#  About $\mathcal{C}^{\infty}$ foliations by holomorphic curves on complex
surfaces

Olivier Thom

###### Abstract.

We study those real $\mathcal{C}^{\infty}$ foliations in complex surfaces
whose leaves are holomorphic curves. The main motivation is to try and
understand these foliations in neighborhoods of curves: can we expect the
space of foliations in a fixed neighborhood to be infinite-dimensional, or are
there some contexts under which every such foliation is holomorphic ?

We give some restrictions and study in more details the geometry of foliations
whose leaves belong to a holomorphic family of holomorphic curves. In
particular, we classify all real-analytic foliations on neighborhoods of
curves which are locally diffeomorphic to foliations by lines, under some non-
degeneracy hypothesis.

##  1\. Introduction

In this paper, we study real $\mathcal{C}^{\infty}$ foliations in complex
surfaces whose leaves are holomorphic curves; we will call such foliations
semiholomorphic for short in this text.

These foliations appear naturally in many different contexts, were studied
from different points of views, sometimes under different names; after giving
some definitions, we will briefly review some problems related to
semiholomorphic foliations and recall some results which might be of interest.

###  1.1. Equations

Let $U\subset\mathbb{C}^{2}$ be an open set and $\mathcal{F}$ a $C^{\infty}$
real codimension 2 foliation on $U$. The foliation $\mathcal{F}$ is called
semiholomorphic if the subsheaf $T\mathcal{F}\subset TU$ consist of
holomorphic directions: $T\mathcal{F}\subset T^{1,0}U$.

Suppose that $U$ is equipped with holomorphic coordinates $(x,y)$ and that the
foliation is smooth and nowhere vertical: $\frac{\partial}{\partial y}\notin
T_{p}\mathcal{F}$ for every $p\in U$. Then the foliation can be described by
the (1,0)-form $\omega=dy-\lambda dx$, where $\lambda\in
C^{\infty}(U,\mathbb{C})$ is the slope. This (1,0)-form satisfies the
integrability condition

(1) $\omega\wedge\overline{\omega}\wedge d\omega=0.$

In terms of the function $\lambda$, this equations writes:

(2)
$\frac{\partial\lambda}{\partial\bar{x}}+\overline{\lambda}\frac{\partial\lambda}{\partial\bar{y}}=0.$

Conversely, a field of holomorphic directions written as the kernel of a
(1,0)-form $\omega$ defines a semiholomorphic foliation if and only if the
integrability condition (1) is satisfied.

Suppose that $\mathcal{F}$ is smooth on $U$ and that $L$ is a real codimension
2 subvariety of $U$ invariant by $\mathcal{F}$. Then at every point $p\in L$,
the tangent space $TL$ is a complex direction. This shows that $L$ is a
complex curve, and that a semiholomorphic foliation is exactly a $C^{\infty}$
foliation by holomorphic curves.

######  Example 1.

Consider the (1,0)-form $\mathrm{Im}(x)dy-\mathrm{Im}(y)dx$. It satisfies
equation (1) so defines a semiholomorphic foliation. This foliation has
singular set $\mathbb{R}\times\mathbb{R}$. The leaf passing through
$(x_{0},y_{0})\notin\mathbb{R}^{2}$ has equation

$\displaystyle y$
$\displaystyle=y_{0}+\frac{\mathrm{Im}(y_{0})}{\mathrm{Im}(x_{0})}(x-x_{0})$
$\displaystyle=\frac{\mathrm{Im}(y_{0})}{\mathrm{Im}(x_{0})}x+\left(\mathrm{Re}(y_{0})-\frac{\mathrm{Im}(y_{0})}{\mathrm{Im}(x_{0})}\mathrm{Re}(x_{0})\right).$

Remark that the leaves of this foliation are exactly the complex affine lines
with real parameters.

###  1.2. Holomorphic motion

Consider an open set $U\subset\mathbb{C}^{2}$ and a semiholomorphic foliation
$\mathcal{F}$ in $U$. Fix a transverse holomorphic fibration $\\{x=cte\\}$ on
$U$ and a local trivialization $(x,y)$ of this fibration. Write $T_{x}$ the
fiber above $x$. Consider also an origin $0\in U$ and suppose that the curve
$\\{y=0\\}$ is a leaf of $\mathcal{F}$.

For each $x$, consider the holonomy transport $\varphi_{x}:T_{0}\rightarrow
T_{x}$ obtained by following the leaves of $\mathcal{F}$. This is a family of
$\mathcal{C}^{\infty}$ diffeomorphisms depending holomorphically on the
parameter $x$. Using the trivialization, we can consider $\varphi_{x}$ as a
family of germs of diffeomorphisms in the variable $y$. Conversely, given any
family $\varphi_{x}$ of germs of diffeomorphisms depending holomorphically on
the parameter $x$, we can construct a semiholomorphic foliation transverse to
the fibration $\\{x=cte\\}$ by taking trajectories of points $y_{0}\in T_{0}$.

To find the equivalence with the previous point of view, consider a point
$(x,y)\in U$ and write $y_{0}$ the point of intersection between $T_{0}$ and
the leaf passing through $(x,y)$. This means that $y=\varphi_{x}(y_{0})$, or
rather $y_{0}=\varphi_{x}^{-1}(y)$. Then the slope of the foliation is

$\lambda(x,y)=\frac{\partial\varphi}{\partial x}(x,\varphi_{x}^{-1}(y)),$

######  Example 2.

In Example 1, the holonomy transport between transversals above $x_{0}=i$ and
$x$ writes

$\varphi_{x}(y)=\mathrm{Re}(y)+x\mathrm{Im}(y).$

This point of view appears naturally in holomorphic dynamics, see for example
[11] and [14]. The main problem studied from this point of view is the
extension of the semiholomorphic foliation in the transverse direction; the
results show that the transversal behaviour is very similar to that of
$\mathcal{C}^{\infty}$ objects (for example, the proof of [14, §2] involves
partitions of unity). The most complete theorem obtained in this direction
seems to be [13]: in $\mathbb{D}\times\mathbb{C}$, any set of disjoint complex
curves $(C_{e})_{e\in E}$ transverse to the fibers $\mathbb{C}$ can be
extended to a semiholomorphic foliation in $\mathbb{D}\times\mathbb{C}$
transverse to the vertical fibration.

###  1.3. Levi-flat hypersurfaces and Ueda theory

An interesting motivation for semiholomorphic foliations comes from Ueda
theory, and more generally the study of Levi-flat hypersurfaces. Recall that a
smooth $\mathcal{C}^{\infty}$ real hypersurface $H$ in a complex surface $S$
is called Levi-flat if the field of complex directions $TH\cap J(TH)$ is
integrable on H, where $J:TS\rightarrow TS$ denotes the operator given by
multiplication by $i$.

It follows that $H$ is foliated by complex curves. In the
$\mathcal{C}^{\infty}$ category, we cannot say anything a priori about the
transverse regularity of this foliation; however, when a Levi-flat
hypersurface $H$ is real-analytic, then by [5] this foliation can be extended
to a holomorphic foliation in a neighborhood of $H$. Note that, if we have a
smooth $\mathcal{C}^{\infty}$ foliation by smooth Levi-flat hypersurfaces
$\mathcal{H}$ in $S$, the field of complex directions $T\mathcal{H}\cap
J(T\mathcal{H})$ defines a smooth semiholomorphic foliation.

This kind of objects appear naturally in the study of neighborhoods of compact
complex curves: by a theorem of T. Ueda [16], for a smooth compact curve $C$
in a complex surface $S$ with torsion normal bundle, either $C$ admits a
system of strictly pseudo-concave neighborhoods, either $C$ admits a
logarithmic 1-form $\omega$ with poles along $C$ and purely imaginary periods.
The foliation defined by this 1-form is a smooth holomorphic foliation
admitting $C$ as a leaf, and with unitary holonomy. It follows that there
exists in $S$ a foliation by Levi-flat hypersurfaces, each of which is the
border of a neighborhood of $C$, and that the foliation defined by $\omega$ is
the semiholomorphic foliation tangent to it. Thus, when $N_{C}$ is torsion,
any semiholomorphic foliation defined in a neighborhood of $C$, tangent to a
foliation by Levi-flat hypersurfaces of this kind, is necessarily holomorphic.

This theorem is also true for smooth compact curves $C$ with generic normal
bundle of degree 0, but in the non-generic cases the question is still open
(typically, the non-generic cases correspond to neighborhood $S$ of $C$ which
are formally but not analytically linearizable).

An interesting question would then be to understand when a neighborhood $S$ of
a compact curve $C$ admits smooth semiholomorphic foliations tangent to $C$
which are not holomorphic, and in particular if these semiholomorphic
foliations are exceptional objects or if any neighborhood $S$ admits one of
them.

###  1.4. Teichmüller theory

Holomorphic motions, and thus semiholomorphic foliations, appear naturally in
the context of Teichmüller theory, giving rise to some intersting examples,
see for example [3] and [12] for more details.

Let us just give one example to explain the link between the two. Consider the
plane $\mathbb{C}$ of the variable $y$, and a polygon $P_{i}$ inside it
equipped with identifications of opposite parallel sides, giving rise to a
translation surface $C_{i}$. Suppose that one side is included in the real
axis and that $P_{i}$ is contained in the upper half-plane $\mathbb{H}_{y}$.
Now, for every $x\in\mathbb{H}$, consider the linear application
$\varphi_{x}\in\mathrm{GL}_{2}(\mathbb{R})$ fixing the real axis and sending
$i$ to $x$. The application $\varphi_{x}$ sends the polygon $P_{i}$ to some
polygon $P_{x}$ defining a translation surface $C_{x}$.

Figure 1. A deformation of translation surfaces

We can consider the space of pairs $(x,y)\in\mathbb{H}\times\mathbb{C}$ as a
complex surface equipped with a holomorphic motion $(\varphi_{x})$; the set of
points $(x,y)$ with $y\in P_{x}$ is stable by the motion and we can see the
union of the translation surfaces $S=\cup_{x}C_{x}$ as a quotient of it. The
semiholomorphic foliation $\mathcal{F}$ defined by $(\varphi_{x})$ descends to
a semiholomorphic foliation on the bundle of translation surfaces $S$. Note
that the application $\varphi_{x}$ writes
$\varphi_{x}(y)=\mathrm{Re}(y)+x\mathrm{Im}(y)$, so that locally the foliation
$\mathcal{F}$ is in fact the one of Example 1.

When the translation surfaces admit a lattice of symmetries $L$, the foliation
descends to a foliation on the surface $S/L$. However by standard arguments,
the lattice $L$ cannot be cocompact. Indeed if $S/L$ were compact, the length
of the shortest loop in $C_{x}$, being a continuous function of $x$, should be
bounded by below. But this length tends to zero for $C_{(ti)}$ when
$\mathbb{R}\ni t\to 0$.

