# Formal Poisson (co)homology of the Lefschetz singularity

Lauran Toussaint, Florian Zeiser

###### Abstract.

We compute the formal Poisson cohomology groups of a real Poisson structure
$\pi$ on ${\mathbb{C}}^{2}$ associated to the Lefschetz singularity
$(z_{1},z_{2})\mapsto z_{1}^{2}+z_{2}^{2}$. In particular we correct an
erroneous computation in the literature. The definition of $\pi$ depends on a
choice of volume form. Using the main result we formally classify all Poisson
structure arising from different choices of volume forms.

##  1\. Introduction

A Poisson structure on a smooth manifold $M$ is a bivector field
$\pi\in\mathfrak{X}^{2}(M)$ such that

$[\pi,\pi]=0,$

where $[\cdot,\cdot]$ denotes the Schouten bracket (see e.g. [LPV13]). As
first observed by Lichnerowicz [Lic77], any Poisson manifold $(M,\pi)$ is
naturally equipped with a differential on the space of mulivector fields

$d_{\pi}:=[\pi,\cdot]:\
{\mathfrak{X}}^{\bullet}(M)\to{\mathfrak{X}}^{\bullet+1}(M).$

The vanishing of the bracket is equivalent to
${\operatorname{d}}_{\pi}^{2}=0$, and the associated cohomology
$H^{\bullet}(M,\pi)$ is called _Poisson cohomology_. The groups
$H^{\bullet}(M,\pi)$ contain important information about the Poisson manifold
$(M,\pi)$. For example:

  * •

$H^{0}(M,\pi)$ consists of functions on $M$ constant along the leaves of the
foliation ${\mathcal{F}}$;

  * •

$H^{1}(M,\pi),[\cdot,\cdot])$ can be seen as the Lie algebra of infinitesimal
outer automorphisms of $(M,\pi)$;

  * •

$H^{2}(M,\pi)$ controls infinitesimal deformations modulo those induced by
diffeomorphisms.

For a detailed exposition of Poisson geometry and Poisson cohomology we refer
the reader to [DZ05, LPV13, CFM21]. Despite their importance, Poisson
cohomology is notoriously hard to compute due to the lack of general methods
for the computation. The difficulty comes from the fact that
$(\mathfrak{X}^{\bullet}(M),\mathrm{d}_{\pi})$ is generally not an elliptic
complex. Assuming that $\pi$ is regular, i.e. all leaves of ${\mathcal{F}}$
have the same dimension, Xu computed the Poisson cohomology of such Poisson
structures in [Xu92]. This, together with the existence of a Mayer-Vietoris
sequence due to Vaisman [Vai94], implies that the fundamental problem for the
computation lies around neighborhoods of singular leaves.

To this end observe that given a Poisson structure $\pi$ on $M$ and a singular
point $p\in M$ of $\pi$, i.e. $\pi_{p}=0$, we have a short exact sequence of
complexes

$\displaystyle
0\to(\mathfrak{X}^{\bullet}_{p}(M),\mathrm{d}_{\pi})\hookrightarrow(\mathfrak{X}^{\bullet}(M),\mathrm{d}_{\pi})\xrightarrow{j^{\infty}_{p}}(\mathfrak{X}^{\bullet}(M)/\mathfrak{X}^{\bullet}_{p}(M)\simeq{\mathbb{R}}[[T^{*}_{p}M]]\otimes\wedge^{\bullet}T_{p}M,\mathrm{d}_{j_{p}^{\infty}\pi})\to
0$

resulting in a long exact sequence in cohomology. Here $j^{\infty}_{p}$ is the
infinite jet map at $p$, and $\mathfrak{X}^{\bullet}_{p}(M)$ denotes
multivector fields whose infinite jet at $p$ is zero. The equivalence for the
quotient is due to Borel’s lemma. A similar short exact sequence can be
obtained when the singular locus of $\pi$ is a higher dimensional submanifold,
see [HZ23].

The short exact sequence above was first used by Abreu & Ginzburg [Gin96] and
Roytenberg [Roy02]. Its usefulness lies in the fact that if $\pi$ vanishes in
a polynomial way then we can compute the cohomology of
$(\mathfrak{X}^{\bullet}_{p}(M),\mathrm{d}_{\pi})$ as if $\pi$ was regular. As
such we are left with the computation of the complex

$({\mathbb{R}}[[T^{*}_{p}M]]\otimes\wedge^{\bullet}T_{p}M,\mathrm{d}_{j_{p}^{\infty}\pi}).$

We call its cohomology the formal Poisson cohomology of $\pi$ at $p\in M$,
denoted by $H^{\bullet}_{F_{p}}(M,\pi)$. The methods used to compute
$H^{\bullet}_{F_{p}}(M,\pi)$ are more algebraic in nature (see e.g. [Mon02,
Pic06, Pel09]).

The Poisson structure under consideration in this paper are defined as
follows. First recall that any fibration
$f:(M^{m},\mu_{M})\to(N^{m-2},\mu_{N})$ between oriented manifolds
$(M,\mu_{M})$ and $(N,\mu_{N})$ defines a Poisson structure
$\pi_{f,\mu_{M},\mu_{N}}$ on $M$ due to [GSV14][Thm 2.6] by:

(1)
$\displaystyle\pi_{f,\mu_{M},\mu_{N}}(\mathrm{d}g,\mathrm{d}h)\mu_{M}:=\mathrm{d}g\wedge\mathrm{d}h\wedge
f^{*}{\mu_{N}}\qquad\text{ for }\quad g,h\in C^{\infty}(M).$

Note that $\pi_{f,\mu_{M},\mu_{N}}$ depends on the fibration $f:M\to N$ and on
the choice of volume forms on $M$ and $N$. Poisson structures defined by
different volume forms admit the same singular foliation ${\mathcal{F}}$ with:

  * •

$2$-dimensional leaves through every $x\in M\setminus\mathrm{Crit}(f)$;

  * •

$0$-dimensional leaves for every $x\in\mathrm{Crit}(f)$.

However, the symplectic form along the leaves will vary for different choices
of volume forms. The study of such Poisson structures has been of increased
interest lately (see e.g. [STV19, BO22]). Our Poisson structure of interest is
induced by a fibration with a _Lefschetz singulartiy_ :

(2) $\displaystyle f:{\mathbb{R}}^{4}\to{\mathbb{R}}^{2}\qquad\text{ with
}\quad
f=(f_{1},f_{2})=\left(x_{1}^{2}-x_{2}^{2}+x_{3}^{2}-x_{4}^{2},\,2(x_{1}x_{2}+x_{3}x_{4})\right)$

######  Definition 1.1.

Let $\pi=\pi_{f,\mu_{M},\mu_{N}}$ be the Poisson structure on
${\mathbb{R}}^{4}$ defined by (1), (2) and the standard volume forms in
${\mathbb{R}}^{4}$ and ${\mathbb{R}}^{2}$, respectively.

In this paper we take the first step in the computation of
$H^{\bullet}({\mathbb{R}}^{4},\pi)$ by computing the formal Poisson cohomology
of $\pi$ at the origin which we denote by
$H^{\bullet}_{F}({\mathbb{R}}^{4},\pi)$. We do so by means of its Poisson
homology, i.e. on a Poisson manifold $(M^{m},\pi)$ we have another
differential on the space of forms

$\displaystyle\delta_{\pi}:\Omega^{\bullet}(M)\to\Omega^{\bullet-1}(M)\qquad\delta_{\pi}:=\iota_{\pi}\circ\mathrm{d}-\mathrm{d}\circ\iota_{\pi}$

introduced by Kozsul [Kos85] and Brylinski [Bry88]. Here $\mathrm{d}$ denotes
the de-Rham differential and $\iota$ the contraction between mulitvector
fields and forms. The homology associated to
$(\Omega^{\bullet}(M),\delta_{\pi})$ is called Poisson homology and denoted by
$H_{\bullet}(M,\pi)$. Given a volume form $\mu\in\Omega^{m}(M)$ on $M$ the
relation between Poisson (co)homology can be encoded as follows. First note
that $\mu$ induces the isomorphism

(3)
$\displaystyle\star:=\mu^{\flat}:\mathfrak{X}^{\bullet}(M)\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}\Omega^{\bullet}(M),\qquad
X\mapsto\iota_{X}(\mu).$

We define a vector field $X_{\mu}\in\mathfrak{X}^{1}(M)$, the _modular vector
field_ of $(M,\pi)$ and $\mu$ (see [Wei97]), by

$\star X_{\mu}:=\mathrm{d}\star(\pi).$

The vector field $X_{\mu}$ is Poisson, i.e. $\mathrm{d}_{\pi}X_{\mu}=0$ and
the class $\textrm{mod}(M,\pi)=[X_{\mu}]\in H^{1}(M,\pi)$ is independent of
the chosen volume form $\mu$. Using the isomorphism $\star$ we obtain the
relation

(4)
$\delta_{\pi}=\star\circ(\mathrm{d}_{\pi}+X_{\mu}\wedge\cdot)\circ\star^{-1},$

between the Poisson differentials $\mathrm{d}_{\pi}$ and $\delta_{\pi}$ (see
[LPV13][Proposition 4.18]). We call $(M,\pi)$ _unimodular_ if
$\textrm{mod}(M,\pi)=0\in H^{1}(M,\pi)$. It is well-known that $(M,\pi)$ is
unimodular iff there exists a volume form $\mu$ with $X_{\mu}=0$. In
particular, for all unimodular Poisson structures we obtain an isomorphism

(5) $\displaystyle
H_{\bullet}(M,\pi)\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}H^{m-\bullet}(M,\pi).$

In particular, for $({\mathbb{R}}^{4},\pi)$ from Definition 1.1 we have

$\star(\pi)=\mathrm{d}f_{1}\wedge\mathrm{d}f_{2},$

and hence $({\mathbb{R}}^{4},\pi)$ is unimodular. Note that the isomorphism
induced by (3) descends via $j^{\infty}_{0}$ to an isomorphism between formal
vector fields and forms, which we denote by

$\displaystyle\mathfrak{X}^{\bullet}_{f}:={\mathcal{R}}\otimes\wedge^{\bullet}{\mathbb{R}}^{4}\qquad\text{
and
}\qquad\Omega^{\bullet}_{f}:={\mathcal{R}}\otimes\wedge^{\bullet}({\mathbb{R}}^{4})^{*}.$

Here ${\mathcal{R}}:={\mathbb{R}}[[x_{1},\dots,x_{4}]]$ denotes the ring of
formal power series. Therefore, (5) descends to the formal setting and we
obtain an isomorphism between formal Poisson (co)homology

$\displaystyle
H_{\bullet}^{F}({\mathbb{R}}^{4},\pi)\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}H^{4-\bullet}_{F}({\mathbb{R}}^{4},\pi),$

which allows us to compute formal Poissson cohomology by means of its
homological counterpart. Since from here on out we refer to Poisson
(co)homology only in the formal setting, we drop the sub- and superscript $F$,
respectively throughout the rest of the paper.

The main result is the following:

######  Theorem 1.2.

The formal Poisson homology groups $H_{\bullet}({\mathbb{R}}^{4},\pi)$ of
$({\mathbb{R}}^{4},\pi)$ are uniquely described in the various degrees as
follows:

  * •

in degree $0$, $H_{0}({\mathbb{R}}^{4},\pi)$ has unique representatives of the
form

$p+\sum_{i=1}^{4}a_{i}x_{i}$

where $p\in{\mathbb{R}}[[f_{1},f_{2}]]$ and
$a_{i}\in{\mathbb{R}}[[x_{2}^{2},x_{4}]]$;

  * •

the group $H_{1}({\mathbb{R}}^{4},\pi)$ has unique representatives of the form

$\sum_{j=1}^{2}p_{j}\zeta_{j}+q_{j}{\operatorname{d}}f_{j}+\sum_{i=1}^{4}{\operatorname{d}}(a_{i}x_{i})+b_{i}x_{i}{\operatorname{d}}f_{1}$

where $p_{j},q_{j}\in{\mathbb{R}}[[f_{1},f_{2}]]$ and
$a_{i},b_{i}\in{\mathbb{R}}[[x_{2}^{2},x_{4}]]$ and

(6)
$\displaystyle\zeta_{1}:=\frac{1}{2}(-x_{3}\mathrm{d}x_{1}+x_{4}\mathrm{d}x_{2}+x_{1}\mathrm{d}x_{3}-x_{2}\mathrm{d}x_{4}),\qquad\zeta_{2}:=\frac{1}{2}(-x_{4}\mathrm{d}x_{1}-x_{3}\mathrm{d}x_{2}+x_{2}\mathrm{d}x_{3}+x_{1}\mathrm{d}x_{4});$

  * •

in degree $2$, $H_{2}({\mathbb{R}}^{4},\pi)$ has unique representatives of the
form

$p\zeta_{1}\wedge\zeta_{2}+q{\operatorname{d}}f_{1}\wedge{\operatorname{d}}f_{2}+\sum_{i=1}^{2}p_{i}{\operatorname{d}}(f_{1}\zeta_{i})+q_{i}{\operatorname{d}}\zeta_{i}+\sum_{i=1}^{4}{\operatorname{d}}(a_{i}x_{i})\wedge{\operatorname{d}}f_{1}$

where $p,p_{i},q,q_{i}\in{\mathbb{R}}[[f_{1},f_{2}]]$ and
$a_{i}\in{\mathbb{R}}[[x_{2}^{2},x_{4}]]$;

  * •

representatives of $H_{3}({\mathbb{R}}^{4},\pi)$ are uniquely described by

$\displaystyle\sum_{i=1}^{2}p_{i}\zeta_{2}\wedge{\operatorname{d}}\zeta_{i}+q_{i}{\operatorname{d}}f_{1}\wedge{\operatorname{d}}\zeta_{i}$

where $p_{i},q_{i}\in{\mathbb{R}}[[f_{1},f_{2}]]$;

  * •

in degree $4$, elements in $H_{4}({\mathbb{R}}^{4},\pi)$ are of the form

$\displaystyle{\mathbb{R}}[[f_{1},f_{2}]]\mu.$

######  Remark 1.3.

The groups $H_{\bullet}({\mathbb{R}}^{4},\pi)$ from Theorem 1.2, or
equivalently $H^{\bullet}({\mathbb{R}}^{4},\pi)$, were erroneously computed in
[BV20]. We outline the problem of their computation in section 4.1.

