# Universal Deformation of a Curve and a Differential

Emma Carberry<EMAIL_ADDRESS>School of Mathematics and
Statistics
University of Sydney
Australia  and  Martin Ulrich Schmidt<EMAIL_ADDRESS>Mathematics Chair III
Universität Mannheim
D-68131 Mannheim, Germany

###### Abstract.

We construct a universal local deformation (Kuranishi family) for pairs
consisting of a compact complex curve and a meromorphic 1-form. Each pair is
assumed to be locally planar, a condition which in particular forces the
periods of the meromorphic differential to be preserved by local deformations.
The hyperelliptic case yields a universal local deformation for the spectral
data of integrable systems such as simply-periodic solutions of the KdV
equation or of the sinh-Gordon equation (cylinders of constant mean
curvature). This is the first of two papers in which we shall develop a
deformation theory of the spectral curve data of an integrable system.

In this paper we construct a universal local deformation of certain pairs
$(X,\Theta)$ where $X$ is a compact one-dimensional complex analytic space and
$\Theta$ is a meromorphic differential on $X$ with prescribed poles. The pairs
are such that away from the poles of $\Theta$, any local primitive $x$ of
$\Theta$ can be complemented by another local function $y$ to give a local
embedding $(x,y)$ of $X$ into the affine plane $\mathbb{C}^{2}$. In a
neighbourhood of a pole, instead an appropriate power of any primitive $x$
provides a local coordinate for the curve. These locally planar pairs
$(X,\Theta)$ are defined in Definition 1.5.

When the curve $X$ is hyperelliptic then a complementary function $y$ may be
provided by a local parameter for the quotient of $X$ by its hyperelliptic
involution. Such hyperelliptic $(X,\Theta)$ arise for example as the spectral
data for simply periodic solutions of KdV or the Sinh-Gordon equation.

The “locally planar” condition has the important consequence that the periods
of $\Theta$ are preserved by local deformations of the data $(X,\Theta)$, as
shown in Remark 1.7. This is particularly pertinent to the spectral curve
applications, as then the periods of the differential $\Theta$ frequently
satisfy restrictive conditions, such as being integer-valued.

These period-preserving deformations can be studied purely locally. There are
only finitely many points $q$ of $X$ where the data $(X,\Theta)$ may deform
nontrivially, namely the plane curve singularities or ramification points of
the local Weierstraß covering $x_{q}$. We show in Theorem 2.2 that every
deformation of locally planar $(X,\Theta)$ is obtained by patching
deformations of finitely many such germs and is uniquely determined by these
germ deformations. Hence for deformations of locally planar $(X,\Theta)$ it is
not necessary to create a global theory along the lines of [ACG11]. Instead we
take a local approach more closely related to the deformation theory of plane
curve singularities [dJP00, GLS07]. This approach is furthermore well adapted
to the spectral curve examples, as in the finite-type situation the Bloch
curve has infinite arithmetic genus but finite geometric genus, and is such
that the deformations permitted for it to remain a spectral curve are non-
trivial only in the neighbourhood of finitely many points. Patching arguments
similar to Theorem 2.2 treat such examples also.

There are two key ways in which the deformation theory of locally planar
$(X,\Theta)$ is simpler than that of the curve alone. The first is the
reduction of the study of deformations of the global object $(X,\Theta)$ to
deformations of finitely many germs. The analogous result of course does not
hold for deformations of the curve alone, as otherwise the complex structure
on a Riemann surface could not be deformed. Secondly, deformations of germs of
$(X,\Theta)$ have a rigidity not enjoyed by deformations of curve
singularities. It is this rigidity that enables the construction of a
universal local deformation, as opposed to merely a semi-universal one.

For plane curve singularities, using the infinitesimal deformations one can
construct a semi-universal local deformation [dJP00, GLS07]. There one cannot
hope for a universal local deformation as the dimension of the space of
infinitesimal deformations is not locally constant. Smoothing out a
singularity of $f(x,y)=0$ decreases the dimension of the Tjurina algebra
$\mathbb{C}\\{x,y\\}/\langle f,\frac{\partial f}{\partial x},\frac{\partial
f}{\partial y}\rangle$.

By contrast, we find that the dimension of infinitesimal deformations
$\mathbb{C}\\{x,y\\}/\langle f,\frac{\partial f}{\partial y}\rangle$ of
$(X,\Theta)$ is locally constant, in fact the infinitesimal deformations form
a vector bundle over the base (Theorem 3.5). This helps us to show that the
local deformation (19) is universal (Theorem 5.1). Our arguments are
furthermore simpler than the construction of semi-universal deformations of
plane curve singularities referenced above in that we do not rely upon
Grauert’s powerful Approximation Theorem [dJP00, Chapter 8.2].

One may ask whether the base space $\mathcal{T}$ of this universal deformation
locally parameterises isomorphism classes of pairs $(X,\Theta)$. It is a
general property of universal deformations that this cannot be the case
whenever $(X,\Theta)$ possesses a non-trivial automorphism (Example 5.3), as
this extends to an isomorphism of fibres over distinct elements of the base
$\mathcal{T}$. We provide also a heuristic argument suggesting that such
isomorphic fibres exist only when the base fibre has a non-trivial
automorphism.

Our interest in this topic arose from wanting to develop a deformation theory
for the spectral data of an integrable system. Spectral curves are typically
endowed with two meromorphic differentials and local primitives of these
provide a locally planar embedding of the curve. Thus the locally planar
condition is automatically satisfied in the integrable systems applications.
In this manuscript we develop the theory of deformations of a curve with a
single differential. In a subsequent paper we shall build upon these results
to deal with deformations of two differentials.

The objects, deformations and morphisms we shall consider are defined in
Section 1. Throughout the paper we incorporate the option of the data
satisfying a reality condition. Section 2 explains that deformations of a
curve $X$ and differential $\Theta$ as above are both generated and uniquely
determined by deformations of finitely many germs of such data. In Section 3
we use the infinitesimal deformations of $(X,\Theta)$ to construct a specific
local deformation (18) of it. Importantly, the infinitesimal deformations of
the fibres (18) all have the same dimension, in fact they form a vector bundle
over the base. Our main result, Theorem 5.1, is that this local deformation
(18) is universal. To prove this we utilise a decomposition of certain
holomorphic functions which is the subject of Section 4. Section 5 completes
the proof of Theorem 5.1. It also contains a discussion of the role of
automorphisms of the base fibre $(X,\Theta)$. We conclude in Section 6 by
extending Theorem 5.1 to the case when $X$ and the curves through which it may
be deformed are hyperelliptic. This immediately yields for example a universal
local deformation for simply periodic solutions of the KdV equation or for
constant mean curvature cylinders in $S^{3},\,\mathbb{R}^{3},H^{3}$ or
$S^{2}\times\mathbb{R}$.

##  1\. Locally planar deformations

In this section we describe the data and the deformations we shall consider.
Our primary objects of study are local deformations of pairs $(X,\Theta)$,
where $X$ is a compact one-dimensional complex space and $\Theta$ is a
meromorphic differential on $X$ with prescribed poles. We make the assumption
that the pair $(X,\Theta)$ is locally planar (see Definition 1.5), which will
have the pleasant consequence that such pairs are locally parameterised by a
smooth manifold. The deformations of locally planar $(X,\Theta)$ (see
Definition 1.6) furthermore preserve the periods of $\Theta$ (Remark 1.7).
Throughout we shall use $x$ to denote a local primitive of $\Theta$.

We begin by recalling the basic notions of deformations of a complex space,
which we will extend to define appropriate deformations of the data we will
discuss. A _deformation_ $X\hookrightarrow Y\twoheadrightarrow\mathcal{S}$ of
a complex space $X$ is a pair of complex spaces $Y$ and $\mathcal{S}$ together
with a flat map $Y\twoheadrightarrow\mathcal{S}$, a marked point
$s_{0}\in\mathcal{S}$ and an isomorphism from $X$ to the preimage of the point
$s_{0}$ in $Y$. The space $X$ is called the _special fibre_ , $Y$ the _total
space_ and $\mathcal{S}$ the _base space_ of the deformation. Flatness ensures
that the fibres of the map $Y\twoheadrightarrow\mathcal{S}$ depend in a
regular way on the points in $\mathcal{S}$. A _morphism_ from a deformation
$X\hookrightarrow Y\twoheadrightarrow\mathcal{S}\ni s_{0}$ to a deformation
$X\hookrightarrow Z\twoheadrightarrow\mathcal{T}\ni t_{0}$ is a commutative
diagram of holomorphic maps

$\begin{array}[]{cclcl}X&\hookrightarrow&Y&\twoheadrightarrow&\mathcal{S}\ni
s_{0}\\\ \|&&\downarrow&&\downarrow\hskip 17.07164pt\downarrow\\\
X&\hookrightarrow&Z&\twoheadrightarrow&\mathcal{T}\ni t_{0}\end{array}$

such that the surjective horizontal maps are flat. For any deformation
$X\hookrightarrow Y\twoheadrightarrow\mathcal{S}\ni s_{0}$ and any open
neighbourhood $O\subset\mathcal{S}$ of $s_{0}$ let $U\subset
Y\twoheadrightarrow\mathcal{S}$ be the preimage of $O$. The resulting
deformation $X\hookrightarrow U\twoheadrightarrow O\ni s_{0}$ is called the
_restriction_ of $X\hookrightarrow Y\twoheadrightarrow\mathcal{S}\ni s_{0}$ to
$O$. To avoid double subscripts we shall denote the base point as $0$;
frequently $\mathcal{S}\subset\mathbb{C}^{r}$ and then without loss of
generality we take the base point to be the origin.

######  Definition 1.1.

A _local deformation_ $X\hookrightarrow Y\twoheadrightarrow\mathcal{S}_{0}$ is
an equivalence class of deformations $X\hookrightarrow
Y\twoheadrightarrow\mathcal{S}$ defined on a neighbourhood of
$0\in\mathcal{S}$, where two such deformations are equivalent if their
restrictions to some neighbourhood of $0\in\mathcal{S}$ are isomorphic.

One often wishes to deform a singularity or other non-generic structure and so
it is also convenient to work locally on the special fibre $X$ and consider
(local) deformations of a space germ $X_{p}$. The above definition of a
deformation is modified in the obvious way with the total space $Y$ and base
space $\mathcal{S}$ being replaced by space germs $Y_{p}$ and
$\mathcal{S}_{0}$ respectively.

Our primary concern is with the case where the complex space $X$ is a curve
and away from finitely many smooth points is locally planar with respect to a
primitive $x$ of $\Theta$ and some other local function $y$, meaning that
$(x,y)$ locally embed $X$ into $\mathbb{C}^{2}$. Within the context of our
main application, namely spectral curves of integrable systems, the locally
planar condition appears naturally.

######  Definition 1.2.

A one-dimensional complex space germ $X_{q}$ is locally planar with respect to
$(x_{q},y_{q})\in\mathcal{O}_{X,q}\times\mathcal{O}_{X,q}$ if these vanish at
$q$ and map $X_{q}$ biregularly onto the zero set of some
$f\in\mathbb{C}\\{x,y\\}$. Let us write $V(f)$ for the space germ given by the
vanishing set of a germ $f\in\mathbb{C}\\{x,y\\}$, and similarily define
$V(F)$ for $F\in\mathbb{C}\\{x,y\\}\hat{\otimes}\mathcal{O}_{\mathcal{S}_{0}}$
as above. A curve $X$ with two regular coordinate functions
$(x,y):X\to\mathbb{C}^{2}$ is called _locally planar_ with respect to $x,y$ at
$p\in X$ if the germ $X_{p}$ is locally planar with respect to
$\left((x-x(p))_{p},(y-y(p))_{p}\right)$ and $X$ is locally planar with
respect to $x,y$ if this holds at every $p\in X$.

An example where this condition fails is that the normalisation of the cusp
$y^{2}=x^{3}$ is not locally planar with respect to the pullbacks of $(x,y)$
by the normalisation map $t\mapsto(t^{2},t^{3})$ at $t=0$, since the local
ring of regular functions of the cusp does not contain any function which
pulls back to the function $t$.

For a germ $\mathcal{S}_{0}$ of a complex space $\mathcal{S}$ at
$0\in\mathcal{S}$ we denote by
$\mathbb{C}\\{x,y\\}\hat{\otimes}\mathcal{O}_{\mathcal{S}_{0}}$ the stalk of
the holomorphic functions on $\mathbb{C}^{2}\times\mathcal{S}$ at
$(x,y,s)=(0,0,0)$ (c.f. [dJP00, Definition 7.3.6]).

######  Definition 1.3.

A _deformation_

$(X_{q},x_{q},y_{q})\hookrightarrow(Y_{q},x_{Y,q},y_{Y,q})\twoheadrightarrow\mathcal{S}_{0}$

of a locally planar space germ $(X_{q},x_{q},y_{q})$ consists of a deformation
$X_{q}\hookrightarrow Y_{q}\twoheadrightarrow\mathcal{S}_{0}$ together with
holomorphic extensions $x_{Y,q},y_{Y,q}\in\mathcal{O}_{Y,q}$ of $x_{q},y_{q}$
such that $(x_{Y,q},y_{Y,q},s)$ maps $Y_{q}$ biregularly onto $V(F)$ with
$F\in\mathbb{C}\\{x,y\\}\hat{\otimes}\mathcal{O}_{\mathcal{S}_{0}}$.

Explicitly, such a deformation is given by
$F\in\mathbb{C}\\{x,y\\}\hat{\otimes}\mathcal{O}_{\mathcal{S}_{0}}$ and a unit
$h\in\mathbb{C}\\{x,y\\}$ such that

(1) $F(x,y,0)=h(x,y)f(x,y).$

Let us write $V(f)$ for the space germ given by the vanishing set of a germ
$f\in\mathbb{C}\\{x,y\\}$ as above. It is proven in [GLS07, Corollary II.1.6]
that all local deformations of $V(f)$ are indeed of the form
$V(f)\hookrightarrow V(F)\twoheadrightarrow\mathcal{S}_{0}$ with
$F\in\mathbb{C}\\{x,y\\}\hat{\otimes}\mathcal{O}_{\mathcal{S}_{0}}$; that is
that all such deformations are embeddable. Hence this definition places no
restriction on the corresponding deformations of the space germs $X_{q}$.

With an eye to our goal of considering deformations of data $(X,\Theta)$ with
$x$ a local primitive of $\Theta$, the morphisms which we consider between
deformations of $(X_{q},x_{q},y_{q})$ respect the maps $x_{q}$ in the
following sense.

######  Definition 1.4.

An $x$-morphism from a deformation

(2)
$(X_{q},x_{q},y_{q})\hookrightarrow(Y_{q},x_{Y,q},y_{Y,q})\twoheadrightarrow\mathcal{S}_{0}$

to a deformation

(3)
$(X_{q},x_{q},y_{q})\hookrightarrow(\widetilde{Y}_{q},\tilde{x}_{\tilde{Y},\tilde{q}},\tilde{y}_{\tilde{Y},\tilde{q}})\twoheadrightarrow\widetilde{\mathcal{S}}_{\tilde{0}}$

is a morphism of deformations

$\begin{array}[]{ccccl}X_{q}&\hookrightarrow&Y_{q}&\twoheadrightarrow&\mathcal{S}_{0}\\\
\|&&\downarrow\psi&&\downarrow\phi\\\
X_{q}&\hookrightarrow&\widetilde{Y}_{q}&\twoheadrightarrow&\widetilde{\mathcal{S}}_{\tilde{0}}\end{array}$

so that $\tilde{x}_{\tilde{Y},\tilde{q}}\circ\,\psi=x_{Y,q}$.

