$TL=j^{\infty}_{p}L=\sum_{j}(Ta_{j})\partial_{x_{j}}\in{\mathbb{K}}[[x-p]]\otimes{\mathbb{K}}^{m}.$

For every smooth real submanifold $O\subset{\mathbb{R}}^{m}$ of codimension
$k$ and $p\in O$, the set $I_{p}(O)$ of germs at $p$ of smooth real functions
in ${\mathbb{R}}^{m}$ vanishing on $O$ forms an ideal in the ring of all germs
at $p$ of smooth real functions in ${\mathbb{R}}^{m}$. The ideal $I_{p}(O)$ is
generated by a $k$-tuple $(f_{1},\ldots,f_{k})$ whose differentials at $p$

(11.1) $df_{1}(p),\ldots,df_{k}(p),$

are linearly independent. Then the set of all Taylor series

$TI_{p}(O)=j^{\infty}_{p}I(O)=\\{Tf(x):f\in
I_{p}(O)\\}\subset{\mathbb{R}}[[x-p]]$

forms an ideal generated by the $k$-tuple $(Tf_{1},\ldots,Tf_{k})$ of the
Taylor series of the generators $(f_{1},\ldots,f_{k})$ and the formal
differentials

$Tdf_{1},\ldots,Tdf_{k},$

when evaluated at $p$, coincide with (11.1), and hence are also linearly
independent. Note that a formal power series in $(x-p)$ can only be evaluated
at $x=p$.

Vice versa, for any ideal $I\subset{\mathbb{R}}[[x-p]]$ generated by $k$
elements $F_{1}\ldots,F_{k}\in I$ with linearly independent differentials

$dF_{1}(p),\ldots,dF_{k}(p),$

there is a smooth real submanifold $O$ through $p$ in ${\mathbb{R}}^{m}$ of
codimension $k$, for which $I=TI_{p}(O)$. Indeed, it suffices to choose a
smooth function germ $f_{j}$ at $p$ with $Tf_{j}=F_{j}$ for each
$j=1,\ldots,k$, and take $O$ defined near $0$ by $f_{1}=\ldots=f_{k}=0$. Here
we use the classical Borel theorem on surjectivity of the Taylor series map
$T$ for smooth functions.

###  11.2. Manifold ideals and formal submanifolds

The above correspondence between submanifolds $O$ and their formal power
series ideals $TI_{p}(O)=j^{\infty}_{p}I_{p}(O)$ motivates the following
abstract definition:

######  Definition 11.1.

An ideal $I\subset{\mathbb{K}}[[x]]$, where $x=(x_{1},\ldots,x_{m})$, is a
manifold ideal if it is generated by $k$ elements $F_{1},\ldots,F_{k}\in I$
with $F_{1}(0)=\ldots=F_{k}(0)$ and linearly independent differentials

$dF_{1}(0),\ldots,dF_{k}(0).$

A manifold ideal $I\subset{\mathbb{K}}[[x]]$ is said to define a formal
submanifold $O$ of ${\mathbb{K}}^{n}$ with its formal ring given by

${\mathbb{K}}[[O]]:={\mathbb{K}}[[x]]/I.$

One major source of formal submanifolds over ${\mathbb{K}}={\mathbb{R}}$ is
provided by $\infty$-jets of real submanifolds $S\subset{\mathbb{R}}^{m}$. If
$I_{p}(S)$ is the ideal of germs at $p\in S$ of smooth functions in a
neighborhood of $p$ in ${\mathbb{R}}^{m}$ vanishing on $S$, then

$j^{\infty}_{p}I(O)=\\{j^{\infty}_{p}f:f\in I(O)\\}\subset{\mathbb{R}}[[x-p]]$

defines a manifold ideal as can be seen by taking $\infty$-jets of a local
system of smooth defining equations of $S$ with independent differentials. We
denote by $j^{\infty}_{p}S\subset{\mathbb{R}}^{m}$ the corresponding formal
submanifold given by the ideal $j^{\infty}_{p}I(O)$.

By the implicit function theorem for formal power series, after possible
reordering coordinates, a manifold ideal $I\subset{\mathbb{K}}[[x]]$ with $k$
generators also admits generators of the form

(11.2)
$x_{j+m-k}-\varphi_{j}(x^{\prime}),\quad\varphi_{j}\in{\mathbb{K}}[[x^{\prime}]],\quad
x^{\prime}=(x_{1},\ldots,x_{m-k}),\quad j=1,\ldots,k,$

and hence the formal power series map

$A(x^{\prime}):=(x^{\prime},\varphi_{1}(x^{\prime}),\ldots,\varphi_{k}(x^{\prime}))\colon({\mathbb{K}}^{m-k},0)\to({\mathbb{K}}^{m},0)$

satisfies

${\rm rank\,}dA(0)=k,\quad F\circ A=0\text{ for all }F\in I.$

In particular, for any $F\in I$, its differential $dF(0)$ vanishes on the
image $dA({\mathbb{K}}^{m-k})\subset{\mathbb{K}}^{m}$, and hence the number of
any generators of $I$ with linearly independent differentials at $0$ is always
$k$. We say that the formal submanifold $O$ defined by $I$ has codimension $k$
and dimension $m-k$.