Using the isomorphism (5) we describe the corresponding cohomology groups in
Section 2. In particular, we can use the result for the second formal
cohomology to classify Poisson structures arising from the fibration in (2)
via (1). We obtain the following result.

######  Corollary 1.4.

Any Poisson structure obtained from a different volume form is formally
equivalent to a Poisson structure obtained from

$\mu_{p}=(c+p)\mu$

for $p\in\langle f_{1},f_{2}\rangle_{{\mathbb{R}}[[f_{1},f_{2}]]}$ and
$c\in{\mathbb{R}}_{+}$ and any $p$ and $c$.

Note that the space of forms $\Omega^{\bullet}$ comes naturally equipped with
another differential, the de-Rham differential ${\operatorname{d}}$. One has
the relation

$\displaystyle\mathrm{d}\circ\delta_{\pi}+\delta_{\pi}\circ\mathrm{d}=0.$

Therefore we have a bidifferential complex
$(\Omega^{\bullet}_{f},\mathrm{d},\delta_{\pi})$ and an induced differential
complex in Poisson homology $(H_{\bullet}({\mathbb{R}}^{4},\pi),\mathrm{d})$.
It’s cohomology are described as follows.

######  Corollary 1.5.

For the complex $(H_{\bullet}({\mathbb{R}}^{4},\pi),\mathrm{d})$ we find

$\displaystyle
H_{DR}^{k}(H_{\bullet}({\mathbb{R}}^{4},\pi),\mathrm{d})=\begin{cases}{\mathbb{R}}&\text{
if }k=0\\\ 0&\text{ else.}\end{cases}$

### Strategy of the proof of Theorem 1.2

Roughly speaking the proof of Theorem 1.2 is as follows. We start by computing
the kernel of $\delta_{\pi}$. In turn this allows us to compute the dimension
of the Poisson homology groups. Together with the Poisson homology groups
these kernels fit into short exact sequences. This allows us to compute the
Hilbert-Poincare series associated to the formal Poisson homology groups, and
hence obtain their dimensions. The last step consists of finding an explicit
set of generators of the right dimension.

In principle all the arguments consist of explicit computations. However, a
recurring complication in these computations is that the following property
does not always hold. Given $\beta\in\Omega^{k}_{f}$, does

${\operatorname{d}}f_{1}\wedge{\operatorname{d}}f_{2}\wedge\beta=0\quad\text{
imply that
}\quad\beta=\sum_{i=1}^{2}{\operatorname{d}}f_{i}\wedge\alpha_{i}\quad\text{
for some }\quad\alpha_{i}\in\Omega^{k-1}_{f}?$

The failure of this property to hold is measured by the groups
$\mathcal{D}^{k}({\operatorname{d}}f_{1},{\operatorname{d}}f_{2})$, first
defined by [Rha54] and generalized by Saito [Sai76]. It turns out that for
$k=1$ this group vanishes. On the other hand
$\mathcal{D}^{2}({\operatorname{d}}f_{1},{\operatorname{d}}f_{2})$ is non-
zero, which plays an important role in the computations.

There appears to be a close relation between the generators of
$\mathcal{D}({\operatorname{d}}f_{1},{\operatorname{d}}f_{2})$ and those of
$H_{\bullet}({\mathbb{R}}^{4},\pi)$. For example,
${\operatorname{d}}\zeta_{i}$ appear both in Theorem 1.2, and as the
generators of
$\mathcal{D}^{2}({\operatorname{d}}f_{1},{\operatorname{d}}f_{2})$, see
Proposition 4.3. Similarly, the appearance of the classes represented by
coefficients $a_{i}$ and $b_{i}$ in ${\mathbb{R}}[[x_{2}^{2},x_{4}]]$ appears
to be related to the description of the division groups. We do not explore
this connection further but leave it as an option question:

######  Question 1.6.

Let $f_{1},\dots,f_{n-2}:{\mathbb{R}}^{n}\to{\mathbb{R}}$ be smooth functions.
What is the precise relation between the groups
$\mathcal{D}^{i}({\operatorname{d}}f_{1},\dots,{\operatorname{d}}f_{n-2})$ and
the formal Poisson (co)homology of the Jacobi-Poisson structure on
${\mathbb{R}}^{n}$ induced by the functions $f_{1},\dots,f_{n-2}$?

### Organization of the paper

Recall from (5) that the formal Poisson homology and cohomology groups are
isomorphic. In Section 2 we start by giving a cohomology description of the
main theorem, and a geometric interpretation of some of the generators. We
also partially describe the Gerstenhaber algebra structure of the groups
$H^{\bullet}({\mathbb{R}}^{4},\pi)$.

In Section 3 we discuss some preliminary notions from Poisson geometry
(Section 3.1), and algebra (Section 3.2). Most notably, the definition of
Jacobi-Poisson structures and an explicit description of $\delta_{\pi}$. Most
of our computations involve power series, and are done degree wise. To this
end we recall, in Section 3.2.1, the definition of Hilbert-Poincare series,
which we later use to compute the rank of the Poisson homology groups in each
homogeneous degree. We also recall some standard facts from algebra concerning
regular sequences in Section 3.2.2, and standard bases in Section 3.2.3.

In Section 4 we recall the notion of division groups, which measure the
failure for the division property to hold. The main result of this section is
Proposition 4.3 giving an explicit description of
$\mathcal{D}^{2}({\operatorname{d}}f_{1},{\operatorname{d}}f_{2})$, the second
division group associated to the coefficients of ${\operatorname{d}}f_{1}$ and
${\operatorname{d}}f_{2}$.

Using these results, Section 5 computes the kernel of the Poisson differential
$\delta_{\pi}$. The results are collected in Proposition 5.1. The proof of
this proposition, which is a long but more or less straightforward
computation, is the core of the paper. In Section 6 we use the description of
$\ker\delta_{\pi}$ to compute the Hilbert-Poincare series of the Poisson
homology groups, see Proposition 6.1.

The proof of Theorem 1.2, which combines the results from Section 5 and
Section 6, is given in Section 7. We check that the generators in Theorem 1.2
are indeed in the kernel. Then, using the Hilbert-Poincare series we see that
the set of generators have the right dimension. The remainder of the proof
consists of showing that none of the generators are in the image of
$\delta_{\pi}$.

Lastly, Section 8 and Section 9 respectively contain the proofs of Corollary
1.4 and Corollary 1.5.

### Acknowledgements

We would like to thank Ioan Mărcut for bringing the problem to our attention
and useful discussions. L. Toussaint is funded by the Dutch Research Council
(NWO) on the project “proper Fredholm homotopy theory” (OCENW.M20.195) of the
research programm Open Competition ENW M20-3. F. Zeiser would like to thank
the Max Planck Institute for Mathematics in Bonn for its hospitality and
financial support during the early stages of this project and the group of
Thomas Rot at Vrije Universiteit Amsterdam for its hospitality and support
during a research visit.

##  2\. An interpretation of the results

In this section we review theorem 1.2 in view of (5), stating the
corresponding result for formal Poisson cohomology and interpreting the
result. In the first section we discuss the foliation ${\mathcal{F}}$
associated with $\pi$ and its relation with $H^{0}({\mathbb{R}}^{4},\pi)$. The
second section is devoted to the Lie algebra $H^{1}({\mathbb{R}}^{4},\pi)$ and
in the third section we look at $H^{2}({\mathbb{R}}^{4},\pi)$ and the
interpretation of some its classes in terms of deformations of $\pi$. Finally,
in the last section we describe the higher Poisson cohomology groups and some
of the additional algebraic structure present for Poisson cohomology.

###  2.1. The foliation and the Casimir functions

For the geometric interpretation let’s take a closer look at the foliation
${\mathcal{F}}$ induced by $\pi$. From the introduction we know that
${\mathcal{F}}$ is closely related to the fibers of the fibration $f$ from
(2). In particular, for $(x,y)\neq 0\in{\mathbb{R}}^{2}$, the fiber
$f^{-1}(x,y)$ is connected and diffeomorphic to a cylinder.

$f^{-1}(x,y)\simeq S^{1}\times{\mathbb{R}}$

The preimage of the origin under $f$ consists of three leaves. The set
$f^{-1}(0)\setminus\\{0\\}$ has two connected components both of which are
diffeomorphic to cylinders. The origin $0\in{\mathbb{R}}^{4}$ is the sole
critical point of $f$ and is therefore a leaf of dimension $0$.

$f^{-1}(0)=S^{1}\times{\mathbb{R}}\cup\\{0\\}\cup S^{1}\times{\mathbb{R}}$

The leaf space ${\mathbb{R}}^{4}/{\mathcal{F}}$ is, as a topological space,
given by ${\mathbb{R}}^{2}$ where we have three distinct points instead of the
origin. However, continuous functions on ${\mathbb{R}}^{4}/{\mathcal{F}}$ can
not distinguish these three points. Viewing $f_{1}$ and $f_{2}$ as the
coordinate functions on the leaf space we obtain for formal functions on the
leaf space precisely what we expect, i.e./ power series in these coordinates.
That is, in degree $0$ we have

$H^{0}({\mathbb{R}}^{4},\pi)={\mathbb{R}}[[f_{1},f_{2}]].$

######  Remark 2.1.

In fact one can use the discussion above to show that we have an isomorphism

$C^{\infty}({\mathbb{R}}^{2})\to H^{0}({\mathbb{R}}^{4},\pi),\quad g\mapsto
g(f_{1},f_{2}).$

Here $H^{0}({\mathbb{R}}^{4},\pi)$ refers to the Poisson cohomology over
smooth multivector fields (see [MZ23]).

###  2.2. The Lie algebra $H^{1}({\mathbb{R}}^{4},\pi)$

To describe the Poisson cohomology group in degree one let $E_{i}$ and
$T_{i}$, $i=1,2$, be the real and imaginary part of the complex vector fields

$\displaystyle E=z\partial_{z}+w\partial_{w}\qquad\text{ and }\qquad
T:=z\partial_{w}-w\partial_{z};$

where $(z,w)=(x_{1}+ix_{2},x_{3}+ix_{4})$.

######  Corollary 2.2.

In degree one $H^{1}({\mathbb{R}}^{4},\pi)$ is the free
$H^{0}({\mathbb{R}}^{4},\pi)$-module generated by

$\displaystyle E_{1},E_{2},T_{1},T_{2}.$

This follows immediately from the relations

(7) $\displaystyle\star
E_{1}=\zeta_{i}\wedge{\operatorname{d}}\zeta_{i},\quad\star
E_{2}=-\zeta_{1}\wedge{\operatorname{d}}\zeta_{2}=\zeta_{2}\wedge{\operatorname{d}}\zeta_{1},\qquad$
(8) $\displaystyle\star
T_{1}=-\frac{1}{4}{\operatorname{d}}f_{i}\wedge{\operatorname{d}}\zeta_{i},\quad\star
T_{2}=-\frac{1}{4}{\operatorname{d}}f_{2}\wedge{\operatorname{d}}\zeta_{1}=\frac{1}{4}{\operatorname{d}}f_{1}\wedge{\operatorname{d}}\zeta_{2}$

The Lie bracket on $H^{1}({\mathbb{R}}^{4},\pi)$ is induced by the Lie bracket
for vector fields. In order to describe the Lie algebra structure, we note
that

$[E_{1},E_{2}]=[T_{1},T_{2}]=[E_{i},T_{j}]=0.$

Hence the brackets in cohomology are fully described by the relations

(9)
$\displaystyle{\mathcal{L}}_{E_{1}}(f_{i})=f_{i},\quad{\mathcal{L}}_{E_{2}}f_{1}=f_{2},\quad{\mathcal{L}}_{E_{2}}f_{2}=-f_{1},\quad{\mathcal{L}}_{T_{i}}f_{j}=0.$

Geometrically, (9) means that the Lie algebra

$\mathfrak{g}_{N}:=\\{p_{i}E_{i}\ |\ p_{i}\in{\mathbb{R}}[[f_{1},f_{2}]]\\}$

describes Poisson vector fields who project non-trivially to the leaf space,
i.e. $E_{1}$ is mapped to the Euler vector field and $E_{2}$ to the rotational
vector field. The vector fields $T_{i}$ preserve the leaves and form a
commutative Lie subalgebra of Poisson vector fields tangent to ${\mathcal{F}}$
which we denote by

$\mathfrak{g}_{T}:=\\{p_{i}T_{i}\ |\ p_{i}\in{\mathbb{R}}[[f_{1},f_{2}]]\\}$

As a Lie algebra, the first formal Poisson cohomology is given by

$H^{1}({\mathbb{R}}^{4},\pi)\simeq\mathfrak{g}_{T}\rtimes\mathfrak{g}_{N}.$

###  2.3. $H^{2}({\mathbb{R}}^{4},\pi)$: Infinitesimal deformations

Combining Theorem 1.2 with the isomorphism from Equation (5), we obtain the
following description of the formal Poisson cohomology in degree $2$. Here
(and in the rest of this section) we use the following notation:

$W_{i}:=\star^{-1}({\operatorname{d}}\zeta_{i}).$

######  Corollary 2.3.

$H^{2}({\mathbb{R}}^{4},\pi)$ has unique representatives of the form

$pE_{1}\wedge E_{2}+q\pi+\sum_{i=1}^{2}p_{i}E_{i}\wedge
T_{1}+q_{i}W_{i}+\star^{-1}\left(\sum_{i=1}^{4}{\operatorname{d}}(a_{i}x_{i})\wedge{\operatorname{d}}f_{1}\right)$

where $p,p_{i},q,q_{i}\in{\mathbb{R}}[[f_{1},f_{2}]]$ and
$a_{i}\in{\mathbb{R}}[[x_{2}^{2},x_{4}]]$.