We pause briefly to describe what data explicitly exhibits an $x$-morphism.
Let $f\in\mathbb{C}\\{x,y\\}$ and
$F,G\in\mathbb{C}\\{x,y\\}\hat{\otimes}\mathcal{O}_{\mathcal{S}_{0}}$ describe
the deformations (2), (3) as in Definition 1.3. Let $\phi$ denote the map from
$\mathcal{S}_{0}$ to $\widetilde{\mathcal{S}}_{0}$ in the $x$-morphism, then
the map $V(F)\to V(G)$ is of the form $(x,y,s)\mapsto(x,u(x,y,s),\phi(s))$
with $u\in\mathbb{C}\\{x,y\\}\hat{\otimes}\mathcal{O}_{\mathcal{S}_{0}}$ and
$u(x,y,0)=y$. In particular there exists a unit
$H\in\mathbb{C}\\{x,y\\}\hat{\otimes}\mathcal{O}_{\mathcal{S}_{0}}$ with

(4) $\displaystyle H(x,y,s)G(x,u(x,y,s),\phi(s))$ $\displaystyle=F(x,y,s).$

Evaluating $u$ at $s=0$ yields $u(x,y,0)=y$. If the defining functions are
normalised in the sense that $f(x,y)=F(x,y,0)=G(x,y,0)$, one also has
$H(x,y,0)=1$.

We now define the global data whose deformations we shall consider. Our
interest in such objects is chiefly motivated by their appearance as the
spectral curve data of an integrable system. Recall that for $p\in X$,
$\Theta_{p}$ denotes the germ of the 1-form $\Theta$ at $p$.

######  Definition 1.5.

Let $X$ be a compact one-dimensional complex space with finitely many smooth
marked points $p_{1},\ldots,p_{K}$ and a meromorphic 1-form $\Theta$. Write
$X^{\circ}:=X\setminus\\{p_{1},\ldots,p_{K}\\}$. We say that
$(X,p_{1},\ldots,p_{K},\Theta)$ or more briefly $(X,\Theta)$ is _locally
planar with prescribed poles_ if

  * (a)

Prescribed poles: For each $p\in\\{p_{1},\ldots,p_{K}\\}$ there exists
$u_{p}\in\mathcal{O}_{X,p}$ which vanishes at $p$ and maps $X_{p}$ biregularly
onto $\mathbb{C}_{0}$ such that $\Theta_{p}=d(u_{p}^{-M_{p}})$.

  * (b)

Locally planar on $X^{\circ}$, compatible with $\mathbf{\Theta}$: For each
$q\in X^{\circ}$ there exist
$(x_{q},y_{q})\in\mathcal{O}_{X,q}\times\mathcal{O}_{X,q}$ which vanish at $q$
and map $X_{q}$ biregularly onto the zero set of some
$f_{q}\in\mathbb{C}\\{x,y\\}$. Further we require that $\Theta_{q}=d(x_{q})$.

If in addition the following condition holds then we say that
$(X,p_{1},\ldots,p_{K},\Theta)$ is _real_.

  * (c)

Real: On $X$ there exists an anti-holomorphic involution $\eta$ which permutes
$p_{1},\ldots,p_{K}$ and acts as:

(5) $\displaystyle x_{\eta(q)}$ $\displaystyle=\eta^{\ast}\bar{x}_{q}$
$\displaystyle y_{\eta(q)}$ $\displaystyle=\eta^{\ast}\bar{y}_{q}$
$\displaystyle u_{\eta(q)}$ $\displaystyle=\eta^{\ast}\bar{u}_{q}$
$\displaystyle f_{\eta(q)}(x,y)$ $\displaystyle=\bar{f}_{q}(\bar{x},\bar{y}).$

We briefly mention some consequences of these conditions. The condition (a)
guarantees that $p_{1},\ldots,p_{K}$ are smooth points of the 1-dimensional
complex space $X$. Further, the germ $u_{p}$ of the local coordinate of $X$ at
$p\in\\{p_{1},\ldots,p_{K}\\}$ is determined by the 1-form $\Theta$ and the
condition that the germ of $\Theta$ at $p$ is equal to $d(u_{p}^{-M_{p}})$ at
$p$ up to multiplication with a $M_{p}$-th root of unity. Finally, the
condition (c) implies that $\eta$ transforms $\Theta$ as
$\eta^{\ast}\Theta=\bar{\Theta}$. We also note that there is a slight abuse of
terminology here in that whilst $X^{\circ}$ is locally planar with respect to
a primitive of $\Theta$ and another coordinate, in fact $X$ does not satisfy
this condition at the points $p_{1},\ldots,p_{K}$. There it is instead locally
biregular to an open set in $\mathbb{C}$.

We now expand our notion of deformations to include the differential $\Theta$
and also to include the points $p_{1},\ldots,p_{K}$ where $X$ instead of being
locally planar is there locally biregular to an open set in $\mathbb{C}$.

######  Definition 1.6.

A _deformation_

(6) $(X,\Theta)\hookrightarrow(Y,\Theta_{Y})\twoheadrightarrow\mathcal{S}.$

_of locally planar $(X,\Theta)$ with prescribed poles_ is given by a
deformation $X\hookrightarrow Y\xtwoheadrightarrow{s}\mathcal{S}$ of the
complex space $X$ together with a meromorphic 1-form $\Theta_{Y}$ on $Y$, such
that the following conditions hold:

  * (a)

Prescribed poles: For each $p\in\\{p_{1},\ldots,p_{K}\\}$, the germ $u_{p}$ of
Definition 1.5 (a) extends to $u_{Y,p}\in\mathcal{O}_{Y,p}$ such that
$(u_{Y,p},s)$ maps $Y_{p}$ biregularly onto
$\mathbb{C}\times\mathcal{S}_{(0,0)}$ and $\Theta_{Y,p}=d(u_{Y,p}^{-M_{p}})$.

  * (b)

Locally planar on $\mathbf{X^{\circ}}$, compatible with $\mathbf{\Theta}$: For
each $q\in X^{\circ}=X\setminus\\{p_{1},\ldots,p_{K}\\}$, the induced
deformation

(7)
$\displaystyle(X_{q},x_{q},y_{q})\hookrightarrow(Y_{q},x_{Y,q},y_{Y,q})\xtwoheadrightarrow{s}\mathcal{S}$

of the locally planar space germ $(X_{q},x_{q},y_{q})$ satisfies
$\Theta_{Y,q}=d(x_{Y,q})$.

  * (c)

Real: If $(X,\Theta)$ is real, i.e. satisfies Definition 1.5 (c), then $\eta$
extends to involutions of $Y$ and $\mathcal{S}$ which commute with the maps
$X\hookrightarrow Y\twoheadrightarrow\mathcal{S}$ and act as

(8) $\displaystyle\begin{aligned}
x_{Y,\eta(q)}&=\eta^{\ast}\bar{x}_{Y,q}&y_{Y,\eta(q)}&=\eta^{\ast}\bar{y}_{Y,q}&u_{Y,\eta(q)}&=\eta^{\ast}\bar{u}_{Y,q}\\\
&&F_{\eta(q)}(x,y,s)&=\bar{F}_{q}(\bar{x},\bar{y},\eta(s)).\end{aligned}$

Here $Y_{p}$ denotes for all $p\in X$ the space germ of $Y$ at $p$ and
$\Theta_{Y,p}$ the germ of the 1-form $\Theta_{Y}$ at $p$. Analogously
$\mathbb{C}\times\mathcal{S}_{(0,0)}$ denotes the space germ of
$\mathbb{C}\times\mathcal{S}$ at $(0,0)\in\mathbb{C}\times\mathcal{S}$.

Again (a) guarantees that $p_{1},\ldots,p_{K}$ are deformed into smooth points
of the fibres of $Y\to\mathcal{S}$ and for each
$p\in\\{p_{1},\ldots,p_{K}\\}$, $u_{Y,p}$ is determined by $\Theta_{Y}$ up to
multiplication by a $M_{p}$-th root of unity. The reality condition (c)
implies that $\Theta_{Y}$ transforms as
$\eta^{\ast}\Theta_{Y}=\bar{\Theta}_{Y}$.

The locally planar condition (b) of Definition 1.6 forces the periods of
$\Theta$ to be locally preserved along fibres of the deformation, as we now
explain.

######  Remark 1.7.

Let $(X,\Theta)\hookrightarrow(Y,\Theta_{Y})\twoheadrightarrow\mathcal{S}$ be
a deformation of locally planar $(X,\Theta)$ with prescribed poles. Suppose
$s:(-\epsilon,\epsilon)\to\mathcal{S}$ and
$\gamma:\mathbb{S}^{1}\times(-\epsilon,\epsilon)\to Y$ are continuous maps
such that the diagram

$\begin{array}[]{rcl}\mathbb{S}^{1}\times(-\epsilon,\epsilon)&\xrightarrow{\gamma}&Y\\\
\downarrow&&\downarrow\\\
(-\epsilon,\epsilon)&\xrightarrow{s}&\mathcal{S}\end{array}$

commutes, where the left vertical map is the projection onto the second factor
and $s(0)=0$. Then for $t\in(-\epsilon,\epsilon)$ the element of $\pi_{1}(Y)$
defined by $\gamma(\cdot,t)$ and the period $\int_{\gamma(\cdot,t)}\Theta_{Y}$
are independent of $t$.

###### Proof.

The continuous map $\gamma$ provides a homotopy between $\gamma(\cdot,a)$ and
$\gamma(\cdot,b)$ and hence the resulting element of $\pi_{1}(Y)$ is
independent of $t$. For the second statement, we note that if $Y$ and $\gamma$
are smooth and $\gamma$ avoids the poles of $\Theta_{Y}$ then the remark is a
consequence of Stokes’ Theorem and the fact that $\Theta_{Y}$ is locally exact
and hence closed. We adapt the proof of Stokes’ Theorem and give an argument
which does not require reformulation when these conditions fail. Fix
$a<b\in(-\epsilon,\epsilon)$. We consider the map
$\beta:[0,2\pi]\times[a,b]\to Y$ with
$\beta(\varphi,t)=\gamma(e^{i\varphi},t)$. For each
$(\varphi,t)\in[0,2\pi]\times[a,b]$ we may take an open rectangular
neighbourhood $Q_{(\varphi,t)}\subset[0,2\pi]\times[a,b]$ such that the
differential $\Theta_{Y}$ has a local integral $x_{Y,\beta(\varphi,t)}$
defined on $\beta(Q_{(\varphi,t)})$. If $\beta(\varphi,t)$ is not a pole of
$\Theta_{Y}$ then this local function is that guaranteed by condition (b) of
Definition 1.6. If $\beta(\varphi,t)$ is a pole then we take
$x_{Y,\beta(\varphi,t)}=u_{Y,p}^{-M_{p}}$ as in condition (a) of Definition
1.6. Taking a finite subcover $Q_{1},\ldots,Q_{N}$, on each $Q_{n}$ we have a
function $\tilde{x}_{n}$ obtained by pulling back the corresponding local
integral of $\Theta_{Y}$. Writing $Q_{n}(t)$ for the slice where the second
coordinate takes the value $t$, for each $t\in[a,b]$ then there are finitely
many $\\{i_{1},\ldots,i_{k}\\}\subset\\{1,\ldots,N\\}$ so that

$\int_{\gamma(\cdot,t)}\Theta_{Y}=\sum_{j=1}^{k}\int_{Q_{i_{j}}(t)}d\tilde{x}_{i_{j}}$

and moreover by construction, this is locally constant in $t$. ∎

Now we shall describe the morphisms of these deformations.

######  Definition 1.8.

A _morphism_ from one deformation
$(X,\Theta)\hookrightarrow(Y,\Theta_{Y})\twoheadrightarrow\mathcal{S}$ to
another
$(X,\Theta)\hookrightarrow(\widetilde{Y},\tilde{\Theta}_{\tilde{Y}})\twoheadrightarrow\widetilde{\mathcal{S}}$
is a morphism

(9)
$\begin{array}[]{cclcl}X&\hookrightarrow&Y&\twoheadrightarrow&\mathcal{S}\\\
\|&&\hskip 2.84526pt\downarrow\psi&&\downarrow\phi\\\
X&\hookrightarrow&\tilde{Y}&\twoheadrightarrow&\tilde{\mathcal{S}}.\end{array}$

of the underlying deformation of $X$ which satisfies
$u_{\tilde{Y},p}\circ\psi_{p}=u_{Y,p}$ for each $p\in\\{p_{1},\ldots,p_{K}\\}$
and such that for each $q\in X^{\circ}$, the induced morphism of deformations
of space germs $(X_{q},x_{q},y_{q})$ is an $x$-morphism.

It immediately follows that $\Theta_{Y}$ is equal to
$\psi^{\ast}\Theta_{\tilde{Y}}$, and so no special provision needs to be made
for the differential in the above definition.

As for deformations of complex spaces, we have the following definition.

######  Definition 1.9.

A _local deformation_
$(X,\Theta)\hookrightarrow(Y,\Theta_{Y})\twoheadrightarrow\mathcal{S}_{0}$ is
an equivalence class of deformations
$(X,\Theta)\hookrightarrow(Y,\Theta_{Y})\twoheadrightarrow\mathcal{S}$ defined
on a neighbourhood of $0$, where two such deformations are equivalent if their
restrictions to some neighbourhood of $0$ are isomorphic.

The notion of morphism extends to local deformations in the obvious way. Our
primary interest lies in the category of _local deformations of locally planar
$(X,\Theta)$_. The main result of this paper is the construction of a
universal object in this category (Theorem 5.1).

##  2\. Patching deformations

Recall that we denote by $q_{l},\,l=1,\ldots,L$ the finitely many points where
either $X$ has a singularity or the local primitive $x$ of $\Theta$ has a
ramification point. The main result of this section is that a deformation of
the corresponding germs $(X_{l},x_{l},y_{l})$ can be uniquely extended to a
local deformation of $(X,\Theta)$. This reduces our study of the compact curve
$X$ and differential $\Theta$ to that of finitely many germs and we shall make
use of this simplification throughout the rest of the paper. To avoid double
indices we have written $l$ rather than $q_{l}$ as the index for the germs.

Let $(X,\Theta)$ be locally planar with prescribed poles at marked points
$p_{1},\ldots,p_{K}$ and
$(X,\Theta)\hookrightarrow(Y,\Theta_{Y})\twoheadrightarrow\mathcal{S}$ a
deformation. The restriction of a deformation to the complement of
sufficiently small open neighbourhoods of $q_{1},\ldots,q_{L}$ is locally
$x$-isomorphic to the trivial deformation. To see this, note first that
condition (a) of Definition 1.6 ensures that the space germs of $Y$ at the
marked points are isomorphic to the space germs of
$\mathbb{C}\times\mathcal{S}$ at $(0,0)$. Furthermore, at smooth points $q\in
X^{\circ}$ where $\Theta_{q}=d(x_{q})$ does not vanish, $x_{q}$ maps $X_{q}$
biregularly onto $\mathbb{C}_{0}$ and $(x_{Y,q},s)$ maps $Y_{q}$ biregularly
onto $\mathbb{C}\times\mathcal{S}_{(0,0)}$. This observation, which we
formalise below, proves the uniqueness part of Theorem 2.2. Note that although
this lemma is stated for locally planar $(X,\Theta)$ with prescribed poles in
order to avoid the introduction of more terminology, the fact that the curve
may be locally embedded in $\mathbb{C}^{2}$ is not used here in any essential
way.

