. (3)

a generalized rotation if the weighted degree of $X$ is bigger than zero and less than one.

3. Computing $\mathfrak{g}_{n}$

We consider a holomorphically nondegenerate model hypersurface $M_{H}$ of finite Catlin multitype in $\mathbb{C}^{3}$ , given by (1.7). Our aim is to find all such hypersurfaces which posses nontrivial $\mathfrak{g}_{n}$.

For easier notation, we will write $P$ instead of $P_{C}$. If $M_{H}$ has nontrivial $\mathfrak{g}_{-\mu_{j}}$ and $X\in\mathfrak{g}_{-\mu_{j}}$, then by Lemma 6.1 in [18] there exist local holomorphic coordinates preserving the multitype (with pluriharmonic terms allowed), such that

[TABLE]

Permuting coordinates, if necessary, we will assume that $j=1$ (hence we allow $\mu_{1}<\mu_{k}$ for some $k$).

It follows from the assumption that we may write $P$ as

[TABLE]

where $P_{j}(z^{\prime},\bar{z}^{\prime})$ are homogeneous real valued polynomials in the variables $z^{\prime}=(z_{2},\dots,z_{n})$ and $P_{m}\neq 0.$

Lemma 3.1**.**

Let $X=i\partial_{z_{1}}$ be in $aut(M_{H},0)$ and $P$ be of the form (3.2). If there is a vector field $Y$ in $aut(M_{H},0)$ such that $[Y,W]=X$, then $m\leq 2$, i.e. $P$ has the form

[TABLE]

Proof.

Suppose, by contradiction, that $m>2.$ We split $Y$ according to the powers of $z_{1}$, writing

[TABLE]

where $Y_{j}$ is of the form

[TABLE]

with ${\varphi_{1}}^{j-1}(z^{\prime})={\varphi_{k}}^{j}(z^{\prime})=0$ for $j<0$ and $Y_{k}\neq 0$. Let us denote

[TABLE]

Note that by weighted homogeneity of $Y$, each coefficient is homogeneous in $z^{\prime}$.

We claim that $2m-1\leq m+k.$ Indeed, if not, applying ${\sf Re}\,Y$ to $P-v$ the first term of the right handside of (3.4) gives

[TABLE]

while all other terms are of maximal power $m+k$ with respect to $z_{1}$, a contradiction.

We will next show that ${\psi}^{k}(z^{\prime})=0$. Indeed, consider the leading term with respect to the variable $z_{1}$ in the tangency equation ${\sf Re}\,Y(P-v)=0$. We obtain

[TABLE]

where $\delta$ is the Kronecker symbol. We observe that, since $m>2,$ ${{\sf Im}\,\psi}^{k}(z^{\prime}){z_{1}}^{k+m}$ cannot contain terms in $x_{1}{y_{1}}^{m+k-1}$ and in ${y_{1}}^{m+k}.$ Hence $\psi^{k}(z^{\prime})=0.$

We will further consider two cases.

1. Let $Y^{\prime}(P_{m})\neq 0.$ We claim that $k=m-1$. Indeed, if $k>m-1$, then we have

[TABLE]

Since $m>2$ and $k>1$, the first term is harmonic in $z_{1}$, but the second one is not. That gives a contradiction. Now, using $k=m-1$, we compare degrees in $z^{\prime}$ in (3.7). By homogeneity it follows that $\varphi_{1}^{k}$ and $P_{m}$ have the same degree. Dividing by $x_{1}^{m-1}$ and looking at coefficients of $y_{1}^{k+1}$ we obtain that $\varphi_{1}^{k}$ is a constant, hence $P_{m}$ is a constant, which gives a contradiction.

2. Let $Y^{\prime}(P_{m})=0.$ We obtain

[TABLE]

Hence the result is a consequence of the following lemma.

Lemma 3.2**.**

There exist uniquely determined complex numbers $\alpha_{1},\dots,\alpha_{m-1}$ such that

[TABLE]

Proof.

