# Threading an atom with light

Rodrigo G. Cortiñas<EMAIL_ADDRESS>Department of Applied Physics
and Physics, Yale University, New Haven, CT 06520, USA  Yale Quantum
Institute, Yale University, New Haven, Connecticut 06520, USA

(March 2023)

###### Abstract

We propose a ponderomotive trapping mechanism for circular Rydberg atoms that
consists in threading the Rydberg orbital with a tightly focused Gaussian
laser beam. The trap exhibits remarkable properties: it can be made
effectively linear, extremely nonlinear, or have the same trapping frequency,
depth, or nonlinearity for distant Rydberg states. The mechanism is capable of
trapping efficiently highly excited Rydberg states with a single active
electron, as well as ground state atoms using a single beam. Alternatively,
the Rydberg atom can be trapped efficiently by the same beam used to excite
it, relaxing optical constraints in well-shielded environments.

Introduction– Rydberg atoms, which are highly excited atoms with principal
quantum number $n\gg 1$, are interesting tools for quantum technologies and
fundamental science [1, 2]. They have long lifetimes and strong dipole moments
originating from their enormous size [3]. For $n\geq 50$, for example, these
atoms are larger than some viruses. These properties have been exploited to
explore the principles of quantum mechanics [4], pioneer quantum computation
and quantum simulation technologies [5, 6, 7], and served as metrological
tools [8, 9].

Recently, laser trapping of Rydberg atoms has been demonstrated [10, 11, 12,
13] and this opened new technological perspectives at the single-atom level
[14]. Rydberg atoms can now be trapped close to surfaces [15, 16, 17], moved
around at will, and used as sensitive single quantum sensors [18, 19]. As a
result, they could serve as a stable ancilla to control bosonic quantum error
correction codes in cavity quantum electrodynamics, and could be kept in
interaction with other atoms for long times to perform quantum computations
and simulations [20, 21, 22, 23, 7].

Among Rydberg atoms, circular Rydberg atoms are particularly interesting [24,
4] due to their exaggerated properties [5, 21]. These are atomic states of
maximal angular and magnetic quantum numbers $l=m=n-1$, usually realized using
alkali metal atoms with a single active electron. These atoms, having no
optically allowed transitions, can be laser trapped by a ponderomotive
potential [25, 26], which means that they are low-intensity seekers. This
constraint led the first experiments [10] to use complex wavefronts, produced
by spatial light modulators, with a dark spot surrounded by laser light. This
puts significant constraints on the optical engineering required for trapping
them, as well as requiring higher laser power needed to trap large $n>80$
states: these traps become shallower as the principal quantum number $n$ grows
until they become anti-trapping [12]. The complex light intensity distribution
typically has “holes” which makes these traps laser-power inefficient. This is
concerning when considering well-shielded cryogenic applications for quantum
simulations [21], microwave-to-optical conversion of quantum information [27,
28, 29, 30, 31], or coupling to superconducting circuits [32, 22], but
especially so when considering trapping tens of thousands of individual
Rydberg atoms for computation, and the ambition of high-fidelity Rydberg-based
quantum gates [23].

We propose here a power-efficient ponderomotive trapping mechanism that holds
these atoms from within the electronic orbital and solely requires a tightly
focused Gaussian beam. This trapping mechanism is original in that it depends
on the shape and geometry of the atomic orbital. In its simplicity, it
provides the flexibility to use the same beam to trap both the ground state
and the circular Rydberg state of an alkali metal atom. Alternatively, a short
wavelength beam, typically used for Rydberg excitation (see [10] for a
possible excitation scheme), can be used as a power-efficient Rydberg trapping
beam. Both techniques relax optical power requirements and engineering
constraints on the submicrometric alignment of the beams: this is hard in a
vacuum chamber with little optical access, and especially in cryostats.

Besides its fundamental interest as a new handle on the atomic orbital, the
practical advantages mentioned above together with a set of properties we will
derive below, make this thread trap interesting technologically. Among its
properties, we will highlight at this point the trap’s tunable nonlinearity.
The high-nonlinearity mode could be used to observe nonlinear quantum dynamics
of a single atom and, in particular, the generation of Schrödinger Kerr-cats
[33, 4]. Note, that this deviation from classical motion has not been observed
for a massive particle and has direct application to autonomous quantum error
correction [34, 35]. The opposite regime of the thread trap, the linear mode,
is preferred for applications of future Rydberg quantum computation,
simulation, and sensing.

Figure 1: Thread trap for circular Rydberg states $n=$ 60, 80, and 100. (a)
Sketch of the thread trap. (b) Radial trapping potential as a function of the
distance between the nucleus and the beam’s propagation axis $\rho$ at $Z=0$.
The focal point of the beam is at the coordinate origin. The trapping laser is
taken at a wavelength of $\lambda=420$ nm, which is a typical Rydberg
excitation laser (see, for example, [10]) and focused to a Gaussian waist of
$w_{0}=\lambda/2$. We show also a schematic of the Gaussian beam within the
atomic orbital in two limiting cases. (c) Axial trapping potential as a
function of the atomic position $Z$ along the beam’s propagation axis and
$\rho=0$. The solid lines correspond to the integration of the laser intensity
over the 3D toroidal wavefunction $\psi_{nc}$. The dashed lines correspond to
a curvilinear integration of the laser intensity along the semiclassical Bohr
orbit.

Circular Rydberg atoms– The atomic orbital of a circular Rydberg state with
principal quantum number $n$ is a toroid centered at radius
$r_{nc}=a_{0}n^{2}$ (where $a_{0}$ is the Bohr radius) with a radial fuzziness
$\Delta r_{nc}$ such that $\Delta r_{nc}/r_{nc}\ll 1$ for $n\gg 1$ [4]. From
this property follows that circular atoms have a long free-space lifetime
$\tau_{nc}\propto n^{5}$ (where $\tau_{50c}\approx 28$ ms) in non-demanding
cryogenic conditions ($\lesssim$1 K). This timescale can be understood
quantitatively in semiclassical terms: a charged electron orbiting at such
distances from the nucleus will emit Larmor radiation at a photon rate
$\approx 1/\tau_{nc}$. The photon can be shown to have an energy
$\hbar\omega_{nc}\propto n^{-3}$ 111This can be shown by a semiclassical
application of Kepler’s third law for planetary motion [4]., which reflects
the dipole transition frequency of the single dominant decay channel of these
states in free space $|nc\rangle\rightarrow|(n-1)c\rangle$ (where
$\omega_{50c}/2\pi\approx 54$ GHz and $\omega_{100c}/2\pi\approx 6$ GHz, for
example) [4]. We estimate [from the pessimistic assumptions that i) all
electron-photon scattering events are causing unwanted transitions and ii)
using the Thomson cross section $\sigma_{c}=8\pi r_{e}^{2}/3$ where $r_{e}$ is
the ‘classical radius for the electron’] that the coherence lifetime in the
thread trap is limited by photon scattering to tens of seconds 222These
considerations are specific to circular atoms and set them apart from low-
angular momentum Rydberg atoms commonly used in state-of-the-art experiments.
See [21] for a related discussion..

The preparation [38] and laser trapping [10] of cold circular atoms has been
recently demonstrated experimentally, allowing validation, to a certain
degree, of the expected theoretical properties. Using Rubidium and having no
optical transitions available, ordinary laser dipole trapping was non-viable.
Instead, a trap based on the ponderomotive potential [25, 26] was used. The
principle of ponderomotive trapping of Rydberg atoms consists in trapping the
dark-seeking Rydberg electron and, by doing so, trapping the entire atom bound
to it. The trapping energy is then obtained by integrating the laser intensity
over the electron wavefunction. The trap we propose here exploits this fact in
the regime where the atomic orbital is larger than the laser intensity
distribution at the focal point.