This follows from the identities

(10) $\displaystyle\star^{-1}(\zeta_{1}\wedge\zeta_{2})$
$\displaystyle\,=-E_{1}\wedge E_{2}$
$\displaystyle\star^{-1}({\operatorname{d}}f_{1}\wedge\zeta_{1})=-4E_{i}\wedge
T_{i}+f_{i}W_{i}\ $ $\displaystyle\text{ and }\
\star^{-1}({\operatorname{d}}f_{1}\wedge\zeta_{2})=4E_{1}\wedge
T_{2}+f_{1}W_{2}=-4E_{2}\wedge T_{1}+f_{2}W_{1}$

To study this result in more detail we use the filtration on
$\mathfrak{X}^{\bullet}_{f}$ induced by $f$. We set

(11)
$\mathfrak{X}^{\bullet}_{f,1}:=\ker\iota_{{\operatorname{d}}f_{1}\wedge{\operatorname{d}}f_{2}}\cap\mathfrak{X}^{\bullet}_{f}\quad\text{
and
}\quad\mathfrak{X}^{\bullet}_{f,2}:=\ker\iota_{{\operatorname{d}}f_{1}}\cap\ker\iota_{{\operatorname{d}}f_{2}}\cap\mathfrak{X}^{\bullet}_{f}$

Note that, since $f_{1}$ and $f_{2}$ are Casimir functions,
${\operatorname{d}}_{\pi}$ preserves $\mathfrak{X}^{\bullet}_{f,1}$ and
$\mathfrak{X}^{\bullet}_{f,2}$, respectively. Infinitesimal deformations which
preserves ${\mathcal{F}}$ are governed by cohomology classes represented by
elements in $\mathfrak{X}^{2}_{f,2}$ up to coboundaries. It follows from the
proof of the degree $2$ part of Theorem 1.2, together with the identity

(12)
$\displaystyle\iota_{\alpha}(\star^{-1}\beta)=(-1)^{k(4-k)}\star^{-1}(\beta\wedge\alpha)$

for $\alpha\in\Omega^{k},\beta\in\Omega^{l}$ and $0\leq k\leq n-l$, that the
only such cohomology classes are of the form $q\pi$ for
$q\in{\mathbb{R}}[[f_{1},f_{2}]]$. All such classes can be realized by a
formal Poisson deformation of $\pi$ of the form

$\pi_{t}:=(1+tq)\pi$

More generally, from (1) we know that any deformation of the volume form $\mu$
induces a deformation of Poisson structures preserving the foliation. By
Corollary 1.4 we conclude that any Poisson structure obtained in this way is
formally equivalent to one as described above.

On the other end of the spectrum of deformation we have those which induce a
symplectic structure almost everywhere. There are two different classes of
such deformations. First, the deformations represented by elements in
$\mathfrak{X}^{2}_{f}/\mathfrak{X}^{2}_{f,1}$. By Corollary 2.3 such
deformations are infinitesimally described by the classes $pE_{1}\wedge
E_{2}$. Indeed, any bivector

$\pi_{t}:=\pi+tpE_{1}\wedge E_{2}$

is a Poisson bivector whose rank is $4$ away from the origin. This can be
easily seen from the identities

$\pi=-8\,T_{1}\wedge T_{2}\quad\text{ and }\quad 16E_{1}\wedge E_{2}\wedge
T_{1}\wedge
T_{2}=(f_{1}^{2}+f_{2}^{2})\partial_{1}\wedge\partial_{2}\wedge\partial_{3}\wedge\partial_{4},$

the fact that $E_{i}$ and $T_{j}$ commute and since
$T_{j}\in\mathfrak{X}^{1}_{f,2}$ by (9). The other deformations we want to
consider here are of the form

$\pi_{1,t}=\pi+tW_{1}\quad\text{ and }\quad\pi_{2,t}=\pi+tW_{2}.$

These bivectors are Poisson and we obtain a symplectic structure on
${\mathbb{R}}^{4}$ for $t>0$ since

$\pi\wedge W_{i}=0\quad\text{ and }\quad W_{i}\wedge
W_{i}=2\partial_{1}\wedge\partial_{2}\wedge\partial_{3}\wedge\partial_{4}.$

###  2.4. Higher degrees and the algebraic structure

Next we consider deformations of the form

(13) $\pi_{i,t}=\pi+tpW_{i}.$

To this end we first need to compute $H^{3}({\mathbb{R}}^{4},\pi)$ and
$H^{4}({\mathbb{R}}^{4},\pi)$.

From Theorem 1.2 we obtain the following:

######  Corollary 2.4.

The formal Poisson cohomology of $\pi$ satisfy:

  * •

the group $H^{3}({\mathbb{R}}^{4},\pi)$ has unique representatives of the form

$\sum_{j=1}^{2}p_{j}E_{j}\wedge W_{1}+q_{j}T_{j}\wedge
W_{1}+\sum_{i=1}^{4}b_{i}x_{i}T_{1}\wedge
W_{1}+\star^{-1}{\operatorname{d}}(a_{i}x_{i})$

where $p_{j},q_{j}\in{\mathbb{R}}[[f_{1},f_{2}]]$ and
$a_{i},b_{i}\in{\mathbb{R}}[[x_{2}^{2},x_{4}]]$;

  * •

the group $H^{4}({\mathbb{R}}^{4},\pi)$ has unique representatives of the form

$\left(p+\sum_{i=1}^{4}a_{i}x_{i}\right)\partial_{1}\wedge\partial_{2}\wedge\partial_{3}\wedge\partial_{4}$

where $p\in{\mathbb{R}}[[f_{1},f_{2}]]$,
$a_{i}\in{\mathbb{R}}[[x_{2}^{2},x_{4}]]$ and where we write
$\partial_{j}=\partial_{x_{j}}$.

The statement follows from the following identities

$\displaystyle\star^{-1}\zeta_{1}=E_{1}\wedge W_{1}=-E_{2}\wedge
W_{2}\quad\text{ and }\quad\star^{-1}\zeta_{2}=E_{1}\wedge W_{2}=E_{2}\wedge
W_{1}$ $\displaystyle\star^{-1}{\operatorname{d}}f_{1}=-4T_{1}\wedge
W_{1}=4T_{2}\wedge W_{2}\quad\text{ and
}\quad\star^{-1}{\operatorname{d}}f_{2}=-4T_{1}\wedge W_{2}=-4T_{2}\wedge
W_{1}$

Going back to the deformations from (13) we obtain, for $i=1$ that:

$\displaystyle[\pi_{1,t},\pi_{1,t}]=8t^{2}p(\partial_{x}pT_{1}\wedge
W_{1}+\partial_{y}pT_{2}\wedge W_{1})$

In order to make the bivector Poisson, we would need to add a second order
element in $t$, i.e. consider a deformation of the form

$\pi_{1,t}=\pi+tpW_{1}+t^{2}W$

for some $W\in\mathfrak{X}^{2}_{f}$ such that

$[\pi,W]=4p(\partial_{x}pT_{1}\wedge W_{1}+\partial_{y}pT_{2}\wedge W_{1}).$

However, the right side represents a non-trivial class in
$H^{3}({\mathbb{R}}^{4},\pi)$ iff $p\in{\mathbb{R}}[[f_{1},f_{2}]]$ is not
constant and hence it is not possible to find such a $W$ for non-constant $p$.
The same argument shows that $\pi_{2,t}$ does not define a deformation of
Poisson structures.

#### The algebraic structure in cohomology

The cohomology groups $H^{\bullet}({\mathbb{R}}^{4},\pi)$ carry a rich
algebraic structure, as both the wedge product and the Schouten bracket

$\displaystyle\wedge:\mathfrak{X}^{k}\times\mathfrak{X}^{l}\to\mathfrak{X}^{k+l}\quad\text{
and
}\quad[\cdot,\cdot]:\mathfrak{X}^{k}\times\mathfrak{X}^{l}\to\mathfrak{X}^{k+l-1}$

descend to cohomology, inducing a Gerstenhaber algebra structure for Poisson
cohomology:

$\left(H^{\bullet}({\mathbb{R}}^{4},\pi),\wedge,[\cdot,\cdot]\right).$

We distinguish two classes of representatives, those with coefficients in
${\mathbb{R}}[[f_{1},f_{2}]]$ and those with coefficients in
${\mathbb{R}}[[x_{2}^{2},x_{4}]]$. For the former, we note that the wedge
product and the bracket can be easily deduced from the various relations
described above. Describing the precise algebraic structure of representatives
with coefficients in ${\mathbb{R}}[[x_{2}^{2},x_{4}]]$ is harder and we do not
attempt to describe them here. We only want to point out that the
corresponding modules, as modules over $H^{0}({\mathbb{R}}^{4},\pi)$ are not
free.

Explicitly, we have:

######  Proposition 2.5.

For cohomology classes represented by some $a_{i},b_{i}\in{\mathbb{R}}$ we
have:

$f_{1}a_{i}x_{i}=-2(x_{2}^{2}+x_{4}^{2})a_{i}x_{i}\qquad\text{ and }\qquad
f_{1}b_{i}x_{i}=-2(x_{2}^{2}+x_{4}^{2})b_{i}x_{i}$

in terms of the representatives in Theorem 1.2. Moreover, we have the
relations

$a(f_{1}x_{1}+f_{2}x_{2})=a(f_{2}x_{1}-x_{2}f_{1})=a(f_{1}x_{3}+f_{2}x_{4})=a(f_{2}x_{3}-f_{1}x_{4})=0$

in cohomology for $a\in{\mathbb{R}}$ and similarly for the classes represented
by $b_{i}$.

###### Proof.

Degree 4: We note that

$\displaystyle
x_{1}f_{1}\mu=-2(x_{2}^{2}+x_{4}^{2})x_{1}\mu+\mathrm{d}f_{1}\wedge\mathrm{d}f_{2}\wedge\nu_{1}$

where

$\displaystyle\nu_{1}:=\frac{1}{4}\big{(}x_{3}\mathrm{d}x_{2}\wedge\mathrm{d}x_{3}-x_{4}\mathrm{d}x_{1}\wedge\mathrm{d}x_{3}+x_{1}\mathrm{d}x_{3}\wedge\mathrm{d}x_{4}\big{)}$

which is closed and hence there exists a primitive
$\alpha_{1}\in\Omega^{1}({\mathcal{R}})$. Similarly we obtain

$\displaystyle
x_{1}f_{2}\mu=-2(x_{2}^{2}+x_{4}^{2})x_{2}\mu+\mathrm{d}f_{1}\wedge\mathrm{d}f_{2}\wedge\nu_{2}$

for the closed two form

$\displaystyle\nu_{2}:=\frac{1}{2}\big{(}-2x_{4}\mathrm{d}x_{1}\wedge\mathrm{d}x_{4}-x_{4}\mathrm{d}x_{1}\wedge\mathrm{d}x_{3}+x_{2}\mathrm{d}x_{3}\wedge\mathrm{d}x_{4}\big{)}.$

Moreover, we have

$\displaystyle(x_{1}f_{1}+x_{2}f_{2})\mu=\mathrm{d}f_{1}\wedge\mathrm{d}f_{2}\wedge\nu$

for the closed two-form

$\displaystyle\nu:=\frac{1}{4}\big{(}x_{4}\mathrm{d}x_{2}\wedge\mathrm{d}x_{4}-x_{3}\mathrm{d}x_{1}\wedge\mathrm{d}x_{4}+x_{1}\mathrm{d}x_{3}\wedge\mathrm{d}x_{4}\big{)}$

The other cases follow along the same lines.