######  Definition 11.2.

Given a formal submanifold $O\subset{\mathbb{K}}^{m}$ of codimension $k$, any
formal power series map

$H\colon({\mathbb{K}}^{m-k},0)\to({\mathbb{K}}^{m},0),\quad
H\in{\mathbb{K}}[[x^{\prime}]]\otimes{\mathbb{K}}^{m},$

satisfying (11.2) is called a (formal) parametrization of $O$.

Again, as a consequence of the implicit function theorem for formal power
series, any two formal parametrizations $A_{1},A_{2}$ of $O$ satisfy

$A_{2}=A_{1}\circ\psi,\quad\psi\in({\mathbb{K}}^{k},0)\to({\mathbb{K}}^{k},0),\quad{\rm
rank\,}d\psi(0)=k,$

for a suitable formal power series map $\psi$.

######  Remark 11.3.

If $S\subset{\mathbb{R}}^{m}$ is a smooth real submanifold with $p\in S$, and
$a\colon({\mathbb{R}}^{s},0)\to({\mathbb{R}}^{m},p)$ a germ of a local smooth
parametrization of $S$, then its Taylor series $A=j^{\infty}_{p}a$ is a formal
parametrization of $j^{\infty}_{p}S$. Vice versa, given any formal
parametrization $A$ of $j^{\infty}_{p}S$, there exists a germ
$a\colon({\mathbb{R}}^{s},0)\to({\mathbb{R}}^{m},p)$ of a local smooth
parametrization of $S$ with $A=j^{\infty}_{0}a$. Indeed, writing $S$ locally
as graph $x^{\prime\prime}=\varphi(x^{\prime})$ in coordinates
$(x^{\prime},x^{\prime\prime})$ and splitting $A$ as
$(A^{\prime},A^{\prime\prime})$ in these coordinates, we can construct $a$ as
$(a^{\prime},\varphi\circ a^{\prime})$ where $a^{\prime}$ is any smooth germ
at $0$ with $A^{\prime}=j^{\infty}_{0}a^{\prime}$.

Since we cannot talk about “points” in a formal submanifold $O$, we only
consider $O$ at the level of formal ideals, i.e. $O$ is identified with its
manifold ideal. We write $I=I(O)$ for the ideal $I\subset{\mathbb{K}}[[x]]$
defining $O$.

Using the language of ideals, we can transfer ideas and concepts from
Algebraic Geometry to formal submanifolds. For example, given two formal
submanifolds $X,Y\subset{\mathbb{K}}^{n}$, their inclusion $X\subset Y$ is
defined as $I(Y)\supset I(X)$. In that situation, it will be useful to have
joint formal parametrizations:

######  Lemma 11.4 (joint formal parametrization).

Let $X\subset Y\subset{\mathbb{K}}^{m}$ be formal submanifolds and $s=\dim Y$,
$s^{\prime}=\dim X$ be their dimensions with
$s^{\prime\prime}:=s-s^{\prime}\geq 0$. Then there exists a formal
parametrization

$A(t^{\prime},t^{\prime\prime})\colon({\mathbb{K}}^{s^{\prime}}\times{\mathbb{K}}^{s^{\prime\prime}},0)\to({\mathbb{K}}^{m},0)$

of $Y$ such that the restriction
$A(t^{\prime},0)=A|_{{\mathbb{K}}^{s^{\prime}}}\colon({\mathbb{K}}^{s^{\prime}},0)\to({\mathbb{K}}^{m},0)$
is a formal parametrization of $X$.

###### Proof.

Let $B\colon({\mathbb{K}}^{s},0)\to({\mathbb{K}}^{m},0)$ be a formal
parametrization of $Y$. Then it is easy to see that the pullback
$B^{*}(I(X))\subset{\mathbb{K}}[[t]]$ is a manifold ideal, hence admitting
generators of the form (11.2). After a formal change of coordinates
$t=(t^{\prime},t^{\prime\prime})\in{\mathbb{K}}^{s^{\prime}}\times{\mathbb{K}}^{s^{\prime\prime}}$,
these generators become coordinate functions in the variable
$t^{\prime\prime}$, leading to the desired property. ∎

Note that, unlike manifold ideals in ${\mathbb{R}}[[x]]$ arising as Taylor
series map images of ideals defined by smooth real submanifolds, a manifold
ideal $I\subset{\mathbb{C}}[[x]]$ corresponds to a complex submanifold through
$0$ in ${\mathbb{C}}^{m}$ if and only if $I$ has convergent power series as
generators.