The thread trap– In Figure 1 (a), we show a sketch of the atom threaded by a
tightly focused Gaussian beam. In Figure 1 (b) and (c) we show the trapping
potential $V$ created taking a laser at wavelength $\lambda=420$ nm. The beam
propagates along the $Z$-axis and we name the radial (cylindrical) coordinate
$\rho$. Due to the subatomic-scale beam needed, the trap is tridimensional as
the circular atom cannot run along the axis of the strongly divergent beam.
The atom is thus localized within a fraction of the Rayleigh length along the
axial direction. It is important to note that, since this trap does not rely
on a dipole resonance, the wavelength $\lambda$ is comparatively
unconstrained. Although the trapping potential depth scales as $\lambda^{2}$
for a given laser power (see below and [25]), the main consideration for
implementation will likely be the diffraction-limited waist at
$\approx\lambda$. Note that the freedom in laser wavelength, beam waist, and
the principal quantum number $n$ provides flexibility and different
configurations of interest. We illustrate this tunability in Figure 1 by
plotting the trapping potential for $n=60,\;80$ and $100$. For $n=60$ we get
the strongest linear confinement, which vanishes as $n$ grows for a fixed beam
waist.

The circular wavefunction 333A directing electric field is needed to impose a
quantization axis. See [10] is found by solving the Schrödinger equation for
the Hydrogen atom (an excellent approximation for circular Rydberg states of
alkali metals), and reads in spherical coordinates [4]

$\psi_{nc}(r,\theta,\phi)=\frac{e^{-r/na_{0}}}{\left(\pi
a_{0}^{3}\right)^{1/2}}\frac{1}{nn!}\left(-\frac{r}{a_{0}n}\sin\theta
e^{i\phi}\right)^{n-1}.$

The integral $V\propto\int I|\psi_{nc}|^{2}d\vec{r}$, over the tightly focused
Gaussian beam of intensity $I$, is used to compute numerically the solid lines
in Figure 1 (b) and (c). In the following, however, we approximate the
circular orbital by a circular one-dimensional Bohr orbit to get analytical
expressions for the properties of the trap. The confining potential computed
under this semiclassical approximation is shown as dashed lines in Figure 1
(b) and (c). The approximation is remarkably good in the parameter regimes
explored here. The ponderomotive potential for an atom whose nucleus is
located at $(X,Y,Z)$ reads then

$\displaystyle V(X,Y,Z)\approx\frac{1}{2\pi}\int_{0}^{2\pi}$ $\displaystyle
d\theta\frac{q_{e}^{2}}{m_{e}\epsilon_{0}c}\frac{\lambda^{2}}{(2\pi
c)^{2}}\frac{\mathcal{P}_{0}}{\pi w^{2}(Z)}$ $\displaystyle\times
e^{-\frac{2}{w^{2}(Z)}\left[\left(X_{n}+X\right)^{2}+\left(Y_{n}+Y\right)^{2}\right]},$

where $q_{e}$ and $m_{e}$ are the charge and mass of the electron,
$\epsilon_{0}$ is the permittivity of free space and $c$ is the speed of light
in vacuum, $\lambda$ is the wavelength of the threading laser,
$\mathcal{P}_{0}$ is the laser power, $X_{n}=r_{nc}\cos\theta$ and
$Y_{n}=r_{nc}\sin\theta$ are the circular Rydberg electron coordinates with
respect to the atomic nucleus, $Z$ is the position of the nucleus (and of the
electron) along the laser propagation axis, and
$w(Z)=w_{0}\sqrt{1+\left(Z/Z_{R}\right)^{2}}$ is the Gaussian waist of the
laser beam, where $Z_{R}=\pi w_{0}^{2}/\lambda$ is the Rayleigh length of the
beam. Here we have placed the origin of the coordinates at the focal point of
the beam.

The mechanical trapping frequency along the axial direction for a circular
Rydberg state $|nc\rangle$ at $\rho=\sqrt{X^{2}+Y^{2}}=0$ is found by
straightforward differentiation to be

$\omega_{z}^{2}=\frac{q_{e}^{2}}{m_{e}\epsilon_{0}c}\frac{\lambda^{2}}{(2\pi
c)^{2}}\frac{2\mathcal{P}_{0}}{\pi
w_{0}^{4}}\frac{\left(2r_{nc}^{2}-w_{0}^{2}\right)e^{-2r_{nc}^{2}/w_{0}^{2}}}{Z_{R}^{2}m_{\mathrm{Rb}}},$

where $m_{\mathrm{Rb}}$ is the mass of 87Rb. The axial nonlinearity from
Taylor expanding the trapping potential around $Z=0$ as $V(\rho=0,Z)\approx
k_{0,z}+k_{2,z}Z^{2}+k_{4,z}Z^{4}$ is

$k_{4,z}=\frac{q_{e}^{2}}{m_{e}\epsilon_{0}c}\frac{\lambda^{2}}{(2\pi
c)^{2}}\frac{\mathcal{P}_{0}}{\pi
w_{0}^{6}}\frac{(2r_{nc}^{4}-4r_{nc}^{2}w_{0}^{2}+w_{0}^{4})e^{-2r_{nc}^{2}/w_{0}^{2}}}{Z_{R}^{4}},$

while the associated Kerr coefficient 444Which is nothing but the fourth order
nonlinearity in units of frequency, obtained by measuring it in units of the
zero point spread of the oscillator., defined as
$K_{z}=\frac{3\hbar}{m_{\mathrm{Rb}}}\frac{k_{4,z}}{k_{2,z}}$, reads

$K_{z}=3\hbar\frac{(2r_{nc}^{4}-4r_{nc}^{2}w_{0}^{2}+w_{0}^{4})}{m_{\mathrm{Rb}}\left(2r_{nc}^{2}-w_{0}^{2}\right)w_{0}^{2}Z_{R}^{2}}.$

Similarly, for the radial direction, we find a closed-form expression in terms
of the modified Bessel function of the first kind and of order zero [41]. It
is then easy to compute the radial small oscillation frequency and the Kerr
coefficient to be

$\omega^{2}_{\rho}=\omega^{2}_{z}\frac{2Z_{R}^{2}}{w_{0}^{2}},\;\;\;\;\;\;\;\;\;K_{\rho}=K_{z}\frac{Z_{R}^{2}}{2w_{0}^{2}}.$

Finally, note that the ponderomotive potential over the closed-shell core of
alkali metals is negligible since it is several thousand times heavier than an
electron ($V\propto 1/\textrm{mass}$).

Figure 2: Properties of the thread trap. We show trap depths, trap
frequencies, and Kerr coefficient in the thread trap configuration for a few
87Rb circular Rydberg states (colors) as a function of the trapping
wavelength. These plots are made using a beam focused to a waist of
$w_{0}=\lambda/2$. The dots correspond to the numerical integration over the
toroidal wavefunction. The solid lines correspond to the semiclassical Bohr
orbit approximation.

Properties of the thread trap– In Figure 2 we show the trap depth, the small
amplitude oscillation frequency and the Kerr nonlinearity of the trap as a
function of the trapping wavelength $\lambda$. We plot these properties for
both the radial and axial trapping axis, and for a few circular Rydberg states
of 87Rb. As solid lines, we plot the semiclassical predictions, and the dots
come from numerical integration of the laser intensity over $|\psi_{nc}|^{2}$.