By comparing terms of … we obtain the value of $\alpha_{1}$. Continuing we obtain other values of $\alpha_{j}$ ∎

This achieves the proof of the lemma. ∎

Lemma 3.3**.**

Let $X=i\partial_{z_{1}}$ be in $aut(M_{H},0)$ and $P$ be of the form (3.2), with $m=2$. If there is a vector field $Y$ in $aut(M_{H},0)$ such that $[Y,W]=X$, then $P_{m}$ is constant, hence $P$ has the form

[TABLE]

*for some nonzero real constant $C$. *

Proof.

Let $k$ be given by (3.4). First assume $k=1$. From coefficients of degree three with respect to $z_{1},$ we obtain,

[TABLE]

If $\psi^{1}(z^{\prime})\neq 0$ in (3.12), then it is a constant, by comparing terms in $y_{1}^{3}.$ Hence, by homogeneity, $P_{2}$ is constant.

Next, assume that $\psi^{1}(z^{\prime})=0.$ Comparing degrees in $z^{\prime}$, we see that $\varphi_{1}^{1}$ and $P_{2}$ have the same degree, or $\varphi_{1}^{1}=0.$ If $\varphi_{1}^{1}\neq 0,$then from the coefficients of $x_{1}y_{1}^{2}$ we obtain that $\varphi_{1}^{1}$ is a constant, hence $P_{2}$ is a constant. On the other hand, $\varphi_{1}^{1}=0$ implies

[TABLE]

which is impossible, since by positivity, the left hand side contains a nonzero balanced term in $z^{\prime}$, while the right hand side has no such terms. since $\deg\varphi_{2}^{1}=\deg P_{2}+1$ weights??.

Now assume that $k\neq 1$. For terms of degree $k+2$ we get

[TABLE]

From the coefficients of $y_{1}^{2+k},$ we see that $\psi^{k}(z^{\prime})$ is a constant. If $\psi^{k}(z^{\prime})\neq 0,$ then by homogeneity $P_{2}$ is a constant. Next, let $\psi^{k}(z^{\prime})=0.$ After dividing by $x_{1}$, the first term is harmonic in $z_{1}$, while the second one is not, unless $k=0$. But we know from the proof of the previous lemma that $k\geq m-1$, hence $k=0$ implies $P_{2}=0$. This achieves the proof of the lemma. ∎

The following lemma considers the case $m=1$. Notice that for $m=1$ the definition of mutitype implies $\mu_{1}=\mu_{2}$, dale uz nevime…

Lemma 3.4**.**

Let $X=i\partial_{z_{1}}$ be in $aut(M_{H},0)$ and $P$ be of the form (3.2) with $m=1$. Let $P_{0}$ contain no harmonic terms. There is a vector field $Y$ in $aut(M_{H},0)$ such that $[Y,W]=X$ if and only if

[TABLE]

for some $l\in\mathbb{N}$, $\alpha\in\mathbb{C}$, or

[TABLE]

*where $\epsilon\in\mathbb{R}$ and $\alpha\in\mathbb{C}$. *

Proof.

$Y$ has to be again of the form

[TABLE]

From ${\sf Re}\,Y(P-v)=0$, using ${\sf Re}\,X(P)=0$, we obtain

[TABLE]

Hence for the constant and linear terms in $z_{1}$ we have

[TABLE]

and

[TABLE]

Let $k$ be as in (3.4) and let first $k>0$. We get

[TABLE]

From the coefficient of $\bar{z}_{1}z_{1}^{k}$, we obtain that the middle term is zero. It follows that ${\sf Re}\,\varphi_{1}^{k}z_{1}^{k+1}P_{1}$ is pluriharmonic, hence $P_{1}$ is constant, which is impossible.

Now let $k=0$. We get

[TABLE]

From the coefficients of $y_{1}$, we get

[TABLE]

This implies that $P_{1}$ is harmonic, namely $P_{1}=c{\sf Re}\,\varphi_{1}^{0}$ for some $c\in\mathbb{R}$. Notice that $\varphi_{1}^{0}=0$ leads to contradiction. Indeed, if $\varphi_{1}^{0}=0$, then $\psi^{0}=0$, since $P_{1}$ cannot be constant. It follows that

[TABLE]

which is impossible, since the left hand side contains a nonzero balanced term in $z^{\prime}$, while the right hand side has no such terms, since $\deg\varphi_{2}^{0}=\deg P_{1}+1$. That gives the contradiction. Next consider the equation for the coefficients of $x_{1}$ in (3.22),