When ${\mathbb{K}}={\mathbb{R}}$, for an ideal $I\subset{\mathbb{R}}[[x]]$, we
also consider its complexification

${\mathbb{C}}I:=\left\\{f+ig:f,g\in I\right\\}\subset{\mathbb{C}}[[x]],$

which is the ideal in ${\mathbb{C}}[[x]]$ generated by $I$, that is invariant
under conjugation. If $I$ is a manifold ideal in ${\mathbb{R}}[[x]]$ with
generators $F_{1},\ldots,F_{k}$, then ${\mathbb{C}}I$ is a manifold ideal in
${\mathbb{C}}[[x]]$ with the same generators. Vice versa, if
$\widetilde{I}\subset{\mathbb{C}}[[x]]$ is a manifold ideal invariant under
the conjugation, it equals the complexification ${\mathbb{C}}I$ of a real
manifold ideal $I\subset{\mathbb{R}}[[x]]$.

For background material and further results on formal submanifolds, the reader
is referred to [BER00, BRZ01, BMR02, BER03, MMZ03, DJL12].

###  11.3. Formal tangent spaces and vector fields

Since we can evaluate formal power series at the origin, we can define the
tangent space $T_{0}O$ for every formal submanifold as follows:

######  Definition 11.5 (tangent space of a formal submanifold).

For a formal submanifold $O\subset{\mathbb{K}}^{n}$, define its tangent space
at $0$ as

$T_{0}O:=\\{\xi\in T_{0}{\mathbb{K}}^{n}:df(\xi)=0\text{ for all }f\in
I(O)\\}\subset T_{0}{\mathbb{K}}^{n}\cong{\mathbb{K}}^{n}.$

Note that $0$ is the only point where we can evaluate formal power series,
hence only the tangent space at $0$ is defined for a formal submanifold.
Another standard notion is that of formal tangent vector fields to a formal
submanifold:

######  Definition 11.6.

If $O\subset{\mathbb{K}}^{n}$ is a formal submanifold, a formal vector field
$L\in{\mathbb{K}}[[x]]\otimes{\mathbb{K}}^{n}$ is said to be tangent to $O$
whenever

$L(I(O))\subset I(O).$

We denote by $D_{O}$ the ${\mathbb{K}}[[x]]$-module of all formal vector
fields tangent to $O$, which is obviously a Lie algebra with respect to the
Lie bracket.

With generators of $I$ chosen as in (11.2), the space $D_{O}$ is spanned,
modulo $I(O)$, by

(11.3) $D_{O}=\text{\rm
span}_{{\mathbb{K}}[[x]]}\big{\\{}\partial_{x_{s}}+\sum_{j}\frac{\partial\varphi_{j}}{\partial
x_{s}}\partial_{x_{j+m-k}}:s=1,\ldots,m-k\big{\\}}+I(O)\cdot{\mathbb{K}}[[x]]\otimes{\mathbb{K}}^{m},$

where as before ${\mathbb{K}}[[x]]\otimes{\mathbb{K}}^{m}$ is identified with
the space of all formal vector fields in ${\mathbb{K}}^{m}$. In particular,
evaluating at $0$, it is easy to see that

(11.4) $D_{O}(0):=\\{L_{0}:L\in D_{O}\\}=T_{0}O.$

More generally, to span $D_{O}$ over the ring ${\mathbb{K}}[[x]]$ by a
collection of formal vector fields, it suffices to span the tangent space at
$0$:

######  Lemma 11.7.

Let $O\subset{\mathbb{K}}^{m}$ be a formal submanifold and
$L^{1},\ldots,L^{l}\in D_{O}$ be such that

$T_{0}O={\sf span}_{\mathbb{K}}\\{L^{1}_{0},\ldots,L^{l}_{0}\\}.$

Then

$D_{O}={\sf
span}_{{\mathbb{K}}[[x]]}\\{L^{1},\ldots,L^{l}\\}+I(O)\cdot{\mathbb{K}}[[x]]\otimes{\mathbb{K}}^{m}.$

###### Proof.

Reordering coordinates if necessary, we may assume that $I(O)$ has generators
as in (11.2), so that (11.3) holds, i.e.