Importantly, to produce this plot we have constrained the laser beam to a
diffraction-limited Gaussian waist of

$w_{0}=\frac{\lambda}{\pi\mathrm{NA}}=\frac{\lambda}{2},$

using a numerical aperture $\mathrm{NA}\approx 0.65$. Note that
$\mathrm{NA}\approx 0.7$ [42] and even $\mathrm{NA}\approx 0.9$ [43] have been
demonstrated in setups like the one required here. Requiring an attractive
potential by $\left.\partial^{2}_{\rho}V\right|_{\rho,Z=0}>0$, the trapping
condition simplifies to $\omega_{z}^{2}\propto(2r_{nc}^{2}-w_{0}^{2})>0$, and
thus reads

$2r_{nc}>\lambda/\sqrt{2},$

and it can be interpreted to mean that the Bohr atom needs to be large enough
to fit a photon (of size $\approx\lambda/\sqrt{2}$) within its orbit. In Table
1 we write trapping conditions for a few Rydberg states.

$|nc\rangle$ | $\lambda$
---|---
$|50c\rangle$ | $<374\mathrm{~{}nm}$
$|60c\rangle$ | $<538\mathrm{~{}nm}$
$|70c\rangle$ | $<733\mathrm{~{}nm}$
$|80c\rangle$ | $<957\mathrm{~{}nm}$
$|90c\rangle$ | $<1212\mathrm{~{}nm}$
Table 1: Trapping wavelengths. Trapping conditions for beams focused to
$w_{0}=\lambda/2$ and different circular states, as derived from
$\lambda<2\sqrt{2}a_{0}n^{2}$. Compare with Figure 2 (c) and (d) where the
trapping frequency nulls.

Note that one can relax this focusing condition by a non-diffraction-limited
beam at a shorter wavelength. To give an example, the condition used here for
a diffraction-limited laser at $\lambda=800$ nm can be achieved by a laser at
$\lambda~{}=~{}$300 nm with a lens of NA $<$0.3.

In Figure 2 (a) and (b) we plot the radial and axial trap depth. We remark the
high-trap depth that the thread trap provides. Comparable ponderomotive traps
for Rydberg atoms [12] are more than five (and up to twenty) times less
efficient. Their trap depth is limited by light “holes” providing typically
barriers of $<4$ µK/mWatt. Power inefficiency will be a bottleneck for
proposals addressing tens of thousands or even millions of atoms in a Rydberg
quantum computer [44, 7], where Rydberg trapping will probably be a
requirement [23].

We see from Figure 2 (c) and (d) that essentially any pair of these circular
Rydberg levels present a focusing diameter $\lambda$ at which the trapping
frequencies are equal. This condition, for consecutive states $n$ and $n\pm 1$
(or $n$ and $n\pm 2$), happens near maximal trap depth. The condition along
the radial direction coincides with the condition along the axial direction.
This is a notable occurrence in the thread trap since it is a property of the
convolution of the Gaussian beam with the circular orbit. Its origin is
unrelated to the conditions of equal polarizability found at particular
wavelengths in ground-state dipole traps [45].

In Figure 2 (e) and (f) the nonlinearities, as measured by the radial
($K_{\rho}$) and longitudinal ($K_{z}$) Kerr coefficient of the trap, are
plotted and shown to cross for different circular states. The vertical
plotting range of the Kerr coefficient is chosen to show the parameter regime
where the perturbative approximation is meaningful while keeping the laser
power at the few milliWatt level ($K_{\rho,z}\ll\omega_{\rho,z}$). Note that,
since $K\propto k_{4}/k_{2}$, the Kerr coefficient is laser power independent.
Another property, that can be read directly from Figure 2 (e) and (f), is that
for essentially any pair of these circular Rydberg states, there exists a
focusing diameter at which the nonlinearities are equal in magnitude (and
opposite in sign). This condition happens close to maximum trap depth for
neighboring states, too. This nonlinear parameter regime, provided
sufficiently low dissipation conditions, will allow the observation of purely
quantum dynamics and, in particular, the generation of Schrödinger Kerr-cats
[4] of a massive oscillator for the first time.

The last property we will mention is that the Kerr coefficient for any given
state changes sign, passing through zero, (almost) simultaneously for the two
trap axes ($K_{\rho}\approx K_{z}=0$). This differs from standard optical
tweezer Gaussian traps or lattices which are constrained to have negative
(softening) Kerr coefficient 555The fundamental difference is that, here, the
trapping potential does not follow from the shape of the beam only but it
depends on the geometry of the wave function. The potential is given by the
convolution between the two.. This happens near the maximum of trap depth too.

Discussion and concluding remarks– We have proposed a trapping mechanism for
Rydberg atoms that relies on the ponderomotive potential of the Rydberg
electron and on the ability to thread a circular atomic orbital with a
Gaussian beam. The trap exploits that the nontrivial shape and geometry of the
Rydberg orbitals can be used as a tool and will provide a new handle on the
atomic wavefunction.

Although threading an atomic orbital with light will be a challenge, it will
be rewarding: freedom in the principal quantum number, beam waist, and the
trapping wavelength allow the creation of deep, tunable traps. The trap can be
used to trap ground states and Rydberg states alike and besides being power-
efficient for Rydberg states, able to trap a cold Rydberg atom with as little
as a milliWatt, the tunability of the energy landscape is interesting and may
be useful for fundamental studies or provide a technical edge for specific
applications.

We have discussed the tread trap mechanism for circular states of an alkali-
metal atom but the idea can, in principle, be extended to non-circular and
alkali-earth Rydberg states too [47, 48, 13]. With a simple Gaussian and
lacking the symmetry of a circular state, low-angular momentum atoms will
yield modest trap depth but one could think of using laser wavefronts other
than a simple Gaussian to exploit the node location in the wavefunctions of
different Rydberg states ($P$, $D$, etc.). For alkali-earth Rydberg atoms with
two active electrons, one electron can be excited to a Rydberg state while the
other is used to dipole-trap the core. This is the configuration of the
experiments reported in [13], were interesting questions about the competition
between the dipolar and ponderomotive potentials arise. We note, however, that
for alkali-earth atoms under both ponderomotive and dipole potentials the trap
properties and the parameter regime available, as well as the trap-induced
loss mechanisms (e.g. photoionization), and their Rydberg lifetime limitations
(e.g. autoionization), are radically different from those expected for
circular Rydberg Rubidium in a thread trap.

Finally, we specify a possible application that we believe is particularly
promising. We point out that highly nonlinear driven systems are rich
platforms for fundamental studies [49, 50]. In particular, we mention that the
interplay of a Kerr nonlinearity and parametric squeezing in a massive quantum
oscillator will bring new opportunities for cold atoms. Squeezing can be
generated by exploiting the property that modulating the power of the trapping
laser will modulate the trap frequency at constant Kerr coefficient. This
nonlinear mechanical parametric oscillator could generate unprecedented
control of quantum tunneling [51], interference in the classically forbidden
region [52, 53] and bosonic encoding of quantum information [54, 35, 55] in
the mechanical degree of freedom of a neutral atom. Much like with interacting
ions [56], the Rydberg interaction provides the opportunity to perform logical
gates between two or more oscillators [57]. As such, this trap may bring the
mechanical motion of cold circular Rydberg Rubidium atoms into a regime where
only highly nonlinear, low-dissipation, superconducting quantum circuits
operate today [58, 35], and the experiment could conceivably be made at room
temperature [59, 60, 61].