[TABLE]

Substituting $P_{1}=c{\sf Re}\,\varphi_{1}^{0}$, from the mixed terms we obtain $c=1$. Denote $P_{1}={\sf Re}\,\varphi_{1}^{0}={\sf Re}\,\alpha z^{\prime l}$. Notice that the degree of $P_{0}$ is $l+1$, by weighted homogeneity, since by the definition of the Catlin multitype $\mu_{1}=\mu_{2}$. For terms of order zero we obtain

[TABLE]

Using the form of $P_{1}$, we obtain

[TABLE]

for some $\beta,\gamma,\delta\in\mathbb{C}$. In particular, $2\varphi^{0}_{2}=\beta z^{\prime l+1}$. We write $P_{0}$ as

[TABLE]

with $A_{j_{0}}\neq 0$. Recall that $j_{0}\neq 0$ by assumption. Substituting into (3.27) and comparing coefficients of $z^{\prime j_{0}}\bar{z}^{\prime 2l+1-j_{0}}$ and $z^{\prime l+1-j_{0}}\bar{z}^{\prime l+j_{0}}$ we obtain

[TABLE]

That gives

[TABLE]

and $\beta=\frac{2}{l+1}\alpha$. It follows that $l$ is odd and $P_{0}=d|z|^{l+1}$ for some $d\in\mathbb{R}$. If $d\neq 0$, we use the explicit forms of $P_{0}$ and $P_{1}$ along with (3.22) and (3.26). By (3.23) we have

[TABLE]

which gives

[TABLE]

From (3.25) we obtain from the equation for the coefficient of $z^{\prime 2l}$

[TABLE]

which gives $l=1$. The converse part of the statement is immediate to verify, using the above calculations (see also Section 4). That finishes the proof.

∎

Next we turn to the second case, $P_{2}\neq 0$. Using scaling in the $z_{1}$ variable, we may assume $P_{2}=1$.

Lemma 3.5**.**

Let $X=i\partial_{z_{1}}$ be in $aut(M_{H},0)$ and $P$ be of the form

[TABLE]

Then there is a vector field $Y$ in $aut(M_{H},0)$ such that $[Y,W]=X$, if and only if $P$ is biholomorphically equivalent, by a change of multitype coordinates, to

[TABLE]

for some $c\in\mathbb{R}\setminus\{0\}$ and $l\in\mathbb{N}$.

Proof.

Since $X=i\partial_{z_{1}},$ we have again

[TABLE]

Without any loss of generality, we can assume that both $P_{1}$ and $P_{0}$ contain no pluriharmonic terms. Note that pluriharmonic terms in $P_{1}$ can be eliminated by a change of variables $z_{1}^{*}=z_{1}+S(z^{\prime})$, where $S$ is a holomorphic polynomial in $z^{\prime}$.

In the first part of the proof, we will show that under this assumption, $P_{1}=0$. Applying ${\sf Re}\,Y$ to $P-v$ gives

[TABLE]

Let $k$ be as in (3.4). Assume first that $k>1$. If $\varphi_{1}^{k}\neq 0$ we get

[TABLE]

which gives a contradiction, since the second term is harmonic in $z_{1}$, while the first one is not. Hence $\varphi_{1}^{k}=0$ and we have, for terms of degree $k+1$ in $z_{1}$,

[TABLE]

Looking at the coefficients of $y_{1}^{k+1}$ we see that $\psi^{k-1}$ is a real or imaginary constant, depending on the parity of $k$. If $\psi^{k-1}\neq 0$, then by homogeneity $P_{1}$ has degree at most one, hence it must be zero. If $\psi^{k-1}=0$, after dividing by $x_{1}$, the right hand side is pluriharmonic in $z_{1}$. It follows that $\delta_{k,2}=0,$ hence $k>2$. But then $\sum_{j=1}^{n}\varphi_{j}^{k}z_{1}^{k}\frac{\partial P_{1}}{\partial z_{j}}$ cannot contain any mixed terms, hence it has to vanish. So

[TABLE]

From the next equation we obtain

[TABLE]

Looking at coefficients of $y_{1}$, we obtain that $2{\sf Re}\,\sum_{j=1}^{n}\varphi_{j}^{k}z_{1}^{k}\frac{\partial P_{0}}{\partial z_{j}}$ is pluriharmonic, hence it has to vanish. That contracicts holomorphic nondegeneracy of $M_{H}$. It follows that as claimed.