(11.5) $D_{O}={\sf
span}_{{\mathbb{K}}[[x]]}\\{X^{s}:s=1,\ldots,m-k\\}+I(O)\cdot{\mathbb{K}}[[x]]\otimes{\mathbb{K}}^{m},\quad
X^{s}=\partial_{x_{s}}+\sum_{j}\frac{\partial\varphi_{j}}{\partial
x_{s}}\partial_{x_{j+m-k}}.$

In particular, we can write

$L^{j}=\sum_{s}A^{j}_{s}X^{s}\mod I(O),$

where $A^{j}_{s}\in{\mathbb{K}}[[x]]$. Since $L^{j}_{0}$ span $T_{0}O$, the
matrix $(A^{j}_{s}(0))$ evaluated at $0$ has rank $m-k$. Reordering $L^{j}$ if
necessary, we may assume the first $(m-k)\times(m-k)$ minor of
$(A^{j}_{s}(0))$ is nonzero. Then the same minor of $(A^{j}_{s})$ is
invertible in ${\mathbb{K}}[[x]]$. Taking the inverse, we conclude that each
$X^{j}$ is contained in the span of $L^{j}$ modulo $I(O)$, which proves the
desired conclusion in view of (11.5). ∎

For a pair of formal submanifolds $O\subset V$, we can approximate vector
fields in $D_{O}$ by those in $D_{V}$ in the following sense:

######  Lemma 11.8.

Let $O\subset V\subset{\mathbb{K}}[[x]]$ be formal submanifolds. Then for
every $L\in D_{O}$ there exists $\widetilde{L}\in D_{V}$ with

(11.6) $\widetilde{L}=L\mod I(O).$

In particular, it follows that $\widetilde{L}\in D_{O}\cap D_{V}$.

###### Proof.

Reordering coordinates
$x=(x^{\prime},x^{\prime\prime})\in{\mathbb{K}}^{m-k}\times{\mathbb{K}}^{k}$
if necessary, we may assume that $I(V)$ has generators as in (11.2), so that
(11.3) holds for $D_{V}$, i.e.

$D_{V}={\sf
span}_{{\mathbb{K}}[[x]]}\\{X^{s}:s=1,\ldots,m-k\\}+I(V)\cdot{\mathbb{K}}[[x]]\otimes{\mathbb{K}}^{m},\quad
X^{s}=\partial_{x_{s}}+\sum_{j}\frac{\partial\varphi_{j}}{\partial
x_{s}}\partial_{x_{j+m-k}}.$

Writing $L=\sum_{j=1}^{m}a_{j}\partial_{x_{j}}$, $a_{j}\in{\mathbb{K}}[[x]]$,
we set $\widetilde{L}:=\sum_{s=1}^{m-k}a_{s}X^{s}\in D_{V}$. Since $O\subset
V$, hence $I(V)\subset I(O)$, we have $L(I(V))\subset I(O)$. Writing this
condition for the generators $x_{j+m-k}-\varphi_{j}(x^{\prime})$ of $(V)$, we
obtain

$L(x_{j+m-k}-\varphi_{j}(x^{\prime}))=a_{j+m-k}-\sum_{s}a_{s}\frac{\partial\varphi_{j}}{\partial
x_{s}}\in I(O)$

and hence

$L-\widetilde{L}=\sum_{s=1}^{m-k}a_{s}\partial_{x_{s}}+\sum_{j=1}^{k}a_{j+m-k}\partial_{x_{j+m-k}}-\sum_{s=1}^{m-k}a_{s}(\partial_{x_{s}}+\sum_{j=1}^{k}\frac{\partial\varphi_{j}}{\partial
x_{s}}\partial_{x_{j+m-k}})\in
I(O)\cdot{\mathbb{K}}[[x]]\otimes{\mathbb{K}}^{m}$

proving (11.6). The inclusion $\widetilde{L}\in D_{O}$ follows immediately
from (11.6) and the definition of $D_{O}$ (Definition 11.6). ∎

###  11.4. Formal CR submanifolds

It is customary to write a real formal power series $f$ in
${{\mathbb{C}}^{n}}\cong{\mathbb{R}}^{2n}$ as series in $z$ and $\bar{z}$:

(11.7)
$f(z,\bar{z})=\sum_{\alpha\beta}c_{\alpha\beta}z^{\alpha}\bar{z}^{\beta}\in{\mathbb{C}}[[z,\bar{z}]],\quad
c_{\alpha\beta}=\overline{c_{\beta\alpha}},\quad z=(z_{1},\ldots,z_{n}),$

which we identify with the complexified series

(11.8)
$\sum_{\alpha\beta}c_{\alpha\beta}z^{\alpha}\zeta^{\beta}\in{\mathbb{C}}[[z,\zeta]],\quad
c_{\alpha\beta}=\overline{c_{\beta\alpha}},\quad
z=(z_{1},\ldots,z_{n}),\;\zeta=(\zeta_{1},\ldots,\zeta_{n}).$

With that identification, the complex conjugation $\sigma$ for series (11.7)
is given by

(11.9)
$\sigma\left({\sum_{\alpha\beta}c_{\alpha\beta}z^{\alpha}\zeta^{\beta}}\right)=\sum_{\alpha\beta}\overline{c_{\alpha\beta}}\zeta^{\alpha}z^{\beta}.$