Degree 2: We make use of the primitives in degree $4$. Consider the
$2$-cocylce given by

$\displaystyle\star
W={\operatorname{d}}(x_{1}f_{1}+2(x_{2}^{2}+x_{4}^{2})x_{1})\wedge{\operatorname{d}}f_{2}.$

Setting $\star X:=-{\operatorname{d}}f_{1}\wedge\nu_{1}$ we obtain that

$W={\operatorname{d}}_{\pi}X.$

Next consider the $2$-cocylce given by

$\displaystyle\star
W={\operatorname{d}}(x_{1}f_{1}+x_{2}f_{2})\mathrm{d}f_{2}.$

If we set $\star X:=-{\operatorname{d}}f_{1}\wedge\nu$ then we obtain that

$W={\operatorname{d}}_{\pi}X.$

The other cases follow similarly using the elements from the proof in degree
$4$.

Degree 3: Consider the $3$-cocylce given by

$\displaystyle\star
T=(x_{1}f_{1}+2(x_{2}^{2}+x_{4}^{2})x_{1})\mathrm{d}f_{1}.$

Let $\alpha_{1}$ be a primitiv of $\nu_{1}$. Then we obtain for $\star
W:=-{\operatorname{d}}f_{1}\wedge\alpha_{1}$ that

$T={\operatorname{d}}_{\pi}W.$

Next consider the $3$-cocylce given by

$\displaystyle\star T=(x_{1}f_{1}+x_{2}f_{2})\mathrm{d}f_{1}.$

Let $\alpha$ be a primitiv of a closed $2$-form $\nu$ and define $\star
W:=-{\operatorname{d}}f_{1}\wedge\alpha$. Then we obtain that

$T={\operatorname{d}}_{\pi}W.$

Similarly we can treat the cocycle

$\displaystyle\star
T={\operatorname{d}}(x_{1}f_{1}+2(x_{2}^{2}+x_{4}^{2})x_{1}).$

Defining $\star W:=-\nu_{1}$ we obtain

$T={\operatorname{d}}_{\pi}W.$

The other cases follow similarly using the primitives from the proof in degree
$4$. ∎

##  3\. Preliminaries

In this section we recall some facts from Poisson geometry and (homological)
algebra. The first section is dedicated to Poisson geometry. In particular, we
recall the definition of Jacobi-Poisson structures and describe the Poisson
structure we are studying. In the second part we recall several notions from
algebra: Hilbert-Poincare series, regular sequences and standard bases.

###  3.1. Jacobi Poisson structures

We recall the notion of Jacobi-Poisson structures as introduce in [Dam89]
attributed to Flaschka and Ratiu. See also [GMP93]. In particular, we give an
explicit formula for $\pi$ from Definition 1.1 and we compute the Poisson
differential $\mathrm{d}_{\pi}$ in the various degrees.

Given a vector space ${\mathbb{R}}^{n}$ with standard volume form $\mu$ and
$n-2$ functions $f_{1},\dots f_{n-2}\in C^{\infty}({\mathbb{R}}^{n})$ one can
define a Poisson structure $\pi$ on ${\mathbb{R}}^{n}$ by the following
relation for the associated Poisson bracket:

$\\{g,h\\}\mu:=\mathrm{d}g\wedge\mathrm{d}h\wedge\mathrm{d}f_{1}\wedge\dots\wedge\mathrm{d}f_{n-2}\qquad\text{
for }g,h\in C^{\infty}({\mathbb{R}}^{n}).$

Poisson structures of this form are called _Jacobi-Poisson_ structures. Note
that the leaves of such Poisson structures have dimension at most $2$. For a
generalization with leaves of higher dimension see [DP12]. For some properties
of Jacobi-Poisson structures and a generalization to manifolds we refer to
[Prz01] and [GSV14]. One property is the following:

######  Lemma 3.1.

Any Jacobi-Poisson structure $\pi\in\mathfrak{X}^{2}({\mathbb{R}}^{n})$ is
unimodular.

###### Proof.

We observe that using (3) we have the identity

$\displaystyle\star(\pi)={\operatorname{d}}f_{1}\wedge\dots\wedge{\operatorname{d}}f_{n-2}.$

Hence using the definition of the modular vector field the result follows. ∎

As a consequence of the Lemma together with (4) we obtain an isomorphism of
complexes:

$\displaystyle\star:(\mathfrak{X}^{\bullet}({\mathbb{R}}^{n}),\mathrm{d}_{\pi})\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}(\Omega^{n-\bullet}({\mathbb{R}}^{n}),\delta_{\pi}).$

#### Poisson structure associated to a Lefschetz fibration

Consider $f:{\mathbb{R}}^{4}\to{\mathbb{R}}^{2}$ as in (2) together with the
standard volume forms $\mu_{4}$ and $\mu_{2}$ on ${\mathbb{R}}^{4}$ and
${\mathbb{R}}^{2}$ respectively, given by

$\displaystyle\mu_{4}=\mathrm{d}x_{1}\wedge\mathrm{d}x_{2}\wedge\mathrm{d}x_{3}\wedge\mathrm{d}x_{4}\qquad\text{
and }\qquad\mu_{2}=\mathrm{d}y_{1}\wedge\mathrm{d}y_{2}.$

We often write $\mu$ instead of $\mu_{4}$. Then $\pi$ from Definition 1.1 is a
Jacobi-Poisson structure and satisfies

$\star(\pi)=\mathrm{d}f_{1}\wedge\mathrm{d}f_{2}.$

Explicitly, the Poisson bivector $\pi$ is given by

$\displaystyle\frac{1}{4}\pi=$ $\displaystyle\ \ \ \
(x_{1}^{2}+x_{2}^{2})\partial_{3}\wedge\partial_{4}-(x_{1}x_{4}-x_{2}x_{3})(\partial_{1}\wedge\partial_{3}+\partial_{2}\wedge\partial_{4})$
$\displaystyle+(x_{3}^{2}+x_{4}^{2})\partial_{1}\wedge\partial_{2}+(x_{1}x_{3}+x_{2}x_{4})(\partial_{2}\wedge\partial_{3}-\partial_{1}\wedge\partial_{4})$

As an immediate consequence we can describe the Poisson differential
$\delta_{\pi}$ on differential forms.

######  Proposition 3.2.

The Poisson differential $\delta_{\pi}$ is given as follows.

  * •

In degree 1 we get for $\alpha\in\Omega^{1}({\mathbb{R}}^{4})$ that

$\displaystyle\delta_{\pi}(\alpha)=$ $\displaystyle\
\iota_{\pi}(\mathrm{d}\alpha)$

  * •

For a $2$-form $\beta\in\Omega^{2}({\mathbb{R}}^{4})$ we obtain the formula

$\displaystyle\delta_{\pi}(\beta)=$
$\displaystyle\iota_{\star^{-1}\mathrm{d}\beta}(\mathrm{d}f_{1}\wedge\mathrm{d}f_{2})-\mathrm{d}\iota_{\pi}(\beta)$

  * •

In degree $3$, the differential of $\gamma\in\Omega^{3}({\mathbb{R}}^{4})$ is
given by

$\displaystyle\delta_{\pi}\gamma=\iota_{\pi}({\operatorname{d}}\gamma)-{\operatorname{d}}\iota_{\star^{-1}\gamma}({\operatorname{d}}f_{1}\wedge{\operatorname{d}}f_{2})$

  * •

For the top degree we obtain for any $g\in C^{\infty}({\mathbb{R}}^{4})$ that

$\displaystyle\delta_{\pi}(g\mu)=\mathrm{d}g\wedge\mathrm{d}f_{1}\wedge\mathrm{d}f_{2}$

The proof follows from the definition of $\delta_{\pi}$ and the identity

$\iota_{V}(\alpha)=(-1)^{k(n-l)}\iota_{\star^{-1}\alpha}(\star V),$

for $V\in\mathfrak{X}^{k}({\mathbb{R}}^{n})$ and
$\alpha\in\Omega^{l}({\mathbb{R}}^{m})$ with $k\leq l\leq m$.

######  Remark 3.3.

One can generalize the formulas for $\delta_{\pi}$ in Proposition 3.2 to
describe the differentials associated to any Jacobi-Poisson structure.

###  3.2. (Homological) Algebra

This section intends to provide the algebraic background needed for the paper.
We first recall the definition of the Hilbert-Poincare series together with an
example which we will use later on. After a brief recap of regular sequences
in the second section, we recall the notion of standard bases in the third
part. As an application, we provide a couple of examples which will be used in
later computations.

####  3.2.1. Hilbert-Poincare series

A reference for the background material in this sequence is [AM69].

Let ${\mathcal{R}}$ be a commutative ring and $C$ a class of
${\mathcal{R}}$-modules. A function $\lambda:C\to{\mathbb{Z}}$ is called
_additive_ if for every short exact sequence of ${\mathcal{R}}$-modules in
$C$:

$0\to M^{\prime}\to M\to M^{\prime\prime}\to
0\qquad\Rightarrow\qquad\lambda(M)=\lambda(M^{\prime})+\lambda(M^{\prime\prime}).$

######  Example 3.4.

For ${\mathcal{R}}={\mathbb{R}}$ and $C$ the class of all finite dimensional
${\mathbb{R}}$-vector spaces, $\lambda=\dim$ is a additive function.

######  Proposition 3.5.

Assume we have an exact sequence of ${\mathcal{R}}$-modules in $C$

$0\to M_{0}\to M_{1}\to\dots\to M_{n}\to 0$

such that all kernels of the homomorphisms are in $C$, then for any additive
function $\lambda$ on $C$:

$\sum_{i=0}^{n}(-1)^{i}\lambda(M_{i})=0.$

Let ${\mathcal{R}}=\oplus_{n=0}^{\infty}{\mathcal{R}}_{n}$ be a Noetherian
graded ring. Note that ${\mathcal{R}}_{0}$ is Noetherian and let
$y_{1},\dots,y_{s}$ be the generators of ${\mathcal{R}}$ as an
${\mathcal{R}}_{0}$-algebra, of degrees $k_{1},\dots,k_{r}$, respectively. Let
${\mathcal{M}}=\oplus_{n=0}^{\infty}{\mathcal{M}}_{n}$ be a finitely-
generated, graded ${\mathcal{R}}$-module. Note in particular that this implies
that all ${\mathcal{M}}_{i}$ are finitely generated
${\mathcal{R}}_{0}$-modules. Moreover, let $\lambda$ be an additive function
on the class of all finitely-generated ${\mathcal{R}}_{0}$-modules. The
_Hilbert-Poincare series_ of $M$ with respect to $\lambda$ is the power series

$HP({\mathcal{M}},t):=\sum_{i=0}^{\infty}\lambda({\mathcal{M}}_{i})t^{i}.$

In general the Hilbert-Poincare series can be described as follows:

######  Theorem 3.6 ([AM69, Thm 11.1]).

$HP({\mathcal{M}},t)$ is a rational function in $t$ of the form

$\frac{p(t)}{\Pi_{j=1}^{r}(1-t^{k_{j}})}\qquad\text{ for }\
p(t)\in{\mathbb{Z}}[t].$

For any graded module ${\mathcal{M}}$ we denote by ${\mathcal{M}}(-a)$ the
module with the degree shifted by $a\in\mathbb{N}$, i.e.

${\mathcal{M}}(-a)_{d}=\begin{cases}{\mathcal{M}}_{d-a}&\text{ if }\ 0\leq
d-a\\\ 0&\text{ else.}\end{cases}$

Hence by definition of the Hilbert-Poincare series we obtain for a degree
shift the relation

$HP({\mathcal{M}}(-a),t)=t^{-a}HP({\mathcal{M}},t)$

Another consequence for the Hilbert-Poincare series using Proposition 3.5 is
the following.

######  Corollary 3.7.

Let ${\mathcal{R}}^{1},\dots,{\mathcal{R}}^{n}$ be Noetherian graded rings
with ${\mathcal{R}}^{1}_{0}=\dots={\mathcal{R}}^{n}_{0}={\mathcal{R}}_{0}$
which are all finitely generated as an ${\mathcal{R}}_{0}$-algebra and let
${\mathcal{M}}^{1},\dots,{\mathcal{M}}^{n}$ be finitely-generated, graded
${\mathcal{R}}^{1},\dots,{\mathcal{R}}^{n}$-modules respectively, fitting into
an exact sequence

$0\to{\mathcal{M}}^{1}\to{\mathcal{M}}^{2}\to\dots\to{\mathcal{M}}^{n}\to 0$

such that all the homomorphisms preserve the grading and all kernels of the
homomorphisms in the various degrees are in $C$. For $\lambda=\mathrm{dim}$ as
in Example 3.4 we obtain using Proposition 3.5 that

$\sum_{i=0}^{n}(-1)^{i}HP({\mathcal{M}}^{i},t)=0.$

An example which we will also use later on is that of formal forms on
${\mathbb{R}}^{n}$:

######  Example 3.8.

Denote by ${\mathcal{R}}:={\mathbb{R}}[[x_{1},\dots,x_{n}]]$ the ring of
formal power series in $n$-variables and by $\Omega^{\bullet}({\mathcal{R}})$
the DGA of differental forms with coefficient in ${\mathcal{R}}$, i.e.