######  Lemma 2.1.

Let $(X,\Theta)$ be locally planar with prescribed poles and denote by
$q_{1},\ldots,q_{L}\in X$ the points at which either $X$ is not smooth or
$\Theta$ has a root. Let

(10) $\displaystyle(X,\Theta)\hookrightarrow(Y,\Theta_{Y})$
$\displaystyle\twoheadrightarrow\mathcal{S},$
$\displaystyle(X,\Theta)\hookrightarrow(Z,\Theta_{Z})$
$\displaystyle\twoheadrightarrow\mathcal{T}$

be deformations such that for each $q_{l},l=1,\ldots,L$ the corresponding
deformations of space germs at $q_{l}$ are $x$-isomorphic with respect to an
isomorphism $\phi_{0}:\mathcal{S}_{0}\to\mathcal{T}_{0}$. Then the
deformations (10) are isomorphic.

###### Proof.

For each of the marked points $p\in\\{p_{1},\ldots,p_{K}\\}$, recall from
Definition 1.6 that the maps
$(u_{Y,p},s):Y_{p}\rightarrow\mathcal{S}\times\mathbb{C}$ and
$(u_{Z,p},t):Z_{p}\rightarrow\mathcal{T}\times\mathbb{C}$ are isomorphisms.
The germ $\psi_{p}:Y_{p}\to Z_{p}$ defined by
$\psi_{p}=(u_{Z,p},t)^{-1}\circ(\phi_{0},\operatorname{id}_{\mathbb{C}})\circ(u_{Y,p},s)$
satisfies $u_{Z,p}\circ\psi_{p}=u_{Y,p}$, so $\psi_{p}$ is an $x$-morphism.
Analogously for each $q\in X^{\circ}\setminus\\{q_{1},\ldots,q_{L}\\}$ since
$(x_{Y,q},s)$ and $(x_{Z,q},t)$ are isomorphisms, replacing $u$ by $x$ yields
$\psi_{q}:Y_{q}\to Z_{q}$ with $x_{Z,q}\circ\psi_{q}=x_{Y,q}$. Since for each
$q\in\\{q_{1},\ldots,q_{L}\\}$ the germs $\psi_{q}:Y_{q}\to Z_{q}$ are
$x$-isomorphisms, all these germs $\psi_{q}:Y_{q}\to Z_{q}$ with $q\in X$ fit
together to form a holomorphic map $\psi$ from an open neighbourhood of the
compact subset $X\subset Y$ into $Z$. The complement of this open
neighbourhood of $X$ is mapped by $Y\twoheadrightarrow\mathcal{S}$ onto the
complement of an open neighbourhood $\mathcal{S}^{\prime}$ of
$0\in\mathcal{S}$, since a proper continuous map to a metrisable space is
closed. We just constructed a morphism from the restriction of the deformation
on the left hand side of (10) to $\mathcal{S}^{\prime}$ to the deformation on
the right hand side. Reversing the roles of these two deformations yields the
inverse morphism. This shows uniqueness of (6) up to isomorphism and
restriction to an open neighbourhood of $0$ in $\mathcal{S}$.∎

We now proceed to the more technically difficult existence part of Theorem
2.2. That is, we now show that given deformations of each of the space germs
$X_{q_{l}}$ one can “patch” these together with this trivial deformation to
yield a deformation of $(X,\Theta)$. To avoid double indices we abbreviate the
index $q_{l}$ for each $l=1,\ldots,L$ by the index $l$ so that for example we
will hitherto write $X_{l}$ for the space germ $X_{q_{l}}$. In particular,
$x_{l}=x_{q_{l}}$, $y_{l}=y_{q_{l}}$, $f_{l}=f_{q_{l}}$ and $F_{l}=F_{q_{l}}$.
Furthermore, we abbreviate the space germs $X_{l}=X_{q_{l}}$ and
$Y_{l}=Y_{q_{l}}$.

######  Theorem 2.2.

Let $(X,\Theta)$ be locally planar with prescribed poles and denote by
$q_{1},\ldots,q_{L}\in X$ the points at which either $X$ is not smooth or
$\Theta$ has a root. For each $l=1,\ldots,L$ suppose we are given a
deformation
$(X_{l},x_{l},y_{l})\hookrightarrow(\widetilde{Y}_{l},x_{\tilde{Y},l},y_{\tilde{Y},l})\twoheadrightarrow\mathcal{S}_{0}$
of the locally planar space germ $(X_{l},x_{l},y_{l})$. Then there exists a
unique local deformation
$(X,\Theta)\hookrightarrow(Y,\Theta_{Y})\twoheadrightarrow\mathcal{S}_{0}$
such that for each $l=1,\dots,L$ the induced deformation
$(X_{l},x_{l},y_{l})\hookrightarrow(Y_{l},x_{Y,l},y_{Y,l})\twoheadrightarrow\mathcal{S}_{0}$
is $x$-isomorphic to the given one.

###### Proof.

We shall prove this theorem by constructing a deformation
$(X,\Theta)\hookrightarrow(Y,\Theta_{Y})\twoheadrightarrow\mathcal{S}$ as in
(6) with the above properties. The uniqueness of (6) is an immediate
consequence of Lemma 2.1. From Definition 1.5, each $X_{l}$ is biregular via
$(x_{l},y_{l})$ to the vanishing set germ $V(f_{l})$ of some
$f_{l}\in\mathbb{C}\\{x,y\\}$ and $\tilde{Y}_{l}$ is biregular to $V(F_{l})$
for some $F_{l}\in\mathbb{C}\\{x,y\\}\hat{\otimes}\mathcal{O}_{\mathcal{S}}$
where $F_{l}(x,y,0)=H_{l}(x,y)f_{l}(x,y)$ for a unit
$H_{l}\in\mathbb{C}\\{x,y\\}$. By replacing $f_{l}$ by $H_{l}f_{l}$ if
necessary, we assume without loss of generality that
$F_{l}(x,y,0)=f_{l}(x,y)$. The space germ $\mathcal{S}_{0}$ is embedded in
some $\mathbb{C}^{k}_{0}$. The holomorphic functions $F_{1},\ldots,F_{L}$
clearly extend to holomorphic functions in
$\mathbb{C}\\{x,y\\}\hat{\otimes}\mathcal{O}_{\mathbb{C}^{k}_{0}}$. This
extension is not unique, but the restriction to the subspace $\mathcal{S}_{0}$
is unique. We prove the Lemma for $\mathcal{S}_{0}=\mathbb{C}^{k}_{0}$ and
construct a deformation with base equal to an open set in $\mathbb{C}^{k}$.
Since the restriction of this deformation to a subspace $\mathcal{S}$ of the
base is a deformation with base $\mathcal{S}$, the general case follows.

We apply the Weierstraß Preparation Theorem [PR94, Chapter I Theorem 1.4] and
replace $F_{1},\ldots,F_{L}$ by Weierstraß polynomials
$F_{1},\ldots,F_{L}\in\mathbb{C}\\{x\\}[y]\hat{\otimes}\mathcal{O}_{\mathcal{S}_{0}}$.
Furthermore, we replace $f_{1},\ldots,f_{L}$ by the corresponding evaluations
$f_{1},\ldots,f_{L}\in\mathbb{C}\\{x\\}[y]$ of these Weierstraß polynomials at
$s=0$. For $l=1,\ldots,L$ the polynomials $F_{l}$ and $f_{l}$ have highest
coefficient $1$ and all other coefficients vanish at $(x,s)=(0,0)$ and $x=0$,
respectively.

We choose an open ball $\mathcal{S}$ around $0\in\mathbb{C}^{k}$ and
$\delta_{1}>0$ such that the polynomials $F_{l}$ are holomorphic on
$(x,y,s)\in B(0,\delta_{1})\times\mathbb{C}\times\mathcal{S}$. For
sufficiently small $\delta_{1}$ there exist disjoint open neighbourhoods
$O_{l}$ of $q_{l}$ in $X$ such that the maps $(x_{l},y_{l})$ in Definition 1.5
(b) extend to biregular maps

(11) $\displaystyle(x_{l},y_{l})$ $\displaystyle:O_{l}\to U_{l},$
$\displaystyle U_{l}=\\{(x,y)\in B(0,\delta_{1})\times\mathbb{C}\mid
f_{l}(x,y)=0\\}.$

The set $\\{q_{1},\ldots,q_{L}\\}$ is comprised of the singular points of $X$
and the smooth points which are roots of $\Theta$. Hence

(12) $\displaystyle U_{l}$ $\displaystyle\rightarrow B(0,\delta_{1}),$
$\displaystyle(x,y)$ $\displaystyle\mapsto x$

is a Weierstraß covering with a single branch point at $(0,0)$. We can take
$\delta_{1}$ sufficiently small and shrink $\mathcal{S}$ so that for all
$l=1,\ldots,L$,

$V_{l}=\left\\{(x,y,s)\in B(0,\delta_{1})\times\mathbb{C}\times\mathcal{S}\mid
F_{l}(x,y,s)=0\right\\}$

is exhibited as a Weierstraß covering

(13) $\displaystyle V_{l}$ $\displaystyle\rightarrow
B(0,\delta_{1})\times\mathcal{S},$ $\displaystyle(x,y,s)$
$\displaystyle\mapsto(x,s)$

and is unbranched over $(x,s)\in\mathcal{A}\times\mathcal{S}$ for the annulus

(14) $\displaystyle\mathcal{A}$
$\displaystyle=B(0,\delta_{1})\setminus\overline{B(0,\delta_{1}/2)}.$

We denote the restrictions of the two coverings (12) and (13) to $\mathcal{A}$
and $\mathcal{A}\times\mathcal{S}$ as

$\displaystyle\widehat{U}_{l}$ $\displaystyle\to\mathcal{A},$
$\displaystyle\widehat{V}_{l}$ $\displaystyle\to\mathcal{A}\times\mathcal{S},$

respectively. In order to extend the given deformation “trivially” to a
deformation of $X$ we now demonstrate the existence of a unique biholomorphic
map $\psi_{l}:\widehat{U}_{l}\times\mathcal{S}\to\widehat{V}_{l}$ of the form
$(x,y(x),s)\mapsto(x,y(x,s),s)$ which is equal to the identity for $s=0$. The
coverings $\widehat{U}_{l}$ and $\widehat{V}_{l}$ are unbranched Weierstraß
coverings with an equal number of sheets. Therefore on $\widehat{U}_{l}$, we
have that $y$ is locally a holomorphic function $y=y(x)$ of $x\in\mathcal{A}$.
Similarly on $\widehat{V}_{l}$, locally $y$ is a holomorphic function
$y=y(x,s)$ with $(x,s)\in\mathcal{A}\times\mathcal{S}$. Furthermore
$y(x,0)=y(x)$. Given a local “branch” of $y(x)$, it extends uniquely to any
simply connected subset of $\mathcal{A}$ containing the original point and
similarly any specific branch of $y(x,s)$ extends uniquely on simply connected
subsets of $\mathcal{A}\times\mathcal{S}$. Let $W_{l_{i}}$ be an open subset
of $\widehat{U}_{l}$ which is mapped biholomorphically by $x$ onto an open
simply connected subset $\mathcal{B}$ of $\mathcal{A}$. The cartesian product
$W_{l_{i}}\times\mathcal{S}$ is also simply connected since $\mathcal{S}$ is
an open ball in $\mathbb{C}^{k}$. Hence there exists a unique biholomorphic
map from $W_{l_{i}}\times\mathcal{S}$ onto a simply connected open subset of
$\widehat{V}_{l}$ which preserves $(x,s)$ and is equal to the identity for
$s=0$.

${W_{l_{i}}\times\mathcal{S}}$${\widehat{V}_{l}}$${\mathcal{B}\times\mathcal{S}}$${\mathcal{B}\times\mathcal{S}}$
${(x,y(x),s)}$${(x,y(x,s),s)}$${(x,s)}$${(x,s).}$

We first cover $\mathcal{A}$ by finitely many simply connected open subsets
$\mathcal{B}$ such that the intersection of any two of them is either empty or
connected. Denoting by $n_{l}$ the number of sheets of the Weierstrass
coverings (12) and (13), there are $n_{l}$ simply connected open subsets
$W_{l_{i}}$ which each map biholomorphically onto $\mathcal{B}$ via the
projection $x$, and taken altogether these cover $\widehat{U}_{l}$. Since
$\widehat{V}_{l}$ is unbranched over $(x,s)\in\mathcal{A}\times\mathcal{S}$,
the maps in the above diagram patch together to form a unique biholomorphic
map

$\displaystyle\psi_{l}:\widehat{U}_{l}\times\mathcal{S}$
$\displaystyle\to\widehat{V}_{l},$ $\displaystyle(x,y(x),s)$
$\displaystyle\mapsto(x,y(x,s),s).$

For each $l=1,\ldots,L$ let $C_{l}$ denote the preimage in $O_{l}\subset X$ of
$U\setminus\widehat{U}_{l}$ with respect to the map (11). Writing
$\widehat{O}_{l}=O_{l}\setminus C_{l}$, we have that the open subsets
$\widehat{O}_{l}\times\mathcal{S}$ of $\big{(}X\setminus(C_{1}\cup\ldots\cup
C_{L})\big{)}\times\mathcal{S}$ are mapped by $\psi_{l}\circ(x_{l},y_{l},s)$
biregularly onto $\widehat{V}_{l}$. This identification of the two “annuli”
allows us to glue $\big{(}X\setminus(C_{1}\cup\ldots\cup
C_{L})\big{)}\times\mathcal{S}$ with $V_{1}\cup\ldots\cup V_{L}$ along the
maps $\psi_{l}\circ(x_{l},y_{l},s)$ for $l=1,\ldots,L$. This defines $Y$
together with the map $Y\twoheadrightarrow\mathcal{S}$. The differential
$\Theta$ on $X$ extends to a differential on
$\big{(}X\setminus(C_{1}\cup\ldots\cup C_{L})\big{)}\times\mathcal{S}$. Its
restriction to $\widehat{O}_{l}\times\mathcal{S}$ agrees with the pullback
under
$\psi_{l}\circ(x_{l},y_{l},s):\widehat{O}_{l}\times\mathcal{S}\rightarrow\widehat{V}_{l}$
of the differential $\Theta$ on $V$. Hence we obtain a global differential
$\Theta_{Y}$ on $Y$ which for $q\notin\\{p_{1},\ldots,p_{k}\\}$ satisfies
$\Theta_{Y,q}=d(x_{Y,q})$. The sets $\\{p_{1},\ldots,p_{k}\\}$ and
$\\{q_{1},\ldots,q_{L}\\}$ are disjoint and so each
$p\in\\{p_{1},\ldots,p_{K}\\}$ has an open neighbourhood $O_{p}$ in $X$ such
that $O_{p}\times\mathcal{S}$ is a neighbourhood of $p$ in $Y$. Hence the germ
$u_{p}$ of Definition 1.5 (a) extends to $u_{Y,p}\in\mathcal{O}_{Y,p}$ such
that $(u_{Y,p},s)$ maps $Y_{p}$ biregularly onto
$\mathbb{C}\times\mathcal{S}_{(0,0)}$ and $\Theta_{Y,p}=d(u_{Y,p}^{-M_{p}})$.
We claim that the map $Y\twoheadrightarrow\mathcal{S}$ is proper and flat. It
is the composition of a Weierstraß map $Y_{l}\twoheadrightarrow
B(0,\delta_{1})\times\mathcal{S}$ and the projection
$B(0,\delta_{1})\times\mathcal{S}\to\mathcal{S}$. By [PR94, Chapter II
Proposition 2.10] and the Weierstraß Isomorphism [GR84, Chapter 2 §4.2],
Weierstraß maps are flat. Furthermore, by [PR94, Chapter II Corollary 2.7]
projections of complex spaces are flat. This shows flatness of
$Y\twoheadrightarrow\mathcal{S}$, since the composition of flat maps is flat
[PR94, Chapter II Proposition 2.6]. It is also proper since $X$ and
$C_{1},\ldots,C_{L}$ are compact. Hence we have constructed a deformation (6)
with the properties in the theorem. ∎

##  3\. Infinitesimal deformations

In this section we will construct a local deformation (19) of a locally planar
$(X,\Theta)$ from the space of isomorphism classes of infinitesimal
deformations of $(X,\Theta)$. Furthermore, we prove that the isomorphism
classes of infinitesimal deformations of the fibres of (18) form a vector
bundle over its base. We will later use this vector bundle structure to show
that the local deformation is universal. The universality will then imply that
the dimension of the isomorphism classes of infinitesimal deformations of the
fibres of any deformation of $(X,\Theta)$ are locally constant.