Then a real formal submanifold $O\subset{\mathbb{C}}^{n}$ has its ideal
$I=I(O)$ in the ring

$\\{f\in{\mathbb{C}}[[z,\zeta]]:\sigma f=f\\}\cong{\mathbb{R}}[[x,y]],\quad
x,y\in{\mathbb{R}}^{n},$

whose complexification

${\mathbb{C}}I=\\{f+ig:f,g\in I\\}$

is a $\sigma$-invariant ideal in ${\mathbb{C}}[[z,\zeta]]$. Conversely, a
$\sigma$-invariant ideal in ${\mathbb{C}}[[z,\zeta]]$ is a complexification of
a real ideal in the ring $\\{f\in{\mathbb{C}}[[z,\zeta]]:\sigma f=f\\}$, and a
$\sigma$-invariant manifold ideal in ${\mathbb{C}}[[z,\zeta]]$ is a
complexification of a real manifold ideal in
$\\{f\in{\mathbb{C}}[[z,\zeta]]:\sigma f=f\\}$. We call

${\mathbb{C}}I={\mathbb{C}}I(O)\subset{\mathbb{C}}[[z,\zeta]]$

the complexified ideal of $O$.

For a complex formal submanifold
$O\subset{{\mathbb{C}}^{n}}\times{{\mathbb{C}}^{n}}$ given by a manifold ideal
$I\subset{\mathbb{C}}[[z,\zeta]]$, define the spaces of $(1,0)$ and $(0,1)$
formal tangent vector fields by

(11.10) ${\mathcal{D}}^{10}_{O}=\left\\{L=\sum
a_{j}(z,\zeta)\partial_{z_{j}}:L(I)\subset
I\right\\},\quad{\mathcal{D}}^{01}_{O}=\left\\{L=\sum
b_{j}(z,\zeta)\partial_{\zeta_{j}}:L(I)\subset I\right\\},$

where $a_{j}(z,\zeta),b_{j}(z,\zeta)\in{\mathbb{C}}[[z,\zeta]]$, and the space
of complex formal tangent vector fields by

${\mathbb{C}}D_{O}=\left\\{L=\sum(a_{j}(z,\zeta)\partial_{z_{j}}+b_{j}(z,\zeta)\partial_{\zeta_{j}}):L(I)\subset
I\right\\},$

while the space $D_{O}$ of real formal tangent vector fields is

$D_{O}=\\{L\in{\mathbb{C}}D_{O}:\sigma L=L\\}.$

If $O$ is a real formal submanifold, i.e. defined by a real submanifold ideal
$I(O)\subset\\{f\in{\mathbb{C}}[[z,\zeta]]:\sigma f=f\\}$, the corresponding
complexification ${\mathbb{C}}I(O)$ in ${\mathbb{C}}[[z,\zeta]]$ is generated
by the $\sigma$-invariant formal power series (11.8) and one has

(11.11)
${\mathcal{D}}^{01}_{O}=\sigma{{\mathcal{D}}^{10}_{O}}=\left\\{\sum\bar{a}_{j}(\zeta,z)\partial_{\zeta_{j}}:\sum
a_{j}(z,\zeta)\partial_{z_{j}}\in{\mathcal{D}}^{10}_{O}\right\\},$

where we keep the notation $\sigma$ for complex formal vector fields given by

$\sigma\left(\sum(a_{j}\partial_{z_{j}}+b_{j}\partial_{\zeta_{j}})\right)=\sum(\sigma
a_{j}\partial_{\zeta_{j}}+\sigma b_{j}\partial_{z_{j}}),$

or, equivalently, by

$\sigma(L)f=\sigma(L(\sigma f)),\quad f\in{\mathbb{C}}[[z,\zeta]].$

It is easy to see that the evaluations

${\mathcal{D}}^{10}_{O}(0)=\left\\{\sum a_{j}(0)\partial_{z_{j}}:\sum
a_{j}(z,\zeta)\partial_{z_{j}}\in{\mathcal{D}}^{10}_{O}\right\\},$
${\mathcal{D}}^{01}_{O}(0)=\left\\{\sum a_{j}(0)\partial_{z_{j}}:\sum
a_{j}(z,\zeta)\partial_{\zeta_{j}}\in{\mathcal{D}}^{01}_{O}\right\\},$

are contained in the corresponding $(1,0)$ and $(0,1)$ tangent spaces at $0$:

$H^{10}_{0}O=H^{10}_{0}{{\mathbb{C}}^{n}}\cap{\mathbb{C}}T_{0}O,\quad
H^{01}_{0}O=H^{01}_{0}{{\mathbb{C}}^{n}}\cap{\mathbb{C}}T_{0}O,$

i.e.