###  1.5. Monge-Ampère foliations

A large class of example is also given by Monge-Ampère foliations, in the
sense of [2]: given a real plurisubharmonic function
$f\in\mathcal{C}^{\infty}(U)$ in some open set $U\subset\mathbb{C}^{2}$,
introduce the complex hessian $\Omega=\partial\overline{\partial}f$: it is a
$(1,1)$-form and whenever $\Omega\wedge\Omega=0$, the field of complex
directions $X$ defined by the equation $i_{X}\Omega=0$ is a semiholomorphic
foliation. In dimension 2, each foliation can be obtained this way, but for
foliations of higher codimension, the two notion are no more equivalent (see
[2] and [7] for more details).

In the article [7], the authors also studied general semiholomorphic
foliations and obtained some interesting results. More precisely, following
the ideas of [1] for Monge-Ampère foliations, they studied Bott’s partial
connection of a semiholomorphic foliation $\mathcal{F}$ which is not
holomorphic, and showed that it induces a connection of negative curvature on
the normal bundle $N^{1,0}\mathcal{F}$. This allows to define an intrinsic
metric of curvature -4 on the leaves of $\mathcal{F}$.

These two facts have interesting consequences, for example that a
semiholomorphic foliation whose leaves are compact is always holomorphic [7,
Example 6.6] (see also [17] for another proof of this fact using different
tools), or that a semiholomorphic foliation whose leaves are parabolic is
necessarily holomorphic [10].

###  1.6. Summary of this article

We begin in section 2.1 by recalling how the antiholomorphic part
$\eta_{\mathcal{F}}$ of Bott’s partial connection gives a hyperbolic singular
metric $|\eta_{\mathcal{F}}|^{2}$ on the leaves of $\mathcal{F}$, as was
already noted in [7]. We then carry on studying the metric
$|\eta_{\mathcal{F}}|^{2}$ to obtain finer results.

It follows from [7] that if a smooth semiholomorphic foliation $\mathcal{F}$
on a surface $S$ has a compact curve $C$, then $C\cdot C<0$. In Theorem 2, we
show also that $C\cdot C\geq 1-g$, with equality when the form
$i^{*}_{C}\eta_{\mathcal{F}}$ has no zeroes (remember that the metric
$|\eta_{\mathcal{F}}|^{2}$ is well-defined, so that even if
$\eta_{\mathcal{F}}$ is not a well-defined 1-form on $C$, its zeroes are well-
defined).

Also, we note in Theorem 3 that under some reasonable hypotheses, the leaves
of a non-holomorpic semiholomorphic foliation should be complete for the
metric $|\eta_{\mathcal{F}}|^{2}$. This reminds the article [4] with the
differences that in the semiholomorpic context, the metric arises naturally
from the foliation, but can be a singular metric.

In section 3 we introduce some other geometric invariants of semiholomorphic
foliations $\mathcal{F}$. Informally, we consider the smallest holomorphic
family of holomorphic curves containing the leaves of $\mathcal{F}$, call it
system of curves defined by $\mathcal{F}$ and call semirank its number of
parameters. These notions are invariant under biholomorphisms, and although a
generic $\mathcal{C}^{\infty}$ foliation will be of infinite semirank, we show
in Proposition 3 that real-analytic semiholomorphic foliations are of semirank
2. Note also that all the examples where the leaves are lines are of semirank
2, so this motivates us to restrict our attention to foliations of semirank 2
for the rest of the article.

This additional hypothesis gives us some new tools to study these objects, for
example we can use the duality between 2-parameter families of curves. This
topic has a long history and we will refer the reader to [8] and the
references therein for more details. In a local context, note that if we fix a
2-parameter family of curves, the set of parameters is a complex surface
$\check{U}$ and that a semiholomorphic foliation gives a real surface
$S_{\mathcal{F}}$ in $\check{U}$. Studying $\mathcal{F}$ then reduces to
studying $S_{\mathcal{F}}$. In particular if $\mathcal{F}$ is the universal
cover of a semiholomorphic foliation in the neighborhood of a curve $C$, then
the fundamental group of $C$ acts by holomorphic automorphisms of the system
of curves, and its action on the dual $\check{U}$ stabilizes the real surface
$S_{\mathcal{F}}$. These conditions are very strong and we will use them
extensively.

As we saw in Theorem 3, we can expect interesting examples to have complete
leaves. We try at the end of this section to describe what a semiholomorphic
foliation in an open set $U\subset\mathbb{C}^{2}$ with complete leaves should
look like in semirank 2, which should be the generic behaviour for a
semiholomorphic foliation of any semirank.

In the last section, we consider foliations whose system of curves is a
projective structure in the sense of [8]. These projective structures are
better understood than general families of curves, and as we saw they can
arise naturally from global contexts. In particular, we classify all real-
analytic foliations $\mathcal{F}$ on neighborhoods of compact curves $C$ which
are locally diffeomorphic to foliations by lines, under the hypothesis that
$i_{C}^{*}\eta_{\mathcal{F}}$ is not identically zero, see Theorems 4 and 5.

###  1.7. Acknowlegments

The author wishes to thank IMPA and PNPD-CAPES for their support. Thanks also
to J. V. Pereira for useful advices and to B. Deroin for pointing out the
importance of the subject, and its links with Teichmüller theory.

##  2\. Bott’s partial connection

###  2.1. Local expression

Suppose in this section that $\mathcal{F}$ is transverse to the fibration
$x=cte$. Consider the form $\omega_{0}=dy-\lambda(x,y)dx$ defining the
semiholomorphic foliation $\mathcal{F}$. Note that

$\displaystyle d\omega_{0}$ $\displaystyle=\frac{\partial\lambda}{\partial
y}dx\wedge dy+\frac{\partial\lambda}{\partial\bar{x}}dx\wedge
d\bar{x}+\frac{\partial\lambda}{\partial\bar{y}}dx\wedge d\bar{y}$
$\displaystyle=\left(\frac{\partial\lambda}{\partial
y}dx\right)\wedge\omega_{0}+\left(\frac{\partial\lambda}{\partial\bar{y}}dx\right)\wedge\overline{\omega_{0}},$

and that for any function $f$, if $\omega=f\omega_{0}$, then

$d\omega=\left(\frac{df}{f}+\frac{\partial\lambda}{\partial
y}dx\right)\wedge\omega+\left(\frac{f^{2}}{|f|^{2}}\frac{\partial\lambda}{\partial\bar{y}}dx\right)\wedge\overline{\omega}.$

If not for the factor $\frac{f^{2}}{|f|^{2}}$, the 1-form
$\eta:=\frac{\partial\lambda}{\partial\bar{y}}dx$ would be well-defined modulo
some multiples of $\omega$ and $\overline{\omega}$, and would define a
$(1,0)$-form on the leaves of $\mathcal{F}$. Since this factor is of modulus
1, the metric on the leaves $|\eta|^{2}=i\eta\wedge\overline{\eta}$ depends
only on the foliation $\mathcal{F}$. We will write this 1-form
$\eta_{\mathcal{F}}$ when the defining form $\omega$ is clear or irrelevant
and call it the anti-holomorphic part of Bott’s connection.

The two following lemmas are immediate.

######  Lemma 1.

The foliation $\mathcal{F}$ is holomorphic if and only if
$\eta_{\mathcal{F}}=0$; it is holomorphic at order 1 along a leaf $L$ if and
only if $i_{L}^{*}\eta_{\mathcal{F}}=0$.

######  Lemma 2.

If $\Phi$ is a holomorphic germ of diffeomorphism around the origin, if $\eta$
and $\widetilde{\eta}$ are the anti-holomorphic parts of Bott’s connection
applied to $\omega$ and $\Phi^{*}\omega$ respectively, then
$\widetilde{\eta}=\Phi^{*}\eta$.

###  2.2. Tangential behaviour

Let us study more explicitely the coefficient
$\frac{\partial\lambda}{\partial\overline{y}}$ in the form
$\eta_{\mathcal{F}}$. First, let us introduce the differential operator

$\partial_{\mathcal{F}}:=\frac{\partial}{\partial
x}+\lambda\frac{\partial}{\partial y}.$

We check that for every function $f:\mathbb{C}^{2}\rightarrow\mathbb{C}$, the
integrability of $\mathcal{F}$ guarantees
$\partial_{\mathcal{F}}\overline{\partial_{\mathcal{F}}}f=\overline{\partial_{\mathcal{F}}}\partial_{\mathcal{F}}f$
so we can introduce the real operator

$\Delta_{\mathcal{F}}=\partial_{\mathcal{F}}\overline{\partial_{\mathcal{F}}}.$

Explicitely, we have
$\Delta_{\mathcal{F}}f=\Delta_{x}f+|\lambda|^{2}\Delta_{y}f+\lambda\frac{\partial^{2}f}{\partial\bar{x}\partial
y}+\bar{\lambda}\frac{\partial^{2}f}{\partial x\partial\bar{y}}$. This
operator corresponds to the Laplacian in restriction to the leaves of
$\mathcal{F}$: indeed if $\\{y=0\\}$ is a leaf of $\mathcal{F}$, then we have
$\Delta_{\mathcal{F}}=\Delta_{x}$ on this leaf.

Put

$a:=\frac{\partial\lambda}{\partial y},\quad
b:=\frac{\partial\lambda}{\partial\overline{y}}.$

The operator $\overline{\partial_{\mathcal{F}}}$ does not commute with
$\frac{\partial}{\partial y}$ (nor with $\frac{\partial}{\partial\bar{y}}$),
more precisely we have

$\left\\{\begin{aligned} \overline{\partial_{\mathcal{F}}}a+\bar{b}b&=0\\\
\overline{\partial_{\mathcal{F}}}b+\bar{a}b&=0.\end{aligned}\right.$

Introduce $\beta=\mathrm{log}(b)=\beta_{1}+i\beta_{2}$ so that the second
equation writes

$\bar{a}=-\overline{\partial_{\mathcal{F}}}\beta,$

and the first equation gives

(3) $\left\\{\begin{aligned}
\Delta_{\mathcal{F}}\beta_{1}&=\mathrm{exp}(2\beta_{1})\\\
\Delta_{\mathcal{F}}\beta_{2}&=0.\end{aligned}\right.$

As we see in [7], this equation gives:

######  Lemma 3.

The metric $|\eta_{\mathcal{F}}|^{2}$ on the leaves is of curvature -4
whenever it is not zero.