$\displaystyle\Omega^{\bullet}({\mathcal{R}})=\wedge^{\bullet}({\mathbb{R}}^{n})^{*}\otimes{\mathcal{R}}$

The degree of $x_{i}$ is $1$ and the degree of $\mathrm{d}x_{i}$ is $-1$ for
all $1\leq i\leq n$. Then we obtain

$\displaystyle
HP_{\Omega^{m}({\mathcal{R}})}(t)=\binom{n}{m}\frac{t^{m}}{(1-t)^{n}}$

####  3.2.2. Regular sequences

Here we follow [Pel09]. Let ${\mathcal{R}}$ be a commutative ring and
${\mathcal{M}}$ be a finitely generated ${\mathcal{R}}$-module. We recall that
we call an element $r\in{\mathcal{R}}$ a _non-zero divisor_ on ${\mathcal{M}}$
if

$\forall m\in{\mathcal{M}}:\ rm=0\quad\Rightarrow m=0,$

and a _${\mathcal{M}}$ -regular sequence_ is a sequence
$r_{1},\dots,r_{q}\in{\mathcal{R}}$ such that $r_{i}$ is a non-zero-divisor in
the quotient

${\mathcal{M}}/\langle r_{1},\dots,r_{i-1}\rangle\cdot{\mathcal{M}},$

where $\langle r_{1},\dots,r_{i-1}\rangle\subset{\mathcal{R}}$ denotes the
ideal generated by $r_{j}$ for $1\leq j\leq i-1$. If
${\mathcal{M}}={\mathcal{R}}$ then we simply call it a regular sequence.

######  Example 3.9.

The polynomials $x_{1}x_{3}+x_{2}x_{4}$ and $x_{1}x_{4}-x_{2}x_{3}$ form a
regular sequence in ${\mathbb{R}}[[x_{1},x_{2},x_{3},x_{4}]]$.

######  Lemma 3.10 (see [Wei94]).

Let $r_{1},\dots,r_{n}$ be a regular sequence in ${\mathcal{R}}$, then the
Koszul complex

$0\to\wedge^{0}({\mathcal{R}}^{n})\xrightarrow{\wedge\alpha}\dots\xrightarrow{\wedge\alpha}\wedge^{n}({\mathcal{R}}^{n})\rightarrow{\mathcal{R}}/\langle
r_{1},\dots,r_{n}\rangle{\mathcal{R}}\to 0$

is exact, where $\alpha=\sum r_{i}e_{i}$ and $(e_{1},\dots,e_{n})$ is a free
basis of ${\mathcal{R}}^{n}$.

######  Corollary 3.11.

The operations $\wedge{\operatorname{d}}f_{1}$ and
$\wedge{\operatorname{d}}f_{2}$ are exact in the sense of the above lemma.

####  3.2.3. Standard bases

The notion of standard bases was introduced by Hironka in [Hir64] and
Buchberger [Buc65]. For more details see e.g [Eis95][chapter 15] or
[GP08][chapter 1]. We only give a brief summary of the most relevant facts for
us. We will use standard bases to compute ideal membership and intersection of
ideals in the ring of power series in an easy way.

######  Remark 3.12.

All our computations are done for the various degrees of homogeneous
polynomials. Hence it would be enough to use Gröbner basis. However, due to
our general setting we decide to recap the theory for power series.

Throughout this subsection let
${\mathcal{R}}:={\mathbb{R}}[[x_{1},\dots,x_{n}]]$. On ${\mathcal{R}}$ we
consider the local ordering obtained from the following monomial ordering (see
[GP08][section 1.2]):

$x^{\alpha}>x^{\beta}\quad\>\Leftrightarrow\quad|\alpha|<|\beta|\ \text{ or }\
|\alpha|=|\beta|\text{ and }\exists 1\leq i\leq n:\
\alpha_{1}=\beta_{1},\dots,\alpha_{i-1}=\beta_{i-1},\alpha_{i}>\beta_{i}.$

For $f\in{\mathcal{R}}$ written as

$f=\sum_{\alpha\in\mathbb{N}^{n}}a_{\alpha}x^{\alpha}$

and any subset $G\subset{\mathcal{R}}$ we define

$LM(f):=\max\\{x^{\alpha}|a_{\alpha}\neq 0\\},\quad
LC(f)=\\{a_{\alpha}|LM(f)=x^{\alpha}\\}\quad\text{ and }\quad L(G):=\langle
LM(g)|g\in G\setminus\\{0\\}\rangle.$

A standard basis is defined as follows.

######  Definition 3.13.

Let ${\mathcal{I}}\subset{\mathcal{R}}$ be an ideal. A standard basis of
${\mathcal{I}}$ is a finite set $G\subset{\mathcal{I}}$ with

$G\subset{\mathcal{I}}\qquad\text{ and }\qquad L({\mathcal{I}})=L(G).$

Moreover we need the following definitions.

######  Definition 3.14.

Let $G\subset{\mathcal{R}}$.

  1. (1)

$f\subset{\mathcal{R}}$ is called reduced with respect to $G$ if no monomial
of $f$ is contained in $LG(G)$.

  2. (2)

$G$ is called reduced if $0\neq G$, for all $f\neq g\in G$ then $LM(f)\nmid
LM(g)$ and for all $g\in G$, $LC(g)=1$ and $g-LM(g)$ is reduced in $G$.

Now we have the following proposition.

######  Proposition 3.15.

Let $f,f_{1},\dots,f_{m}\in{\mathcal{R}}$ then there exist
$q_{1},\dots,q_{m},r\in{\mathcal{R}}$ such that

$f=\sum_{i=1}^{m}q_{i}f_{i}+r$

and $r$ is reduced with respect to $\\{f_{1},\dots,f_{m}\\}$ and for all
$i=1,\dots m$ we have $LM(q_{i}f_{i})\leq LM(f)$.

Moreover, if $\\{f_{1},\dots,f_{m}\\}$ is reduced, then $r$ is unique.

###### Proof.

The existence of $r$ follows from [GP08][Theorem 6.4.1]. Uniqueness follows by
comparing the monomials of two different $r$’s. ∎

For $f_{1},\dots f_{m}\in{\mathcal{R}}$ as above we call $r$ the normal form
of $f\in{\mathcal{R}}$, i.e.

$NF(f|\\{f_{1},\dots f_{m}\\}):=r.$

######  Corollary 3.16.

Let ${\mathcal{I}}\subset{\mathcal{R}}$ be an ideal, $G\subset{\mathcal{I}}$ a
reduced standard basis of ${\mathcal{I}}$. Then for any $f\in{\mathcal{R}}$:

$f\in{\mathcal{I}}\qquad\Leftrightarrow\qquad NF(f,G)=0.$

###### Proof.

For a proof see [GP08][Lemma 1.6.7]. ∎

The following applications will be useful later. Let
${\mathcal{R}}:={\mathbb{R}}[[x_{1},x_{2},x_{3},x_{4}]]$ and let
${\mathcal{J}}\subset{\mathcal{R}}$ be the ideal defined by

(14)
$\displaystyle{\mathcal{J}}={\mathcal{J}}({\operatorname{d}}f_{1},{\operatorname{d}}f_{2})=\langle
G:=\\{x_{1}^{2}+x_{2}^{2},x_{3}^{2}+x_{4}^{2},x_{1}x_{3}+x_{2}x_{4},x_{1}x_{4}-x_{2}x_{3}\\}\rangle_{\mathcal{R}}.$

We have the following two results.

######  Lemma 3.17.

Consider the module ${\mathcal{M}}$ over ${\mathbb{R}}[[x_{2}^{2},x_{4}]]$
defined by

$\displaystyle{\mathcal{M}}:=\langle
x_{1},x_{2},x_{3},x_{4}\rangle_{{\mathbb{R}}[[x_{2}^{2},x_{4}]]}.$

Then:

${\mathcal{J}}\cap{\mathbb{R}}[[f_{1},f_{2}]]={\mathcal{J}}\cap({\mathbb{R}}[[f_{1},f_{2}]]+{\mathcal{M}})=(f_{1}^{2}+f_{2}^{2}){\mathbb{R}}[[f_{1},f_{2}]]\qquad\text{
and }\qquad{\mathbb{R}}[[f_{1},f_{2}]]\cap{\mathcal{M}}=\\{0\\}.$

###### Proof.

Note that $G$ is reduced, hence we can use Corollary 3.16. Moreover,
${\mathcal{J}}$ is generated by homogeneous polynomials. Hence it is enough to
check the statement for homogeneous degree polynomials. For odd homogeneous
degree $n=2m+1$ we only have contributions from ${\mathcal{M}}$. We write
$h\in{\mathcal{M}}$ as

$h=\sum_{j=1}^{4}x_{j}\sum_{i=0}^{m}a_{i}^{j}x_{2}^{2i}x_{4}^{2(m-i)}.$

Then we obtain by exchanging $x_{1}x_{4}$ with $x_{2}x_{3}$ that

$\displaystyle r:=NF(h|G)=$
$\displaystyle\,a_{m}^{1}x_{1}x_{2}^{2m}+x_{3}\left(\sum_{i=0}^{m}a_{i}^{3}x_{2}^{2i}x_{4}^{2(m-i)}+\sum_{i=0}^{m-1}a_{i}^{1}x_{2}^{2i+1}x_{4}^{2(m-i)-1}\right)$
$\displaystyle\,+\sum_{i=0}^{m}\left(a_{i}^{2}x_{2}^{2i+1}x_{4}^{2(m-i)}+a_{i}^{4}x_{2}^{2i}x_{4}^{2(m-i)+1}\right).$

Hence we have

$h\in{\mathcal{J}}\quad\Leftrightarrow\quad r=0\quad\Leftrightarrow\quad h=0.$

For the even case $n=2m>0$ we observe that

$f_{1}^{2}+f_{2}^{2}=(x_{1}^{2}+x_{2}^{2})^{2}+(x_{3}^{2}+x_{4}^{2})^{2}+2(x_{1}x_{3}+x_{2}x_{4})^{2}-2(x_{1}x_{4}-x_{2}x_{3})^{2},$

hence it is enough to consider $g\in{\mathbb{R}}[[f_{1},f_{2}]]$ of the form

$g=g_{1}f_{1}^{m}+g_{2}f_{1}^{m-1}f_{2}.$

We have

(15) $\displaystyle
r_{g}=NF(g|G)=g_{1}(-2)^{m}(x_{2}^{2}+x_{4}^{2})^{m}+g_{2}(-1)^{m-1}2^{m}\left(x_{1}x_{2}^{2m-1}+x_{3}\frac{(x_{2}^{2}+x_{4}^{2})^{m}-x_{2}^{2m}}{x_{4}}\right).$

We write an element $h\in{\mathcal{M}}$ as

$h=\sum_{j=1}^{4}x_{j}\sum_{i=0}^{m-1}a_{i}^{j}x_{2}^{2i}x_{4}^{2(m-i)-1}.$

Note that

$\displaystyle r_{h}=$
$\displaystyle\,x_{3}\sum_{i=0}^{m-1}\left(a_{i}^{3}x_{2}^{2i}x_{4}^{2(m-i)-1}+a_{i}^{1}x_{2}^{2i+1}x_{4}^{2(m-i-1)}\right)+\sum_{i=0}^{m-1}\left(a_{i}^{2}x_{2}^{2i+1}x_{4}^{2(m-i)-1}+a_{i}^{4}x_{2}^{2i}x_{4}^{2(m-i)+1}\right)$

and we conclude

$g+h\in{\mathcal{J}}\quad\Leftrightarrow\quad
r_{g}+r_{h}=0\quad\Leftrightarrow\quad g=h=0.$

∎

######  Lemma 3.18.

Let $g\in{\mathbb{R}}[[f_{1},f_{2}]]$ and
$(p,(q_{1},q_{2}))\in{\mathbb{R}}[[x_{2}]]\oplus{\mathbb{R}}[[x_{2},x_{4}]]^{2}$
be such that

$\displaystyle g\cdot\mu=$
$\displaystyle\,\mathrm{d}f_{1}\wedge\mathrm{d}(x_{1}p+x_{3}q_{1}+q_{2})(\mathrm{d}x_{1}\wedge\mathrm{d}x_{4}+\mathrm{d}x_{2}\wedge\mathrm{d}x_{3})\mod{\mathcal{J}}\cdot\mu,$

where $\mu$ is the standard volume form on ${\mathbb{R}}^{4}$. Then we have:

$\displaystyle
x_{2}p+x_{4}q_{1},q_{2}\in{\mathbb{R}}[[x_{2}^{2}+x_{4}^{2}]]\qquad\text{ and
}\qquad g\in(f_{1}^{2}+f_{2}^{2}){\mathbb{R}}[[f_{1},f_{2}]],$

and the same result holds if we replace $\mathrm{d}f_{1}$ by
$\mathrm{d}f_{2}$.