Acknowledgments – RGC thanks Jean-Michel Raimond, Michel Devoret, and Brice
Ravon for critical remarks on an early version of this manuscript. RGC
acknowledges discussions with W. D. Phillips, H. Wu, M. Brune, Y. Machu, M.
Favier, C. Sayrin, S. Gleyzes, J. Thompson, V. Sivak, W. Adamczyk, S. Zacarias
and P. and V. Kurilovich. Support from the Center for Quantum Dynamics on
Modular Quantum Devices (NSF Grant No. CHE-2124511) and from the Yale Quantum
Institute is gratefully acknowledged.

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# On Normal Forms of Singular Levi-Flat Real Analytic Hypersurfaces

Arturo Fernández-Pérez  Instituto de Matemática Pura e Aplicada, IMPA Estrada
Dona Castorina, 110, 22460-320. Rio de Janeiro, RJ, Brazil<EMAIL_ADDRESS>
###### Abstract.

Let $F(z)=\mathcal{R}e(P(z))+h.o.t$ be such that $M=(F=0)$ defines a germ of
real analytic Levi-flat at $0\in\mathbb{C}^{n}$, $n\geq{2}$, where $P(z)$ is a
homogeneous polynomial of degree $k$ with an isolated singularity at
$0\in\mathbb{C}^{n}$ and Milnor number $\mu$. We prove that there exists a
holomorphic change of coordinate $\phi$ such that
$\phi(M)=(\mathcal{R}e(h)=0)$, where $h(z)$ is a polynomial of degree $\mu+1$
and $j^{k}_{0}(h)=P$.

###### Key words and phrases:

Levi-flat hypersurfaces - Holomorphic foliations

###### 2010 Mathematics Subject Classification:

Primary 32V40 - 37F75

##  1\. Introduction and Statement of the results

Let $M$ be a germ at $0\in\mathbb{C}^{n}$ of a real codimension one
irreducible analytic set. For the sake of simplicity we will denote germs and
representative of germs by the same letter. Since $M$ is real analytic of
codimension one and irreducible, it can be defined in $\mathbb{C}^{n}$ by
$(F=0)$, where $F$ is an irreducible germ of real analytic function. The
singular set of $M$ is defined by $sing(M)=(F=0)\cap(dF=0)$ and its smooth
part $(F=0)\backslash(dF=0)$ will denoted by $M^{*}$. The Levi distribution
$L$ on $M^{*}$ is defined by $L_{p}:=ker(\partial{F}(p))\subset
T_{p}M^{*}=ker(dF(p))$, for any $p\in M^{*}$.

######  Definition 1.1.

We say $M$ is Levi-flat if the Levi distribution on $M^{*}$ is integrable.

######  Remark 1.2.

The integrability condition of $L$ implies that $M^{*}$ is tangent to a real
codimension one foliation $\mathcal{L}$. Since the hyperplanes $L_{p}$, $p\in
M^{*}$, are complex, the leaves of $\mathcal{L}$ are complex codimension one
holomorphic submanifolds immersed on $M^{*}$.

######  Remark 1.3.

If the hypersurface $M$ is defined by $(F=0)$ then the Levi distribution $L$
on $M$ can be defined by the real analytic 1-form
$\eta=i(\partial{F}-\bar{\partial}F)$, which will be called the Levi 1-form of
$F$. The integrability condition is equivalent to
$(\partial{F}-\bar{\partial}F)\wedge\partial\bar{\partial}F|_{M^{*}}=0$

In the case of a real analytic smooth Levi-flat hypersurface $M$ in
$\mathbb{C}^{n}$, its local structure is very well understood, according to E.
Cartan, around each $p\in M$ we can find local holomorphic coordinates
$z_{1},\ldots,z_{n}$ such that $M=\\{\mathcal{R}e(z_{1})=0\\}$.

More recently D. Burns and X. Gong [B-G] have proved an analogous result in
the case $M=F^{-1}(0)$ Levi-flat, where
$F:(\mathbb{C}^{n},0)\rightarrow(\mathbb{R},0)$, $n\geq 2$, is a germ of real
analytic function such that

$F(z_{1},\ldots,z_{n})=\mathcal{R}e(z_{1}^{2}+\ldots+z_{n}^{2})+h.o.t.$

They show that there exists a germ of biholomorphism
$\phi:(\mathbb{C}^{n},0)\rightarrow(\mathbb{C}^{n},0)$ such that
$\phi(M)=(\mathcal{R}e(z_{1}^{2}+\ldots+z_{n}^{2})=0)$.

In [C-L], the authors prove the above result by using the theory of
holomorphic foliations. In this paper we are interested in finding similar
normal forms in a situation more general. Our main result is the following:

######  Theorem 1.

Let $M=F^{-1}(0)$, where $F:(\mathbb{C}^{n},0)\rightarrow(\mathbb{R},0)$,
$n\geq 2$, be a germ of irreducible real analytic function such that

  1. (1)

$F(z_{1},\ldots,z_{n})=\mathcal{R}e(P(z_{1},\ldots,z_{n}))+h.o.t$, where $P$
is a homogeneous polynomial of degree $k$ with an isolated singularity at
$0\in\mathbb{C}^{n}$.

  2. (2)

The Milnor number of $P$ at $0\in\mathbb{C}^{n}$ is $\mu$.

  3. (3)

$M$ is Levi-flat.

Then there exists a germ of biholomorphism
$\phi:(\mathbb{C}^{n},0)\rightarrow(\mathbb{C}^{n},0)$ such that
$\phi(M)=(\mathcal{R}e(h)=0)$, where $h(z)$ is a polynomial of degree $\mu+1$
and $j^{k}_{0}(h)=P$.

######  Remark 1.4.

In particular, we obtain the result of [B-G].

######  Theorem 2.

In the same spirit we have the following generalization: Let $M=F^{-1}(0)$,
where $F:(\mathbb{C}^{n},0)\rightarrow(\mathbb{R},0)$, $n\geq 3$, be a germ of
irreducible real analytic function such that

  1. (1)

$F(z_{1},\ldots,z_{n})=\mathcal{R}e(Q(z_{1},\ldots,z_{n}))+h.o.t$, where $Q$
is a quasihomogeneous polynomial of degree $d$ with an isolated singularity at
$0\in\mathbb{C}^{n}$.

  2. (2)

$M$ is Levi-flat.

Then there exists a germ of biholomorphism
$\phi:(\mathbb{C}^{n},0)\rightarrow(\mathbb{C}^{n},0)$ such that

$\phi(M)=(\mathcal{R}e(Q(z)+\sum_{j}c_{j}e_{j}(z))=0),$

where $e_{1},\ldots,e_{s}$ are the elements of the monomial basis of the local
algebra of $Q$ such that $deg(e_{j})>d$ and $c_{j}\in\mathbb{C}$.

Acknowledgments

I want to express my thanks to my advisor Alcides Lins Neto, for valuable
conversations. The author would like to thank also IMPA, where this work was
developed. I also want to thank the referee for his suggestions.

##  2\. Preliminaries

Let us fix some notations that will be used from now on.

  1. (1)

$\mathcal{O}_{n}$ : The ring of germs of holomorphic functions at
$0\in\mathbb{C}^{n}$.
$\mathcal{O}(U)=$ set of holomorphic functions in the open set
$U\subset\mathbb{C}^{n}$.

  2. (2)

$\mathcal{O}^{*}_{n}=\\{f\in\mathcal{O}_{n}/f(0)\neq 0\\}$.
$\mathcal{O}^{*}(U)=\\{f\in\mathcal{O}(U)/f(z)\neq 0,\forall z\in U\\}$.