${\mathcal{D}}^{10}_{O}(0)\subset
H^{10}_{0}O,\quad{\mathcal{D}}^{01}_{O}(0)\subset H^{01}_{0}O.$

The opposite inclusions do not hold in general and, in the case $O$ is real-
analytic, are equivalent to $O$ being CR in a neighborhood of $0$. This
motivates the following definition (cf. [DJL12, p. 105]:

######  Definition 11.9.

A real formal submanifold $O\subset{{\mathbb{C}}^{n}}$ is said to be CR if

$H^{10}_{0}O\subset{\mathcal{D}}^{10}_{O}(0),$

or, equivalently, if every vector in $H^{10}_{0}O$ extends to a formal vector
field in ${\mathcal{D}}^{10}_{O}$.

In view of (11.11), Definition 11.9 can be restated for $(0,1)$ vector fields,
i.e. a real formal submanifold $O\subset{\mathbb{C}}^{n}$ is CR if and only if
${\mathcal{D}}^{10}_{O}(0)=H^{10}_{0}O$ or, equivalently,
${\mathcal{D}}^{01}_{O}(0)=H^{01}_{0}O$. We quote the following structure
result for formal CR submanifolds from Della Salla-Juhlin-Lamel [DJL12, Lemma
1]:

######  Lemma 11.10.

Let $O\subset{{\mathbb{C}}^{n}}$ be a formal CR submanifold of codimension $d$
with its complexified ideal
$I={\mathbb{C}}I(O)\subset{\mathbb{C}}[[z,\zeta]]$. Then there exists integers
$d_{1},d_{2},d_{3}\geq 0$ with $n=d_{1}+d_{2}+d_{3}$, $d=2d_{1}+d_{3},$ such
that after renumbering coordinates $z$ and $\zeta$ if necessary, we can write

$z=(z^{1},z^{2},z^{3}),\;\zeta=(\zeta^{1},\zeta^{2},\zeta^{3})\in{\mathbb{C}}^{d_{1}}\times{\mathbb{C}}^{d_{2}}\times{\mathbb{C}}^{d_{3}},$

and choose a set of generators of $I$ given by the components of

$z^{1}-\varphi(z^{2},z^{3}),\quad\zeta^{1}-\sigma\varphi(\zeta^{2},\zeta^{3}),\quad\zeta^{2}-R(z^{2},z^{3},\zeta^{3})$

for some vector-valued formal power series
$\varphi\in{\mathbb{C}}[[z^{2},z^{3}]]^{d_{1}}$,
$R\in{\mathbb{C}}[[z^{2},z^{3},\zeta^{3}]]$.

It is well-known that real-analytic CR submanifolds $O$ are generic in their
local intrinsic complexifications defined as minimal complex submanifolds
(locally) containing $O$, see e.g. [BER99, Chapter I]. We shall need the
following formal analogue:

######  Corollary 11.11.

Let $O\subset{{\mathbb{C}}^{n}}$ be a formal CR submanifold. Then the maximal
formal holomorphic ideal $I^{\prime}\subset{\mathbb{C}}[[z]]$ contained in the
complexification ${\mathbb{C}}I(O)=I(O)+iI(O)\subset{\mathbb{C}}[[z,\zeta]]$,
i.e.

$I^{\prime}:={\mathbb{C}}I(O)\cap{\mathbb{C}}[[z]],$

is a complex manifold ideal, (where ${\mathbb{C}}[[z]]$ is identified with the
subring of all power series in ${\mathbb{C}}[[z,\zeta]]$ independent of
$\zeta$).

###### Proof.

Using the notation and generators of $I(O)$ in Lemma 11.10, it follows that
$I^{\prime}$ is generated over ${\mathbb{C}}[[z]]$ by the components of
$z^{1}-\varphi(z^{2},z^{3})$, hence is a manifold ideal as claimed. ∎

######  Definition 11.12.

The intrinsic complexification of a formal CR submanifold
$O\subset{{\mathbb{C}}^{n}}$ is the complex formal submanifold
$V\subset{{\mathbb{C}}^{n}}$ defined by the ideal

$I^{\prime}={\mathbb{C}}I(O)\cap{\mathbb{C}}[[z]].$

To any formal complex submanifold $V\subset{{\mathbb{C}}^{n}}$ given by an
ideal $I(V)\subset{\mathbb{C}}[[z]]$, there is also the associated real formal
submanifold $V^{\mathbb{R}}\subset{{\mathbb{C}}^{n}}$ given by the real formal
ideal $I(V^{\mathbb{R}})\subset\\{f\in{\mathbb{C}}[[z,\zeta]]:f=\sigma f\\}$
whose complexification ${\mathbb{C}}I(V^{\mathbb{R}})$ is generated by