###### Proof.

By a direct computation we obtain that the right hand side of the statement
equals:

$\displaystyle
2\Big{(}-x_{2}q_{1}-x_{3}(x_{1}\partial_{2}p+x_{3}\partial_{2}q_{1}+\partial_{2}q_{2})+x_{1}(x_{3}\partial_{4}q_{1}+\partial_{4}q_{2})+x_{4}p\Big{)}\mu.$

Note that again it is enough to check the statement for homogeneous
polynomials. We set

$p=p_{0}x_{2}^{n-1},\quad
q_{1}=\sum_{i=0}^{n-1}a_{i}x_{2}^{i}x_{4}^{n-1-i},\quad
q_{2}=\sum_{i=0}^{n}b_{i}x_{2}^{i}x_{4}^{n-i}.$

Then we obtain for the right hand side:

$\displaystyle r_{R}:=NF($ $\displaystyle-
x_{2}q_{1}-x_{3}(x_{1}\partial_{2}p+x_{3}\partial_{2}q_{1}+\partial_{2}q_{2})+x_{1}(x_{3}\partial_{4}q_{1}+\partial_{4}q_{2})+x_{4}p|G)=$
$\displaystyle\,-\sum_{i=0}^{n-1}a_{i}x_{2}^{i+1}x_{4}^{n-1-i}+(n-1)p_{0}x_{2}^{n-1}x_{4}+\sum_{i=0}^{n-1}ia_{i}x_{2}^{i-1}x_{4}^{n-i+1}-\sum_{i=0}^{n}ib_{i}x_{2}^{i-1}x_{3}x_{4}^{n-i}$
$\displaystyle\,-\sum_{i=0}^{n-1}(n-i-1)a_{i}x_{2}^{i+1}x_{4}^{n-1-i}+\sum_{i=0}^{n-2}(n-i)b_{i}x_{2}^{i+1}x_{3}x_{4}^{n-i-2}+b_{n-1}x_{1}x_{2}^{n-1}+p_{0}x_{2}^{n-1}x_{4}$
$\displaystyle=$
$\displaystyle\,np_{0}x_{2}^{n-1}x_{4}+a_{1}x_{4}^{n}-a_{n-1}x_{2}^{n}-2a_{n-2}x_{2}^{n-1}x_{4}-\sum_{i=0}^{n-3}\left((n-i)a_{i}-(i+2)a_{i+2}\right)x_{2}^{i+1}x_{4}^{n-1-i}$
$\displaystyle-
b_{1}x_{3}x_{4}^{n-1}+b_{n-1}x_{1}x_{2}^{n-1}+\sum_{i=0}^{n-2}\left((n-i)b_{i}-(i+2)b_{i+2}\right)x_{2}^{i+1}x_{3}x_{4}^{n-i-2}.$

If $n=2m+1$ is odd then $g$ does not contribute and we obtain the condition

$\displaystyle(2m+1-i)a_{i}=-(i+2)a_{i+2},\ 0\leq i\leq 2m-2,\quad
a_{1}=a_{2m}=0,\quad 2a_{2m-1}=(2m+1)p_{0}$
$\displaystyle(i+2)b_{i+2}=(2m+1-i)b_{i},\ 0\leq i\leq 2m-1,\quad
b_{1}=b_{2m}=0,$

which imply that $p=q_{1}=q_{2}=0$.

For $n=2m$ is even we note that

$f_{1}^{2}+f_{2}^{2}=(x_{1}^{2}+x_{2}^{2})^{2}+(x_{3}^{2}+x_{4}^{2})^{2}+2(x_{1}x_{3}+x_{2}x_{4})^{2}-2(x_{1}x_{4}-x_{2}x_{3})^{2}.$

Hence it is enough to consider $g=g_{1}f_{1}^{m}+g_{2}f_{1}^{m-1}f_{2}$ as in
the previous lemma. From (15) we obtain the conditions

$\displaystyle
2(m-i)a_{2i}=2(i+1)a_{2(i+1)},\quad(2(m-i)-1)a_{2i+1}-(2i+3)a_{2i+3}=(-2)^{m-1}\binom{m}{i+1}g_{1},\
$

for $0\leq i\leq m-2$ and

$\displaystyle a_{1}=-(-2)^{m-1}g_{1},\quad a_{2m-1}=(-2)^{m-1}g_{1},\quad
2mp_{0}=2a_{2m-2},$

as well as

$\displaystyle 2(i+1)b_{2(i+1)}$ $\displaystyle\,=2(m-i)b_{2i},\qquad 0\leq
i\leq m-1,$ $\displaystyle(2i+3)b_{2i+3}$
$\displaystyle\,=(2(m-i)-1)b_{2i+1}+(-2)^{m-1}\binom{m}{i+1}g_{2},\quad 0\leq
i\leq m-2,$ $\displaystyle b_{1}=(-2)^{m-1}g_{2}\quad$ $\displaystyle\,\quad
b_{2m-1}=-(-2)^{m-1}g_{2},$

which implies the statement. The result for $\mathrm{d}f_{2}$ follows along
the same lines. ∎

##  4\. Division groups

Consider the space $\Omega^{\bullet}(M)$ of differential forms on a smooth
manifold $M$. It is possible to divide differential forms in the following
sence: let $\alpha\in\Omega^{1}(M)$ be nowhere vanishing then for any
$\beta\in\Omega^{\bullet}(M)$,

$\beta\wedge\alpha=0\quad\implies\quad\beta=\mu\wedge\alpha,$

for some $\mu\in\Omega^{\bullet-1}(M)$. This property is extremely useful when
manipulating forms but does not hold in general for the exterior algebra of a
module. To measure the failure, de Rham [Rha54] introduced the following
groups.

Consider a free ${\mathcal{R}}$-module ${\mathcal{M}}$ with basis
$(e_{1},\dots,e_{n})$. Fix elements
$\alpha_{1},\dots,\alpha_{k}\in{\mathcal{M}}$. The quotient

${\mathcal{D}}^{p}(\alpha_{1},\dots,\alpha_{k}):=\\{\beta\in\Lambda^{p}{\mathcal{M}}\mid\beta\wedge\alpha_{1}\wedge\dots\wedge\alpha_{k}=0\\}/\sum_{i=1}^{k}\left(\alpha_{i}\wedge\Lambda^{p-1}{\mathcal{M}}\right),$

is called the $p$-th division group associated to
$(\alpha_{1},\dots,\alpha_{k})$.

We make use of a result by Saito [Sai76] about these groups. To state the
results we first define the following. Use the basis elements to write

$\alpha_{1}\wedge\dots\wedge\alpha_{k}=\sum_{1\leq i_{1}<\dots<i_{k}<\leq
n}a_{i_{1}\cdots i_{k}}e_{i_{1}}\wedge\dots\wedge e_{i_{k}},$

with $a_{i_{1}\cdots i_{k}}\in{\mathcal{R}}$, and define
${\mathcal{J}}={\mathcal{J}}(\alpha_{1},\dots,\alpha_{k})\subset{\mathcal{R}}$,
the ideal generated by the coefficients $a_{i_{1},\cdots,i_{k}}$.

######  Definition 4.1.

The depth $\mathrm{depth}_{{\mathcal{R}}}(I,{\mathcal{M}})$ of an ideal
$I\subset{\mathcal{R}}$ on a finitely generated module ${\mathcal{M}}$ is the
supremum of the lengths of all ${\mathcal{M}}$-regular sequences of elements
of $I$.

######  Proposition 4.2 ([Sai76]).

The division groups satisfy:

${\mathcal{D}}^{p}(\alpha_{1},\dots,\alpha_{k})=0,$

for all $0\leq p<\operatorname{depth}({\mathcal{J}},{\mathcal{R}})$.

Now, let us return to the setting of Lefschetz singularities. We introduce the
notation:

$\displaystyle\beta_{1}:=\mathrm{d}\zeta_{1}=\mathrm{d}x_{1}\wedge\mathrm{d}x_{3}-\mathrm{d}x_{2}\wedge\mathrm{d}x_{4}\quad\text{
and
}\quad\beta_{2}:=\mathrm{d}\zeta_{2}=\mathrm{d}x_{1}\wedge\mathrm{d}x_{4}+\mathrm{d}x_{2}\wedge\mathrm{d}x_{3},$

where $\zeta_{i}$ were defined in (6). We have the following result.

######  Proposition 4.3.

The depth of
${\mathcal{J}}={\mathcal{J}}({\operatorname{d}}f_{1},{\operatorname{d}}f_{2})$
is $2$, and there is an isomorphism:

$\displaystyle I:\ \
{\mathbb{R}}\oplus{\mathbb{R}}[[x_{2}]]\oplus{\mathbb{R}}[[x_{2},x_{4}]]^{2}\
$
$\displaystyle\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}\qquad\mathcal{D}^{2}({\operatorname{d}}f_{1},{\operatorname{d}}f_{2})$
$\displaystyle(c,p,(q_{1},q_{2}))\qquad$
$\displaystyle\,\mapsto[c\beta_{1}+(px_{1}+q_{1}x_{3}+q_{2})\beta_{2}]$

###  4.1. Consequence of Proposition 4.3

In this subsection we explain the problem of the claimed computation for the
formal Poisson cohomology for Lefschetz singularities in [BV20].

Let us first recall the definition of an isolated complete intersection
singularity. Set ${\mathcal{R}}={\mathbb{R}}[[x_{1},\dots,x_{n}]]$ and
consider polynomials $p_{1},\dots,p_{k}\in{\mathbb{R}}[x_{1},\dots,x_{n}]$
with zero constant term. Define
${\mathcal{I}}={\mathcal{I}}(p_{1},\dots,p_{k})$ to be the ideal generated by

$\displaystyle\langle p_{1},\dots,p_{k},\det(\partial_{j_{i}}p_{i})_{1\leq
i\leq k}\rangle.$

######  Definition 4.4.

$(p_{1},\dots,p_{k})$ has an isolated complete intersection singularity (ICIS)
if $(p_{1},\dots,p_{k})$ is a regular sequence in ${\mathcal{R}}$ and
${\mathcal{R}}/{\mathcal{I}}$ is a finite dimensional ${\mathbb{R}}$-vector
space.

Let us define the ideal ${\mathcal{J}}={\mathcal{J}}(p_{1},\dots,p_{k})$ by
its generators

$\displaystyle{\mathcal{J}}:=\langle\det(\partial_{j_{i}}p_{i})_{1\leq i\leq
k}\rangle_{{\mathcal{R}}}.$

The following is due to [Loo84][Proof of Proposition (4.4)]:

######  Proposition 4.5.

If $(p_{1},\dots,p_{k})$ is an ICIS, then the depth of the ideal
${\mathcal{J}}$ in ${\mathcal{R}}$ is $k+1$.

For the computations in [BV20] it is assumed that $(f_{1},f_{2})$ is an ICIS
(p. 16, last paragraph). This assumption is crucial for the computations as
done in [Pel09]. However, combining 4.3 and Proposition 4.5 implies that
$(f_{1},f_{2})$ is not an ICIS. In fact, as we shall see, the non-vanishing of
$\mathcal{D}^{2}$ seems to be main factor for some of the cohomology groups
being non-free modules over $H^{0}$.

###  4.2. Proposition 4.3: The proof and a Corollary

We start with the proof of Proposition 4.3. To see that
$\operatorname{depth}({\mathcal{J}},{\mathcal{R}})=2$ observe that

$\beta_{i}\wedge{\operatorname{d}}f_{1}\wedge{\operatorname{d}}f_{2}=0,\quad
i=1,2.$

Furthermore, the coefficients of $\beta_{i}$ are homogenenous of degree $0$,
whereas the coefficients of ${\operatorname{d}}f_{i}$ are homogeneous of
degree $1$. This implies that $\beta_{1}$ and $\beta_{2}$ define a non-zero
classes in
${\mathcal{D}}^{2}({\operatorname{d}}f_{1},{\operatorname{d}}f_{2})$. Hence,
by Proposition 4.2 we have
$\operatorname{depth}({\mathcal{J}},{\mathcal{R}})\leq 2$. Hence Example 3.9
implies that $\operatorname{depth}({\mathcal{J}},{\mathcal{R}})=2$.