Equation (3) will imply that $b$ is very regular along the leaves; in fact we
can prove this directly using the point of view of holomorphic motions. Denote
as before $\varphi_{x}$ the holonomy transport of $\mathcal{F}$ between fibers
of $\\{x=cte\\}$; the antiholomorphic part of Bott’s connection is given by

$\frac{\partial\lambda}{\partial\bar{y}}(x,y)=\frac{\partial^{2}\varphi}{\partial
x\partial
y}\left(x,\varphi_{x}^{-1}(y)\right)\frac{\partial\varphi_{x}^{-1}}{\partial\bar{y}}(y)+\frac{\partial^{2}\varphi}{\partial
x\partial\bar{y}}\left(x,\varphi_{x}^{-1}(y)\right)\overline{\frac{\partial\varphi_{x}^{-1}}{\partial
y}(y)}.$

In restriction to a leaf $\\{y=0\\}$, we can approximate $\varphi_{x}$ by its
linear part:
$\varphi_{x}(y)=l_{x}(y)+O(|y|^{2})=u(x)y+v(x)\bar{y}+O(|y|^{2})$, where $u$
and $v$ are holomorphic functions of the parameter $x$. Note that

$l_{x}^{-1}(y)=\frac{\overline{u}}{|u|^{2}-|v|^{2}}y-\frac{v}{|u|^{2}-|v|^{2}}\bar{y}.$

Thus we get in first order approximation

$\lambda(x,y)=u^{\prime}(x)\frac{\overline{u}y-v\bar{y}}{|u|^{2}-|v|^{2}}+v^{\prime}(x)\frac{u\bar{y}-\bar{v}y}{|u|^{2}-|v|^{2}}+O(|y|^{2}),$

and

$\frac{\partial\lambda}{\partial\bar{y}}(x,0)=\frac{uv^{\prime}-u^{\prime}v}{|u|^{2}-|v|^{2}}(x).$

In particular, we see that the border of the leaf $\\{y=0\\}$ is given by the
real analytic curve $\\{|u|^{2}=|v|^{2}\\}$; we can also deduce the following:

######  Lemma 4.

On a leaf of $\mathcal{F}$, the set of points $x$ where $b(x)=0$ is either
discrete or the whole leaf. Moreover, if $x$ is an isolated zero of $b$, then
$\frac{\partial b}{\partial\bar{x}}(x)=0$.

###  2.3. Global setting

In this section, $S$ is a complex surface and $\mathcal{F}$ a smooth
semiholomorphic foliation on $S$; we fix a covering $S=\cup U_{i}$, local
coordinates $(x_{i},y_{i})$ with $\mathcal{F}$ transverse to the fibration
$x_{i}=cte$, and a (1,0)-form
$\omega_{i}=f_{i}\cdot(dy_{i}-\lambda_{i}dx_{i})$ defining $\mathcal{F}$ on
each $U_{i}$.

The form $\eta_{\mathcal{F}}$ depends on the choice of the representant
$\omega_{i}$ of $\mathcal{F}$, but as we have seen in section 2.1, it is a
section of a line bundle $T^{1,0}\mathcal{F}\otimes L$ where the transition
functions of $L$ are $\mathcal{C}^{\infty}$ functions of modulus 1. Putting
together lemmas 1, 2, 3 and 4, we obtain the following:

######  Proposition 1.

The metric $|\eta_{\mathcal{F}}|^{2}$ is intrinsically defined as a metric on
the leaves of $\mathcal{F}$. On each leaf, this metric is either identically
zero, or only has isolated zeroes. If it is not identically zero, then it has
curvature -4 away from its zeroes.

The following corollary, and Theorem 1 were already consequences of [7] and
[10].

######  Corollary 1.

Let $S$ be a complex surface with a smooth holomorphic fibration $\pi$ towards
an elliptic curve. Then every smooth semiholomorphic foliation transverse to
$\pi$ is holomorphic.

######  Theorem 1.

Suppose $S$ is a neighborhood of an elliptic curve $C$ and there exists a
singular $\mathcal{C}^{\infty}$ foliation $\mathcal{H}$ by compact Levi-flat
hypersurfaces such that $C$ is invariant by $\mathcal{H}$, the foliation
$\mathcal{H}$ is smooth outside $C$ and every leaf of $\mathcal{H}$ is the
border of a neighborhood of $C$. Suppose moreover that for every sequence of
points $p_{n}\in S$ with $p_{n}\rightarrow p_{\infty}\in C$ and such that the
sequence $T_{p_{n}}\mathcal{H}$ has a limit $H_{\infty}$, then
$T_{p_{\infty}}C\subset H_{\infty}$.

Then $\mathcal{H}$ is tangent to a holomorphic foliation.

###### Proof.

The field of complex directions $\mathcal{F}=T^{1,0}\mathcal{H}$ is integrable
so $\mathcal{F}$ is a semiholomorphic foliation. Under the hypotheses, for
every local fibration $\\{x_{i}=cte\\}$ transverse to $C$ on an open set
$U_{i}\subset S$, this fibration is also transverse to $\mathcal{H}$ in a
neighborhood of $C\cap U_{i}$. Thus it is transverse to the foliation
$\mathcal{F}$, which implies in particular that $\mathcal{F}$ is smooth in a
neighborhood of $C$.

When $|\eta_{\mathcal{F}}|^{2}$ is not identically zero, it is a metric of
curvature $-4$, so we only need to prove that every leaf of $\mathcal{F}$ is
uniformized by $\mathbb{C}$. Choose a Kähler form $\omega$ on $S$ so that we
can compute the curvature of the leaves using $\omega$. Consider a smooth
$\mathcal{C}^{\infty}$ foliation $\mathcal{G}$ transverse to $\mathcal{F}$.
For any base point $p_{0}\in C$, if we denote by $G_{0}$ the leaf of
$\mathcal{G}$ passing through $p_{0}$, we get an application
$f:G_{0}\times\widetilde{C}\rightarrow\widetilde{S}$ following the leaves of
$\mathcal{F}$, where $\widetilde{C}$ is the universal cover of $C$ and
$\widetilde{S}$ that of $S$. Consider the map $f(y_{0},\cdot):C\cap
U_{i}\rightarrow L_{0}\cap U_{i}$ on an small open set $U_{i}$, where
$y_{0}\in G_{0}$ and $L_{0}$ is the leaf of $\mathcal{F}$ passing through
$y_{0}$: by continuity of the metric induced by $\omega$ on the leaves of
$\mathcal{F}$, this application is a quasi-isometry, and we can choose
constants of quasi-isometry which do not depend on $y_{0}$. Since there are a
finite number of these $U_{i}$, we see that $f(y_{0},\cdot)$ is a quasi-
isometry between $\widetilde{C}$ and the universal cover $\widetilde{L}$ of
the leaf $L$ of $\mathcal{F}$ passing through $y_{0}$, for any $y_{0}\in
G_{0}$. We conclude that any leaf is uniformized by $\mathbb{C}$ as needed. ∎

Remark that the proof is somewhat more general: we only need a smooth
semiholomorphic foliation whose leaves stay inside a neighborhood of $C$.

We will see in the explicit examples of section 4.1 that for a semiholomorphic
foliation with a compact leaf, the normal and tangent bundles of this leaf are
closely related; in general we can prove the following.

######  Theorem 2.

Suppose $C$ is a compact leaf of genus $g$ of a smooth semiholomorphic
foliation $\mathcal{F}$ on a surface $S$, and
$i_{C}^{*}\eta_{\mathcal{F}}\not\equiv 0$. Write $n$ the number of zeroes of
the form $i_{C}^{*}\eta_{\mathcal{F}}$, counted with multpilicity. Then
$C\cdot C=1-g+\frac{n}{2}$.

Note in particular that Camacho-Sad’s theorem is false for smooth
semiholomorphic foliations.

###### Proof.

The approximation at first order along $C$ of $\mathcal{F}$ is a smooth
semiholomorphic foliation on the normal bundle of $C$ in $S$. The proposition
only depends on this first-order approximation, so we can suppose that $S$ is
the normal bundle of $C$.

Consider local charts $(x_{i},y_{i})$ with

$(x_{j},y_{j})=\Phi_{ji}(x_{i},y_{i})=(\alpha_{ji}(x_{i}),\beta_{ji}(x_{i})y_{i}).$

Consider local $(1,0)$-forms $\omega_{i}=dy_{i}-\lambda_{i}dx_{i}$ defining
$\mathcal{F}$, and the corresponding antiholomorphic parts of Bott’s
connection
$\eta_{i}=\frac{\partial\lambda_{i}}{\partial\overline{y_{i}}}dx_{i}$. From
Lemma 2 we know that

$i_{C}^{*}(\Phi_{ji}^{*}\eta_{j})=\frac{\beta_{ji}^{2}}{|\beta_{ji}|^{2}}i_{C}^{*}\eta_{i}.$

Thus if we write
$b_{i}=\frac{\partial\lambda_{i}}{\partial\overline{y_{i}}}|_{C}$, we get

$b_{j}\circ\alpha_{ji}=\frac{\beta_{ji}^{2}}{|\beta_{ji}|^{2}\alpha^{\prime}_{ji}}b_{i}.$

The cocycle $|\beta_{ji}|^{-2}$ is real positive so its sections do not
vanish; the cocycle $\beta_{ji}$ defines the normal bundle and
$\alpha^{\prime}_{ji}$ the tangent bundle. On the other hand we see by Lemma 4
that $b_{i}$ is a $\mathcal{C}^{\infty}$ function with holomorphic zeroes. It
follows that $\mathrm{deg}(\Omega^{1}_{C}\otimes N^{2})=n$ where $n$ is the
number of zeroes of the section $b$, hence the result. ∎

######  Example 3.

Consider a germ of surface $(S,C)$ along a compact curve $C$ of genus $g\geq
2$, and suppose there exists on $S$ a smooth semiholomorphic foliation
$\mathcal{F}$ leaving $C$ invariant with $i_{C}^{*}\eta_{\mathcal{F}}$ nowhere
vanishing. From the theorem we know that $C\cdot C=1-g$.

Consider then two curves $c_{1},c_{2}$ in $S$ cutting $C$ transversally at two
points $p_{1}\neq p_{2}$, and the ramified covering
$(\widetilde{S},\widetilde{C})$ of order 2 over $(S,C)$ ramifying along
$c_{1}$ and $c_{2}$. The genus of $\widetilde{C}$ is $\widetilde{g}=2g$; its
self-intersection is $\widetilde{C}\cdot\widetilde{C}=2-2g=2-\widetilde{g}$,
so that $\mathrm{deg}(\Omega^{1}_{C}\otimes
N_{C}^{2})=(2\widetilde{g}-2)+2(2-\widetilde{g})=2$. We see from Lemma 2 that
$i_{\widetilde{C}}^{*}\widetilde{\eta}$ does not vanish at points different
from $p_{1},p_{2}$, and a similar computation shows that it admits simple
zeroes at $p_{1}$ and $p_{2}$.

Note that, by [7, Lemma 5.1], the self-intersection $C\cdot C$ should always
be negative under the assumption $i_{C}^{*}\eta\not\equiv 0$. With this in
mind, the theorem above seems incomplete, and for some reason, the form
$\eta_{\mathcal{F}}$ cannot have an arbitrarily high number of zeroes for $C$
fixed. It is not clear what values are admissible for $C\cdot C$ in
$[1-g,-1]$, but to get an idea of this, it seems necessary to find examples
which are not ramified coverings.

######  Theorem 3.

Suppose that $L$ is a leaf of $\mathcal{F}$ with compact adherence in $S$.
Suppose moreover that $i_{C}^{*}\eta_{\mathcal{F}}$ is not identically zero
for any leaf $C$ in the adherence $\overline{L}$. Then $L$ is complete for the
metric $|\eta_{\mathcal{F}}|^{2}$.

###### Proof.

Suppose $i_{L}^{*}\eta_{\mathcal{F}}$ is not identically zero. Then the set of
points on $L$ for which the metric degenerates is discrete, and we want to
prove that for any geodesic $\gamma:\mathbb{R}^{+}\rightarrow L$ which tends
to the border of $L$, the length
$\ell(\gamma):=\int_{0}^{\infty}|\eta(\gamma^{\prime}(t))|dt$ is infinite.