######  Definition 3.1.

An infinitesimal deformation is a deformation whose base space is the complex
space germ $\mathcal{I}_{\epsilon}$ consisting of a single point with local
ring (i.e. holomorphic functions)
$\Pi=\mathbb{C}\\{\epsilon\\}/\langle\epsilon^{2}\rangle\simeq\mathbb{C}[\epsilon]/\langle\epsilon^{2}\rangle$.

Informally, $\mathcal{I}_{\epsilon}$ may be considered as a point with a
single tangent direction. For an analytic curve germ $V(f)$ in the plane, the
Tjurina algebra $\mathbb{C}\\{x,y\\}\big{/}\langle f,\tfrac{\partial
f}{\partial x},\tfrac{\partial f}{\partial y}\rangle$ parameterises
isomorphism classes of infinitesimal deformations of $V(f)$ [dJP00, Theorem
10.1.5]. The natural analog holds for $x$-isomorphism classes:

######  Lemma 3.2.

The $x$-isomorphism classes of the infinitesimal deformations of a space germ
$V(f)$ with $f\in\mathbb{C}\\{x,y\\},\,f(0,0)=0$ are in one-to-one
correspondence with

(15) $\displaystyle\mathbb{C}\\{x,y\\}\big{/}\langle f,\tfrac{\partial
f}{\partial y}\rangle.$

###### Proof.

An infinitesimal deformation of $V(f)$ is given by $V(F)$ for
$F(\epsilon)=f+\epsilon g$ and $g\in\mathbb{C}\\{x,y\\}$. Suppose we are given
two infinitesimal deformations of $V(f)$, defined by
$F_{1}(\epsilon)=f+\epsilon g_{1}$ and $F_{2}(\epsilon)=f+\epsilon g_{2}$
respectively. An $x$-morphism between them has the form

$\begin{array}[]{cclcl}V(f)&\hookrightarrow&V(F_{1})&\twoheadrightarrow&\mathcal{I}_{\epsilon}\\\
\|&&\hskip 2.84526pt\downarrow\psi&&\downarrow\phi\\\
V(f)&\hookrightarrow&V(F_{2})&\twoheadrightarrow&\mathcal{I}_{\epsilon}.\end{array}$

Note however that any automorphism $\phi$ of $\mathcal{I}_{\epsilon}$ can be
lifted to an automorphism of the infinitesimal deformation. Hence for the
purpose of classifying $x$-isomorphism classes of the infinitesimal
deformations, we may assume that $\phi$ is the identity. An $x$-morphism for
which $\phi$ is the identity is explicitly described, as in (4), by
$u,H\in\mathbb{C}\\{x,y\\}$ such that

$(1+\epsilon H(x,y))(f(x,y+\epsilon u(x,y))+\epsilon g_{2}(x,y+\epsilon
u(x,y))=f(x,y)+\epsilon g_{1}(x,y)$

holds in $\mathbb{C}\\{x,y\\}\otimes\Pi$. The $\epsilon^{0}$-term and the
$\epsilon^{1}$-term yield

$\displaystyle f(x,y)$ $\displaystyle=f(x,y),$ $\displaystyle
H(x,y)f(x,y)+u(x,y)\frac{\partial f}{\partial y}(x,y)+g_{2}(x,y)$
$\displaystyle=g_{1}(x,y).$

We conclude that the two infinitesimal deformations
$F_{1}(\epsilon)=f+\epsilon g_{1}$ and $F_{2}(\epsilon)=f+\epsilon g_{2}$ are
isomorphic if and only if $g_{1}=g_{2}$ in $\mathbb{C}\\{x,y\\}/\langle
f,\frac{\partial f}{\partial y}\rangle$. ∎

We now describe how the space of isomorphism classes of infinitesimal
deformations of a locally planar $(X,\Theta)$ induces a local deformation. Let
$(X,\Theta)$ be locally planar and as above let $q_{1},\ldots,q_{L}\in X$ be
the points at which either $X$ is not smooth or $\Theta$ has a root. Take
$f_{l}\in\mathbb{C}\\{x,y\\}$ such that $X_{l}=X_{q_{l}}$ is biregular to
$V(f_{l})$ via $(x_{l},y_{l})$. By Lemma 3.2 and Theorem 2.2, taking a basis
$g_{l}$ of each $\mathbb{C}\\{x,y\\}/\langle f_{l}\rangle$ can be thought of
as a choice of basis for the space of infinitesimal deformations of
$(X,\Theta)$. To see that the spaces $\mathbb{C}\\{x,y\\}/\langle
f_{l}\rangle$ are finite dimensional and to choose an appropriate basis of
each, we begin with some observations.

The Weierstraß Preparation Theorem [PR94, Chapter I Theorem 1.4] tells us that
each space $\mathbb{C}\\{x,y\\}/\langle f_{l}\rangle$ is unchanged when we
replace $f_{l}$ by the unique Weierstraß polynomial
$f_{l}\in\mathbb{C}\\{x\\}[y]$ whose degree is the order $d_{l}$ of $f_{l}$ in
$y$ at $0$ and which has highest coefficient $1$ and lower coefficients
vanishing at $x=0$. For sufficiently small $\delta_{1}>0$ these Weierstraß
polynomials $f_{1},\ldots,f_{L}$ are holomorphic on $x\in B(0,\delta_{1})$ and
belong to $\mathbb{C}[y]\hat{\otimes}\mathcal{O}_{B(0,\delta_{1})}$. The
corresponding complex space $U_{l}$ is a Weierstraß covering over $x_{l}\in
B(0,\delta_{1})$ with $d_{l}$ sheets. Condition (b) of Definition 1.5
guarantees that for sufficiently small $\delta_{1}$ the maps $(x_{l},y_{l})$
extend to give a biregular morphism of an open neighbourhood $O_{l}\subset X$
of each $q_{l}$ to $U_{l}$, as in (11).

Applying the Weierstraß Divison Theorem [PR94, Chapter I Theorem 1.1], we have
that the quotient $\mathbb{C}\\{x,y\\}/\langle f_{l}\rangle$ is the same as
$\mathbb{C}\\{x\\}[y]/\langle f_{l}\rangle$. This implies
$\mathbb{C}\\{x,y\\}/\langle f_{l},\frac{\partial f_{l}}{\partial
y}\rangle=\mathbb{C}\\{x\\}[y]/\langle f_{l},\frac{\partial f_{l}}{\partial
y}\rangle$. Moreover, $\frac{\partial f_{l}}{\partial y}$ does not vanish
identically on the zero locus of $f_{l}$ in $(x,y)\in\mathbb{C}^{2}$ and
$\mathbb{C}\\{x\\}[y]/\langle f_{l},\frac{\partial f_{l}}{\partial y}\rangle$
is finite dimensional.

For each $l=1,\ldots,L$, choose tuples $g_{l}=(g_{l,1},\ldots,g_{l,r_{l}})$ of
$g_{l,i}(x,y)\in\mathbb{C}[y]\hat{\otimes}\mathcal{O}_{B(0,\delta_{1})}$,
polynomial with respect to $y$ of degree less than $d_{l}$, which induce a
basis of

$\mathbb{C}\\{x\\}[y]/\langle f_{l},\tfrac{\partial f_{l}}{\partial
y}\rangle\simeq\mathbb{C}[y]\hat{\otimes}\mathcal{O}_{B(0,\delta_{1})}/\langle
f_{l},\tfrac{\partial f_{l}}{\partial y}\rangle.$

We may choose small open balls $\mathcal{T}_{l}\subset\mathbb{C}^{r_{l}}$,
such that for all $t_{l}\in\mathcal{T}_{l}$, the roots of the discriminant of
the polynomial $f_{l}+t_{l}\cdot g_{l}$ with respect to $y_{l}$ belong to
$x\in B(0,\delta_{1}/2)$. Each basis element $g_{l}$ defines a deformation

$U_{l}\rightarrow V_{l}\rightarrow\mathcal{T}_{l}$

where

(16) $\displaystyle G_{l}(x,y,t_{l})$
$\displaystyle=f_{l}(x,y)+t_{l,1}g_{l,1}(x,y)+\ldots+t_{l,r_{l}}g_{l,r_{l}}(x,y)$
$\displaystyle=:f_{l}(x,y)+t_{l}\cdot g_{l}(x,y)$

and

(17) $\displaystyle V_{l}$ $\displaystyle=\\{(x,y,t_{l})\in
B(0,\delta_{1})\times\mathbb{C}\times\mathcal{T}_{l}\mid
G_{l}(x,y,t_{l})=0\\}.$

As in the proof of Theorem 2.2, each $V_{l}$ is a Weierstraß covering with
$d_{l}$ sheets over $x\in B(0,\delta_{1})\times\mathcal{T}_{l}$, unbranched
over $\mathcal{A}\times\mathcal{T}_{l}$ with $\mathcal{A}$ the annulus in
(14). Write $\mathcal{T}=\mathcal{T}_{1}\times\ldots\times\mathcal{T}_{L}$. We
may glue $\big{(}X\setminus(C_{1}\cup\ldots\cup
C_{L})\big{)}\times\mathcal{T}$ with $V_{1}\cup\ldots\cup V_{L}$, as was done
in the proof of Theorem 2.2, in such a way that $d(x_{1}),\ldots,d(x_{L})$
extend to a global meromorphic 1-form obeying the conditions (a)-(b) of
Definition 1.6.

######  Definition 3.3.

We say that the deformation

(18) $(X,\Theta)\hookrightarrow(Z,\Theta_{Z})\twoheadrightarrow\mathcal{T}$

described above is the deformation of $(X,\Theta)$ defined by the choice of
basis $g_{l},l=1,\ldots,L$ for the isomorphism classes of infinitesimal
deformations of $(X,\Theta)$. The corresponding local deformation of
$(X,\Theta)$ is denoted

(19)
$(X,\Theta)\hookrightarrow(Z,\Theta_{Z})\twoheadrightarrow\mathcal{T}_{0}.$

We shall show in Theorem 5.1 that this local deformation is universal. In
particular, the isomorphism class of (18) or equivalently the local
deformation (19) do not depend on the choice of the basis $g_{l}$.

If we additionally impose the reality condition (c) of Definition 1.5, then
the maps $(x_{l},y_{l}):X_{l}\to\mathbb{C}^{2}$ obey (5). We define an action
$l\mapsto\eta l$ of $\eta$ on $\\{1,\ldots,L\\}$ such that
$\eta(q_{l})=q_{\eta l}$. Due to (5) the unique Weierstraß polynomials
$f_{1},\ldots,f_{L}$ obey

$f_{\eta l}(x,y)=\bar{f}_{l}(\bar{x},\bar{y}).$

We choose the bases $g_{1},\ldots,g_{L}$ in such a way that they obey:

$g_{\eta l}(x,y)=\bar{g}_{l}(\bar{x},\bar{y}).$

In particular, for $\eta_{l}=l$ the coefficients of $g_{l}$ take real values
for real $x_{l}\in B(0,\delta_{1})$. Consequently the deformation (18) obeys
condition (c) in Definition 1.6 with $\eta(t_{l})=\bar{t}_{\eta l}$ also
denoted by $\eta$.

Our next task is to show in Theorem 3.5 that the spaces of isomorphism classes
of infinitesimal deformations of the fibres of (18) form a vector bundle over
$\mathcal{T}$. In the proof we shall use the following description of the
regular forms (Rosenlicht differentials).

######  Lemma 3.4.

Let $f\in\mathbb{C}[y]\hat{\otimes}\mathcal{O}_{W}$ be a Weierstraß polynomial
of degree $d$ with respect to $y$, whose coefficients are holomorphic
functions on the open subset $x\in W\subset\mathbb{C}$. If the roots of the
discriminant of $f$ are isolated in $W$, then the regular forms of the complex
space $U=\\{(x,y)\in W\times\mathbb{C}\mid f(x,y)=0\\}$ have the following
form:

$\omega=g(x,y)\Theta\big{/}\frac{\partial f}{\partial y}(x,y),\,\text{with
}g\in\mathcal{O}_{O}.$

###### Proof.

By definition, a regular form is a meromorphic form whose product with any
local holomorphic function has no residues [Ser88, Chapter IV §3] and [Ros52].
The statement of the lemma is a local statement. Therefore it suffices to show
that for each $(x_{0},y_{0})\in O$ there exists a ball
$B(x_{0},\epsilon)\subset W$ such that the lemma holds on the restriction of
the Weierstraß covering $O\to W$ to the preimage $\tilde{O}$ of
$B(x_{0},\epsilon)$. In particular we may choose $B(x_{0},\epsilon)$ small
enough such that the all connected components of $\tilde{O}$ contain only one
point $(x_{0},y_{0})$ over $x_{0}$. Since the holomorphic functions on
different connected components are independent, it suffices to prove the
statement of the lemma for each connected component of $\tilde{O}$ separately.
Consequently we may assume that $\tilde{O}\to B(x_{0},\epsilon)$ is a
Weierstraß covering with only one point $(x_{0},y_{0})$ over $x_{0}$. In this
case the residue of $h\omega$ at $(x_{0},y_{0})$ is the residue at $x_{0}$ of
the meromorphic 1-form on $B(x_{0},\epsilon)$, which is the sum of $h\omega$
over the sheets of $\tilde{O}\to B(x_{0},\epsilon)$. For $\omega$ which are
regular at $(x_{0},y_{0})$ and for all $h\in\mathcal{O}_{\tilde{O}}$ the sum
of $h\omega$ over the $d$ sheets of $\tilde{O}\to B(x_{0},\epsilon)$ has no
residue at $x_{0}$. This implies that this sum of $h\omega$ is holomorphic at
$x_{0}$ as a 1-form on $x\in W$. Hence it suffices to prove that a meromorphic
function $g=\frac{\partial f}{\partial y}\frac{\omega}{\Theta}$ on $\tilde{O}$
is holomorphic at $(x_{0},y_{0})$, if and only if for all
$h\in\mathcal{O}_{\tilde{O}}$ the sum of $hg/\frac{\partial f}{\partial y}$
over the $d$ sheets of $\tilde{O}\to B(x_{0},\epsilon)$ is holomorphic at
$x_{0}$.