$I(V)\oplus\sigma I(V)=\\{f+\sigma g:f,g\in
I(V)\\}\subset{\mathbb{C}}[[z,\zeta]],$

and using these generators, it follows that

${\mathbb{C}}D_{V^{\mathbb{R}}}=D_{V^{\mathbb{R}}}^{10}\oplus
D_{V^{\mathbb{R}}}^{01}\subset{\mathbb{C}}[[z,\zeta]]\otimes{{\mathbb{C}}^{n}}.$

In the sequel, we slightly abuse the notation writing $V$ for both complex
formal submanifold and its associated real formal submanifold
$V^{\mathbb{R}}$. We have the following consequence of Lemma 11.10:

######  Corollary 11.13.

If $V$ is the intrinsic complexification of a formal CR submanifold
$O\subset{{\mathbb{C}}^{n}}$, it follows that $O$ is generic in $V$ in the
sense that

(11.12) $T_{0}O+JT_{0}O=T_{0}V,$

where $J$ is the standard complex structure acting on the standard basis
vectors by

$J\partial_{z_{j}}=i\partial_{z_{j}},\quad
J\partial_{\zeta_{j}}=-i\partial_{\zeta_{j}},\quad j=1,\ldots,n.$

As an application, we obtain a convenient expression of tangent vector fields
to the complexification $V$ in terms of vector fields tangent to both $O$ and
$V$:

######  Corollary 11.14.

If $O\subset V\subset{{\mathbb{C}}^{n}}$ are formal submanifolds satisfying
(11.12), it follows that

(11.13) $D_{V}=(D_{O}\cap D_{V})+J(D_{O}\cap D_{V})\mod I(V).$

In particular, (11.13) holds when $O$ is a formal CR submanifold and $V$ is
the intrinsic complexification of $O$.

###### Proof.

In view of (11.4), there exists a collection of vector fields
$L^{1},\ldots,L^{s}\in D_{O}$ such that

$T_{0}O=\text{\rm span}_{{\mathbb{C}}}\\{L^{1}_{0},\ldots,L^{s}_{0}\\}.$

By Lemma 11.6, there exists approximations
$\widetilde{L}^{1},\ldots,\widetilde{L}^{s}\in D_{O}\cap D_{V}$ with
$\widetilde{L}^{j}=L^{j}\mod I(O)$. In particular, the values
$\widetilde{L}^{1}_{0},\ldots,\widetilde{L}^{s}_{0}$ also span $T_{0}O$. Then
by Corollary 11.13,

$T_{0}V=\text{\rm
span}_{{\mathbb{C}}}\\{L^{1}_{0},\ldots,L^{s}_{0},JL^{1}_{0},\ldots,JL^{s}_{0}\\}.$

Finally, since $V$ is a complex formal submanifold, each $JL^{j}$ belongs to
$D_{V}$, and the desired conclusion follow from Lemma 11.7 applied to $V$. ∎

###  11.5. Formal orbits

We make use of a result of Baouendi-Ebenfelt-Rothschild [BER03, Proposition
5.1] (an adaptation to the formal case of an unpublished proof of Nagano’s
theorem by the author) to obtain a formal variant of Nagano’s theorem:

######  Proposition 11.15 (existence of a formal orbit).

Let $\mathfrak{g}$ be Lie algrebra of formal vector fields over ${\mathbb{K}}$
in $n$ indeterminates $x=(x_{1},\ldots,x_{n})$. Then there exists an unique
formal submanifold $O\subset{\mathbb{K}}^{n}$ with $\dim
O=\dim\mathfrak{g}(0)$ such that any vector field $L\in\mathfrak{g}$ is
tangent to $O$, i.e. $L(I(O))\subset I(O)$. Moreover if
$O^{\prime}\subset{\mathbb{K}}^{n}$ is any other formal manifold such that all
vector fields in $\mathfrak{g}$ are tangent to $O^{\prime}$, then $O\subset
O^{\prime}$.

Here $\mathfrak{g}(0)\subset T_{0}{\mathbb{R}}^{n}$ denotes the subspace
consisting of all values $L_{0}$ at $0$ of $L\in\mathfrak{g}$. We call the
formal submanifold $O$ in Proposition 11.15 the orbit of $\mathfrak{g}$.

######  Remark 11.16.

Since every value $L_{0}$ is contained in $T_{0}O$, the equality $\dim
O=\dim\mathfrak{g}(0)$ holds if and only if $T_{0}O=\mathfrak{g}(0)$.