Since the adherence of $L$ is compact, we can cover it by a finite number of
open sets $U_{i}$ with $\eta_{i}=b_{i}dx_{i}$ on $U_{i}$. Remark first that
for $U_{i}$ small enough, we can suppose that $\gamma$ passes through an
infinite number of these $U_{i}$. On each leaf, the set of points where
$b_{i}=0$ is discrete, so we can find an $\varepsilon>0$ such that
$B_{i}:=\\{|b_{i}|<\varepsilon\\}$ is relatively compact in $U_{i}$ on each
leaf. If $\gamma$ passes an infinite number of times through these $B_{i}$,
then $\gamma$ must make an infinite number of times the path from within
$B_{i}$ to the border of $U_{i}$, and since $|b_{i}|\geq\varepsilon$ on the
complement $U_{i}\setminus B_{i}$, we must have $\ell(\gamma)=\infty$.

Thus we can suppose that $\gamma$ only meets a finite number of these $B_{i}$.
Then after a finite time $T>0$, the geodesic $\gamma$ only stays in the
regions $|b_{i}|\geq\varepsilon$. The result follows easily. ∎

##  3\. Semirank

###  3.1. Definitions

We will define the semirank using the construction of jet spaces; let us
recall this construction in our setting. Consider a holomorphic manifold $X$
equipped with a holomorphic field of holomorphic 2-planes $\mathcal{P}$.

We can consider the field of 2-planes $\mathcal{P}$ as a rank 2 subbundle of
$TX$; let $X^{\prime}=\mathbb{P}\mathcal{P}$ be the $\mathbb{P}^{1}$-bundle
obtained by projectivizing the rank 2 linear bundle $\mathcal{P}$. Then
$X^{\prime}$ comes with a fibration $p:X^{\prime}\rightarrow X$, a field of
3-planes $p^{*}\mathcal{P}$ and a field of 2-planes $\mathcal{P}^{\prime}$
inside $p^{*}\mathcal{P}$ such that the value of $\mathcal{P}^{\prime}$ at a
point $(x,\lambda)\in X^{\prime}$ is given by $\mathrm{Ker}(dv-\lambda
du)\subset p^{*}\mathcal{P}$ if $(u,v)$ are linear variables on
$\mathcal{P}_{x}$ with $\lambda$ corresponding to the direction in the kernel
of $dv-\lambda du$.

The manifold $X^{\prime}$ satisfies the property that every holomorphic curve
$C$ in $X$ tangent to $\mathcal{P}$ can be uniquely lifted to a curve
$C^{\prime}$ in $X^{\prime}$ tangent to $\mathcal{P}^{\prime}$ with
$p(C^{\prime})=C$.

Beginning by $(X_{0},\mathcal{P}_{0})=(U,TU)$, we can apply this construction
inductively to obtain a sequence $(X_{n},\mathcal{P}_{n})$ of
$(n+2)$-dimensional manifolds equipped with holomorphic fields of 2-planes and
fibrations $p_{n}:X_{n}\rightarrow X_{n-1}$ with fibers $\mathbb{P}^{1}$.
Consider $\pi_{n}:X_{n}\rightarrow U$ the composition of the $p_{n}$. By
construction, each holomorphic curve in $U$ can be uniquely lifted to a curve
in $X_{n}$ tangent to $\mathcal{P}_{n}$.

Now, suppose that $\mathcal{F}$ is a semiholomorphic foliation on $U$. It can
be considered as a family of holomorphic curves with two real parameters; as
such it can be lifted to each $X_{n}$ as a family of curves tangent to
$\mathcal{P}_{n}$ and defines on each $X_{n}$ a real 4-dimensional submanifold
$Y_{n}$ of $X_{n}$. Note that any point $p\in U$ around which $\mathcal{F}$ is
smooth can be uniquely lifted to the point in $X_{n}$ corresponding to the
lift of the leaf of $\mathcal{F}$ passing through $p$. By abuse of notation,
we will still write $p$ this point whenever the foliation is clear from the
context.

######  Definition 1.

We define the semirank of a semiholomorphic foliation $\mathcal{F}$ around a
point $p\in U$ as the lowest integer $n$ such that the germ of $Y_{n}$ at $p$
is not Zariski-dense in $X_{n}$.

By convention, the semirank is infinite if there exists no such integer. Note
that holomorphic foliations correspond to semirank 1.

######  Proposition 2.

Suppose that $\mathcal{F}$ is of semirank $n<\infty$ around a point $p$. Then
the Zariski closure $Z$ of $Y_{n}$ in $X_{n}$ is a hypersurface generically
transverse to the fibers of $p_{n}:X_{n}\rightarrow X_{n-1}$

###### Proof.

Suppose on the contrary that $Z$ is tangent to the fibration $p_{n}$ or is not
a hypersurface. Then $Z_{n-1}:=p_{n}(Z)$ is a strict subvariety of $X_{n-1}$.
By construction, $Z_{n-1}$ contains $Y_{n-1}$, thus $Y_{n-1}$ is not Zariski-
dense in $X_{n-1}$, contradicting the minimality of $n$. ∎

From this proposition, we see that $TZ\cap\mathcal{P}_{n}$ defines a foliation
by curves in $Z$ tangent to $\mathcal{P}_{n}$ which can be projected to a
holomorphic family of holomorphic curves in $U$ with $n$ parameters containing
the leaves of $\mathcal{F}$. By construction, this n-parameter holomorphic
family of curves is uniquely determined by $\mathcal{F}$.

######  Definition 2.

If $\mathcal{F}$ is of semirank $n$, the n-parameter holomorphic family of
curves given by the Zariski closure of $Y_{n}$ in $X_{n}$ is called the system
of curves $\mathscr{S}$ defined by $\mathcal{F}$. We will also say that
$\mathcal{F}$ is tangent to $\mathscr{S}$.

######  Example 4.

For the foliation $\mathrm{Im}(x)dy-\mathrm{Im}(y)dx$, the induced system of
curves is the set of complex affine lines.

To put this system into equations, take some coordinates
$(x,y,\lambda_{1},\ldots,\lambda_{n})$ of $X_{n}$ centered around the point
given by the leaf of $\mathcal{F}$ passing through the origin, so that
$\lambda_{k}$ is the coordinate of the $\mathbb{P}^{1}$-fiber of $X_{k}$
corresponding to the direction $d\lambda_{k-1}/dx$. Since the Zariski closure
$Z$ of $Y_{n}$ is generically transverse to the fibration $p_{n}$, we can
express it around a generic point as the graph of a holomorphic application

$\lambda_{n}=F(x,y,\lambda_{1},\ldots,\lambda_{n-1}).$

In this case, the system of curves is given by solutions of the differential
equation

$y^{(n)}=F(x,y,y^{\prime},\ldots,y^{(n-1)}),$

where a curve is written as a function $y(x)$ and derivations are made with
respect to the variable $x$. Note that some points can behave singularly for
the system of curves, even if $\mathcal{F}$ is smooth. This happens exactly
when $Z$ is tangent to the fibration $p_{n}:X_{n}\rightarrow X_{n-1}$. We say
that the family $\mathscr{S}$ is smooth when $Z$ is smooth and transverse to
the fibration $p_{n}$.

######  Example 5.

The system of curves associated to the foliation
$\mathrm{Re}(1+x^{2})dy-2x\mathrm{Re}(y)dx$ is the set of parabolas
$\\{y=ax^{2}+b\\}$. The differential equation satisfied by these curves is
$xy^{\prime\prime}=y^{\prime}$ so $Z$ has equation
$x\lambda_{2}-\lambda_{1}=0$ and is not transverse to the vector field
$\frac{\partial}{\partial\lambda_{2}}$ above $x=0$.

The proof of the following proposition is an adaptation of [5, Chapitre I,
§I.4] in the context of semiholomorphic foliations.

######  Proposition 3.

Let $\mathcal{F}$ be a real-analytic semiholomorphic foliation. Then the
semirank of $\mathcal{F}$ is at most 2.

###### Proof.

Suppose that $(x,y)$ are coordinates around the origin such that $\mathcal{F}$
is nowhere vertical. Consider coordinates $(x,y,\lambda_{1},\lambda_{2})$ of
$X_{2}$ where $\lambda_{i}$ is a coordinate of the $\mathbb{P}^{1}$-fiber of
$X_{i}$. Let $\lambda(x,y)$ be the slope of $\mathcal{F}$. The variety $Y_{2}$
satisfies the equations

$\left\\{\begin{aligned} \lambda_{1}&=\lambda(x,y)\\\
\lambda_{2}&=\frac{\partial\lambda}{\partial
x}(x,y)+\lambda(x,y)\frac{\partial\lambda}{\partial
y}(x,y).\end{aligned}\right.$

Consider the 4 real parameters $(x_{1},x_{2},v_{1},v_{2})$ with
$l=l_{1}+il_{2}$ for each $l=x,v$, and $y(x,v)$ the solution of the equations

$\left\\{\begin{aligned} \frac{\partial y}{\partial x}&=\lambda(x,y)\\\
\frac{\partial y}{\partial\bar{x}}&=0\\\ y(0,v)&=v\end{aligned}\right.$

and put

$\varphi(x,v)=(x,y,\lambda(x,y),\partial_{x}\lambda(x,y)+\lambda(x,y)\partial_{y}\lambda(x,y)),$

where we wrote $y=y(x,v)$ for short. Now remark that $\varphi$ is a
parametrization of $Y_{2}$, and since $\lambda$ satisfies equation (2), the
antiholomorphic derivative $\frac{\partial\varphi}{\partial\bar{x}}$ is zero.

Thus $\varphi$ is holomorphic in $x$, and real-analytic in $v_{1},v_{2}$; as
such it can be extended to a germ of holomorphic application
$\varphi:(x,v_{1},v_{2})\to X_{2}$ defined in a neighborhood $V$ of
$(\mathbb{C}\times\mathbb{R}\times\mathbb{R},0)$ in $(\mathbb{C}^{3},0)$. Then
$\varphi(V)$ is a complex 3-dimensional subvariety of $X_{2}$ containing
$Y_{2}$, which concludes the proof. ∎

Remark that when $\mathcal{F}$ is a semiholomorphic foliation and $L$ is a
leaf of $\mathcal{F}$, the first order approximation of $\mathcal{F}$ along
$L$ is always real analytic, thus holomorphic or of semirank 2. If its system
of curves is locally given by $\lambda^{\prime}=F(x,y,\lambda)$, the first
order approximation is the limit of $h_{t}^{*}\mathcal{F}$ when $t$ tends to
$0$, where $h_{t}$ is the dilatation $h_{t}(x,y)=(x,ty)$. The system of curves
defined by the first order approximation is thus of the form
$\lambda^{\prime}=a_{1}(x)y+a_{2}(x)\lambda$; it is of order less than 3 in
$\lambda$, so it is a projective structure. However, in general this
projective structure will have singular points along $L$.

After these generalities, we will restrict ourselves to foliations of semirank
2.