Let $f(y)$ be a complex polynomial of degree $d$ with highest coefficient $1$
and pairwise different roots $y_{1},\ldots,y_{d}$. For any polynomial $g(y)$
of degree less than $d$ we have the identity

(20) $g(y)=\sum_{i=1}^{d}g(y_{i})\prod_{j\neq i}(y-y_{j})\big{/}\frac{\partial
f}{\partial y}(y_{i}).$

to see this, observe that the values of both sides at $y=y_{1},\ldots,y_{d}$
coincide. Since both sides are polynomials with respect to $y$ of degree less
than $d$ they are equal. In particular, the sum
$\sum_{i=1}^{d}g(y_{i})/\frac{\partial f}{\partial y}(y_{i})$ is equal to the
coefficient of the monomial $y^{d-1}$ in $g(y)$. For the monomials
$g(y)=1,y,\ldots,y^{d-1}$ this implies

(21) $\sum_{i=1}^{d}y_{i}^{n}\big{/}\frac{\partial f}{\partial
y}(y_{i})=\left\\{\begin{array}[]{ll}0&\mbox{ if }0\leq n<d-1\\\ 1&\mbox{ if
}n=d-1.\end{array}\right.$

We conclude that the same is true if the coefficients of $f$ are holomorphic
functions depending on $x\in W$, provided the roots of the discriminant of $f$
are isolated in $W$. Due to the Weierstraß Divison Theorem [PR94, Chapter I
Theorem 1.1] each holomorphic function on $\tilde{O}$ may be written as a
polynomial with respect to $y$ of degree less than $d$ whose coefficients are
holomorphic functions depending on $x\in B(x_{0},\epsilon)$. Furthermore, the
meromorphic functions on $\tilde{O}$ can be written as polynomials with
respect to $y$ of degree less than $d$, whose coefficients are meromorphic
functions on $x\in B(x_{0},\epsilon)$. Hence the sum of such a function $g$
over the sheets of the Weierstraß covering $\tilde{O}\to B(x_{0},\epsilon)$ is
holomorphic at $(x_{0},y_{0})$, if and only if the coefficient corresponding
to $y^{d-1}$ is holomorphic at $x_{0}$. We multiply $g$ with the monomials
$y^{0},y^{1},\ldots,y^{d-1}$ and subtract from the product multiples of $f$
such that the difference again has degree less than $d$ with respect to $y$.
Therefore $\sum_{i=1}^{d}g(y_{i})y_{i}^{n}/\frac{\partial f}{\partial
y_{i}}(y_{i})$ is holomorphic at $x_{0}$ for holomorphic $g$ and
$n=1,\ldots,d-1$. Conversely we may conclude inductively that all coefficients
of $g$ are holomorphic at $x_{0}$, if
$\sum_{i=1}^{d}g(y_{i})y_{i}^{n}/\frac{\partial f}{\partial y_{i}}(y_{i})$ are
holomorphic at $x_{0}$ for $n=0,\ldots,d-1$. ∎

######  Theorem 3.5.

Let $g_{l},\,l=1,\ldots,L$ be a basis for the isomorphism classes of
infinitesimal deformations of locally planar $(X,\Theta)$ and

$(X,\Theta)\hookrightarrow(Z,\Theta_{Z})\twoheadrightarrow\mathcal{T}$

be the induced deformation of $(X,\Theta)$, as in (18). The isomorphism
classes of infinitesimal deformations of the fibres of (18) comprise a vector
bundle over $\mathcal{T}$. On a neighbourhood of $0\in\mathcal{T}$, the
$g_{l}$ are linearly independent sections and hence provide a trivialisation
of this bundle.

###### Proof.

From Theorem 2.2, it is equivalent to show that the spaces of $x$-isomorphism
classes of infinitesimal deformations of the $Z(t)_{l}$ form a vector bundle
over the base $\mathcal{T}$, where $X(t)$ denotes the fibre of the projection
over $t$ and $X(t)_{l}$ its germ at the point $q_{l}(t)$ resulting from
deforming the point $q_{l}\in X$. Lemma 3.2 tells us that for each $t_{l}$,
these are given by $\mathbb{C}\\{x,y\\}/\langle
G_{l}(x,y,t_{l}),\frac{\partial G_{l}(x,y,t_{l})}{\partial y}\rangle$, where
$G_{l}$ was defined in (16). Here $G_{l}(x,y,t_{l})$ is for each fixed $t_{l}$
considered as a function of $x$ and $y$. Writing

(22) $U_{l}(t_{l})=\\{(x,y)\in B(0,\delta_{1})\times\mathbb{C}\mid
G_{l}(x,y,t_{l})=0\\},$

since $U_{l}(t_{l})$ is an open subset of the vanishing set of
$G_{l}(x,y,t_{l})$, these $x$-isomorphism classes are given by the elements of

(23) $\mathcal{O}_{U_{l}(t_{l})}\big{/}\frac{\partial G_{l}}{\partial
y}(x,y,t_{l})\mathcal{O}_{U_{l}(t_{l})}.$

We shall demonstrate the theorem then by showing that for $l=1,\ldots,L$ and
small $t_{l}\in\mathcal{T}_{l}$, the $g_{l}$ form a basis of the above
quotient space.

Let $\tilde{V}_{l}$ be the subspace of $V_{l}$ (17) defined by the vanishing
of $\frac{\partial G_{l}}{\partial y}(x,y,t_{l})$ and note that
$\tilde{V}_{l}$ is the support of the quotient sheaf (23). By shrinking
$\tilde{V}_{l}$ if necessary we may assume that each $U_{l}(t_{l})$ is
contained in a compact set on which $\frac{\partial G_{l}}{\partial
y}(x,y,t_{l})$ is holomorphic and hence this derivative has only finitely many
roots on $U_{l}(t_{l})$. By the construction of the deformation (18) each
$V_{l}$ is isomorphic to an open subspace of $Z$ and under this isomorphism
the union $\tilde{V}_{1}\cup\ldots\tilde{V}_{L}$ defines a closed subspace
$\tilde{Z}$ of $Z$. The map $\tilde{Z}\twoheadrightarrow\mathcal{T}$ is finite
and we shall show that it is again flat. As proven in [PR94, Chapter II
Proposition 2.10], a finite holomorphic map
$\tilde{Z}\twoheadrightarrow\mathcal{T}$ to a reduced complex analytic space
is flat precisely when for $l=1,\ldots,L$ and $t=(t_{1},\ldots,t_{L})$ the sum
of the dimensions of (23) over the finitely many points
$(x,y,t_{l})\in\tilde{V}_{l}$ is locally constant on $t\in\mathcal{T}$.

Let $\mathcal{S}(t)$ be the locally free subsheaf of the sheaf of meromorphic
functions on the fibres $X(t)$ of (18) which is generated by $(\frac{\partial
G_{l}}{\partial y}(x,y,t_{l}))^{-1}$ near $q_{l}$ for all $l=1,\ldots,L$ and
generated by $1$ away from $q_{1},\ldots,q_{L}$. The aforementioned sum of
dimensions of (23) is equal to

(24) $\dim
H^{0}(X(t),\mathcal{S}(t)/\mathcal{O}_{X(t)})=:\deg(\mathcal{S}(t)).$

We remark that a finitely generated subsheaf of the sheaf of meromorphic
functions is termed a generalised divisor and is a natural extension of the
notion of a divisor to one-dimensional complex analytic spaces with
singularities; the above is the usual definition of the degree of a
generalised divisor [KLSS20]. The isomorphism of Lemma 3.4 extends to give an
isomorphism

$\displaystyle\mathcal{S}(t)$ $\displaystyle\rightarrow\Omega^{1}_{X(t)}(D),$
$\displaystyle h$ $\displaystyle\mapsto h\Theta$

where $D$ is the divisor $\sum_{k=1}^{K}(M_{k}+1)p_{k}$ and
$\Omega^{1}_{X(t)}(D)$ denotes the sheaf of meromorphic 1-forms $\omega$ on
the fibre $X(t)$, which are regular away from $p_{1},\ldots,p_{K}$ and have
for $k=1,\ldots,K$ poles of order at most $M_{k}+1$ at the smooth point
$p_{k}$. Consider the long exact cohomology sequence of

$0\rightarrow\mathcal{O}_{X(t)}\rightarrow\mathcal{S}(t))\rightarrow\mathcal{S}(t)/\mathcal{O}_{X(t)}\rightarrow
0.$

The alternating sum of the dimensions vanishes and using that the first
cohomology of the quotient sheaf vanishes since it has finite support, we
obtain that

$\deg(\mathcal{S}(t))=2g-2+K+M\text{, where }M=\sum_{i=1}^{K}M_{i}.$

The Semi-Continuity Theorem also states [PR94, Chapter III Theorem 4.7d] that
the direct image of the structure sheaf under a proper flat map is a vector
bundle. Applying this result here yields that the direct image of the
structure sheaf of $\tilde{Z}$ with respect to
$\tilde{Z}\twoheadrightarrow\mathcal{T}$ is a holomorphic vector bundle on
$\mathcal{T}$ of rank $2g-2+M+K$. Since $\tilde{Z}$ is the support of the
quotient sheaves (23), this is the statement that the isomorphism classes of
infinitesimal deformations form a vector bundle. Then any choice of sections,
such as the $g_{l}$, which form a basis in the fibre over $0\in\mathcal{T}$
form a basis of all fibres in a neighbourhood of $0\in\mathcal{T}$. ∎

##  4\. Decomposition of holomorphic functions

In this section we prove a decomposition result which we shall use in our
proof that (19) is a universal local deformation. Universality means that
given a deformation
$(X,\Theta)\hookrightarrow(Y,\Theta_{Y})\twoheadrightarrow\mathcal{S}$, then
after shrinking $\mathcal{S}$ if necessary, there exists a unique holomorphic
map $\phi:\mathcal{S}\rightarrow\mathcal{T}$ such that the pullback of (19)
under $\phi$ is isomorphic to the given deformation.

As explained in Theorem 2.2, such an isomorphism is determined by
$x$-isomorphisms

$\begin{array}[]{ccccl}(X_{l},x_{l},y_{l})&\hookrightarrow&(Y_{l},x_{Y,l},y_{Y,l})&\twoheadrightarrow&\mathcal{S}_{0}\\\
\|&&\downarrow&&\downarrow\\\
(X_{l},x_{l},y_{l})&\hookrightarrow&(Z_{l},x_{Z,l},y_{Z,l})&\twoheadrightarrow&\mathcal{T}_{0}\end{array}$

of deformations of the corresponding germs $X_{l}=X_{q_{l}}$ for
$l=1,\ldots,L$. Explicitly, these $x$-morphisms are comprised for each
$l=1,\ldots,L$ of a triple $(\phi_{l},u_{l},H_{l})$, as in (4), such that

$H_{l}(x,y,s)G_{l}(x,u_{l}(x,y,s),\phi_{l}(s))=F_{l}(x,y,s),$

where $G_{l}$ was defined in (16) and $(x_{Y_{l}},y_{Y_{l}})$ defines a
biregular map from $Y_{l}$ to $V(F_{l})$. We shall differentiate (4) with
respect to $s\in\mathcal{S}$ and derive a vector field for such triples and
then apply the Picard-Lindelöf Theorem. The differential of the left hand side
in (4) consists of a linear combination of $G_{l},\frac{\partial
G_{l}}{\partial u_{l}}$ and $g_{l}$, . In order to derive the vector field, we
prove in the next lemma that the derivative of the right hand side in (4) has
a unique such decomposition.

We supplement the choice of $\delta_{1}>0$ in the proof of Theorem 2.2 by a
$\delta_{2}>0$ chosen so that for each $l=1,\ldots,L$ the complex space
$U_{l}$ in (11) is a subset of the multidisc $(x,y)\in B(0,\delta_{1})\times
B(0,\delta_{2})$. Furthermore, we choose the balls
$\mathcal{T}_{l}\subset\mathbb{C}^{r_{l}}$ sufficiently small such that
$V_{l}$ (17) is a subset of the multidisc $(x,y,t_{l})\in
B(0,\delta_{1})\times B(0,\delta_{2})\times\mathcal{T}_{l}$. Then let
$\mathrm{H}$ denote the Banach space of bounded holomorphic functions on
$(x,y)\in B(0,\delta_{1})\times B(0,\delta_{2})$ with the supremum norm
$\|\cdot\|_{\infty}$. Recalling that $d_{l}$ is the degree with which $x$
exhibits $V_{l}$ as a Weierstraß covering over
$B(0,\delta_{1})\times\mathcal{T}_{l}$, define $\mathrm{H}_{l}$ to be the
subset of $H$ consisting of those elements which are polynomials in $y$ of
degree less than $d_{l}$.

######  Lemma 4.1.

There exist $\epsilon_{1},\epsilon_{2}>0$ such that for each fixed
$t_{l}\in\mathcal{T}_{l}$ and $u\in\mathrm{H}_{l}$ with $|t_{l}|<\epsilon_{1}$
and $\|u-y\|_{\infty}<\epsilon_{2}$, every $h\in\mathrm{H}$ has a unique
decomposition on $(x,y)\in B(0,\delta_{1})\times B(0,\delta_{2})$ as

(25) $h(x,y)=a(x,y)G_{l}(x,u(x,y),t_{l})+b(x,y)\frac{\partial G_{l}}{\partial
u}(x,u(x,y),t_{l})+c\cdot g_{l}(x,u(x,y)),$

where $(a,b,c)\in\mathrm{H}\times\mathrm{H}_{l}\times\mathbb{C}^{r_{l}}$.
Furthermore $(a,b,c)$ depends holomorphically on such
$(h,u,t_{l})\in\mathrm{H}\times\mathrm{H}_{l}\times\mathcal{T}_{l}$.

###### Proof.

Since $u$ is a polynomial with respect to $y$, for small $\|u-y\|_{\infty}$
the derivative $\frac{\partial u}{\partial y}-1$ is small. Furthermore, by
Banach’s Fixed Point Theorem, for all $(x,y)\in B(0,\delta_{1})\times
B(0,\delta_{2})$ the map $v\mapsto y-(u(x,v)-v)$ has on $B(y,\delta_{2})$ a
unique fixed point $v(x,y)$ obtained by iterating this map with the initial
input $v=0$. Therefore the map $(x,y)\mapsto(x,u(x,y))$ is a biholomorphism
from $B(0,\delta_{1})\times B(0,\delta_{2})$ onto an open subset
$U\subset\mathbb{C}^{2}$, with inverse map $\varphi:(x,y)\mapsto(x,v(x,y))$.
The function $\tilde{h}(x,y)=h(x,v(x,y))$ is holomorphic on $U$ and bounded.
In the proof of Theorem 3.5 it was shown that for sufficiently small
$t_{l}\in\mathcal{T}_{l}$, the $g_{l}$ form a basis of
$\mathcal{O}_{U_{l}(t_{l})}\big{/}\frac{\partial G_{l}}{\partial
y}(x,y,t_{l})\mathcal{O}_{U_{l}(t_{l})}$. (23). Hence there exists a unique
$c\in\mathbb{C}^{{r_{l}}_{l}}$ such that

$\left(\tilde{h}(x,y)-c\cdot g_{l}(x,y)\right)\big{/}\frac{\partial
G_{l}}{\partial y}(x,y,t_{l})$

is holomorphic on the complex space $U_{l}(t_{l})$ (22). Here we assume that
$\delta_{1}>0$ is for our choice of $\delta_{2}>0$ sufficiently small that
$U_{l}(t_{l})$ is contained in $U$. The space $U_{l}(t_{l})$ is by definition
a $d_{l}$ -fold Weierstraß covering over $x\in B(0,\delta_{1})$. Hence
$\varphi(U_{l}(t_{l}))\subset B(0,\delta_{1})\times B(0,\delta_{2})$ is also a
$d_{l}$ -fold covering over $x\in B(0,\delta_{1})$. In particular there exists
a unique $F\in\mathrm{H_{l}}$ such that

$\varphi\left(U_{l}(t_{l})\right)=\\{(x,y)\in B(0,\delta_{1})\times
B(0,\delta_{2})\mid F(x,y)=0\\}.$

We conclude that the function

(26) $\left(h(x,y)-c\cdot g_{l}(x,u(x,y))\right)\big{/}\frac{\partial
G_{l}}{\partial u}(x,u(x,y),t_{l})$

is holomorphic on $\varphi(U_{l}(t_{l}))$. Then by the Weierstraß Division
Theorem [PR94, Chapter I Theorem 1.1], on the zero set of $F$ this function is
equal to a unique $b\in\mathrm{H}_{l}$. Hence

$h(x,y)-c\cdot g_{l}(x,u(x,y))-b(x,y)\frac{\partial G_{l}}{\partial
u}(x,u(x,y),t_{l})$

vanishes on

$\varphi(U_{l}(t_{l}))=\\{(x,y)\in B(0,\delta_{1})\times B(0,\delta_{2})\mid
G_{l}(x,u(x,y),t_{l})=0\\}.$

Since $\varphi^{-1}$ is a biholomorphism from this space to the reduced space
$U_{l}(t_{l}))$, $\varphi(U_{l}(t_{l}))$ is reduced. Therefore there exists a
unique $a\in\mathrm{H}$ so that $(a,b,c)$ solve (25).