  3. (3)

$\mathcal{M}_{n}=\\{f\in\mathcal{O}_{n}/f(0)=0\\}$ maximal ideal of
$\mathcal{O}_{n}$.

  4. (4)

$\mathcal{A}_{n}$ : The ring of germs at $0\in\mathbb{C}^{n}$ of complex
valued real analytic functions.

  5. (5)

$\mathcal{A}_{n\mathbb{R}}$ : The ring of germs at $0\in\mathbb{C}^{n}$ of
real valued real analytic functions. Note that $F\in\mathcal{A}_{n}$ is in
$\mathcal{A}_{n\mathbb{R}}$ if and only if $F=\bar{F}$.

  6. (6)

$Diff(\mathbb{C}^{n},0)$ : The group of germs at $0\in\mathbb{C}^{n}$ of
holomorphic diffeomorphisms
$f:(\mathbb{C}^{n},0)\rightarrow(\mathbb{C}^{n},0)$ with the operation of
composition.

  7. (7)

$j^{k}_{0}(f)$ : The $k$-jet at $0\in\mathbb{C}^{n}$ of $f\in\mathcal{O}_{n}$.

######  Definition 2.1.

Two germs $f,g\in\mathcal{O}_{n}$ are said to be right equivalent, if there
exists $\phi\in Diff(\mathbb{C}^{n},0)$ such that $f\circ\phi^{-1}=g$.

The local algebra of $f\in\mathcal{O}_{n}$ is by definition

$A_{f}=\mathcal{O}_{n}/(\partial{f}/\partial{z}_{1},\ldots,\partial{f}/\partial{z}_{n}).$

######  Definition 2.2.

Define by $\mu(f,0):=dimA_{f}$, the Milnor number of $f$ at
$0\in\mathbb{C}^{n}$.

Morse Lemma can now be rephrased by saying that if $0\in\mathbb{C}^{n}$ is an
isolated singularity of $f$ with Milnor number $\mu(f,0)=1$ then $f$ is right
equivalent to its second jet. The next lemma is a generalization of Morse’s
Lemma. We refer to [A-G-V], pg.121.

######  Lemma 2.3.

Suppose $0\in\mathbb{C}^{n}$ is an isolated singularity of
$f\in\mathcal{M}_{n}$ with Milnor number $\mu$. Then $f$ is right equivalent
to $j^{\mu+1}_{0}(f)$.

###  2.1. The complexification

In this section we state some general facts about complexification of germs of
real analytic functions.

Given $F\in\mathcal{A}_{n}$; we can write its Taylor series at
$0\in\mathbb{C}^{n}$ as

(2.1) $F(z)=\sum_{\mu,\nu}F_{\mu\nu}z^{\mu}\bar{z}^{\nu},$

where $F_{\mu\nu}\in\mathbb{C}$, $\mu=(\mu_{1},\ldots,\mu_{n})$,
$\nu=(\nu_{1},\ldots,\nu_{n})$, $z^{\mu}=z_{1}^{\mu_{1}}\ldots
z_{n}^{\mu_{n}}$,
$\bar{z}^{\nu}=\bar{z}_{1}^{\nu_{1}}\ldots\bar{z}_{n}^{\nu_{n}}$. When
$F\in\mathcal{A}_{n\mathbb{R}}$ then the coefficients $F_{\mu\nu}$ satisfy

$\bar{F}_{\mu\nu}=F_{\nu\mu}.$

The complexification $F_{\mathbb{C}}\in\mathcal{O}_{2n}$ of $F$ is defined by
the series

(2.2) $F_{\mathbb{C}}(z,w)=\sum_{\mu,\nu}F_{\mu\nu}z^{\mu}w^{\nu}.$

If $F\in\mathcal{A}_{n\mathbb{R}}$, $F(0)=0$ and $M=F^{-1}(0)$ defines a Levi-
flat, the complexification $\eta_{\mathbb{C}}$ of its Levi 1-form
$\eta=i(\partial{F}-\bar{\partial}F)$ can be written as

$\eta_{\mathbb{C}}=i(\partial_{z}F_{\mathbb{C}}-\partial_{w}F_{\mathbb{C}})=i\sum_{\mu,\nu}(F_{\mu\nu}w^{\nu}d(z^{\mu})-F_{\mu\nu}z^{\mu}d(w^{\nu})).$

The complexification $M_{\mathbb{C}}$ of $M$ is defined as
$M_{\mathbb{C}}=F^{-1}_{\mathbb{C}}(0)$ and its smooth part is
$M^{*}_{\mathbb{C}}=M_{\mathbb{C}}\backslash(dF_{\mathbb{C}}=0)$. The
integrability condition of $\eta=i(\partial{F}-\bar{\partial}F)|_{M^{*}}$
implies that $\eta_{\mathbb{C}}|_{M^{*}_{\mathbb{C}}}$ is integrable.
Therefore $\eta_{\mathbb{C}}|_{M^{*}_{\mathbb{C}}}=0$ defines a foliation
$\mathcal{L}_{\mathbb{C}}$ on $M^{*}_{\mathbb{C}}$ that will be called the
complexification of $\mathcal{L}$.

######  Definition 2.4.

The algebraic dimension of $sing(M)$ is the complex dimension of the singular
set of $M_{\mathbb{C}}$.

Consider a germ at $0\in\mathbb{C}^{2}$ of real analytic Levi-flat $M=(F=0)$,
where $F$ is irreducible in $\mathcal{A}_{2\mathbb{R}}$. Let $F_{\mathbb{C}}$,
$M_{\mathbb{C}}=(F_{\mathbb{C}}=0)\subset(\mathbb{C}^{4},0)$ and
$M_{\mathbb{C}}^{*}$ be as before. We will assume that the power series that
defines $F_{\mathbb{C}}$ converges in a neighborhood of
$\bar{\bigtriangleup}=\\{(z,w)\in\mathbb{C}^{4}/|z|,|w|\leq 1\\}$, so that
$F_{\mathbb{C}}(z,\bar{z})=F(z)$ for all $|z|\leq 1$.

Let $V:=M_{\mathbb{C}}^{*}\backslash
sing(\eta_{\mathbb{C}}|_{M^{*}_{\mathbb{C}}})$ and denote $L_{p}$ the leaf of
$\mathcal{L}_{\mathbb{C}}$ through $p$, where $p\in V$. In this situation we
have the following important Lemma of [C-L].

######  Lemma 2.5.

In the above situation, for any $p=(z_{0},w_{0})\in V$ the leaf $L_{p}$ is
closed in $M^{*}_{\mathbb{C}}$.

In the proof of theorem 1 we will use the following result of [C-L].

######  Theorem 2.6.

Let $M=F^{-1}(0)$ be a germ of an irreducible real analytic Levi-flat
hypersurface at $0\in\mathbb{C}^{n}$, $n\geq{2}$, with Levi 1-form
$\eta=i(\partial{F}-\bar{\partial}F)$. Assume that the algebraic dimension of
$sing(M)\leq 2n-4$. Then there exists an unique germ at $0\in\mathbb{C}^{n}$
of holomorphic codimension one foliation $\mathcal{F}_{M}$ tangent to $M$, if
one of the following conditions is fulfilled:

  1. (1)

$n\geq 3$ and
$cod_{M_{\mathbb{C}}^{*}}(sing(\eta_{\mathbb{C}}|_{M_{\mathbb{C}}^{*}}))\geq
3$.

  2. (2)

$n\geq 2$,
$cod_{M_{\mathbb{C}}^{*}}(sing(\eta_{\mathbb{C}}|_{M_{\mathbb{C}}^{*}}))\geq
2$ and $\mathcal{L}_{\mathbb{C}}$ has a non-constant holomorphic first
integral.