###  3.2. Duality

Suppose given a holomorphic family of curves $\mathscr{S}$ with $2$ parameters
in an open set $U\subset\mathbb{C}^{2}$, represented by the hypersurface
$Z\subset X_{2}$ and suppose that $Z$ is smooth and transverse to the
fibration $p_{2}:X_{2}\rightarrow X_{1}$. We will consider $Z$ as a germ
around the origin $0\in X_{2}$. Write $\mathcal{G}$ the smooth holomorphic
foliation of $Z$ given by $\mathscr{S}$, and $\check{U}$ the contraction of
$Z$ in the direction $\mathcal{G}$. The space $\check{U}$ is a holomorphic
surface parametrizing the curves of $\mathscr{S}$; we call it the dual of the
system of curves $\mathscr{S}$.

This dual comes with a 2-parameter family of curves: for each point $z\in U$,
consider the $\mathbb{P}^{1}$-fiber $F_{z}$ of $X_{1}$ over $z$. Since $Z$ is
transverse to the fibration $p_{2}$, the preimages of the $F_{z}$ on $Z$ form
a smooth foliation by curves $\check{\mathcal{G}}$ on $Z$. This foliation is
transverse to $\mathcal{G}$, so that it is projected to a 2-parameter family
of smooth curves on $\check{U}$.

This family is the dual family of $\mathscr{S}$ on $\check{U}$; since $U$ is
the contraction of $Z$ in the direction $\check{\mathcal{G}}$, we see that
$\mathcal{G}$ and $\check{\mathcal{G}}$ play symmetric roles on $Z$ so that
the bidual of $\mathscr{S}$ is $\mathscr{S}$ itself.

To any semiholomorphic foliation $\mathcal{F}$ tangent $\mathscr{S}$, we can
associate a real surface $S_{\mathcal{F}}\subset\check{U}$ by looking at its
leaves as points in $\check{U}$ (or equivalently, by projecting $Y_{2}$ to
$\check{U}$). Of course, $\mathcal{F}$ is holomorphic if and only if
$S_{\mathcal{F}}$ is a holomorphic curve.

Most of the time, this duality is incomplete, in the sense that both $U$ and
$\check{U}$ are small open sets. One can expect that when either $U$ or
$\check{U}$ are globally well-defined, the situation is much more rigid. For
example, when the family $\mathscr{S}$ is a projective structure (meaning that
through each point and each direction passes a curve of $\mathscr{S}$), then
$\check{U}$ is a neighborhood of a $\mathbb{P}^{1}$ of self-intersection 1,
and the automorphism group of the family $\mathscr{S}$ is finite-dimensional,
see [8] for generic automorphisms, and the Zusatz of [6, Satz 4] to show that
automorphisms are finitely determinated.

Let us consider for one moment the most particular case: when $\mathscr{S}$ is
the family of affine lines. We can thus suppose that $U=\mathbb{P}^{2}$ and
$\check{U}=\check{\mathbb{P}}^{2}$, but the foliation is a priori only a
foliation in the neighborhood of a point $0\in\mathbb{P}^{2}$. Then
$S_{\mathcal{F}}$ is a germ of a real surface in $\check{\mathbb{P}}^{2}$ and
we can use [15] to form its dual
$\check{S}_{\mathcal{F}}\subset\mathbb{P}^{2}$. In general,
$\check{S}_{\mathcal{F}}$ is the hypersurface given by the enveloppe of the
family of curves $\mathcal{F}$. The construction of [15] is in fact very
general: we can do it for any system of curves $\mathscr{S}$. In this case,
the definition of the dual of a real subvariety $M\subset U$ is

$\check{M}=\\{L\in\check{U}\>|\>\text{$L$ intersects $M$ at a point $p$ with
}T_{p}L+T_{p}M\neq\mathbb{C}^{2}\\}.$

As stated above, the dual of a real surface is generically a real
hypersurface. More precisely, we can say the following:

######  Lemma 5.

Suppose $\mathscr{S}$ is a smooth 2-parameter system of curves in
$U\subset\mathbb{C}^{2}$ and $S$ is a real surface in $U$. We have the
following possibilities for the dual $\check{S}$:

  1. (1)

$\check{S}$ is a point, and $S$ is a complex curve in the family
$\mathscr{S}$.

  2. (2)

$\check{S}$ is a complex curve, and $S$ is also a complex curve.

  3. (3)

$\check{S}$ is a real surface, in which case the intersection between curves
of $\mathscr{S}$ and the surface $S$ define a real 2-parameter family of
curves $\mathscr{S}_{\mathbb{R}}$ on $S$; when $S$ is real analytic,
$\mathscr{S}$ is the complexification of $\mathscr{S}_{\mathbb{R}}$.

  4. (4)

$\check{S}$ is a real hypersurface.

###### Proof.

The projection $p_{2}:X_{2}\rightarrow X_{1}$ induces a biholomorphism between
the germs $(Z,0)$ and $(X_{1},0)$ so we will work with $X_{1}$. In particular,
the families $\mathscr{S}$ and $\check{\mathscr{S}}$ give two smooth
transverse foliations $\mathcal{G}$ and $\check{\mathcal{G}}$ on $X_{1}$, and
the contact structure $\mathcal{P}_{1}$ is given by
$\mathcal{P}_{1}=T\mathcal{G}\oplus T\check{\mathcal{G}}$. Consider also the
two projections $p=p_{1}:X_{1}\rightarrow U$ and
$p^{\prime}:X_{1}\rightarrow\check{U}$.

Now, as in [15], we define the lift $p^{*}S\subset X_{1}$ of $S$ as the set of
those $(z,\lambda)\in X_{1}$, where $z$ is a coordinate in $U$ and $\lambda$ a
coordinate of the fiber, such that $z\in S$, and $\lambda$ is the complex
direction of a real tangent vector in $T_{z}S$. Note that
$\check{S}=p^{\prime}(p^{*}S)$. As discussed in [15, §4], the lift $p^{*}S$
can be non-transverse to the projection $p^{\prime}$ so that $\check{S}$ can
be a priori very singular, in which case we could have difficulties defining
the second lift $p^{\prime*}\check{S}$. However, as soon as it is well-
defined, we have the equality of germs $p^{*}S=p^{\prime*}\check{S}$. In
particular, if $\check{S}$ is a point, then $p^{*}S$ is a leaf of
$\mathcal{G}$, and $S$ is a complex curve in the family $\mathscr{S}$. We see
also that when $\check{S}$ is a holomorphic curve, then $p^{*}S$ is the
holomorphic legendrian lift of $\check{S}$, so that its projection
$S=p(p^{*}S)$ is also a holomorphic curve.

When $\check{S}$ is a real surface which is not holomorphic, the lift $p^{*}S$
is a real 3-dimensional subvariety of $X_{1}$ which is everywhere non-
transverse to both $p$ and $p^{\prime}$. In particular, the fibers of
$p^{\prime}$ define a foliation by real curves $\mathcal{G}_{\mathbb{R}}$ on
$p^{*}S$. This foliation is generically transverse to the projection $p$ so
$p(\mathcal{G}_{\mathbb{R}})$ is a foliation by real curves on $S$. By
definition, the leaves of $\mathcal{G}_{\mathbb{R}}$ are included in leaves of
$\mathcal{G}$ so that the leaves of the induced foliation on $S$ are included
in curves of the system $\mathscr{S}$. When $S$ is real analytic, we can
suppose modulo biholomorphism that $S=\mathbb{R}^{2}$ around a generic point,
and in this case it is clear that $\mathscr{S}$ is the complexification of
$\mathscr{S}_{\mathbb{R}}$. ∎

Note however that if $\mathcal{F}$ is a smooth semiholomorphic foliation in an
open set $U$ and $S_{\mathcal{F}}$ is its dual surface, then the bidual
$\check{S}_{\mathcal{F}}$ will not intersect $U$. We will explain this in more
details in the next section, but for now, let us study some cases when
$\check{S}_{\mathcal{F}}$ does intersect $U$, which is very exceptional, as we
see in the following result:

######  Proposition 4.

Let $\mathscr{S}$ be a smooth system of curves in an open set
$U\subset\mathbb{C}^{2}$ and $\mathcal{F}$ a semiholomorphic foliation tangent
to $\mathscr{S}$ with a singular point $p\in U$. If $p$ is an isolated
singular point, then $\mathcal{F}$ is the pencil of curves of $\mathscr{S}$
passing through $p$; it is holomorphic and $\check{S}_{\mathcal{F}}=\\{p\\}$.
Suppose $p\in U$ is a non-isolated singular point of $\mathcal{F}$. Then the
bidual $\check{S}_{\mathcal{F}}$ is a real surface passing through $p$,
$\mathrm{Sing}(\mathcal{F})=\check{S}_{\mathcal{F}}$, the system of curves
$\mathscr{S}$ induces a real system of curves $\mathscr{S}_{\mathbb{R}}$ on
$\check{S}_{\mathcal{F}}$ and when $\mathcal{F}$ is real analytic,
$\mathscr{S}$ is the complexification of $\mathscr{S}_{\mathbb{R}}$.

###### Proof.

Since $\mathcal{F}$ is a foliation, the real codimension of its singular set
$\Sigma$ is at least 2. Consider a leaf $L_{0}$ of $\mathcal{F}$ intersecting
$\Sigma$ at a point $p_{0}$. By definition, $\mathcal{F}$ has another leaf
$L_{1}$ intersecting $L_{0}$ at $p_{0}$. Since $L_{0}$ and $L_{1}$ are curves
in $\mathscr{S}$ and $\mathscr{S}$ is smooth, their interesction at $p_{0}$ is
transversal, and any leaf $L$ of $\mathcal{F}$ close to $L_{1}$ intersects
$L_{0}$ at a point $p\in\Sigma$ close to $p_{0}$. If this point $p$ is always
equal to $p_{0}$, then all of the leaves pass through $\\{p_{0}\\}$ and
$\Sigma=\\{p_{0}\\}$. In this case we see that
$\check{\Sigma}=S_{\mathcal{F}}$ is the holomorphic curve of the family
$\check{\mathscr{S}}$ corresponding to $p_{0}$ so that $\mathcal{F}$ is
holomorphic.

Else the set of these points $p$ is a curve on $\Sigma$ and in particular
$L\cap\Sigma$ is a curve. We can say the same thing for all leaves, so that
each leaf $L$ intersects $\Sigma$ along a curve. Since the intersection points
between leaves are points, $\Sigma$ cannot be a real curve and it must be a
real surface. Necessarily, these intersections define a real 2-parameter
family of real curves on $\Sigma$, that is, a real system of curves
$\mathscr{S}_{\mathbb{R}}$. In particular, there is an open set $V$ in the
real tangent bundle $T\Sigma$ such that for $(p,v)\in V$, there is a leaf $L$
of $\mathcal{F}$ intersecting $\Sigma$ at $p$ in the real direction $v$. By
definiton of the dual of $\Sigma$, we have $\check{\Sigma}=S_{\mathcal{F}}$,
and thus $\Sigma=\check{S}_{\mathcal{F}}$.