It remains to prove that such triples $(a,b,c)$ depend holomorphically on
$(u,h,t)$. For sufficiently small $\|u-y\|_{\infty}$ and $(x,y)\in
B(0,\delta_{1})\times B(0,\delta_{2})$, Banach’s Fixed Point Theorem
guarantees that the unique fixed point of $v\mapsto y-(u(x,v)-v)$ is obtained
by iterating this map starting with $v=0$. These iterations depend
holomorphically on $(x,y)$ and converge uniformly in $\mathrm{H}$ to a map $v$
such that $(x,y)\mapsto(x,v(x,y))$ is the inverse of $(x,y)\mapsto(x,u(x,y))$.
Therefore $v\in\mathrm{H}$ depends holomorphically on $u$ and
$\tilde{h}(x,y)=h(x,v(x,y))$ depends holomorphically on $u$ and $h$.

For each $l=1,\ldots,L$ the function $\frac{\partial f_{l}}{\partial y}(x,y)$
has only one root at $(0,0)$ on $U_{l}$ (11). We know from Lemma 3.4 that a
holomorphic function $g$ belongs to the ideal $\langle f_{l},\frac{\partial
f_{l}}{\partial y}\rangle$ of
$\mathbb{C}[y]\hat{\otimes}\mathcal{O}_{B(0,\delta_{1})}$ if for all
$f\in\mathcal{O}_{U_{l}}$ the residue of $gf(\frac{\partial f_{l}}{\partial
y})^{-2}\Theta$ on $X_{l}$ vanishes. Moreover, for $f\in\frac{\partial
f_{l}}{\partial y}\mathcal{O}_{U_{l}}$ this pairing vanishes. Therefore we
obtain the following non-degenerate pairing

(27) $\displaystyle\mathcal{O}_{U_{l}}\big{/}\frac{\partial f_{l}}{\partial
y}\mathcal{O}_{U_{l}}\times\mathcal{O}_{U_{l}}\Theta\big{/}\frac{\partial
f_{l}}{\partial y}\mathcal{O}_{U_{l}}\Theta$
$\displaystyle\rightarrow\mathbb{C}$ $\displaystyle,(f,g)$
$\displaystyle\mapsto\operatorname*{Res}_{U_{l}}gf(\frac{\partial
f_{l}}{\partial y})^{-2}\Theta.$

Using this pairing, the basis $g_{l,1},\ldots,g_{l,{r_{l}}}$ of
$\mathcal{O}_{U_{l}}\big{/}\frac{\partial f_{l}}{\partial
y}\mathcal{O}_{U_{l}}$ induces a dual basis
$h_{l,1},\ldots,h_{l,{r_{l}}}\in\mathbb{C}[y]\hat{\otimes}\mathcal{O}_{B(0,\delta_{1})}$
of $\mathcal{O}_{U_{l}}\Theta\big{/}\frac{\partial f_{l}}{\partial
y}\mathcal{O}_{U_{l}}\Theta$. The ${r_{l}}\times{r_{l}}$ matrix

$B_{ij}(t)=\operatorname*{Res}_{U_{l}(t_{l})}\left(g_{l,i}h_{l,j}\Theta\big{/}\left(\frac{\partial
G_{l}}{\partial y}\right)^{2}\right).$

depends holomorphically on small $t_{l}\in\mathcal{T}_{l}$ and stays nearby
${1\\!\\!\\!\>\mathrm{l}}$. In particular $B$ is invertible. For small
$t_{l}\in\mathcal{T}_{l}$, the proof of Theorem 3.5 shows that the basis
$h_{l,1},\ldots,h_{l,{r_{l}}}$ of
$\mathcal{O}_{U_{l}}\Theta\big{/}\frac{\partial f_{l}}{\partial
y}\mathcal{O}_{U_{l}}\Theta$ defines also a basis of
$\mathcal{O}_{U_{l}(t_{l})}/\frac{\partial f_{l}}{\partial
y}\mathcal{O}_{U_{l}(t_{l})}$. Taking the residue on $U_{l}(t_{l})$, the
pairing (27) remains non-degenerate and with respect to the bases
$g_{l},h_{l}$ is given by the matrix $B(t)$. Hence $c$ is given by

$c_{i}=\sum_{j=1}^{r}B^{-1}_{ij}\operatorname*{Res}_{U_{l}(t_{l})}\left(\tilde{h}h_{l,j}\Theta\big{/}\left(\frac{\partial
G_{l}}{\partial y}\right)^{2}\right)$

and depends holomorphically on $h$, $u$ and $t_{l}$.

Given $x\in B(0,\delta_{1})$, for sufficiently small $\delta_{1}$, $|t_{l}|$
and $\|u-y\|_{\infty}$, the polynomial $y\mapsto G_{l}(x,u(x,y),t_{l})$ has
exactly $d_{l}$ roots in $y\in B(0,\delta_{2})$. These are also the roots of
the Weierstraß polynomial $y\mapsto F(x,y)$. For $n=0,\ldots,d_{l}-1$ the sum
of the $n$-th powers of these roots are equal to

$\displaystyle\operatorname*{Res}_{y\in B(0,\delta_{2})}\frac{\partial
G_{l}}{\partial u}(x,u(x,y),t_{l})\frac{\partial u}{\partial
y}(x,y)y^{n}dy\big{/}G_{l}(x,u(x,y),t_{l}).$

For the same $x\in B(0,\delta_{1})$ the coefficients of the Weierstraß
polynomials $F$ are polynomials with respect these residues [Mac79, (2.14’)].
In particular, these coefficients depend holomorphically on $(x,u,t_{l})\in
B(0,\delta_{1})\times\mathrm{H}\times\mathcal{T}_{l}$.

It remains to show that the coefficients of $b$ depend holomorphically on
$(x,h,u,t_{l})\in
B(0,\delta_{1})\times\mathrm{H}\times\mathrm{H}\times\mathcal{T}_{l}$. For any
function $g:\mathbb{C}\to\mathbb{C}$ the right hand side of (20) defines a
polynomial of degree $d-1$, which takes at the roots $y_{1},\ldots,y_{d}$ of
$f(y)$ the same values as $g$. Hence for such $g$ the difference of the left
hand side minus the right hand side of (20) is always a multiple of $f$. We
claim that the coefficients of the right hand side are polynomials in the
coefficients of the polynomials $f$ and

(28) $\sum_{i=1}^{d}g(y_{i})y_{i}^{n}\big{/}\frac{\partial f}{\partial
y}(y_{i})\quad 0\leq n\leq d-1.$

In order to prove the claim we may assume that $g(y)$ is a polynomial of
degree $d-1$. Due to (21) the highest coefficient, in front of $y^{d-1}$, is
equal to (28) for $n=0$. Inductively the difference of $y^{n}g(y)$ minus a
multiple of $f(y)$ has degree less than $d$. Consequently the coefficient of
$g$ in front of $y^{d-1-n}$ is equal to (28) minus a polynomial with respect
to the coefficients of $f$ and higher coefficients of $g$. This proves the
claim.

Given $x\in B(0,\delta_{1})$ we apply this claim to the function $g(y)$ given
by (26). For given $x\in B(0,\delta_{1})$ the corresponding expression (28) is
equal to

$\operatorname*{Res}_{y\in B(0,\delta_{2})}\left(h(x,y)-c\cdot
g_{l}(x,u(x,y))\right)\frac{\partial u}{\partial
y}(x,y)y^{n}dy\big{/}G_{l}(x,u(x,y),t_{l}).$

Hence the coefficients of $F$ and of $b$ depend holomorphically on
$(x,h,u,t_{l})\in
B(0,\delta_{1})\times\mathrm{H}\times\mathrm{H}\times\mathcal{T}_{l}$.

Finally, given $(u,h,t_{l})$, $c$ and $b$, $a(x,y)$ is equal to

$\left(h(x,y)-b(x,y)\frac{\partial G_{l}}{\partial u}(x,u(x,y),t_{l})+c\cdot
g_{l}(x,u(x,y))\right)\big{/}G_{l}(x,u(x,y),t_{l}).$

By choice of $b$ and since $U_{l}(t_{l})$ (22) is reduced, this expression
belongs to $\mathrm{H}$ and depends holomorphically on
$(h,u,t_{l})\in\mathrm{H}\times\mathrm{H}\times\mathcal{T}_{l}$. ∎

##  5\. Existence of a universal deformation

In this section we shall prove our main result, which is the following
theorem.

######  Theorem 5.1.

Let $g_{1},\ldots,g_{L}$ be a basis for the isomorphism classes of
infinitesimal deformations of a locally planar pair $(X,dx)$ and

$(X,\Theta)\hookrightarrow(Z,\Theta_{Z})\twoheadrightarrow\mathcal{T}$

the corresponding deformation, constructed in (18). Then given any deformation
$(X,\Theta)\hookrightarrow(Y,dx_{Y})\twoheadrightarrow\mathcal{S}$, after
reducing the base $\mathcal{S}$ if necessary, there exists a unique
holomorphic map $\phi:\mathcal{S}\to\mathcal{T}$ such that the pullback of
(18) with respect to $\phi$ is isomorphic to the given deformation.

Thus the local deformation

$(X,\Theta)\hookrightarrow(Z,\Theta_{Z})\twoheadrightarrow\mathcal{T}_{0}$

of (19) is a universal object in the category of local deformations under
morphism.

In particular then, (19) is independent of the choice of basis $g_{l}$.

###### Proof.

Let $(X,dx)$ be locally planar with prescribed poles, and adopt the notation
of Definition 1.5. Let
$(X,\Theta)\hookrightarrow(Y,dx_{Y})\twoheadrightarrow\mathcal{S}$ be a
deformation of $(X,dx)$. It was shown in Theorem 2.2 that the corresponding
local deformation
$(X,\Theta)\hookrightarrow(Y,dx_{Y})\twoheadrightarrow\mathcal{S}_{0}$ is
uniquely determined by deformations
$(X_{q},x_{q},y_{q})\hookrightarrow(Y_{q},x_{Y,q},y_{Y,q})\twoheadrightarrow\mathcal{S}_{0}$
where $q_{1},\ldots,q_{L}$ denote the points where $x_{l}$ is not a local
coordinate. These points contain the singularities of $X$ together with the
smooth points of $X$ which are roots of $\Theta$. It was argued at the
beginning of the proof of Theorem 2.2 that we may without loss of generality
take a representative deformation whose base $\mathcal{S}$ is an open ball
around $0\in\mathbb{C}^{k}$. The deformations of the space germs may be
represented as

(29) $\displaystyle Y_{l}$ $\displaystyle=\\{(x,y,s)\in B(0,\delta_{1})\times
B(0,\delta_{2})\times\mathcal{S}\mid F_{l}(x,y,s)=0\\}$

with $F_{l}\in\mathbb{C}\\{x,y\\}\hat{\otimes}\mathcal{O}_{\mathcal{S}_{0}}$.
Furthermore, by the Weierstraß Preparation Theorem [PR94, Chapter I Theorem
1.4] we may assume that each $F_{l}(x,y,s)$ is a Weierstraß polynomial of
degree $d_{l}$ in
$\mathbb{C}\\{x\\}[y]\hat{\otimes}\mathcal{O}_{\mathcal{S}}$.

The theorem calls for a holomorphic map $\phi:\mathcal{S}\to\mathcal{T}$ such
that the pullback of (18) is isomorphic to the given deformation. By Theorem
2.2, this is equivalent to the construction of holomorphic maps
$\phi=(\phi_{1},\ldots,\phi_{L}):\mathcal{S}\to\mathcal{T}=\mathcal{T}_{1}\times\ldots\times\mathcal{T}_{L}$
such that for all $l=1,\ldots,L$ the pullback of $X_{l}\hookrightarrow
Z_{l}\twoheadrightarrow\mathcal{T}_{l}$ with respect to $\phi_{l}$ is
$x$-isomorphic to $X_{l}\hookrightarrow Y_{l}\twoheadrightarrow\mathcal{S}$.
As in (4), alongside the maps $\phi_{1},\ldots,\phi_{L}$ we shall construct
units $H_{1},\ldots,H_{L}$ and holomorphic functions $u_{1},\ldots,u_{L}$ such
that for all $l=1,\ldots,L$ we have

(30) $H_{l}(x,y,s)G_{l}(x,u_{l}(x,y,s),\phi_{l}(s))=F_{l}(x,y,s)$

for all $s\in\mathcal{S}$. These maps define an $x$-morphism from the
deformation $X_{l}\hookrightarrow Y_{l}\twoheadrightarrow\mathcal{S}$ to the
pullback of $X_{l}\hookrightarrow Z_{l}\twoheadrightarrow\mathcal{T}_{l}$ with
respect to $\phi$. The derivative of (30) with respect to $s$ gives

$\displaystyle\frac{\partial H_{l}}{\partial s}$
$\displaystyle(x,y,s)G_{l}(x,u_{l}(x,y,s),\phi_{l}(s))$ (31)
$\displaystyle+H_{l}(x,y,s)\frac{\partial G_{l}}{\partial
u_{l}}(x,u_{l}(x,y,s),\phi_{l}(s))\frac{\partial u_{l}}{\partial s}(x,y,s)$
$\displaystyle+H_{l}(x,y,s)\frac{\partial\phi_{l}}{\partial s}(s)\cdot
g_{l}(x,u_{l}(x,y,s))\hskip 56.9055pt=\frac{\partial F_{l}}{\partial
s}(x,y,s).$

For fixed $s\in\mathcal{S}$ the $x$-morphism $(x,y)\mapsto(x,u_{l}(x,y,s))$
should be an isomorphism. Therefore there should exist
$v_{l}\in\mathbb{C}\\{x,y,s_{1},\ldots,s_{K}\\}$ such that
$v_{l}(x,u_{l}(x,y,s),s)=y$ and such that $(x,y)\mapsto(x,v_{l}(x,y,s))$ is
the inverse of $(x,y)\mapsto(x,u_{l}(x,y,s))$. The Weierstraß Division Theorem
[PR94, Chapter I Theorem 1.1] proves that the map
$(x,y)\mapsto(x,u_{l}(x,y,s))$ can be represented uniquely as a polynomial
with respect to $y$ of degree less than $d_{l}$ . If the distance
$\|u_{l}-y\|_{\infty}$ of this polynomial $u_{l}$ to $y$ is sufficiently
small, then $|\frac{\partial u_{l}}{\partial y}-1|$ is bounded on $(x,y)\in
B(0,\delta_{1})\times B(0,\delta_{2})$ by $\frac{1}{2}$ and for all fixed
$x\in B(0,\delta_{1})$ the map $y\mapsto u_{l}(x,y,s)$ has bounded inverse
$y\mapsto v_{l}(x,y)$ from an open subset of $\mathbb{C}$ onto
$B(0,\delta_{2})$.