Moreover, in both cases the foliation $\mathcal{F}_{M}$ has a non-constant
holomorphic first integral $f$ such that $M=(Re(f)=0)$.

##  3\. Proof of theorem 1

Let $M=F^{-1}(0)\subset(\mathbb{C}^{n},0)$ be a Levi-flat, where
$F(z)=\mathcal{R}e(P(z))+h.o.t$ with $P$ be a homogeneous polynomial of degree
$k\geq 2$ with an isolated singularity at $0\in\mathbb{C}^{n}$ and Milnor
number $\mu$. We want to prove that there exists $\phi\in
Diff(\mathbb{C}^{n},0)$ such that $\phi(M)=(\mathcal{R}e(h)=0)$, where $h$ is
a polynomial of degree $\mu+1$.

The idea is to use theorem 2.6 to prove that there exists a germ
$f\in\mathcal{O}_{n}$ such that the foliation $\mathcal{F}$ defined by $df=0$
is tangent to $M$ and $M=(\mathcal{R}e(f)=0)$. The foliation $\mathcal{F}$ can
viewed as an extension to a neighborhood of $0\in\mathbb{C}^{n}$ of the Levi
foliation $\mathcal{L}$ on $M^{*}$.

Suppose for a moment that $M=(\mathcal{R}e(f)=0)$ and let us conclude the
proof. Without lost of generality, we can suppose that $f$ is not a power in
$\mathcal{O}_{n}$. In this case $\mathcal{R}e(f)$ is irreducible (cf. [C-L]).
This implies that $\mathcal{R}e(f)=U.F$, where $U\in\mathcal{A}_{n\mathbb{R}}$
and $U(0)\neq 0$. Let $\sum_{j\geq k}f_{j}$ be the taylor series of $f$, where
$f_{j}$ is a homogeneous polynomial of degree $j$, $j\geq k$. Then

$\mathcal{R}e(f_{k})=j^{k}_{0}(\mathcal{R}e(f))=j^{k}_{0}(U.F)=U(0).\mathcal{R}e(P(z_{1},\ldots,z_{n})).$

Hence $f_{k}(z_{1},\ldots,z_{n})=U(0).P(z_{1},\ldots,z_{n})$. We can suppose
that $U(0)=1$, so that

(3.1) $\displaystyle f(z)=P(z)+h.o.t$

In particular, $\mu=\mu(f,0)=\mu(P,0)$, $f\in\mathcal{M}_{n}$, because $P$ has
isolated singularity at $0\in\mathbb{C}^{n}$. Hence by lemma 2.3, $f$ is right
equivalent to $j^{\mu+1}_{0}(f)$, i.e. there exists $\phi\in
Diff(\mathbb{C}^{n},0)$ such that $h:=f\circ\phi^{-1}=j^{\mu+1}_{0}(f)$.
Therefore, $\phi(M)=(\mathcal{R}e(h)=0)$ and this will conclude the proof of
theorem 1.

Let us prove that we can apply theorem 2.6. We can write

$F(z)=\mathcal{R}e(P(z_{1},\ldots,z_{n}))+H(z_{1},\ldots,z_{n}),$

where $H:(\mathbb{C}^{n},0)\rightarrow(\mathbb{R},0)$ is a germ of real-
analytic function and $j^{k}_{0}(H)=0$. For simplicity, we assume that $P$ has
real coefficients. Then we get the complexification

$F_{\mathbb{C}}(z,w)=\frac{1}{2}(P(z)+P(w))+H_{\mathbb{C}}(z,w)$

and $M_{\mathbb{C}}=F^{-1}_{\mathbb{C}}(0)\subset(\mathbb{C}^{2n},0)$. In the
general case, replacing $P(w)=\sum a_{j}w^{j}$ by
$\tilde{P}(w)=\sum\bar{a}_{j}w^{j}$, we will recover each step of proof.

Since $P(z)$ has an isolated singularity at $0\in\mathbb{C}^{n}$, we get
$sing(M_{\mathbb{C}})=\\{0\\}$, and so the algebraic dimension of $sing(M)$ is
$0$. On other hand, the complexification of
$\eta=i(\partial{F}-\bar{\partial}F)$ is

$\eta_{\mathbb{C}}=i(\partial_{z}F_{\mathbb{C}}-\partial_{w}F_{\mathbb{C}}).$

Recall that $\eta|_{M^{*}}$ and $\eta_{\mathbb{C}}|_{M^{*}_{\mathbb{C}}}$
define $\mathcal{L}$ and $\mathcal{L}_{\mathbb{C}}$. Now we compute
$sing(\eta_{\mathbb{C}}|_{M^{*}_{\mathbb{C}}})$. We can write
$dF_{\mathbb{C}}=\alpha+\beta$, with

$\alpha=\sum_{j=1}^{n}\frac{\partial{F_{\mathbb{C}}}}{\partial{z}_{j}}dz_{j}:=\frac{1}{2}\sum_{j=1}^{n}(\frac{\partial{P}}{\partial{z}_{j}}(z)+A_{j})dz_{j}$

and

$\beta=\sum_{j=1}^{n}\frac{\partial{F_{\mathbb{C}}}}{\partial{w}_{j}}dw_{j}:=\frac{1}{2}\sum_{j=1}^{n}(\frac{\partial{P}}{\partial{w}_{j}}(w)+B_{j})dw_{j},$

where
$\frac{1}{2}\sum_{j=1}^{n}A_{j}dz_{j}=\sum_{j=1}^{n}\frac{\partial{H_{\mathbb{C}}}}{\partial{z}_{j}}dz_{j}$
and
$\frac{1}{2}\sum_{j=1}^{n}B_{j}dw_{j}=\sum_{j=1}^{n}\frac{\partial{H_{\mathbb{C}}}}{\partial{w}_{j}}dw_{j}$.

Then $\eta_{\mathbb{C}}=i(\alpha-\beta)$, and so

(3.2)
$\displaystyle\eta_{\mathbb{C}}|_{M^{*}_{\mathbb{C}}}=(\eta_{\mathbb{C}}+idF_{\mathbb{C}})|_{M^{*}_{\mathbb{C}}}=2i\alpha|_{M^{*}_{\mathbb{C}}}=-2i\beta|_{M^{*}_{\mathbb{C}}}.$

In particular, $\alpha|_{M^{*}_{\mathbb{C}}}$ and
$\beta|_{M^{*}_{\mathbb{C}}}$ define $\mathcal{L}_{\mathbb{C}}$. Therefore
$sing(\eta_{\mathbb{C}}|_{M^{*}_{\mathbb{C}}})$ can be splited in two parts.
Let $M_{1}=\\{(z,w)\in
M_{\mathbb{C}}|\frac{\partial{F}_{\mathbb{C}}}{\partial{w_{j}}}\neq 0$ for
some $j=1,\ldots,n\\}$ and $M_{2}=\\{(z,w)\in
M_{\mathbb{C}}|\frac{\partial{F}_{\mathbb{C}}}{\partial{z_{j}}}\neq 0$ for
some $j=1,\ldots,n\\}$, note that $M_{\mathbb{C}}=M_{1}\cup M_{2}$; if we
denote by

$X_{1}:=M_{1}\cap\\{\frac{\partial{P}}{\partial{z}_{1}}(z)+A_{1}=\ldots=\frac{\partial{P}}{\partial{z}_{n}}(z)+A_{n}=0\\}$

and

$X_{2}:=M_{2}\cap\\{\frac{\partial{P}}{\partial{w}_{1}}(w)+B_{1}=\ldots=\frac{\partial{P}}{\partial{w}_{n}}(w)+B_{n}=0\\},$

then $sing(\eta_{\mathbb{C}}|_{M^{*}_{\mathbb{C}}})=X_{1}\cup X_{2}$. Since
$P\in\mathbb{C}[z_{1},\ldots,z_{n}]$ has an isolated singularity at
$0\in\mathbb{C}^{n}$, we conclude that
$cod_{M^{*}_{\mathbb{C}}}sing(\eta_{\mathbb{C}}|_{M^{*}_{\mathbb{C}}})=n$.