Finally, note that the curves of the system $\mathscr{S}_{\mathbb{R}}$ are all
contained in leaves of $\mathcal{F}$, so that the complexified of
$\mathscr{S}_{\mathbb{R}}$ is a complex system of curves containing all the
leaves of $\mathcal{F}$. Since $\mathcal{F}$ is not holomorphic, it must be
equal to $\mathscr{S}$. ∎

######  Proposition 5.

Let $C$ be an elliptic curve and $S$ a 2-dimensional neighborhood of $C$.
Suppose there exists a smooth semiholomorphic foliation $\mathcal{F}$ of
semirank 2 in $S$ admitting $C$ as a leaf. Suppose also that the system of
curves $\mathscr{S}$ defined by $\mathcal{F}$ is smooth in the neighborhood of
the curve $C$, and the dual system $\check{\mathscr{S}}$ is a projective
structure. Then $\mathcal{F}$ is holomorphic.

###### Proof.

Consider the universal cover $(U,\tilde{C})$ of $(S,C)$, equipped with the
pullback semiholomorphic foliation, and the induced system of curves which we
will still denote by $\mathcal{F}$ and $\mathscr{S}$. If the dual system
$\check{\mathscr{S}}$ is a projective structure, then the dual $V\supset U$ of
$\check{U}$ contains a compact curve $L\simeq\mathbb{P}^{1}$; the curve $L$ is
the compactification $L=\tilde{C}\cup\\{\infty\\}$, the surface $V$ is a
neighborhood of $L$, we have $L\cdot L=1$, and the system $\mathscr{S}$
extends naturally as a 2-parameter family of deformations of $L$.

It follows that any curve $L^{\prime}\in\mathscr{S}$ close to $L$ intersects
$L$ at one point, so that if the foliation $\mathcal{F}$ is smooth, then
$L^{\prime}\cap L\in L\setminus\tilde{C}=\\{\infty\\}$. We see that
$\mathcal{F}$ is exactly the pencil of curves $L^{\prime}\in\mathscr{S}$ which
intersect $L$ at infinity, so that it is indeed holomorphic. ∎

###  3.3. Complete local models

As we saw in Theorem 3, in many interesting cases, the leaves will be complete
for the metric $|\eta_{\mathcal{F}}|^{2}$. It would then be interesting to
study local models of foliations $\mathcal{F}$ in open sets
$U\subset\mathbb{C}^{2}$ such that the leaves of $\mathcal{F}$ are complete.

######  Example 6.

Consider the foliation given by the 1-form
$\omega=\mathrm{Im}(x)dy-\mathrm{Im}(y)dx$. This is in fact a singular
foliation on the whole of $\mathbb{P}^{2}(\mathbb{C})$. Its singular set is
equal to $\mathrm{Sing}(\mathcal{F})=\mathbb{P}^{2}(\mathbb{R})$. Each complex
line $L$ tangent to the foliation is cut in two pieces by
$\mathrm{Sing}(\mathcal{F})$, and each piece equipped with the metric
$|\eta|^{2}$ is equal to Poincaré’s half plane.

This example is in fact very special: as we saw in Proposition 4, it
corresponds to the case when the dual of $\check{S}_{\mathcal{F}}$ is a real
surface, which is a degenerate case. In the rest of this section, we suppose
that we are in the generic case when $\check{S}_{\mathcal{F}}$ is a real
hypersurface.

By the duality, the hypersurface $\check{S}_{\mathcal{F}}$ is the enveloppe of
the family of curves parametrized by $S_{\mathcal{F}}$: each leaf $L$ of
$\mathcal{F}$ intersect $\check{S}_{\mathcal{F}}$ along a curve $\gamma\subset
L$, and is tangent to $\check{S}_{\mathcal{F}}$ along $\gamma$ (see [15] for
more details); it follows that the leaves of $\mathcal{F}$ are tangent to
$\check{S}_{\mathcal{F}}$ but we cannot extend them outside the curve $\gamma$
or different leaves will intersect $L$. This behaviour should be the generic
behaviour for complete models, even when the foliation is not of semirank 2.

More precisely, consider an open set $V\subset\mathbb{C}^{2}$, a smooth
2-parameter family of curves $\mathscr{S}$, and a germ of semiholomorphic
foliation $\mathcal{F}$ along a leaf $d_{0}$, tangent to $\mathscr{S}$ and
giving a germ of real surface $(S_{\mathcal{F}},d_{0})$ in the dual surface
$\check{V}$. Now suppose that the family $\mathscr{S}$ can be extended to a
smooth system still denoted $\mathscr{S}$ on some open set $U\supset V$, and
that the dual $\check{S}_{\mathcal{F}}$ is a real hypersurface in $U$ which
intersects $d_{0}$ along a real curve. Inside the germ of surface $(U,d_{0})$,
the hypersurface $\check{S}_{\mathcal{F}}$ has an interior which can be
considered the biggest domain of definition of $\mathcal{F}$, and each leaf of
$\mathcal{F}$ cuts tangencially $\check{S}_{\mathcal{F}}$ along a real curve.
Note that, using notations of section 1.2, this real curve has equation
$\\{|u|^{2}=|v|^{2}\\}$ so that a generic geodesic along a leaf which tends to
$\check{S}_{\mathcal{F}}$ has infinite length. Thus, when the real curve
$\check{S}_{\mathcal{F}}\cap d_{0}$ is compact, $\mathcal{F}$ can be extended
to a semiholomorphic foliation in the interior of $\check{S}_{\mathcal{F}}$
whose leaves are complete. Conversely, note that if leaves of $\mathcal{F}$ do
not adhere to the hypersurface $\check{S}_{\mathcal{F}}$ then they are not
complete.

Figure 2. The hypersurface $H=\check{S}_{\mathcal{F}}$ is tangent to leaves of
$\mathcal{F}$.

######  Definition 3.

We call germ of smooth non-degenerated local complete model along a leaf $L$
the triples $(U,V,\mathcal{F})$ where $V\subset U\subset\mathbb{C}^{2}$ are
germs of open sets along $L$, $\mathcal{F}$ is a germ of smooth
semiholomorphic foliation on $V$ tangent to a system of curves $\mathscr{S}$,
$L\cap V$ is a leaf of $\mathcal{F}$, the system $\mathscr{S}$ can be extended
to a smooth system of curves on $U$, and the border $H=\partial V$ is a germ
of compact hypersurface which satisfies $H=\check{S}_{\mathcal{F}}$ as germs
along $L\cap H$.

If a smooth semiholomorphic foliation tangent to a regular system of curves on
a global surface has complete leaves, then its universal covering is either of
the form described in Proposition 4, or is a smooth non-degenerated local
complete model along each leaf.

##  4\. Foliations tangent to projective structures

###  4.1. Examples of foliations by lines

We try to construct some examples of foliations $\mathcal{F}$ on germs of
surfaces $(S,C)$ around a hyperbolic compact complex curve $C$, such that $C$
is a leaf of $\mathcal{F}$. To do so, we will take $\mathcal{F}$ locally
modelled on the foliation $\mathcal{F}_{0}$ given by the $(1,0)$-form
$\omega_{0}=\mathrm{Im}(x)dy-\mathrm{Im}(y)dx$ on $\mathbb{P}^{2}(\mathbb{C})$
around the leaf $L_{0}=\\{y=0\\}$. Equivalently, we want to find a group $G$
of germs of diffeomorphisms of the surface $\mathbb{P}^{2}(\mathbb{C})$ around
an open set $V_{0}\subset L_{0}$, such that the quotient of $V_{0}$ by
$G|_{V_{0}}$ is diffeomorphic to $C$. Of course, if we want the foliation
$\mathcal{F}_{0}$ to give a foliation $\mathcal{F}$ on the quotient, the group
$G$ must send leaves of $\mathcal{F}_{0}$ to leaves of $\mathcal{F}_{0}$. This
means in particular that $G$ must preserve the set of affine lines in
$\mathbb{P}^{2}(\mathbb{C})$, so $G$ is a subgroup of
$\mathrm{PSL}_{3}(\mathbb{C})$.

The group $G$ must also preserve the singular set $\mathbb{P}^{2}(\mathbb{R})$
of $\mathcal{F}_{0}$, so $G\subset\mathrm{PSL}_{3}(\mathbb{R})$. Note that by
the duality explained in section 3.2, the fact for $G$ to preserve
$\mathbb{P}^{2}(\mathbb{R})$ is equivalent for its action on the dual
$\check{\mathbb{P}}^{2}(\mathbb{C})$ to preserve the dual surface
$S_{\mathcal{F}}$. This means exactly that $G$ sends leaves of
$\mathcal{F}_{0}$ to leaves of $\mathcal{F}_{0}$, i.e. the group of
automorphisms of $\mathcal{F}_{0}$ is $\mathrm{PSL}_{3}(\mathbb{R})$. Write an
element $M\in\mathrm{SL}_{3}(\mathbb{R})$ as

$M=\begin{bmatrix}a_{11}&a_{12}&a_{13}\\\ a_{21}&a_{22}&a_{23}\\\
a_{31}&a_{32}&a_{33}\end{bmatrix}.$

If we want $M$ to stabilize $L_{0}$, we must have $a_{21}=a_{23}=0$, and
$M|_{L_{0}}$ is given in an affine coordinate $x$ by

$M|_{L_{0}}(x)=\frac{a_{11}x+a_{13}}{a_{31}x+a_{33}}.$

Hence the construction: take any fuchsian subgroup
$G_{0}\subset\mathrm{PSL}_{2}(\mathbb{R})$ such that the quotient of the half-
plane $\mathbb{H}_{0}\subset L_{0}$ by $G_{0}$ is a hyperbolic compact curve
$C$. Write $a_{11}(g),a_{13}(g),a_{31}(g),a_{33}(g)$ the coefficients of the
elements $g\in G_{0}$. Choose any extension $G$ of $G_{0}$ to
$\mathrm{PSL}_{3}(\mathbb{R})$ (i.e. $a_{22}(g)=1$, and
$(a_{12}(g),a_{32}(g))$ is a cocycle for the group $G_{0}$); the most simple
extension being of course $a_{12}(g)=a_{32}(g)=0$ and $a_{22}(g)=1$. Then the
quotient of a neighborhood $U$ of $\mathbb{H}_{0}$ in
$\mathbb{P}^{2}(\mathbb{C})$ by $G$ is a surface $S$ containing a curve $C$
quotient of $\mathbb{H}_{0}$, and the foliation $\mathcal{F}_{0}$ descends to
a smooth semiholomorphic foliation $\mathcal{F}$ on $S$ having $C$ as a leaf.

######  Theorem 4.

Let $C=\mathbb{H}_{0}/G_{0}$ be a compact curve of genus $g\geq 2$, where
$G_{0}\subset\mathrm{PSL}_{2}(\mathbb{R})$ is a Fuchsian subgroup. Let
$\mathscr{M}$ denote the moduli space of neighborhoods $(S,C,\mathcal{F})$ of
$C$ in complex surfaces equipped with a smooth semiholomorphic foliation
$\mathcal{F}$ locally diffeomorphic to $\mathrm{Im}(x)dy-\mathrm{Im}(y)dx$,
modulo biholomorphism. The construction explained above induces a bijection

$\mathscr{M}\simeq H^{1}(G_{0},\mathbb{R}^{2}).$

###### Proof.

As we explained above, every example comes from a cocycle
$Z^{1}(G_{0},\mathbb{R}^{2})$, so that we only need to prove that equivalence
modulo biholomorphism for $(S,C,\mathcal{F})$ corresponds to equivalence
modulo coboundaries for cocycles. We keep the notations above in this proof.