Now let us assume that $k=1$. For the third order terms in $z_{1}$ we obtain

[TABLE]

This determines $\varphi_{1}^{1}$ and $\psi^{1}$, which are thus constant. Looking at terms of second order in $z_{1}$ we obtain from (3.34) and (3.31)

[TABLE]

Looking at coefficients of $y_{1}^{2}$, we obtain that $P_{1}$ is pluriharmonic, hence $P_{1}=0$.

We will further assume that $P_{1}=0$. We obtain for terms linear in $z_{1}$,

[TABLE]

which gives equations for coefficients of $x_{1}$ and $y_{1}$. Namely

[TABLE]

and

[TABLE]

Using $\varphi_{j}^{1}=\alpha z^{\prime}$ for some $\alpha\in\mathbb{C}$ and the fact that $P_{0}$ contains no harmonic terms, it follows that

[TABLE]

hence

[TABLE]

It follows that $P_{0}$ has a complex reproducing field, hence $P_{0}$ is a balanced polynomial. as claimed. That finishes the proof. ∎

4. Proofs of Theorems 1.1 - 1.9.

In this section we complete the proofs of the results stated in the introduction, up to Theorem 1.9. Theorem 1.1 is an immediate consequence of Theorem 1.2.

Proof of Theorem 1.2. We apply Lemma 3.1 - 3.5. Note that by Lemma 3.1 and Lemma 3.3, we obtain either $\mu_{1}=\frac{1}{2}$, or $\mu_{1}=\mu_{2}$. Hence we can assume without any loss of generality that $i\partial_{z_{1}}\in\text{\rm aut}(M_{H},0)$. Using suitable scaling, rotation and adding harmonic terms leads to the form given in the statement of the theorem. ∎

Theorem 1.7 follows immediately from Theorem 4.7 in [18]. Combining Theorem 1.2, 1.4 and 1.7 with the results of [18] leads to Theorem 1.8.

Proof of Theorem 1.9. If $\mathfrak{g}_{1}\neq 0$, then by Theorem 4.7 in [18] we have

[TABLE]

for some pair $\Lambda=(\lambda_{1},\lambda_{2})$ (not necessarily equal to the multitype weight). Let $K$ be the number of nonzero terms in the sum. We order the multiindices and write $P$ as

[TABLE]

Consider the hyperquadric in $\mathbb{C}^{2K+1}$ defined by

[TABLE]

and the mappping $f:\mathbb{C}^{3}\to\mathbb{C}^{2K+1}$ given by $\eta=w$ and $\zeta_{j}=z^{\alpha_{j}}$ for $j=1,\dots,2K$.

It is immediate to verify that the vector field in $aut(M_{H},0)$

[TABLE]

is $f$-related to the infinitesimal automorphism of the above hyperquadric given by

[TABLE]

Next, if $\mathfrak{g}_{n}\neq 0$ we consider the two cases from Theorem 1.2. We have $K=3$ in both cases. In the first case, we define $f$ by $\eta=w,\ \zeta_{1}=z_{1},\zeta_{2}=z_{2}^{l}$. We verify that the vector field

[TABLE]

in $aut(M_{H},0)$ is $f$-related to the infinitesimal automorphism of the hyperquadric,

[TABLE]

given by

[TABLE]

The second case is completely analogous. The case of $\mathfrak{g}_{c}$ follows from Theorem 1.2 in [19]. This finishes the proof. ∎

5. Dimension of $\text{\rm aut}(M_{H},0)$.

In this section we will again assume that $M_{H}$ is a holomorphically nondegenerate model given by (1.7). We will first prove two auxiliary lemmata. Let us denote $\mathfrak{g}_{t}=\mathfrak{g}_{-\mu_{1}}\oplus\mathfrak{g}_{-\mu_{2}}$, the part of $\text{\rm aut}(M_{H},0)$ containing complex tangential shifts.