We note first that the map is well-defined. Let
$\omega\in\Omega^{2}({\mathcal{R}})$ be of the form

$\displaystyle\omega=\sum_{i<j}\omega_{ij}\mathrm{d}x_{i}\wedge\mathrm{d}x_{j}\qquad\text{
with }\omega_{ij}\in{\mathcal{R}}.$

The wedge product of $\omega$ with $\mathrm{d}f_{1}\wedge\mathrm{d}f_{2}$ is
described by the function

(16)
$\displaystyle\star^{-1}(\frac{1}{4}\omega\wedge\mathrm{d}f_{1}\wedge\mathrm{d}f_{2})=$
$\displaystyle\quad\
(x_{1}^{2}+x_{2}^{2})\omega_{34}-(x_{1}x_{4}-x_{2}x_{3})(\omega_{13}+\omega_{24})$
$\displaystyle\,+(x_{3}^{2}+x_{4}^{2})\omega_{12}+(x_{1}x_{3}+x_{2}x_{4})(\omega_{23}-\omega_{14}).$

We note that for $\gamma\in\Omega^{1}({\mathcal{R}})$ of the form

$\displaystyle\gamma=\sum_{i}\gamma_{i}\mathrm{d}x_{i}$

we obtain that

$\displaystyle\frac{1}{2}\mathrm{d}f_{1}\wedge\gamma=$ $\displaystyle\ \ \ \
(x_{1}\gamma_{2}+x_{2}\gamma_{1})\mathrm{d}x_{1}\wedge\mathrm{d}x_{2}+(x_{3}\gamma_{4}+x_{4}\gamma_{3})\mathrm{d}x_{3}\wedge\mathrm{d}x_{4}$
(17)
$\displaystyle+(x_{1}\gamma_{3}-x_{3}\gamma_{1})\mathrm{d}x_{1}\wedge\mathrm{d}x_{3}+(x_{4}\gamma_{2}-x_{2}\gamma_{4})\mathrm{d}x_{2}\wedge\mathrm{d}x_{4}$
$\displaystyle+(x_{1}\gamma_{4}+x_{4}\gamma_{1})\mathrm{d}x_{1}\wedge\mathrm{d}x_{4}-(x_{2}\gamma_{3}+x_{3}\gamma_{2})\mathrm{d}x_{2}\wedge\mathrm{d}x_{3}.$

Similarly we have for $\delta\in\Omega^{1}({\mathcal{R}})$ that

$\displaystyle\frac{1}{2}\mathrm{d}f_{2}\wedge\delta=$ $\displaystyle\ \ \ \
(x_{2}\delta_{2}-x_{1}\delta_{1})\mathrm{d}x_{1}\wedge\mathrm{d}x_{2}+(x_{4}\delta_{4}-x_{3}\delta_{3})\mathrm{d}x_{3}\wedge\mathrm{d}x_{4}$
(18)
$\displaystyle+(x_{2}\delta_{3}-x_{4}\delta_{1})\mathrm{d}x_{1}\wedge\mathrm{d}x_{3}+(x_{1}\delta_{4}-x_{3}\delta_{2})\mathrm{d}x_{2}\wedge\mathrm{d}x_{4}$
$\displaystyle+(x_{2}\delta_{4}-x_{3}\delta_{1})\mathrm{d}x_{1}\wedge\mathrm{d}x_{4}+(x_{1}\delta_{3}-x_{4}\delta_{2})\mathrm{d}x_{2}\wedge\mathrm{d}x_{3}.$

We want to show that for any $\omega\in Z^{2}$ we can find $\gamma$ and
$\delta$ such that

(19)
$\displaystyle\bar{\omega}=\omega+\mathrm{d}f_{1}\wedge\gamma+\mathrm{d}f_{2}\wedge\delta$

is of the form as described in Proposition 4.3.

Let us assume that $\omega\in Z^{2}$ or equivalently that

$\displaystyle\star^{-1}(\omega\wedge\mathrm{d}f_{1}\wedge\mathrm{d}f_{2})=0.$

Idea: We show in steps how we can simplify (parts) of the coefficients of
$\omega$ as follows:

  1. (1)

We make choices for (parts) of the coefficients of $\gamma_{i}$ in (17) and
$\delta_{j}$ in (18) to show that these simplifications can be achieved.

  2. (2)

We continue in the next step with a choice for $\gamma_{i}$ and $\delta_{j}$
based upon the simplification for $\omega$ which we verified in the previous
steps.

Notation: To keep notation simple:

  * •

We write $I\in\mathbb{N}_{0}^{4}$ with a subscript to indicate which entries
vary over non-zero entries, e.g. $I_{24}$ stands for $(0,i_{2},0,i_{4})$ with
$i_{2},i_{4}\in\mathbb{N}_{0}$. We simply write $I$ for $I_{1234}$;

  * •

For $r\in{\mathcal{R}}$ the expression $r(I)$ refers to the coefficient of the
term $x_{1}^{i_{1}}x_{2}^{i_{2}}x_{3}^{i_{3}}x_{4}^{i_{4}}$ for $r$;

  * •

If $J=(j_{1},j_{2},j_{3},j_{4})$ happens to be such that at least one of the
$j_{i}$ is negative we set $r(J):=0$.

We note first that the coefficients $\omega_{12}$ and $\omega_{34}$ satisfy:

$\displaystyle\omega_{12}(I_{34})=0\qquad\text{ and
}\qquad\omega_{34}(I_{12})=0.$

This follows immediately from (16) since the only non-zero elements in the
subrings ${\mathbb{R}}[[x_{1},x_{2}]]$ and ${\mathbb{R}}[[x_{3},x_{4}]]$ can
come from the corresponding parts of $\omega_{34}$ and $\omega_{12}$,
respectively.

Claim 1: We may assume that the coefficients $\omega_{12}$ and $\omega_{34}$
satisfy:

(20) $\displaystyle\omega_{12}=\omega_{34}=0.$

We can choose

$\displaystyle\delta_{1}(I):=$
$\displaystyle\,\frac{1}{2}\omega_{12}(I+e_{1})+\gamma_{2}(I)+(\gamma_{1}+\delta_{2})(I+e_{1}-e_{2}),$
(21) $\displaystyle\delta_{2}(I_{234}):=$
$\displaystyle\,-\frac{1}{2}\omega_{12}(I_{234}+e_{2})-\gamma_{1}(I_{234}),\qquad\delta_{4}(I_{124}):=-\frac{1}{2}\omega_{34}(I_{124}+e_{4})-\gamma_{3}(I_{124})$
$\displaystyle\delta_{3}(I):=$
$\displaystyle\,\frac{1}{2}\omega_{34}(I+e_{3})+\gamma_{4}(I)+(\gamma_{3}+\delta_{4})(I+e_{3}-e_{4}).$

By comparing the coefficients in (17), (18) and (19) this implies the claim.

Claim 2: We may assume that

$\displaystyle\omega_{13}+\omega_{24}=0=\omega_{23}-\omega_{14}\quad\text{
i.e.\ }\quad r=0.$

By Example 3.9 the sequence
$\\{x_{1}x_{3}+x_{2}x_{4},x_{1}x_{4}-x_{2}x_{3}\\}\subset{\mathcal{R}}$ is
regular. Since ${\mathcal{R}}$ is a unique factorization domain, the vanishing
of (16) together with (20) implies the existence of $r\in{\mathcal{R}}$ such
that

$\displaystyle\omega_{13}+\omega_{24}=(x_{1}x_{3}+x_{2}x_{4})r\quad\text{ and
}\quad\omega_{23}-\omega_{14}=(x_{1}x_{4}-x_{2}x_{3})r.$

We choose

$\displaystyle\gamma_{1}(I+e_{1}):=(\gamma_{3}+\delta_{4})(I+e_{3})-\delta_{2}(I+e_{1})-\frac{1}{2}r.$

Summing up the $\mathrm{d}x_{1}\wedge\mathrm{d}x_{3}$ and the
$\mathrm{d}x_{2}\wedge\mathrm{d}x_{4}$-components of (17) and (18) yields

$\displaystyle(x_{1}x_{3}+x_{2}x_{4})\big{(}(\gamma_{3}+\delta_{4})(I+e_{3})-(\gamma_{1}+\delta_{2})(I+e_{1})\big{)}.$

Similarly, we obtain for the $\mathrm{d}x_{1}\wedge\mathrm{d}x_{4}$-components
minus the $\mathrm{d}x_{2}\wedge\mathrm{d}x_{3}$-components that

$\displaystyle-(x_{1}x_{4}-x_{2}x_{3})\big{(}(\gamma_{3}+\delta_{4})(I+e_{3})-(\gamma_{1}+\delta_{2})(I+e_{1})\big{)}$

implying the claim.

Claim 3: We may assume that

$\displaystyle\omega_{13}\in{\mathbb{R}}.$

Note that the sum of the $\mathrm{d}x_{1}\wedge\mathrm{d}x_{3}$-components
from (17) and (18) is given by

$\displaystyle
x_{1}\gamma_{3}-x_{3}\gamma_{1}+x_{2}\delta_{3}-x_{4}\delta_{1}=x_{1}\gamma_{3}(I_{124})-x_{3}\gamma_{1}(I_{234})+x_{2}\gamma_{4}-x_{4}\gamma_{2}+x_{1}x_{3}(\delta_{2}(I+e_{1})-\delta_{4}(I+e_{3})).$

Hence by taking

$\displaystyle\gamma_{2}(I_{4}):=$
$\displaystyle\,\frac{1}{2}\omega_{13}(I_{4}+e_{4}),\qquad\gamma_{4}(I_{24}):=-\frac{1}{2}\omega_{13}(I_{24}+e_{2})+\gamma_{2}(I_{24}+e_{2}-e_{4}),$
(22) $\displaystyle\gamma_{3}(I_{124}):=$
$\displaystyle\,-\frac{1}{2}\omega_{13}(I_{124}+e_{1})-\gamma_{4}(I_{124}+e_{1}-e_{2})+\gamma_{2}(I_{124}+e_{1}-e_{4}),$
$\displaystyle\gamma_{1}(I_{234}):=$
$\displaystyle\,\frac{1}{2}\omega_{13}(I_{234}+e_{3})+\gamma_{4}(I_{234}+e_{3}-e_{2})-\gamma_{2}(I_{234}+e_{3}-e_{4}),$
$\displaystyle\delta_{4}(I+e_{3}):=$
$\displaystyle\,\frac{1}{2}\omega_{13}(I+e_{1}+e_{3})+\delta_{2}(I+e_{1})+\gamma_{4}(I+(1,-1,1,0))-\gamma_{2}(I+(1,0,1,-1))$

we obtain the claim.

To simplify $\omega_{14}$ we add the
$\mathrm{d}x_{1}\wedge\mathrm{d}x_{4}$-components of (17) and (18), which
yields

(23)
$x_{1}\gamma_{4}+x_{4}\gamma_{1}+x_{2}\delta_{4}-x_{3}\delta_{1}=\begin{array}[]{l}(x_{1}^{2}+x_{2}^{2})\gamma_{4}(I_{124}+e_{1})+(x_{1}x_{3}+x_{2}x_{4})\gamma_{4}(I+e_{3})\\\
-(x_{3}^{2}+x_{4}^{2})\gamma_{2}(I+e_{3})+(x_{1}x_{4}-x_{2}x_{3})(\gamma_{3}(I+e_{3})+\gamma_{2}(I_{24}+e_{2})).\end{array}$

Note that all the different coefficients are independent of each other.
Moreover, we have

(24)
$\displaystyle{\mathcal{J}}={\mathcal{R}}\cdot(x_{3}^{2}+x_{4}^{2})+{\mathcal{R}}\cdot(x_{1}x_{3}+x_{2}x_{4})+{\mathcal{R}}\cdot(x_{1}x_{4}-x_{2}x_{3})+{\mathbb{R}}[[x_{1},x_{2},x_{4}]]\cdot(x_{1}^{2}+x_{2}^{2}).$

Hence the result follows from:

Claim 4: The following map is an isomorphism of ${\mathbb{R}}$-vector spaces:

$\begin{array}[]{rccc}m:&{\mathbb{R}}[[x_{2}]]\oplus{\mathbb{R}}[[x_{2},x_{4}]]^{2}&\xrightarrow{\raisebox{-1.99997pt}[0.0pt][0.0pt]{\smash{$\sim$}}}&{\mathcal{R}}/{\mathcal{J}}\\\
&(p,(q_{1},q_{2}))&\mapsto&[px_{1}+q_{1}x_{3}+q_{2}].\end{array}$

We first show that the map $m$ is injective. In order to do this we use that
the linear map:

$\displaystyle
l:{\mathbb{R}}[[x_{1},x_{2}]]\oplus{\mathbb{R}}[[x_{3},x_{4}]]\oplus{\mathcal{R}}^{2}$
$\displaystyle\to\qquad\qquad\qquad\qquad\qquad\qquad\qquad{\mathcal{J}}$
$\displaystyle(p_{1},p_{2},(r_{1},r_{2}))\ \qquad$
$\displaystyle\mapsto\,(x_{1}^{2}+x_{2}^{2})p_{1}+(x_{3}^{2}+x_{4}^{2})p_{2}+(x_{1}x_{3}+x_{2}x_{4})r_{1}+(x_{1}x_{4}-x_{2}x_{3})r_{2}$

is surjective with kernel given by

$(0,0,((x_{1}x_{4}-x_{2}x_{3})\cdot r,-(x_{1}x_{3}+x_{2}x_{4})\cdot
r))\qquad\text{ for }r\in{\mathcal{R}}.$

This can be seen from Example 3.9 and the relations

$\displaystyle
x_{3}(x_{1}^{2}+x_{2}^{2})-x_{1}(x_{1}x_{3}+x_{2}x_{4})+x_{2}(x_{1}x_{4}-x_{2}x_{3})=0,$
$\displaystyle
x_{4}(x_{1}^{2}+x_{2}^{2})-x_{2}(x_{1}x_{3}+x_{2}x_{4})-x_{1}(x_{1}x_{4}-x_{2}x_{3})=0,$
$\displaystyle
x_{1}(x_{3}^{2}+x_{4}^{2})-x_{3}(x_{1}x_{3}+x_{2}x_{4})-x_{4}(x_{1}x_{4}-x_{2}x_{3})=0,$
$\displaystyle
x_{2}(x_{3}^{2}+x_{4}^{2})-x_{4}(x_{1}x_{3}+x_{2}x_{4})+x_{3}(x_{1}x_{4}-x_{2}x_{3})=0.$

Let
$(p,(q_{1},q_{2}))\in{\mathbb{R}}[[x_{2}]]\oplus{\mathbb{R}}[[x_{2},x_{4}]]^{2}$
such that $m(p,(q_{1},q_{2}))=0$, i.e.