The holonomy of $\mathcal{F}$ in $S$ defines a permutation of the leaves,
which is exactly the action of $G$ on the dual surface $S_{\mathcal{F}}$; in
particular, from this point of view the holonomy is given by elements of
$\mathrm{PSL}_{3}(\mathbb{R})$. It follows that this holonomy is well-defined
modulo conjugacy in $\mathrm{PSL}_{3}(\mathbb{R})$; we can suppose that this
conjugacy fixes $L_{0}$ and the group $G_{0}=G|_{L_{0}}$. The action by
conjugacy of the subgroup of $\mathrm{PSL}_{3}(\mathbb{R})$ fixing $L_{0}$ and
$G_{0}$ is exactly the action of coboundaries on cocycles, so that the theorem
is proved. ∎

Note that, if we have coordinates $[x:y:z]$ on $\mathbb{P}^{2}$, in the affine
chart $z=1$ the action of the group given by the zero cocycle writes

$M(x,y)=\left(\frac{a_{11}x+a_{13}}{a_{31}x+a_{33}},\frac{y}{a_{31}x+a_{33}}\right)=:(\alpha(x),\beta(x)y).$

The tangent bundle of the leaf $L_{0}=\\{y=0\\}$ will be given by
$\alpha^{\prime}(x)=\frac{a_{11}a_{33}-a_{13}a_{31}}{(a_{31}x+a_{33})^{2}}$
and its normal bundle by $\beta(x)$. We see here explicitely that the tangent
bundle and the normal bundle are closely related, as stated in Theorem 2.

Note also that the cocycles only intervene at higher order.

###  4.2. Generic foliations by lines

######  Theorem 5.

Consider a real-analytic semiholomorphic foliation by lines $\mathcal{F}$
defined in an open set $U\subset\mathbb{P}^{2}(\mathbb{C})$ neighborhood of
the leaf $L_{0}$. Suppose there is a subgroup $G<\mathrm{PSL}_{3}(\mathbb{C})$
stabilizing $\mathcal{F}$ and $L_{0}$, such that the restriction to $L_{0}$ is
injective, $G|_{L_{0}}$ is a Fuchsian group and the quotient of $L_{0}$ by $G$
is a compact Riemann surface of genus $g\geq 2$.

Suppose moreover that $\mathcal{F}$ is not holomorphic at first order along
$L_{0}$, i.e. $i_{L_{0}}^{*}\eta_{\mathcal{F}}\not\equiv 0$. Then
$\mathcal{F}$ is biholomorphic to the foliation given by the $(1,0)$-form
$\mathrm{Im}(x)dy-\mathrm{Im}(y)dx$.

The foliation $\mathcal{F}$ defines a real surface $S=S_{\mathcal{F}}$ in
$\check{\mathbb{P}}^{2}$, and the action of $G$ on the dual space
$\check{\mathbb{P}}^{2}$, denoted by $\check{G}<\mathrm{PSL}_{3}(\mathbb{C})$
stabilizes $S$. Since $G$ stabilizes a line $L_{0}$, the group $\check{G}$
stabilizes a point $p_{0}$. If $g\in G$ is hyperbolic in restriction to
$L_{0}$, then its action $\varphi\in\mathrm{PSL}_{3}(\mathbb{C})$ on
$\check{\mathbb{P}}^{2}$ is such that $d_{p_{0}}\varphi$ is hyperbolic. Most
real surfaces do not have a lot of symmetries in
$\mathrm{PSL}_{3}(\mathbb{C})$, we begin by examining those having one
hyperbolic symmetry.

######  Lemma 6.

Let $(S,p_{0})\subset\check{\mathbb{P}}^{2}$ be a smooth germ of real-analytic
surface such that $T_{p_{0}}S$ is not complex. Suppose that
$\varphi\in\mathrm{PSL}_{3}(\mathbb{C})$ stabilizes $S$ and $d_{p_{0}}\varphi$
is hyperbolic. Then either $S$ is a real affine plane, or $S$ is equipped with
a real codimension 1 foliation whose leaves are invariant by $\varphi$.
Moreover, these leaves are intersections between $S$ and real affine planes
passing through $p_{0}$; the foliation has exactly two separatrices at
$p_{0}$, and they are tangent to eigenvectors of $d_{p_{0}}\varphi$.

###### Proof.

Since $\varphi(p_{0})=p_{0}$, we can write $\varphi$ in some coordinates

$\varphi=\begin{bmatrix}a_{11}&a_{12}&0\\\ a_{21}&a_{22}&0\\\
a_{31}&a_{32}&1\end{bmatrix}\in\mathrm{GL}_{3}(\mathbb{C}).$

Now, the differential $d_{p_{0}}\varphi=\begin{bmatrix}a_{11}&a_{12}\\\
a_{21}&a_{22}\end{bmatrix}$ is hyperbolic, and thus diagonalizable:
$d_{p_{0}}\varphi\sim\begin{bmatrix}\lambda&0\\\ 0&\mu\end{bmatrix}$, for
example with $|\lambda|>|\mu|$. By hypothesis, $T_{p_{0}}S$ is not a complex
direction, so there is a conjugacy $\psi$ with
$\psi\in\mathrm{PSL}_{3}(\mathbb{C})$ fixing $p_{0}$, such that
$T_{p_{0}}\psi(S)=p_{0}+\mathbb{R}^{2}$. This implies
$d_{p_{0}}(\psi\varphi\psi^{-1})(\mathbb{R}^{2})=\mathbb{R}^{2}$, and since
$\lambda$ and $\mu$ are not complex conjugates, we see that $\lambda$ and
$\mu$ are real.

There are two cases to examine: either $\varphi$ is diagonalizable and in some
coordinates,

$\varphi=\begin{bmatrix}\lambda&0&0\\\ 0&\mu&0\\\ 0&0&1\end{bmatrix},$

or

$\varphi=\begin{bmatrix}\lambda&0&0\\\ 0&1&0\\\ 0&1&1\end{bmatrix}.$

In both cases, if we consider coordinates $(x,y)$ centered at $p_{0}$ with
$d_{0}\varphi(x,y)=(\lambda x,\mu y)$, if we write $x=x_{1}+ix_{2}$,
$y=y_{1}+iy_{2}$, then for any equations of the tangent plane $T_{p_{0}}S$

$\left\\{\begin{aligned} l_{11}x_{1}+l_{12}x_{2}+l_{13}y_{1}+l_{14}y_{2}&=0\\\
l_{21}x_{1}+l_{22}x_{2}+l_{23}y_{1}+l_{24}y_{2}&=0,\end{aligned}\right.$

by the equation $d\varphi(T_{p_{0}}S)=T_{p_{0}}S$, we get

$\left\\{\begin{aligned} (\lambda/\mu-1)(l_{11}x_{1}+l_{12}x_{2})&=0\\\
(\mu/\lambda-1)(l_{23}y_{1}+l_{24}y_{2})&=0.\end{aligned}\right.$

We deduce that when $T_{p_{0}}S$ is not complex, it has equations
$\\{x_{2}/x_{1}=\mathrm{tan}(\alpha_{1}),y_{2}/y_{1}=\mathrm{tan}(\alpha_{2})\\}$
for some constants $\alpha_{1},\alpha_{2}\in S^{1}$. The application
$(x,y,z)\mapsto(e^{-i\alpha_{1}}x,y,z)$ stabilizes $\varphi$ so we can suppose
$\alpha_{1}=0$ in both cases. In the diagonalizable case, we can also suppose
$\alpha_{2}=0$.

Suppose now that we are in the diagonalizable case: in some coordinates
$(x,y)$ centered at $p_{0}$, we have $\varphi(x,y)=(\lambda x,\mu y)$. Remark
first that the real functions $x_{2}/x_{1}$ and $y_{2}/y_{1}$ are stable by
$\varphi$ and induce real foliations by invariant curves on $S$ whenever they
are not constant on $S$. We can write $S$ as a graph

$\left\\{\begin{aligned} x_{2}&=x_{1}f_{1}(x_{1},y_{1})\\\
y_{2}&=y_{1}f_{2}(x_{1},y_{1}).\end{aligned}\right.$

The functions $f_{1},f_{2}$ are unique, and since $S$ is stable by $\varphi$,
we get for every $x_{1},y_{1}$,

$\left\\{\begin{aligned} f_{1}(x_{1},y_{1})&=f_{1}(\lambda x_{1},\mu y_{1})\\\
f_{2}(x_{1},y_{1})&=f_{2}(\lambda x_{1},\mu y_{1}).\end{aligned}\right.$

There are two cases: either the diffeomorphism $\varphi|_{\mathbb{R}^{2}}$ has
a non-constant real-analytic first integral or not. In the first case, every
first integral is of the form $f(x_{1}^{p}y_{1}^{q})$ for some integers $p,q$,
so that the two foliations induced by $x_{2}/x_{1}$ and $y_{2}/y_{1}$ on $S$
are in fact equal. This means that every leaf of this common foliation is a
component of the intersection between $S$ and a real affine plane
$\\{x_{2}=\mathrm{tan}(\theta)x_{1},y_{2}=\mathrm{tan}(\alpha)y_{1}\\}$. Note
that, since this foliation has a first integral $x_{1}^{p}y_{1}^{q}$, it has
two separatrices at $p_{0}$, and the complexification of the tangents at
$p_{0}$ of these separatrices are the eigenvectors of $\varphi$ corresponding
to $\lambda$ and $\mu$. In the second case, every first integral is constant,
so that $S$ is included in a real affine plane
$\\{x_{2}=\mathrm{tan}(\theta)x_{1},y_{2}=\mathrm{tan}(\alpha)y_{1}\\}$.

In the non-diagonalizable case, $\varphi$ can be expressed in some coordinates
$(x,y)$ centered at $p_{0}$ as

$\varphi(x,y)=\left(\frac{\lambda x}{1+y},\frac{y}{1+y}\right).$

In the coordinates $(z,w)=(x/y,1/y)$, this expression becomes
$\varphi(z,w)=(\lambda z,w+1)$. As before, we notice that the functions
$z_{2}/z_{1}$ and $w_{2}$ are invariant by $\varphi$.

The tangent plane to $S$ can be parametrized by
$(s,t)\in\mathbb{R}^{2}\mapsto(x,y)=(s,te^{i\alpha_{2}})$, so that in a
neighborhood of $p_{0}$, $S$ is close to the surface parametrized by
$(z,w)=(se^{-i\alpha_{2}}/t,e^{-i\alpha_{2}}/t)$. In particular, if
$\alpha_{2}\neq\pm\pi/2$, in a neighborhood of $p_{0}$ we can write
$S\setminus\\{p_{0}\\}$ as a graph

$\left\\{\begin{aligned} z_{2}&=z_{1}f_{1}(z_{1},w_{1})\\\
w_{2}&=f_{2}(z_{1},w_{1}).\end{aligned}\right.$

As in the diagonalizable case, the functions $f_{i}$ are unique and $S$ is
stabilized by $\varphi$ so we get

$\left\\{\begin{aligned} f_{1}(z_{1},w_{1})&=f_{1}(\lambda z_{1},w_{1}+1)\\\
f_{2}(z_{1},w_{1})&=f_{2}(\lambda z_{1},w_{1}+1).\end{aligned}\right.$

The local first integrals of $\varphi|_{\mathbb{R}^{2}}$ are all of the form
$f(z_{1}\mathrm{exp}(-w_{1}\mathrm{log}(\lambda)))$. Note that
$z_{1}\mathrm{exp}(-w_{1}\mathrm{log}(\lambda))=(x/y)\mathrm{exp}(-\mathrm{log}(\lambda)/y)$
so that these first integrals are not real analytic at $(x,y)=(0,0)$. It
follows that $z_{2}/z_{1}$ and $w_{2}$ are constant on $S$. Since
$[x:y:1]=[z:1:w]$, these equations indeed define real affine subspaces.