Lemma 5.1**.**

Let there exist two regular vector fields in $\mathfrak{g}_{t}$ whose values at [math] are linearly independent over $\mathbb{R}$, but dependent over $\mathbb{C}$. Then $M_{H}$ is biholomorphic to

[TABLE]

for some $C\in\mathbb{R}$, $\alpha\in\mathbb{C}$ and homogeneous polynomial $Q$ without harmonic terms.

Proof.

Let $Z_{1}$, $Z_{2}$ be such vector fields. Without loss of generality, we may assume that $Z_{1}=i\partial_{z_{1}}$ and

[TABLE]

Note that this form is attained using transformations preserving multitype coordinates (with pluriharmonic terms allowed). The commutator of $Z_{1}$, $Z_{2}$ either vanishes, or lies in $\mathfrak{g}_{-1}$, i.e. it is a real multiple of $\partial_{w}$. This leads to \psi(z_{1},z_{2})=Cz_{1}+\alpha z_{2}^{l}\ for some $\alpha\in\mathbb{C}$, $C\in\mathbb{R}$ and $l\in\mathbb{N}$. From ${\sf Re}\,Z_{1}(P)=0$ it follows that

[TABLE]

From ${\sf Re}\,Z_{2}(P)=0$ we obtain that ${\sf Re}\,\dfrac{\partial P}{\partial z_{1}}$ is pluriharmonic. It follows that

[TABLE]

which finishes the proof. ∎

Recall that by the results of [18] and [19], $\dim\mathfrak{g}_{c}\leq 1$ and $\dim\mathfrak{g}_{1}\leq 1$. The following lemma considers the case when both components are nontrivial.

Lemma 5.2**.**

Assume that $\dim\mathfrak{g}_{c}=\dim\mathfrak{g}_{1}=1$. Then $\dim\mathfrak{g}_{0}=3$.

Proof.

Let $Z\in\mathfrak{g}_{0}$ be a rotation and $Y\in\mathfrak{g}_{1}$ be nonzero vector field. By Theorem 4.7 in [18], $Y$ has the form

[TABLE]

where the $\varphi_{j}$ have the complex reproducing field property

[TABLE]

It is immediate to verify that in Jordan normal form the linear vector field $\sum_{j=1}^{2}\varphi_{j}(z)\partial_{z_{j}}$ is diagonal with real coefficients. We can thus consider multitype coordinates in which $Y$ is a real multiple of

[TABLE]

We claim that in such coordinates, $Z$ is also diagonal. Indeed, let $X\in\mathfrak{g}_{c}$ be a nonzero vector field. The commutator of $X$ and $Y$ is of weight bigger than one, so $[X,Y]=0$, by the results of [18]. It follows that the pair $(\lambda_{1},\lambda_{2})$ is linearly independent with the multitype weights $(\mu_{1},\mu_{2})$. If $\mu_{1}\neq\mu_{2}$, any rotation is diagonal. So we may assume $\mu_{1}=\mu_{2}$, which implies $\lambda_{1}\neq\lambda_{2}$. The commutator of $Z$ with $Y$ has to be a real multiple of $Y$. Computing the commutator, it follows immediately that $Z$ is diagonal. Next, the rotations with real coefficients have dimension one, the coefficients being given by $\lambda_{1}-\mu_{1}$ and $\lambda_{2}-\mu_{2}$. Let us write $P$ in the form

[TABLE]

If there exist in addition two linearly independent imaginary rotations, we have $\alpha_{1}=\hat{\alpha}_{1}$, $\alpha_{2}=\hat{\alpha}_{2}$ whenever $A_{\alpha.\hat{\alpha}}\neq 0$ From the real rotation and weighted homogeneity we obtain a unique solution for $\alpha_{1},\alpha_{2},\hat{\alpha}_{1},\hat{\alpha}_{2}$. That contradicts holomorphic nondegeneracy of $M_{H}$. On the other hand, there is an imaginary rotation with coefficients $i\lambda_{1},i\lambda_{2}$. Hence there are two linearly independent rotations, and therefore $\dim\mathfrak{g}_{0}=3.$ ∎

Lemma 5.3**.**

There exist multitype coordinates in which every rotation is linear. Moreover,

[TABLE]

Proof.