If $n\geq 3$, we can directly apply Theorem 2.6 and the proof ends. In the
case $n=2$, we are going to prove that $\mathcal{L}_{\mathbb{C}}$ has a non-
constant holomorphic first integral.

We begin by a blow-up at $0\in\mathbb{C}^{4}$. Let
$F(x,y)=\mathcal{R}e(P(x,y))+h.o.t$ and $M=F^{-1}(0)$ Levi-flat. Its
complexification can be written as

$F_{\mathbb{C}}(x,y,z,w)=\frac{1}{2}P(x,y)+\frac{1}{2}P(z,w)+H_{\mathbb{C}}(x,y,z,w).$

We take the exceptional divisor $D=\mathbb{P}^{3}$ of the blow-up
$\pi:(\tilde{\mathbb{C}}^{4},\mathbb{P}^{3})\rightarrow(\mathbb{C}^{4},0)$
with homogeneous coordinates $[a:b:c:d]$,
$(a,b,c,d)\in\mathbb{C}^{4}\backslash\\{0\\}$. The intersection of the strict
transform $\tilde{M}_{\mathbb{C}}$ of $M_{\mathbb{C}}$ by $\pi$ with the
divisor $D=\mathbb{P}^{3}$ is the surface

$Q=\\{[a:b:c:d]\in\mathbb{P}^{3}/P(a,b)+P(c,d)=0\\},$

which is an irreducible smooth surface.

Consider for instance the chart $(W,(t,u,z,v))$ of $\tilde{\mathbb{C}}^{4}$
where

$\pi(t,u,z,v)=(t.z,u.z,z,v.z)=(x,y,z,w).$

We have

$F_{\mathbb{C}}\circ\pi(t,u,z,v)=z^{k}(\frac{1}{2}P(t,u)+\frac{1}{2}P(1,v)+zH_{1}(t,u,z,v)),$

where $H_{1}(t,u,z,v)=H(tz,uz,z,vz)/z^{k+1}$, which implies that

$\tilde{M}_{\mathbb{C}}\cap
W=(\frac{1}{2}P(t,u)+\frac{1}{2}P(1,v)+zH_{1}(t,u,z,v)=0)$

and so $Q\cap W=(z=P(t,u)+P(1,v)=0)$.

On the other hand, as we have seen in $(3.2)$, the foliation
$\mathcal{L}_{\mathbb{C}}$ is defined by $\alpha|_{M^{*}_{\mathbb{C}}}=0$,
where

$\alpha=\frac{1}{2}\frac{\partial{P}}{\partial{x}}dx+\frac{1}{2}\frac{\partial{P}}{\partial{y}}dy+\frac{\partial{H}_{\mathbb{C}}}{\partial{x}}dx+\frac{\partial{H}_{\mathbb{C}}}{\partial{y}}dy.$

In particular, we get

$\pi^{*}(\alpha)=z^{k-1}(\frac{1}{2}\frac{\partial{P}}{\partial{x}}(t,u)zdt+\frac{1}{2}\frac{\partial{P}}{\partial{y}}(t,u)zdu+\frac{1}{2}kP(t,u)dz+z\theta),$

where
$\theta=\pi^{*}(\frac{\partial{H}_{\mathbb{C}}}{\partial{x}}dx+\frac{\partial{H}_{\mathbb{C}}}{\partial{y}}dy)/z^{k}$.

Hence, $\tilde{\mathcal{L}}_{\mathbb{C}}$ is defined by

(3.3)
$\displaystyle\alpha_{1}=\frac{1}{2}\frac{\partial{P}}{\partial{x}}(t,u)zdt+\frac{1}{2}\frac{\partial{P}}{\partial{y}}(t,u)zdu+\frac{1}{2}kP(t,u)dz+z\theta.$

Since $Q\cap W=(z=P(t,u)+P(1,v)=0)$, we see that $Q$ is
$\tilde{\mathcal{L}}_{\mathbb{C}}$-invariant. In particular, $S:=Q\backslash
sing(\tilde{\mathcal{L}}_{\mathbb{C}})$ is a leaf of
$\tilde{\mathcal{L}}_{\mathbb{C}}$. Fix $p_{0}\in S$ and a transverse section
$\sum$ through $p_{0}$. Let $G\subset Diff(\sum,p_{0})$ be the holonomy group
of the leaf $S$ of $\tilde{\mathcal{L}}_{\mathbb{C}}$. Since $dim(\sum)=1$, we
can think that $G\subset Diff(\mathbb{C},0)$. Let us prove that $G$ is finite
and linearizable.

At this part we use that the leaves of $\tilde{\mathcal{L}}_{\mathbb{C}}$ are
closed (see lemma 2.5).

Let $G^{\prime}=\\{f^{\prime}(0)/f\in G\\}$ and consider the homomorphism
$\phi:G\rightarrow G^{\prime}$ defined by $\phi(f)=f^{\prime}(0)$. We assert
that $\phi$ is injective. In fact, assume that $\phi(f)=1$ and by
contradiction that $f\neq id$. In this case $f(z)=z+a.z^{r+1}+\ldots$, where
$a\neq 0$. According to [L], the pseudo-orbits of this transformation
accumulate at $0\in(\sum,0)$, contradicting that the leaves of
$\tilde{\mathcal{L}}_{\mathbb{C}}$ are closed. Now, it suffices to prove that
any element $g\in G$ has finite order (cf. [M-M]). In fact, if
$\phi(g)=g^{\prime}(0)$ is a root of unity then $g$ has finite order because
$\phi$ is injective. On the other hand, if $g^{\prime}(0)$ was not a root of
unity then $g$ would have pseudo-orbits accumulating at $0\in(\sum,0)$ (cf.
[L]). Hence, all transformations of $G$ have finite order and $G$ is
linearizable.

This implies that there is a coordinate system $w$ on $(\sum,0)$ such that
$G=\langle w\rightarrow\lambda w\rangle$, where $\lambda$ is a
$d^{th}$-primitive root of unity (cf. [M-M]). In particular, $\psi(w)=w^{d}$
is a first integral of $G$, that is $\psi\circ g=\psi$ for any $g\in G$.

Let $Z$ be the union of the separatrices of $\mathcal{L}_{\mathbb{C}}$ through
$0\in\mathbb{C}^{4}$ and $\tilde{Z}$ be its strict transform under $\pi$. The
first integral $\psi$ can be extended to a first integral
$\varphi:\tilde{M}_{\mathbb{C}}\backslash\tilde{Z}\rightarrow\mathbb{C}$ be
setting

$\varphi(p)=\psi(\tilde{L}_{p}\cap\sum),$

where $\tilde{L}_{p}$ denotes the leaf of $\tilde{\mathcal{L}}_{\mathbb{C}}$
through $p$. Since $\psi$ is bounded (in a compact neighborhood of
$0\in\sum$), so is $\varphi$. It follows from Riemann extension theorem that
$\varphi$ can be extended holomorphically to $\tilde{Z}$ with
$\varphi(\tilde{Z})=0$. This provides the first integral and finishes the
proof of theorem 1.