If $\alpha_{2}$ is equal to $\pm\pi/2$, we can write $S$ as a graph

$\left\\{\begin{aligned} z_{1}&=z_{2}f_{1}(z_{2},w_{2})\\\
w_{1}&=f_{2}(z_{2},w_{2}),\end{aligned}\right.$

and we get

$\left\\{\begin{aligned} f_{1}(z_{2},w_{2})&=f_{1}(\lambda z_{2},w_{2})\\\
f_{2}(z_{2},w_{2})&=f_{2}(\lambda z_{2},w_{2})-1.\end{aligned}\right.$

From the second equation we get that $f_{2}|_{\\{z_{2}=0\\}}=\infty$ so that
in fact $\alpha_{2}\neq\pm\pi/2$. ∎

###### Proof of Theorem 5.

Aiming at a contradiction, we suppose in this proof that the dual surface $S$
is not contained in any affine real plane. The group $G|_{L_{0}}$ is generated
by $2g$ hyperbolic elements $h_{i}$, and two different $h_{i}$ have different
pairs of fixed points on the border $\partial L_{0}$. These fixed points are
the eigenvectors of the differentials $d_{p_{0}}\varphi_{i}$ of the actions of
$h_{i}$ on $\check{\mathbb{P}}^{2}$.

Consider 3 among them: $\varphi_{1},\varphi_{2},\varphi_{3}$. The foliations
defined by them on $S$ have different separatrices, so are different. Thus by
a generic point $p\in S$ pass three curves contained in real affine planes
also passing through $p_{0}$. Consider the projection
$\pi:(\mathbb{P}^{2}(\mathbb{C}),p_{0})\setminus\\{p_{0}\\}\rightarrow\mathbb{P}^{3}(\mathbb{R})$.
Since $S$ is not contained in any real plane, the space $\pi(S)$ is a surface;
and we just saw that through a generic point $p\in\pi(S)$ pass three different
affine lines contained in $\pi(S)$. By [9, §16.5], this implies that $\pi(S)$
is a real affine plane, so that $S$ is contained in a real affine hyperplane
$H$. This hyperplane $H$ is obviously unique, so that it is stable by
$d_{p_{0}}\varphi$ for each $\varphi\in G$. It follows that the unique complex
direction tangent to $H$ is stable by each $d_{p_{0}}\varphi$, and that all of
the elements of $G$ share an eigenvector. This is obviously not possible for
the fundamental group of a smooth compact curves. ∎

In the proof of this result, the hypothesis of real analyticity is only used
in Lemma 6 to show that when two real functions on $S$ are invariant by an
automorphism $\varphi$, they must define the same foliation. This result is
false in the $\mathcal{C}^{\infty}$ context. Indeed, consider a diffeomorphism
$\varphi(x_{1},y_{1})=(\lambda x_{1},\mu y_{1})$ with for example
$\lambda>1>\mu>0$ and $\lambda^{p}\mu^{q}=1$ for some integers
$p,q\in\mathbb{N}^{*}$.

Write $l=\mathrm{log}(\lambda),m=\mathrm{log}(\mu)$ and consider the functions
$r=x^{p}y^{q}$ and $\theta=\frac{1}{m-l}\mathrm{log}\left(\frac{y}{x}\right)$.
Then $r\circ\varphi=r$ and $\theta\circ\varphi=\theta+1$; the foliation
$\\{r=cte\\}$ is stable by $\varphi$, has two separatrices
$L_{x}=\\{y=0\\},L_{y}=\\{x=0\\}$ and these separatrices cut the plane
$\mathbb{R}^{2}$ in four invariant quadrants
$\mathbb{R}^{2}\setminus(L_{x}\cup L_{y})=\bigcup Q_{i}$.

If $f$ is a function defined in a quadrant $Q_{i}$ which is invariant by
$\varphi$, its restriction to a curve $\\{r=c\\}$ is periodic in $\theta$, and
we can develop it in Fourier series:

$f|_{Q_{i}}(r,\theta)=\sum_{n\in\mathbb{Z}}c_{n}^{(i)}(r)e^{in\theta}.$

We know by Fourier theory that $f|_{Q_{i}}$ is $\mathcal{C}^{\infty}$ if and
only if $|c_{n}^{(i)}(r)|=o(n^{k})$ for each $k\in\mathbb{N}$. The question is
then to find those tuples $(f_{1},f_{2},f_{3},f_{4})$ which can be glued to a
$\mathcal{C}^{\infty}$ function $f$ on $\mathbb{R}^{2}$. Remark here that if
$|c_{n}^{(i)}(r)|=o(r^{k})$ for each $i,n,k$, we can indeed glue the functions
$f_{i}$ to a function $f$: when $(x,y)$ tends to one of the separatrices
$L_{x}$ or $L_{y}$, a quick computation shows that all the derivatives
$\frac{\partial^{k_{1}+k_{2}}f_{i}}{\partial x^{k_{1}}\partial y^{k_{2}}}$
tend to zero.

This shows that there are indeed germs of real surfaces invariant by an
automorphism $\varphi$ which contradict Lemma 6 in the $\mathcal{C}^{\infty}$
context.

###  4.3. Another example

We can try to find other examples of semiholomorphic foliations whose leaves
are curves of a projective structure. In general, projective structures could
have isolated symmetries, so we will only consider symmetries which are
exponentials of infinitesimal automorphisms, as studied in [8, §6.2]. The only
groups of infinitesimal automorphisms which can give rise to a compact curve
of genus $g\geq 2$ are $\mathfrak{sl}_{2}(\mathbb{C})$ and
$\mathfrak{sl}_{3}(\mathbb{C})$. The only projective structure (modulo
biholomorphisms) having a symmetry group $\mathfrak{sl}_{3}(\mathbb{C})$ is
the family of lines in $\mathbb{P}^{2}$, which we already considered. As for
$\mathfrak{sl}_{2}(\mathbb{C})$, it only occurs as infinitesimal symmetries
for the structure $\mathscr{S}$ whose curves have equations $y=y(x)$ in a
neighborhood $U$ of the origin with

$y^{\prime\prime}=(xy^{\prime}-y)^{3}.$

However the action of the group is the linear action of
$G=\mathrm{SL}_{2}(\mathbb{C})$ so it fixes a point. Thus we cannot build an
example of neighborhood of a compact curve of genus $g\geq 2$ from this
structure.

Consider then the dual family $\check{\mathscr{S}}$ of curves of this
projective structure: it is defined in the surface $S_{0}$ described in [8,
§5.4]. Each point $p\in U$ gives rise to a rational curve $L_{p}\subset
S_{0}$, these curves all have self-intersection 1 in $S_{0}$, and they form
the dual family. We can consider $S_{0}$ as a neighborhood of $L_{0}$. Since
this family is the dual of $\mathscr{S}$ , it also has symmetry group
$\check{G}=\mathrm{SL}_{2}(\mathbb{C})$; in $U$ the automorphism group fixed
the origin $0\in U$, so that the action of $\check{G}$ on $S_{0}$ stabilizes
the curve $L_{0}$, and acts as Möbius transforms on it.

Recall from [8] that $S_{0}$ is a double cover of
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ along the diagonal $\Delta$, so that the
family $\check{\mathscr{S}}$ corresponds to graphs of Möbius functions tangent
to $\Delta$. In some coordinates $(x,y)$, a curve $L$ in $\check{\mathscr{S}}$
has equation

$y=\frac{a(x-x_{0})}{\sqrt{1+a^{2}(x-x_{0})}},$

where the two parametres $x_{0}$ and $a$ correspond respectively to the point
of intersection between $L$ and $L_{0}$, and to the slope of $L$ at this
point.

Now consider the real plane $\\{(x_{0},a)\in\mathbb{R}^{2}\\}\subset U$, which
corresponds to a semiholomorphic foliation $\mathcal{F}$ in $S_{0}$ whose
leaves are precisely those curves for which $a$ and $x_{0}$ are real. We can
check that when $a$ and $x_{0}$ are real, we can recover $x_{0}$ from the
equation

$x_{0}=\frac{\mathrm{Im}(\bar{x}y^{2})-|y|^{2}\mathrm{Im}(x)\sqrt{1-\frac{\mathrm{Im}(y^{2})}{\mathrm{Im}(x)}}}{\mathrm{Im}(y^{2})}=:f(x,y).$

In particular, the real function $f$ is constant along the leaves of
$\mathcal{F}$ so that its level hypersurfaces are Levi-flat hypersurfaces
foliated by leaves of $\mathcal{F}$. We deduce that the $(1,0)$-form $\partial
f$ defines the foliation $\mathcal{F}$, and after some computations that the
expression of the slope $\lambda=dy/dx$ is

$\lambda=\frac{\mathrm{Im}(y^{2})}{2\mathrm{Im}(x)}\frac{\bar{y}\sqrt{1-\frac{\mathrm{Im}(y^{2})}{\mathrm{Im}(x)}}+y\left(1-\frac{\mathrm{Im}(y^{2})}{2\mathrm{Im}(x)}\right)}{|y|^{2}\sqrt{1-\frac{\mathrm{Im}(y^{2})}{\mathrm{Im}(x)}}+\left(\mathrm{Re}(y^{2})-\bar{y}^{2}\frac{\mathrm{Im}(y^{2})}{2\mathrm{Im}(x)}\right)}.$

The surface $S_{0}$ is defined over $\mathbb{R}$ and is the complexified of
the real-analytic surface $S_{\mathbb{R}}=\\{(x,y)\in\mathbb{R}^{2}\\}$. As
before, $L_{0}\setminus S_{\mathbb{R}}$ is the union of two hyperbolic planes;
consider one of them $\mathbb{H}_{0}\subset L_{0}$ and a small neighborhood
$V\subset S_{0}$ of $\mathbb{H}_{0}$. One can check that if $V$ is small
enough, $\mathcal{F}$ is smooth in $V$, and that its automorphism group is
$\mathrm{SL}_{2}(\mathbb{R})$ acting as Möbius transformations on
$\mathbb{H}_{0}$.

We conclude that for any compact curve $C$ of genus $g\geq 2$ there is exactly
one example (up to biholomorphism) of a surface $(S,C)$ equipped with a smooth
semiholomorphic foliation $\mathcal{F}$ coming from the construction above.
Remember that the construction above only gives those examples which come in
family: there might be exceptional examples coming from automorphisms which
are not exponentials of infinitesimal automorphisms.

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