###### Proof of Proposition 11.15.

In case ${\mathbb{K}}={\mathbb{C}}$, the ${\mathbb{C}}[[x]]$-submodule
$\widetilde{\mathfrak{g}}\subset{\mathbb{C}}[[x]]\otimes{\mathbb{C}}^{n}$
generated by $\mathfrak{g}$ is a Lie algebra satisfying the assumptions of
[BER03, Proposition 5.1], which yields the existence and uniqueness of a
$\widetilde{\mathfrak{g}}$-orbit $O$. Then the same $O$ is also a
$\mathfrak{g}$-orbit, proving the statement for ${\mathbb{K}}={\mathbb{C}}$.
The uniqueness follows since any $\mathfrak{g}$-orbit is also a
$\widetilde{\mathfrak{g}}$-orbit. Finally if $O^{\prime}$ is as in the
proposition, then all vector fields in $\widetilde{\mathfrak{g}}$ are tangent
to $O^{\prime}$ and $O\subset O^{\prime}$ by the second statement of [BER03,
Proposition 5.1].

In case ${\mathbb{K}}={\mathbb{R}}$, consider the complexification

$\\{X+iY:X,Y\in\mathfrak{g}\\}\subset{\mathbb{C}}[[x]]\otimes{\mathbb{C}}^{n},$

which is a Lie subalgebra over ${\mathbb{C}}$ that is invariant under
conjugations. By the statement for ${\mathbb{K}}={\mathbb{C}}$, there exists a
$\widetilde{\mathfrak{g}}$-orbit $\widetilde{O}\subset{\mathbb{C}}^{n}$, which
is a formal submanifold over ${\mathbb{C}}$. Since $\widetilde{\mathfrak{g}}$
is invariant under conjugations and $L\overline{f}=\overline{\overline{L}f}$
for $L\in\widetilde{\mathfrak{g}}$, $f\in I(\widetilde{O})$, it follows that
$\overline{I(\widetilde{O})}$ is also a submanifold ideal defining a formal
submanifold of the same dimension as $\widetilde{O}$, to which
$\widetilde{\mathfrak{g}}$ is also tangent. Hence by the uniqueness of the
$\widetilde{\mathfrak{g}}$-orbit,
$\overline{I(\widetilde{O})}=I(\widetilde{O})$, which implies that
$I(\widetilde{O})$ is the complexification of a real manifold ideal
$I=I(O)\subset{\mathbb{R}}[[x]]$ defining a formal manifold
$O\subset{\mathbb{R}}^{n}$. Then any
$L\in\mathfrak{g}\subset\widetilde{\mathfrak{g}}$ satisfies
$L(I(\widetilde{O}))\subset I(\widetilde{O})$ and hence $L(I(O))\subset I(O)$
since $L$ is real and $I(O)=I(\widetilde{O})\cap{\mathbb{R}}[[x]]$. If
$O^{\prime}$ is any other $\mathfrak{g}$-orbit, its complexification
$\widetilde{O}^{\prime}$ is a $\widetilde{\mathfrak{g}}$-orbit, hence
$\widetilde{O}^{\prime}=\widetilde{O}$ by the uniqueness for
${\mathbb{K}}={\mathbb{C}}$, and then intersecting with ${\mathbb{R}}[[x]]$
yields $O^{\prime}=O$. Finally, if $O^{\prime}$ is as in the proposition, all
vector fields in $\widetilde{g}$ are tangent to its complexification
$\widetilde{O}^{\prime}$, hence by the case ${\mathbb{K}}={\mathbb{C}}$ of the
proposition, $\widetilde{O}\subset\widetilde{O}^{\prime}$, i.e.
$I(\widetilde{O})\supset I(\widetilde{O}^{\prime})$. Intersecting with
${\mathbb{R}}[[x]]$, we obtain $I(O)\supset I(O^{\prime})$, i.e. $O\subset
O^{\prime}$ as claimed. ∎

Taking generators of $O$ of the form (11.2) and corresponding basis of $D_{O}$
given by (11.3), and using the assumption that $\mathfrak{g}$ is a
${\mathbb{K}}[[x]]$-submodule, one can prove the following property of the
orbit:

######  Lemma 11.17.

Let $\mathfrak{g}$ be as in Proposition 11.15 that is also a
${\mathbb{K}}[[x]]$-module. Then every formal vector field that is tangent to
the orbit $O$ of $\mathfrak{g}$ is necessarily contained in $\mathfrak{g}$
modulo $O$, in fact

$D_{O}=\mathfrak{g}\mod I(O).$

This ends the Appendix section.

## Acknowledgements

The author is deeply grateful to Professor Joe Kohn, to whose memory this work
is dedicated, and without whose constant interest and encouragement this work
might not have been possible. The author would also like to thank G. Bharali,
D.W. Catlin, J.P. D’Angelo, S. Fu, A.C. Nicoara and E.J. Straube for inspiring
discussions on the subject of this work, B. Lamel and J. Lebl for sharing
Example 5.4 and X. Huang for his interest and patience in verifying many
details of the proofs.

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