We apply Lemma 4.1 for $l=1,\ldots,L$ and small $s\in\mathcal{S}$ to

$\displaystyle h(x,y)$ $\displaystyle=H_{l}^{-1}(x,y,s)\frac{\partial
F_{l}}{\partial s}(x,y,s),$ $\displaystyle u(x,y)$
$\displaystyle=u_{l}(x,y,s),$ $\displaystyle t_{l}$
$\displaystyle=\phi_{l}(s)$

and obtain from (5) and (25) that the equations we wish to solve are

(32) $\displaystyle\frac{\partial H_{l}}{\partial s}(x,y,s)$
$\displaystyle=a(x,y)H_{l}(x,y,s),$ $\displaystyle\frac{\partial
u_{l}}{\partial s}(x,y,s)$ $\displaystyle=b(x,y),$
$\displaystyle\frac{\partial\phi_{l}}{\partial s}(s)$ $\displaystyle=c.$

These equations define for small $s\in\mathcal{S}$ a smooth vector field along
the straight line connecting $0$ with $s$. For all sufficiently small
$s\in\mathcal{S}$ we apply the Picard-Lindelöf Theorem and integrate these
vector fields along these straight lines up to $s$, hence obtaining at $s$ a
solution of (30). For sufficiently small $s$ the corresponding
$u_{1},\ldots,u_{L}$ have small distance $\|u_{l}-y_{l}\|_{\infty}$ from
$y_{l}$ so that $(x_{l},y_{l})\mapsto(x_{l},u_{l}(x_{l},y_{l},s))$ are
$x$-isomorphisms. Consequently, on a neighbourhood of $0$ the given
deformation is $x$-isomorphic to the pullback of (18) with respect to $\phi$.

It remains only to prove the uniqueness of the map $\phi$. Any map
$\phi:\mathcal{S}\to\mathcal{T}$ with respect to which the pullback of (18) is
isomorphic to the given deformation is described by solutions
$(H_{l},u_{l},\phi_{l})_{l\in\\{1,\ldots,L\\}}$ of (30) obeying (5). Hence
uniqueness follows from Lemma 4.1 and the Picard-Lindelöf Theorem. ∎

######  Corollary 5.2.

Let $(X,dx)$ be a locally planar pair, $g_{1},\ldots,g_{L}$ a basis for the
infinitesimal deformations of $(X,dx)$ and
$(X,\Theta)\hookrightarrow(Z,\Theta_{Z})\twoheadrightarrow\mathcal{T}$ the
deformation (18). Then after shrinking the base $\mathcal{T}$ if necessary,
(18) represents the local universal deformation of all fibres of
$Z\twoheadrightarrow\mathcal{T}$.

###### Proof.

By Theorem 3.5, there is an open neighbourhood of the base point
$0\in\mathcal{T}$ such that the local sections $g_{1},\ldots,g_{l}$ are at
each point a basis for the isomorphism classes of infinitesimal deformations
of the corresponding fibre of $Z\twoheadrightarrow\mathcal{T}$. Clearly then
by Theorem 5.1 the restriction of the deformation
$(X,\Theta)\hookrightarrow(Z,\Theta_{Z})\twoheadrightarrow\mathcal{T}$ to this
open subset respresents the local universal deformation of all its fibres. ∎

We shall henceforth take $\mathcal{T}$ to be this reduced neighbourhood, so
that it enjoys the property of Corollary 5.2. If in Theorem 5.1 we impose the
reality condition (c) on the fibres of (6), then the involution $\eta$ extends
to an involution of (6) and (18). In this case Theorem 5.1 implies that the
morphism $\phi$ commutes with $\eta$. Furthermore $\phi$ maps the fixed point
set of $\eta$ in $\mathcal{S}$ into the fixed point set of $\eta$ in
$\mathcal{T}$. In particular, the restriction of (18) to the fixed point set
of $\eta$ in $\mathcal{T}$ defines a universal local deformation of locally
planar $(X,\Theta)$ obeying the reality condition.

We find it convenient below to extend Definitions 1.6 and 1.8 in the obvious
way to encompass germs of locally planar pairs $(V(f),\Theta)$; the
isomorphism classes of these pairs are clearly the same as the $x$-isomorphism
classes of germs $V(f)$.

We now turn to the question of whether $\mathcal{T}$ locally parameterises the
isomorphism classes of locally planar $(X,\Theta)$ with prescribed poles
(Definition 1.5). In fact it does not, as is easily revealed by considering
the case when the special fibre has a non-trivial automorphism. Since the
deformation $X\hookrightarrow Y\twoheadrightarrow\mathcal{T}$ is universal,
the automorphism of $X$ induces automorphisms of $Y$ and $\mathcal{T}$, with
respect to which the above maps are equivariant. The fibres over distinct
points of the base space are mapped to one another under the automorphism of
the total space and are hence isomorphic. Using Theorem 2.2, the universal
deformation is described by the corresponding universal deformations of space
germs $V(f_{l})\hookrightarrow V(G_{l})\twoheadrightarrow\mathcal{T}_{l}$ for
$l=1,\ldots,L$. We present now an example which demonstrates how a symmetry in
$f_{l}$ induces an isomorphism between the fibres over different elements of
$\mathcal{T}$.

######  Example 5.3.

Taking $f(x,y)=y^{3}-x^{2}$, as before we write $V(f)$ for the space germ
defined by $f$ at $(0,0)$. Using Lemma 3.2, the isomorphism classes of
infinitesimal deformations (15) of $(V(f),dx)$ are parameterised by
${\mathbb{C}\\{x,y\\}}/{\langle f,f_{y}\rangle}$, for which $1,x,y,xy$
provides a basis. The corresponding deformation
$(V(f),\Theta)\hookrightarrow(Z,\Theta)\rightarrow\mathcal{T}$ is given by

$G(x,y,t_{1},t_{2},t_{3},t_{4})=y^{3}-x^{2}+t_{1}+t_{2}x+t_{3}y+t_{4}xy$

and as shown above this represents a universal local deformation. Writing
$q=e^{\frac{2}{3}\pi i}$ and using the notation of (4), the pairs
$(H,u)=(1,qy)$ and $(H,u)=(1,q^{2}y)$ each define isomorphisms of
$(V(f),\Theta)$. They lift to isomorphisms between the fibres of
$(Z,\Theta)\rightarrow\mathcal{T}$ over $(t_{1},t_{2},t_{3},t_{4})$,
$(t_{1},t_{2},qt_{3},qt_{4})$ and $(t_{1},t_{2},q^{2}t_{3},q^{2}t_{4})$. Note
that the presence of these isomorphisms does not contradict the uniqueness
property of the universal local deformation; uniqueness demands that there is
no non-trivial $x$-isomorphism of $Z\rightarrow\mathcal{T}$ which acts
trivially on the fibre $V(f)$ over $0\in T$ but the above are allowable since
they do not restrict to the identity automorphism of $V(f)$.

In this example, the isomorphism classes of locally planar pairs $(X,dx)$ in a
neighbourhood of $(V(f),\Theta)$ are locally parameterised not by
$\mathcal{T}$ but by $\mathcal{T}/\mathbb{Z}_{3}$, giving locally the
structure of an orbifold whose finite group action is the induced action of
the automorphism group of the central fibre. This will be made explicit later
in this section by the construction of a 3-fold cover
$\operatorname{Dis}:\mathcal{T}\rightarrow\mathbb{C}^{4}\cong{\mathbb{C}\\{x,y\\}}/{\langle
f,f_{y}\rangle}$, where this map is the symmetrisation of the branch values of
$x$.

It is straightforward to generalise the above example and see that non-trivial
automorphisms of the pair $(X_{l},dx)$ force distinct fibres of
$(Z_{l},\Theta)\rightarrow\mathcal{T}_{l}$ to be isomorphic. We expect
furthermore that whenever $(Z_{l},\Theta)\rightarrow\mathcal{T}_{l}$ has such
fibres, the base space $(X_{l},dx)$ possesses a non-trivial automorphism. We
give below a heuristic argument which we anticipate could be expanded into a
proof of this statement.

We shall describe the fibres of the universal deformation (18) as local
coverings with respect to the regular funtions $x_{q}$, and show that this
family is locally parameterised by the branch values of the local functions
$x$. Simplest are those points of $\mathcal{T}$ for which the corresponding
fibre of (18) is smooth and contains only simple ramification points of $x$.
We begin by showing that such points of $\mathcal{T}$ are generic.

######  Lemma 5.4.

Let $X(t)\subset Z$ denote for all $t\in\mathcal{T}$ the fibres of (18). Then
the following set is open and dense in $\mathcal{T}$:

$\mathcal{T}^{\ast}:=\\{t\in\mathcal{T}\mid X(t)\text{ is smooth and contains
only simple roots of }\Theta_{Z}\\}$

###### Proof.

By Theorem 2.2, it suffices to prove the statement for the analogous subsets
$\mathcal{T}_{l}^{\ast}\subset\mathcal{T}_{l}$. Then have
$\mathcal{T}^{\ast}=\mathcal{T}_{l}^{\ast}\times\cdots\times\mathcal{T}_{L}^{\ast}$.
We shall show that for each $l=1,\ldots,L$ the set

$\mathcal{T}_{l}^{\circ}:=\\{t_{l}\in\mathcal{T}_{l}\mid U_{l}(t_{l})\text{ is
smooth}\\}$

is the complement of an analytic proper subset of $\mathcal{T}_{l}$ and that

$\mathcal{T}_{l}^{\ast}:=\\{t_{l}\in\mathcal{T}_{l}^{\circ}\mid
U_{l}(t_{l})\text{ contais only simple roots of }\Theta_{l}\\}$

is the complement of an analytic proper subset of $\mathcal{T}_{l}^{\circ}$.
We note first that

$\frac{\partial G_{l}}{\partial y}dy+\frac{\partial G_{l}}{\partial
x}\Theta=0,$

and on $\mathcal{T}_{l}^{\circ}$ neither the partial derivatives nor the
differentials have a common root. Hence we may equivalently characterise
$\mathcal{T}_{l}^{\ast}$ as

$\mathcal{T}_{l}^{\ast}:=\\{t_{l}\in\mathcal{T}_{l}^{\circ}\mid
U_{l}(t_{l})\text{ contais only simple roots of }\frac{\partial
G_{l}}{\partial y}\\}.$

The complement of $\mathcal{T}_{l}^{\circ}$ is the the set of all
$t_{l}\in\mathcal{T}_{l}$ such that $\frac{\partial G_{l}}{\partial x}$ and
$\frac{\partial G_{l}}{\partial y}$ have a common root on $U_{l}(T)$. The
complement of $\mathcal{T}_{l}^{\ast}$ in $\mathcal{T}_{l}^{\circ}$ is the set
of all $t_{l}\in\mathcal{T}_{l}^{\circ}$ such that $\frac{\partial
G_{l}}{\partial y}$ and $\frac{\partial^{2}G_{l}}{\partial^{2}y}$ have a
common root on $U_{l}(T)$. Thus these complements are each given by the
vanishing of a resultant and hence the Weierstraß Preparation Theorem [PR94,
Chapter I Theorem 1.4] implies they are analytic subsets of $\mathcal{T}_{l}$
and $\mathcal{T}_{l}^{\circ}$, respectively. It remains to prove that they are
proper subsets. For this, consider the deformation

$V(f_{l})\hookrightarrow
V(f_{l}-s)\twoheadrightarrow\mathcal{S}:=\\{s\in\mathbb{C}\\}.$

We have hitherto used the notation $V(f)$ for the germ of the vanishing set of
$f$ at $(0,0)$; here we also allow it to indicate an open neighbourhood of
$(0,0)$ representing this germ. The fibre over $s$ contains a singularity if
and only if $s$ is a critical value of $f$. By Sard’s Theorem these critical
values have measure zero, and the deformation contains in every neighbourhood
of $s=0$ a fibre without singularities. Since this deformation is the pullback
of $X_{l}\hookrightarrow Z_{l}\twoheadrightarrow\mathcal{T}_{l}$ with respect
to a unique map from $\mathcal{S}$ to $\mathcal{T}_{l}$, every neighbourhood
of $0\in\mathcal{T}_{l}$ contains an element of $\mathcal{T}_{l}^{\circ}$.
This shows that $\mathcal{T}_{l}^{\circ}$ is open and dense in
$\mathcal{T}_{l}$.

Now we give a similar argument for the analytic subset
$\mathcal{T}_{l}^{\ast}$ of $\mathcal{T}_{l}^{\circ}$. Suppose, for
$f_{l}\in\mathbb{C}\\{x,y\\}$, that $x:V(f_{l})\rightarrow\mathbb{C}$ has a
smooth ramification point of order $m\geq 2$ at $(0,0)$. We may assume that
$f_{l}$ is a polynomial of degree $m+1$ with respect to $y$ with highest
coefficient equal to 1 and vanishing second highest coefficient:

$\displaystyle f_{l}(x,y)$
$\displaystyle=y^{m+1}+a_{m-1}(x)y^{m-1}+\ldots+a_{0}(x).$

All coefficients $a_{0},\ldots,a_{m-1}\in\mathbb{C}\\{x\\}$ vanish at $x=0$
and since $V(f_{l})$ is smooth, furthermore $a_{0}^{\prime}(0)\neq 0$. We
consider the deformation

$V(f_{l})\hookrightarrow
V(f_{l}-s_{0}-s_{1}y)\twoheadrightarrow\mathcal{S}:=\\{(s_{0},s_{1})\in\mathbb{C}^{2}\\}.$

For sufficiently small $s_{0}$, the inverse function theorem guarantees that
the coefficient $a_{0}(x)-s_{0}$ has a unique small root
$x(s_{1})\in\mathbb{C}$. Then for $s_{1}\neq a_{1}(x(s_{0}))$,
$V(f_{l}-s_{0}-s_{1}y)$ contains no ramification point of order $m$, since all
coefficients $a_{0}-s_{0},a_{1}-s_{1},a_{2},\ldots,a_{m-1}$ must vanish at
such ramification points. Again the given deformation is a pullback of
$X_{l}\hookrightarrow Z_{l}\twoheadrightarrow\mathcal{T}_{l}$ and all
$t_{l}\in\mathcal{T}_{l}^{\circ}$ whose fibres contain a ramification point of
order $m$ have nearby fibres with ramification points of order $m-1$. Due to
Theorem 3.5 the deformation
$(X_{l},\Theta_{l})\hookrightarrow(Z_{l},\Theta_{l})\twoheadrightarrow\mathcal{T}_{l}$
represents a universal deformation of all its fibres. Hence we can iterate the
lowering of the ramification order. This completes the proof that
$\mathcal{T}_{l}^{\ast}$ is open and dense in $\mathcal{T}_{l}^{\circ}$ and
thus in $\mathcal{T}_{l}$. ∎

Let be a $X_{l}=V(f_{l})$ be a complex space germ and
$(X_{l},dx_{l})\hookrightarrow(Z_{l},dx)\twoheadrightarrow\mathcal{T}_{l}$
denote the universal deformation defined in (16). It deforms higher order
ramification points of the map $x_{l}$ (i.e. higher order roots of
$\frac{\partial f_{l}}{\partial y}$ at $(0,0)$) into lower order ramification
points. On the fibres of $t_{l}\in\mathcal{T}_{l}^{\ast}$ the map $x_{l}$ has
only first order ramification points, and as proven in the above lemma,
$\mathcal{T}_{l}^{\ast}$ is open and dense in $\mathcal{T}$.

We will show that the fibres $U_{l}(t_{l})$ of each
$\mathcal{Z}_{l}\rightarrow\mathcal{T}_{l}$ are locally parameterised by the
(symmetrisation of the) branch points of
$x_{l}:U_{l}(t_{l})\rightarrow\mathbb{C}$. We begin by considering only smooth
fibres with simple ramification points.