The first part of the statement is proved in Proposition 3.9 in [18]. In the normalization of Proposition 2.6 in [18], it is immediate to check that the nonzero vector fields given by

[TABLE]

where $a,c\in\mathbb{R},\beta\in\mathbb{C}$ do not belong to $\mathfrak{g}_{0}$. Hence the space of rotations is at most four dimensional, and $\dim\mathfrak{g}_{0}\leq 5$.

∎

Proof of Theorem 1.10.  First we consider the case when $\dim\mathfrak{g}_{n}>0.$ By Theorem 1.2 $M_{H}$ is given by (1.4) or (1.10). The vector fields in $\text{\rm aut}(M_{H},0)$ of (1.4) are described explicitly in [23], which shows that the dimension is equal to 10. A completely analogous computation gives all vector fields in $\text{\rm aut}(M_{H},0)$ for (1.10), and shows that in this case, $\dim\text{\rm aut}(M_{H},0)=9.$ Indeed, we have $\dim\mathfrak{g}_{-\frac{1}{2}}=2$ and $\dim\mathfrak{g}_{\frac{1}{2}}=2$. Further, since $\mu_{1}\neq\mu_{2}$, all rotations have to be diagonal, which gives immediately $\dim\mathfrak{g}_{0}=3$. By the results of [19], we have $\dim\mathfrak{g}_{c}=0$. Since $\dim\mathfrak{g}_{-1}=\dim\mathfrak{g}_{1}=1$, we obtain $\dim\text{\rm aut}(M_{H},0)=9$.

Next we consider the case when $\dim\mathfrak{g}_{n}=0$ and show that in this case

[TABLE]

We will consider all possible dimensions of $\mathfrak{g}_{t}$, namely $0,1,2,3$. Note that if $\dim\mathfrak{g}_{t}=4$, by Lemma 5.1 it follows that $\mu_{1}=\mu_{2}=\frac{1}{2}$, hence $M_{H}$ is a Levi nondegenerate hyperquadric.

Let us first assume that $\dim\mathfrak{g}_{t}=0$. Then by Theorem 1.2, Lemma 5.2 and 5.3 we obtain immediately $\dim\text{\rm aut}(M_{H},0)\leq 7$.

Next assume that $\dim\mathfrak{g}_{t}=1$. If $\dim\mathfrak{g}_{c}+\dim\mathfrak{g}_{1}=2$, then by Lemma 5.2 $\dim\mathfrak{g}_{0}\leq 3$, and since $\dim\mathfrak{g}_{n}=0$, this leads to $\dim\text{\rm aut}(M_{H},0)\leq 7$. Now let $\dim\mathfrak{g}_{1}+\dim\mathfrak{g}_{c}\leq 1$. Without any loss of generality, we can assume $Y=\partial_{z_{1}}\in\mathfrak{g}_{t}.$ By the partial normalization (Proposition 2.6 in [18]), the five dimensional linear subspace of vector fields of the form

[TABLE]

where $\alpha,\beta\in\mathbb{C},d\in\mathbb{R}$ has trivial intersection with $\mathfrak{g}_{0}$. This follows from Lemma 5.3 and the fact that if $Z\in\text{\rm aut}(M_{H},0)$, the commutator of $Z$ and $Y$ has to be a real multiple of $Y$. It implies $\dim\mathfrak{g}_{0}\leq 4$. Since $\dim\mathfrak{g}_{n}=0$, we obtain $\dim\text{\rm aut}(M_{H},0)\leq 7$.