##  4\. Quasihomogeneous polynomials

In this section, we state some general facts about normal forms of
quasihomogeneous polynomials.

######  Definition 4.1.

The Newton support of germ $f=\sum a_{ij}x^{i}y^{j}$ is defined as
$supp(f)=\\{(i,j):a_{ij}\neq 0\\}$.

######  Definition 4.2.

A holomorphic function $f:(\mathbb{C}^{n},0)\rightarrow(\mathbb{C},0)$ is said
to be quasihomogeneous of degree $d$ with indices
$\alpha_{1},\ldots,\alpha_{n}$, if for any $\lambda\in\mathbb{C}$ and
$(z_{1},\ldots,z_{n})\in\mathbb{C}^{n}$, we have

$f(\lambda^{\alpha_{1}}z_{1},\ldots,\lambda^{\alpha_{n}}z_{n})=\lambda^{d}f(z_{1},\ldots,z_{n}).$

The index $\alpha_{s}$ is also called the weight of the variable $z_{s}$.

In the above situation, if $f=\sum a_{k}x^{k}$, $k=(k_{1},\ldots,k_{n})$,
$x^{k}=x_{1}^{k_{1}}\ldots x^{k_{n}}$, then
$supp(f)\subset\Gamma=\\{k:a_{1}k_{1}+\ldots+a_{n}k_{n}=d\\}$. The set
$\Gamma$ is called the diagonal. Usually one takes $\alpha_{i}\in\mathbb{Q}$
and $d=1$.

One can define the quasihomogeneous filtration of the ring $\mathcal{O}_{n}$.
It consists of the decreasing family of ideals
$\mathcal{A}_{d}\subset\mathcal{O}_{n}$,
$\mathcal{A}_{d^{\prime}}\subset\mathcal{A}_{d}$ for $d<d^{\prime}$. Here
$\mathcal{A}_{d}=\\{Q:$ degrees of monomials from $supp(Q)$ are $deg(Q)\geq
d\\}$; (the degree is quasihomogeneous).

When $\alpha_{1}=\ldots=\alpha_{n}=1$, this filtration coincides with the
usual filtration by the usual degree.

######  Definition 4.3.

A function $f$ is called semiquasihomogeneous if $f=Q+F^{\prime}$, where $Q$
is quasihomogeneous of degree $d$ of finite multiplicity and
$F^{\prime}\in\mathcal{A}_{d^{\prime}}$, $d^{\prime}>d$.

We will use the following result (cf. [A]).

######  Theorem 4.4.

Let $f$ be a semiquasihomogeneous function, $f=Q+F^{\prime}$ with
quasihomogeneous $Q$ of finite multiplicity. Then $f$ is right equivalent to
the function $Q+\sum_{j}c_{j}e_{j}(z)$, where $e_{1},\ldots,e_{s}$ are the
elements of the monomial basis of the local algebra $A_{Q}$ such that
$deg(e_{j})>d$ and $c_{j}\in\mathbb{C}$.

######  Example 4.5.

If $f=Q+F^{\prime}$ is semiquasihomogeneous and $Q(x,y)=x^{2}y+y^{k}$, then
$f$ is right equivalent to $Q$. Indeed, the base of the local algebra
$\mathcal{O}_{2}/(xy,x^{2}+ky^{k-1})$ is $1,x,y,y^{2},\ldots,y^{k-1}$ and lies
below the diagonal $\Gamma$. Here $\mu(Q,0)=k+1$.

##  5\. Proof of theorem 2

Let $M=F^{-1}(0)$ be a germ at $0\in\mathbb{C}^{n}$, $n\geq 3$ of real
analytic Levi-flat hypersurface, where $F(z)=\mathcal{R}e(Q(z))+h.o.t$ and $Q$
is a quasihomogeneous polynomial of degree $d$ with an isolated singularity at
$0\in\mathbb{C}^{n}$. It is easily seen that $sing(M_{\mathbb{C}})=\\{0\\}$
and $cod_{M^{*}_{\mathbb{C}}}sing(\mathcal{L}_{\mathbb{C}})\geq 3$. The
argument is essentially the same of the proof of theorem 1. In this way, there
exists an unique germ at $0\in\mathbb{C}^{n}$ of holomorphic codimension one
foliation $\mathcal{F}_{M}$ tangent to $M$, moreover $\mathcal{F}_{M}$:
$dh=0$, $h(z)=Q(z)+h.o.t$ and $M=(\mathcal{R}e(h)=0)$. Acoording to theorem
4.4, there exists $\phi\in Diff(\mathbb{C}^{n},0)$ such that
$h\circ\phi^{-1}(w)=Q(w)+\sum_{k}c_{k}e_{k}(w)$, where $c_{k}$ and $e_{k}$ as
above. Hence

$\phi(M)=(\mathcal{R}e(Q(w)+\sum_{k}c_{k}e_{k}(w))=0).$

##  6\. Applications

Here we give some applications of theorem 1.

######  Example 6.1.

$Q(x,y)=x^{2}y+y^{3}$ is a homogeneous polynomial of degree $3$ with an
isolated singularity at $0\in\mathbb{C}^{2}$ and Milnor number $\mu(Q,0)=4$.
According to [A-G-V] pg. 184, any germ $f(x,y)=x^{2}y+y^{3}+h.o.t$ is right
equivalent to $x^{2}y+y^{3}$.

In particular, if $F(z)=\mathcal{R}e(x^{2}y+y^{3})+h.o.t$ and $M=(F=0)$ is a
germ of real analytic Levi-flat at $0\in\mathbb{C}^{2}$, theorem 1 implies
that there exists a holomorphic change of coordinate such that

$M=(\mathcal{R}e(x^{2}y+y^{3})=0).$

######  Example 6.2.

If $Q(x,y)=x^{5}+y^{5}$ then $f(x,y)=Q(x,y)+h.o.t$ is right equivalent to
$x^{5}+y^{5}+c.x^{3}y^{3}$, where $c\neq 0$ is a constant (cf. [A-G-V], pg.
194). Let $F(z)=\mathcal{R}e(x^{5}+y^{5})+h.o.t$ be such that $M=(F=0)$ is
Levi-flat, theorem 1 implies that there exists a holomorphic change of
coordinate such that

$M=(\mathcal{R}e(x^{5}+y^{5}+c.x^{3}y^{3})=0).$

References

  * [A]

V.I. Arnold: “Normal Form of functions in the neighbourhood of degenerate
critical points”, UNM 29:2 (1974), 11-49, RMS 29:2 19-48.

  * [A-G-V]

V.I. Arnold, S.M. Gusein-Zade, A.N. Varchenko: “Singularities of Differential
Maps”, Vol. I, Monographs in Math., vol. 82, Birkhäuser, $1985$.

  * [B-G]

D. Burns, X. Gong: “Singular Levi-flat real analytic hypersurfaces”, Amer. J.
Math. 121, $(1999)$, pp. 23-53.

  * [C-L]

D. Cerveau, A. Lins Neto: “Local Levi-Flat hypersurfaces invariants by a
codimension one holomorphic foliation”. To appear in Amer. J. Math.

  * [L]

F. Loray: “Pseudo-groupe d’une singularité de feuilletage holomorphe en
dimension deux”. Avaliable in
http://hal.archives-ouvertures.fr/ccsd-00016434

  * [M-M]

J.F. Mattei, R. Moussu: “Holonomie et intégrales premières”, Ann. Ec. Norm.
Sup. 13, $(1980)$, pg. 469-523.