######  Lemma 5.5.

We may locally define a map on ${\mathcal{T}_{l}}^{\ast}$ by sending
$t_{l}\in{\mathcal{T}_{l}}^{\ast}$ to the branch points of
$x_{l}:U_{l}(t_{l})\rightarrow\mathbb{C}$. There are
$r_{l}=\dim\left(\mathcal{O}_{U_{l}(t_{l})}\big{/}\frac{\partial
G_{l}}{\partial y}(x,y,t_{l})\mathcal{O}_{U_{l}(t_{l})}\right)$ such branch
points and this defines a local diffeomorphism

${\mathcal{T}_{l}}^{\ast}\rightarrow\mathbb{C}^{r_{l}}.$

###### Proof.

There is no global ordering of the branch points, but locally on
${\mathcal{T}_{l}}^{\ast}$ we may choose any ordering and take the
corresponding map. Any curve in ${\mathcal{T}_{l}}^{\ast}$ can be lifted to a
curve $s\mapsto(x(s),y(s),t_{l}(s))$ with $x(0)$ a branch point. Then
differentiating $G(x,y,t)=0$ with respect to $t$ gives

$\dot{x}(s)=-\dot{t}_{l}(s)\cdot g_{l}(x(s),y(s))\big{/}\frac{\partial
G_{l}(x(s),y(s),t(s))}{\partial x}$

and for $t_{l}\in\mathcal{T}_{l}^{\ast}$ the denominator is non-vanishing.
Hence it suffices to show that the $r$ vectors obtained by evaluating
$g_{l}=(g_{l,1},\ldots g_{l,r_{l}})$ at the $r$ ramification points of $x_{l}$
are linearly independent and that $r=r_{l}$. As stated in (23), the
$x$-isomorphism classes of infinitesimal deformations of $U_{l}(t_{l})$ are
given by the elements of $\mathcal{O}_{U_{l}(t_{l})}\big{/}\frac{\partial
G_{l}}{\partial y}(x,y,t_{l})\mathcal{O}_{U_{l}(t_{l})}$. For
$t_{l}\in\mathcal{T}_{l}^{\ast}$, we define a local map

$\mathcal{O}_{U_{l}(t_{l})}\big{/}\frac{\partial G_{l}}{\partial
y}(x,y,t_{l})\mathcal{O}_{U_{l}(t_{l})}\rightarrow\mathbb{C}^{r}$

by sending a function to its values at the (simple) ramification points of
$x_{l}$, where the ordering of these may be fixed locally. We shall argue that
this is an isomorphism and hence in particular conclude that $r=r_{l}$. Note
that these branch points are precisely the (simple) roots of $\frac{\partial
G_{l}}{\partial y}$. The map is injective since the ideal in
$\mathcal{O}_{U_{l}(t_{l})}$ generated by $\frac{\partial G_{l}}{\partial y}$
consists precisely of the functions vanishing at these ramification points. It
is surjective by Theorem 3.26.7 of [For81]. Then from Theorem 3.5 we see that
for $t_{l}\in{\mathcal{T}_{l}}^{\ast}$ the $r_{l}$ vectors obtained by
evaluating $g_{l}=(g_{l,1},\ldots g_{l,r_{l}})$ at the ramification points are
linearly independent. ∎

In order to define a global analogue of the local map in the above lemma, we
symmetrise with respect to permutations of the branch points. For
$t_{l}\in\mathcal{T}_{l}^{\ast}$, the branch points of $x_{l}$ are precisely
the roots of $\frac{\partial G_{l}}{\partial y}$ on $U_{l}(t_{l})$ and so they
coincide with the roots of the discriminant of the polynomial $G_{l}$ with
respect to $y_{l}$ (holding $t_{l}$ fixed). This discriminant
$\operatorname{Dis}_{l}(x_{l},t_{l})$ is a Weierstraß polynomial of degree
$r_{l}$ with respect to $x_{l}$, with coefficients depending holomorphically
on $t_{l}\in\mathcal{T}_{l}$. The elementary symmetric functions of the roots
of $\operatorname{Dis}_{l}(x_{l},t_{l})$ are given by its coefficients. Hence
these coefficients define a global holomorphic map on $\mathcal{T}_{l}^{\ast}$
which by Lemma 5.4 we may extend holomorphically to $\mathcal{T}_{l}$. We
write this map also as $\operatorname{Dis}_{l}$.

######  Remark 5.6.

  1. (1)

The map

$\operatorname{Dis}_{l}:\mathcal{T}_{l}\rightarrow\mathbb{C}^{r_{l}}$

is a covering map whose ramification points all lie in
$\mathcal{T}_{l}\setminus{\mathcal{T}_{l}}^{\ast}$.

  2. (2)

If the fibres of (18) over $t_{l}\neq t^{\prime}_{l}\in\mathcal{T}_{l}$ are
isomorphic then
$\operatorname{Dis}_{l}(x_{l},t_{l})=\operatorname{Dis}_{l}(x_{l},t^{\prime}_{l})$.
Hence if every sufficiently small neighbourhood of $t\in\mathcal{T}_{l}$
contains exactly $n$ distinct $t_{l}$ with isomorphic fibres, then the map
$\operatorname{Dis}_{l}$ has a ramification point of order $n$ at $t$.

  3. (3)

If there is an isomorphism between the fibres of (18) over $t_{l}\neq
t^{\prime}_{l}\in{\mathcal{T}_{l}}^{\ast}$ then it extends to an isomorphism
between the fibres of all nearby pairs
$t_{l}\not=t_{l}^{\prime}\in\mathcal{T}_{l}^{\ast}$ with
$\operatorname{Dis}_{l}(x_{l},t_{l})=\operatorname{Dis}_{l}(x_{l},t_{l}^{\prime})$.

###### Proof.

  1. (1)

Lemma 5.5 guarantees that the differential of $\operatorname{Dis}_{l}$ is
invertible on $\mathcal{T}_{l}^{\ast}$.

  2. (2)

If the fibres over $t_{l}\neq t^{\prime}_{l}\in\mathcal{T}_{l}$ are isomorphic
then the values of $x_{l}$ at the branch points of $x_{l}$ must agree and so
$\operatorname{Dis}_{l}(x_{l},t_{l})=\operatorname{Dis}_{l}(x_{l},t^{\prime}_{l})$.

  3. (3)

This is an immediate consequence of Theorem 3.5.

∎

The above arguments provide some evidence for our conjecture that if every
neighbourhood of $t\in\mathcal{T}$ has $n$ distinct points over which the
fibres of (18) are isomorphic then the fibre over $t$ has an automorphism of
order $n$. In order to prove this conjecture it would suffice to show that

  1. (1)

the $n$ points can be assumed to lie in $\mathcal{T}^{\ast}$

  2. (2)

the local automorphism of $\mathcal{T}^{\ast}$ induced by permuting the fibres
of $\operatorname{Dis}_{l}$ extends to an automorphism of $\mathcal{T}$,
fixing the corresponding branch point of $\operatorname{Dis}_{l}$.

##  6\. Deformations of hyperelliptic curves with one differential

Hitherto we have considered deformations of locally planar pairs $(X,\Theta)$
with prescribed poles, comprised of a curve and a single differential
satisfying Definition 1.5. In a forthcoming work we shall extend our results
to triples $(X,\Theta,dy)$ which satisfy a natural analogue of Definition 1.5,
namely that poles of $dy$ are also prescribed and that the locally planar
structure is compatible with both $\Theta$ and $dy$. There is a particular
case, arising in a number of integrable systems, for which the existence of a
local universal deformation of such triples $(X,\Theta,dy)$ follows quite
easily from our results above. Namely, we consider in this section the case of
locally planar pairs $(X,\Theta)$ where $X$ possesses a global meromorphic
function $y$ of degree two, and hence is hyperelliptic. Instead of directly
requiring the existence of the global function $y$ of degree two, we impose an
involution $\sigma$ upon $(X,\Theta)$ and the local functions $y_{q}$ of
Definition 1.5, such that the quotient $X/\sigma$ has arithmetic genus zero.

More precisely, we consider locally planar $(X,\Theta)$ with local functions
$x_{q},y_{q}$ for each $q\in X^{\circ}$ such that $X$ possesses a holomorphic
involution $\sigma$ which interchanges $p_{1},\ldots,p_{K}$, satisfies

(33) $\displaystyle\sigma^{\ast}\Theta$ $\displaystyle=-\Theta$
$\displaystyle\sigma^{\ast}x_{q}$ $\displaystyle=-x_{\sigma(q)}$
$\displaystyle\sigma^{\ast}y_{q}$ $\displaystyle=y_{\sigma(q)}$

and is such that each $y_{q}$ is a local parameter of the quotient $X/\sigma$.

In particular then $X/\sigma$ is smooth, since it has a local parameter about
every point in $X^{\circ}/\sigma$ whilst the remaining points
$p_{1},\ldots,p_{K}\in X$ are smooth and remain so in the quotient as $\sigma$
has only isolated fixed points. We shall only consider the case when $X$ is
hyperelliptic with hyperelliptic involution $\sigma$ (i.e.
$X/\sigma\simeq\mathbb{P}^{1}$). As arithmetic genus is preserved in families
and the quotient is smooth, hyperellipticity will be preserved under our
deformations.

For the the initial $(X,\Theta)$ we choose particular local parameters
$y_{q}$. Let again $q_{1},\ldots,q_{L}\in X^{\circ}$ denote the points where
$x_{q}$ is not a local coordinate. These points are interchanged by $\sigma$.
The corresponding action of $\sigma$ on the index set $\\{1,\ldots,L\\}$ is
denoted by $l\mapsto\sigma l$. Again we abbreviate the indices $q_{l}$ by
indices $l$ and the indices $q_{\sigma(l)}$ by indices $\sigma l$. In
particular $x_{\sigma l}=-x_{l}$ and $y_{\sigma l}=y_{l}$. Nearby $q_{l}\neq
q_{\sigma l}$ we choose parameters such that

$\displaystyle x_{l}$ $\displaystyle=y_{l}^{r+2}$ $\displaystyle x_{\sigma l}$
$\displaystyle=-y_{\sigma l}^{r+2}$ $\displaystyle f_{l}(x_{l},y_{l})$
$\displaystyle=y_{l}^{r+2}-x_{l}$ $\displaystyle f_{\sigma l}(x_{\sigma
l},y_{\sigma l})$ $\displaystyle=y_{\sigma l}^{r+2}+x_{\sigma l}$

with some $r\in\mathbb{N}$ (depending on $l$). Nearby $q_{l}=q_{\sigma l}$ we
may instead assume

$\displaystyle x_{l}^{2}$ $\displaystyle=y_{l}^{r+2}$ $\displaystyle
f_{l}(x_{l},y_{l})$ $\displaystyle=y_{l}^{r+2}-x_{l}^{2}.$

######  Definition 6.1.

A deformation of a hyperelliptic locally planar $(X,\Theta)$ with local
functions $y_{q}$ as above is a deformation
$(X,\Theta)\hookrightarrow(Y,\Theta_{Y})\twoheadrightarrow\mathcal{S}$ such
that

  1. (1)

the involution $\sigma$ extends to an involution of $Y$ which acts trivially
on $\mathcal{S}$, commutes with the maps $X\hookrightarrow
Y\twoheadrightarrow\mathcal{S}$ and satisfies

(34) $\displaystyle\sigma^{\ast}\Theta_{Y}$ $\displaystyle=-\Theta_{Y}$
$\displaystyle\sigma^{\ast}x_{Y,q}$ $\displaystyle=-x_{Y,\sigma(q)}$
$\displaystyle\sigma^{\ast}y_{Y,q}$ $\displaystyle=y_{Y,\sigma(q)},$

  2. (2)

at $q\in X^{\circ}$, $(y_{Y,q},s)$ are local parameters for the quotient
$Y/\sigma$.

We could consider more general deformations with non-trivial actions of
$\sigma$ on $\mathcal{S}$, but only the fibres of
$Y\twoheadrightarrow\mathcal{S}$ over the fixed points of $\sigma$ in
$\mathcal{S}$ obey (33).

The morphisms of these deformations are the morphisms $\psi$ of deformations
of $(X,\Theta)$ (see Definition 1.8 ) such that $\psi$ commutes with $\sigma$.
Finally, a local deformation
$(X,\Theta,y_{k})\hookrightarrow(Y,\Theta_{Y},y_{Y,q})\twoheadrightarrow\mathcal{S}_{0}$
is an equivalence class of deformations
$(X,\Theta,y_{k})\hookrightarrow(Y,\Theta_{Y},y_{Y,q})\twoheadrightarrow\mathcal{S}$,
under this notion of isomorphism.

Theorem 2.2 carries over and shows that a local deformation of hyperelliptic
$(X,\Theta,y_{k})$ is equivalent to a deformation of each $X_{l}=X_{q_{l}}$
preserving the involution $\sigma$. In order to describe these deformations,
we decompose the algebra $\mathbb{C}\\{x,y\\}$ into a direct sum of symmetric
and anti-symmetric elements with respect to the involution $\sigma^{\ast}x=-x$
and $\sigma^{\ast}y=y$:

$\displaystyle g$ $\displaystyle=g^{+}+g^{-}$ $\displaystyle g^{+}$
$\displaystyle=\tfrac{1}{2}(g+\sigma^{\ast}g)$ $\displaystyle g^{-}$
$\displaystyle=\tfrac{1}{2}(g-\sigma^{\ast}g).$

Since $\sigma$ acts trivially on the base, we choose for $\sigma l=l$ a basis
of the symmetric part of $\mathbb{C}\\{x,y\\}/\langle f_{l},\frac{\partial
f_{l}}{\partial y}\rangle$. In all cases we choose the basis in (22) to be of
the form

$\displaystyle(g_{l,1},\ldots,g_{l,r})=(g_{\sigma l,1},\ldots,g_{\sigma l,r})$
$\displaystyle=(1,y_{l},\ldots,y_{l}^{r-1}).$

This basis again defines a particular deformation denoted by

(35)
$(X,\Theta,_{q})\hookrightarrow(Z,\Theta_{Z},y_{Y,q})\twoheadrightarrow\mathcal{T},$

and the corresponding local deformation is independent of the choice of
$\sigma$-symmetric basis $g_{l}$. Our arguments carry over to the
hyperelliptic case, yielding the following analog to Theorem 5.1:

######  Corollary 6.2.

Let $(X,\Theta,y_{q})$ be hyperelliptic and locally planar as described above.
Then (35) defines a universal local deformation
$(X,\Theta,_{q})\hookrightarrow(Z,\Theta_{Z},y_{Y,q})\twoheadrightarrow\mathcal{T}_{0}$
of $(X,\Theta,y_{q})$.

If we additionally require $(X,\Theta,y_{q})$ to be real with respect to an
anti-holomorphic involution $\eta$, as in Definition 1.5(c), then $\eta$ lifts
to an involution of (35). By restricting to the fixed point set
$\mathcal{T}^{\mathrm{R}}$ of $\eta$ acting on $\mathcal{T}$ describes a
universal local deformation of real hyperelliptic locally planar data
$(X,\Theta,y_{q})$. In particular this includes the periodic solutions of the
KdV equation and of the sinh-Gordon equation. In [GS95] the space of spectral
curves is investigated as a covering with respect to the local parameters
$t\in\mathcal{T}$. For real periodic solutions of the KdV equation they are
globally one-to-one [MOk75, Kor08]. In [HKS16] these deformations are applied
to the periodic solutions of the sinh-Gordon equation. We shall discuss these
and other examples in more detail in forthcoming work.

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