Next, let $\dim\mathfrak{g}_{t}=2$ and the values of two generators of $\mathfrak{g}_{t}$ at [math] are complex independent. We claim that $\dim\mathfrak{g}_{0}\leq 2$. Indeed, let $Z_{1}\in\mathfrak{g}_{-\mu_{1}}$, $Z_{2}\in\mathfrak{g}_{-\mu_{2}}$ be two vector fields, whose values at [math] are complex independent. Since their commutator is of weight $-\mu_{1}-\mu_{2}>-1$, it must vanish. We may therefore assume $Z_{1}=i\partial_{z_{1}}$, $Z_{2}=i\partial_{z_{2}}$. Hence $P$ is a function of $x_{1},x_{2}$. Again, this form is attained by a transformation preserving multitype coordinates (with pluriharmonic terms allowed). Computing commutators of $Z_{1}$ and $Z_{2}$ with a rigid element of $\mathfrak{g}_{0}$ of the form

[TABLE]

we see that the coefficients $a,b,c,d$ must be real. It follows that

[TABLE]

The last term is pluriharmonic and the remaining part is a function of $x_{1},x_{2}$. It follows that the pluriharmonic term vanishes, and we obtain

[TABLE]

It follows that $a,b,c,d$ are determined uniquely by $P$, up to a real mutiple. Note that the left hand side is nonconstant, otherwise $M_{H}$ is holomorphically degenerate. Hence $\dim\mathfrak{g}_{0}\leq 2$. It follows that $\dim\mathfrak{g}\leq 7$.

If $\mathfrak{g}_{t}$ contains two complex dependent at [math] vector fields, we use Lemma 5.1 and consider a rotation

[TABLE]

Let first $C\neq 0$. After scaling in $z_{1}$ and absorbing the mixed term into $x_{1}^{2}$, we may assume

[TABLE]

where $Q$ is different from $|z_{2}|^{l}$, since $\dim\mathfrak{g}_{n}=0$. We obtain

[TABLE]

From the coefficient of $y_{1}$ we obtain ${\sf Im}\,(c\dfrac{\partial Q}{\partial z_{2}})=0$, which implies $ic\partial_{z_{2}}\in\mathfrak{g}_{t}$. It follows that $c=0$. Hence

[TABLE]

The first three terms have different powers of $z_{1}$, so they must all be pluriharmonic. It follows that $a$ is purely imaginary and $b=d=0$. Hence $\dim\mathfrak{g}_{0}\leq 2$ which implies $\dim\mathfrak{g}\leq 6$.

Let now $C=0$, i.e.

[TABLE]

Without any loss of generality, we will assume that $Q$ contains no harmonic terms, and no $(1,l)$ terms, which can be absorbed into the first term by a change of variables $z^{*}_{1}=z_{1}+\gamma z_{2}.$ We have

[TABLE]

From the equation for the coefficient of $z_{1}\bar{z}_{1}z_{2}^{l-1}$ we obtain that $c=0$. From terms of order zero in $z_{1}$, looking at the coefficient of $\bar{z}_{2}z_{2}^{l}$ we obtain $b=0$, since $Q$ contains no $(1,l)$ terms. From the coefficient of $z_{1}\bar{z}_{2}^{l}$ we obtain $a+l\bar{d}=0$. This gives $\dim\mathfrak{g}_{0}\leq 3$ and $\dim\mathfrak{g}\leq 7$.

If $\dim\mathfrak{g}_{t}=3$, we use Lemma 5.1 to conclude that $P$ must be biholomorphic to

[TABLE]

where $l>2$. Since $\mu_{1}\neq\mu_{2}$, the rotations have to be diagonal. It is immediate to verify that the only rotations are real multiples of $iz_{1}\partial_{z_{1}}$. Hence $\dim\mathfrak{g}_{0}=2$ and $\dim\mathfrak{g}\leq 7$.

To finish the proof, we give examples of models with $\text{\rm aut}(M_{H},0)=7,6,5,4,3,2.$ The hypersurface given by

[TABLE]

admits the same rotations as the sphere, hence $\dim\mathfrak{g}_{0}=5$ and $\dim\text{\rm aut}(M_{H},0)=7$.

Six dimensional $\text{\rm aut}(M_{H},0)$ occurs for

[TABLE]

which has $\dim\mathfrak{g}_{t}=3$ and $\dim\mathfrak{g}_{0}=2$.

Dimension five occurs for

[TABLE]

where $\dim\mathfrak{g}_{0}=3$.

Dimension four occurs for

[TABLE]

where $\dim\mathfrak{g}_{t}=2.$

Dimension three occurs for

[TABLE]

with $\dim\mathfrak{g}_{t}=1.$

Dimension two occurs for a generic model, e.g.,

[TABLE]

This finishes the proof of the theorem. ∎