$\exists(p_{1},p_{2},(r_{1},r_{2}))\in{\mathbb{R}}[[x_{1},x_{2}]]\oplus{\mathbb{R}}[[x_{3},x_{4}]]\oplus{\mathcal{R}}^{2}:\quad
l(p_{1},p_{2},(r_{1},r_{2}))=px_{1}+q_{1}x_{3}+q_{2}.$

Comparing the terms in ${\mathbb{R}}[[x_{1},x_{2}]]$ and
${\mathbb{R}}[[x_{3},x_{4}]]$ we immediately obtain

$p=0\qquad q_{1}(I_{4})=0,\qquad q_{2}(I_{2})=0\qquad
q_{2}(I_{4})=0\quad\text{ and }\quad p_{1}=p_{2}=0.$

Hence we can write the image of $m$ uniquely as

$x_{2}x_{3}\tilde{q}_{1}+x_{2}x_{4}\tilde{q}_{2}\qquad\text{ for some
}\tilde{q}_{1},\tilde{q}_{2}\in{\mathbb{R}}[[x_{2},x_{4}]].$

But such an element is in the image of $l$ iff it is zero, hence $m$ is indeed
injective.

We conclude the proof by comparing dimensions of the domain and codomain of
$m$ for a fixed homogeneous degree in ${\mathcal{R}}/{\mathcal{J}}$. We denote
by ${\mathcal{R}}_{d}$ and ${\mathcal{J}}_{d}$ the corresponding subvector
spaces generated by polynomials of homogeneous degree $d\in\mathbb{N}_{0}$.
Using the fact that for $n\in N$ and $d\in\mathbb{N}_{0}$ we have

$\displaystyle\dim{\mathbb{R}}[[x_{1},\dots,x_{n}]]_{d}=\binom{n-1+d}{n-1},$

and the description of ${\mathcal{J}}$ via $l$ we obtain for the dimension of
${\mathcal{J}}_{d}$ for $2\leq d$ that

$\displaystyle\text{ for }2\leq d<4:$
$\displaystyle\quad\dim{\mathcal{J}}_{d}=2\binom{d-1}{1}+2\binom{d+1}{3},$ and
else :
$\displaystyle\quad\dim{\mathcal{J}}_{d}=2\binom{d-1}{1}+2\binom{d+1}{3}-\binom{d-1}{3}.$

Finally, a direct computation implies that for any $d\in\mathbb{N}$ we have

$\dim{\mathcal{R}}_{d}/{\mathcal{J}}_{d}=\dim{\mathcal{R}}_{d}-\dim{\mathcal{J}}_{d}=2(d+1),$

which proves the claim by comparing it with
$\dim(m^{-1}(\\{{\mathcal{R}}_{d}/{\mathcal{J}}_{d}\\}))$. Hence we proved
Proposition 4.3.

######  Corollary 4.6.

The isomorphism from Proposition 4.3 satisfies the relations:

$\displaystyle I^{-1}([f_{1}\cdot I(0,p,(q_{1},q_{2}))])$
$\displaystyle\,=-2(0,x_{2}^{2}p,(x_{2}x_{4}p+(x_{2}^{2}+x_{4}^{2})q_{1},(x_{2}^{2}+x_{4}^{2})q_{2}))$
$\displaystyle I^{-1}([f_{2}\cdot I(0,p,(q_{1},q_{2}))])$
$\displaystyle\,=2(0,q_{2}(I_{2})x^{I_{2}+e_{2}},(x_{4}q_{2}+x_{2}^{2}x_{4}^{-1}(q_{2}-q_{2}(I_{2})x^{I_{2}}),-(x_{2}p+x_{4}q_{1})(x_{2}^{2}+x_{4}^{2})))$

for any
$(p,(q_{1},q_{2}))\in{\mathbb{R}}[[x_{2}]]\oplus{\mathbb{R}}[[x_{2},x_{4}]]^{2}$.
Additionally, we have

$\displaystyle[f_{1}\beta_{1}-f_{2}\beta_{2}]$
$\displaystyle\,=0=[f_{2}\beta_{1}+f_{1}\beta_{2}].$

In particular, for any
$(p,(q_{1},q_{2}))\in{\mathbb{R}}[[x_{2}]]\oplus{\mathbb{R}}[[x_{2},x_{4}]]^{2}$
which satisfy

$px_{2}+x_{4}q_{1}=a(x_{2}^{2}+x_{4}^{2})^{m}\qquad\text{ and }\qquad
q_{2}=b(x_{2}^{2}+x_{4}^{2})^{m}$

for some $a,b\in{\mathbb{R}}$, we obtain that

$[(x_{1}p+x_{3}q_{1})\beta_{2}]=[c_{1}f_{1}^{m-1}f_{2}\beta_{2}]\quad\text{and}\quad[q_{2}\beta_{2}]=[c_{2}f_{1}^{m}\beta_{2}],$

for $c_{1}:=(-2)^{-m}a$ and $c_{2}:=2^{-m}b$.

###### Proof.

The first two equations follow by a direct computation using (23) and its
induced partition for ${\mathcal{J}}$ as stated in (24). The last two
equalities are obtained from a direct computation and choices for $\gamma$ and
$\delta$ as described in (22) and (21). ∎

##  5\. Kernel of the Poisson differential

The aim of this section is to compute the kernel of the Poisson differential
$\delta_{\pi}:\Omega^{\bullet}_{f}\to\Omega^{\bullet-1}_{f}$. We prove the
following:

######  Proposition 5.1.

For the kernel of the differential $\delta_{\pi}$ from Proposition 3.2 we
obtain the following results for formal differential forms. The statements are
for $g,g_{i}\in{\mathcal{R}}$ and
$p,p_{i},q_{i}\in{\mathbb{R}}[[f_{1},f_{2}]]$.

  * •

For $\alpha\in\Omega^{1}_{f}$ we obtain that

(25)
$\alpha\in\ker\delta_{\pi}\quad\Leftrightarrow\quad\alpha=\mathrm{d}g_{0}+\sum_{i=1}^{2}p_{i}\zeta_{i}+g_{i}\mathrm{d}f_{i}.$

  * •

In degree $2$ we get for $\omega\in\Omega^{2}_{f}$ that

(26)
$\displaystyle\omega\in\ker\delta_{\pi}\quad\Leftrightarrow\quad\omega=p\zeta_{1}\wedge\zeta_{2}+g\mathrm{d}f_{1}\wedge\mathrm{d}f_{2}+\sum_{i=1}^{2}\mathrm{d}f_{1}\wedge
p_{i}\zeta_{i}+\mathrm{d}(q_{i}\zeta_{i})+\mathrm{d}g_{i}\wedge\mathrm{d}f_{i}.$

  * •

For a formal $3$-form $\gamma\in\Omega^{3}_{f}$ we obtain

(27) $\displaystyle\gamma\in\ker\delta_{\pi}\quad\Leftrightarrow\quad\gamma=$
$\displaystyle\
\mathrm{d}f_{1}\wedge\mathrm{d}f_{2}\wedge\mathrm{d}g+\sum_{i=1}^{2}q_{i}\epsilon_{i}+\mathrm{d}f_{1}\wedge\mathrm{d}\big{(}p_{i}\zeta_{i}\big{)}$

where $\epsilon_{i}=\star E_{i}$.

  * •

In the top degree we have for $g\in{\mathcal{R}}$ that

(28) $\displaystyle g\mu\in\ker\delta_{\pi}\quad\Leftrightarrow\quad
g\in{\mathbb{R}}[[f_{1},f_{2}]].$

For top degree forms this is a consequence of [Pel09][Proposition 3.1], since
$\mathcal{D}^{1}=0$. In the other degrees, Pelap uses the vanishing of
$\mathcal{D}^{2}$, which doesn’t hold in our case. Therefore, we need an
adaptation which takes the non-triviality of $\mathcal{D}^{2}$ into account.
The sections below provide proofs for the various degree and are ordered
according to their logical dependence:

$\text{ degree 1 }\Rightarrow\text{ degree 3}\Rightarrow\text{ degree 2}.$

Therefore, the proof for degree 1 is at the core of the proof of Proposition
5.1, which we obtain by an induction argument. Another important ingredient
for the proof in any degree is a good understanding of the group
$\mathcal{D}^{2}$, as described in Proposition 4.3 and Corollary 4.6. We have
two more observations:

  * •

First we note that the standard scalar multiplication
$m_{t}:{\mathbb{R}}^{4}\to{\mathbb{R}}^{4}$ on ${\mathbb{R}}^{4}$ satisfies

$\delta_{\pi}\circ m_{t}^{*}=m_{t}^{*}\circ\delta_{\pi}.$

Hence $\delta_{\pi}$ maps homogeneous degree coefficients to homogeneous
degree coefficients and it is enough to prove the statements for coefficients
of a fixed homogeneous degree.

  * •

It is easy to verify the implications $\Leftarrow$ in Proposition 5.1 using
Proposition 3.2. Therefore, we really only need to show that $\Rightarrow$
holds. To do so it is enough to show that

(29) $\ker\delta_{\pi}\mod V=\\{0\\},$

where $V$ denotes the vector space generated by elements on the right hand
side of Proposition 5.1. Whenever we look at this quotient we refer to it by
$\operatorname{mod}V$.

That being said, we are ready to start the proof of Proposition 5.1.

###  5.1. Proof of Degree $1$

For $\alpha\in\Omega^{1}_{f}$ recall from Proposition 3.2 that

$\displaystyle\delta_{\pi}(\alpha)$
$\displaystyle={\operatorname{d}}\alpha\wedge{\operatorname{d}}f_{1}\wedge{\operatorname{d}}f_{2}.$

Hence it suffices to prove that

${\operatorname{d}}\alpha\wedge{\operatorname{d}}f_{1}\wedge{\operatorname{d}}f_{2}=0\quad\Leftrightarrow\quad\alpha={\operatorname{d}}g_{0}+\sum_{i=1}^{2}p_{i}(f_{1},f_{2})\zeta_{i}+g_{i}{\operatorname{d}}f_{i}.$

If the coefficients of $\alpha$ are of homogeneous degree $n+1$, i.e.
$m_{t}^{*}\alpha=t^{n+2}\alpha$, the following lemma will allow us to derive
the statement by induction over even and odd degrees, respectively.

######  Lemma 5.2.

For $n=0$ there exist $c_{1},c_{2}\in{\mathbb{R}}$ such that

${\operatorname{d}}(\alpha-(c_{1}\zeta_{1}+c_{2}\zeta_{2}))=0.$

For $1\leq n=2m$ there exist $c_{1},c_{2}\in{\mathbb{R}}$ such that

${\operatorname{d}}(\alpha-(c_{1}f_{1}^{m}+c_{2}f_{1}^{m-1}f_{2})\zeta_{2})=\alpha_{1}\wedge{\operatorname{d}}f_{1}+\alpha_{2}\wedge{\operatorname{d}}f_{2}.$

Moreover, for $1\leq n=2m+1$ we have

${\operatorname{d}}\alpha=\alpha_{1}\wedge{\operatorname{d}}f_{1}+\alpha_{2}\wedge{\operatorname{d}}f_{2}.$

In all cases we may assume

$m_{t}^{*}\alpha_{i}=t^{n}\alpha_{i}\qquad\text{ for }\quad i=1,2.$

###### Proof.

From Proposition 4.3 we obtain that

${\operatorname{d}}\alpha\wedge{\operatorname{d}}f_{1}\wedge{\operatorname{d}}f_{2}=0\quad\Rightarrow\quad{\operatorname{d}}\alpha=I(c,p,(q_{1},q_{2}))+\sum_{i=1}^{2}\alpha_{i}\wedge{\operatorname{d}}f_{i}$

for some
$c\in{\mathbb{R}},p\in{\mathbb{R}}[[x_{2}]],q\in{\mathbb{R}}[[x_{2},x_{4}]]^{2}$
and $\alpha_{i}\in\Omega^{1}_{f}$. Note that since $\alpha$ has coefficient of
homogeneous degree $n+1$ we may assume that the coefficients of $I(c,p,q)$ and
$\alpha_{i}$ are homogeneous of degree $n$ and $n-1$, respectively. Wedging
the equation for ${\operatorname{d}}\alpha$ with ${\operatorname{d}}f_{1}$ and
applying ${\operatorname{d}}$ yields

(30)
$\displaystyle\mathrm{d}f_{1}\wedge\mathrm{d}I(c,p,q)=\mathrm{d}f_{1}\wedge\mathrm{d}f_{2}\wedge\mathrm{d}\alpha_{2}.$

If $n=0$, then the statement follows immediately from the fact that
$\mathrm{d}\zeta_{i}=\beta_{i}$. Hence we assume $n>0$. Then (30) implies the
hypothesis of Lemma 3.18 for $g\cdot\mu=0$, which in turn implies that
$x_{2}p+x_{4}q_{1},q_{2}\in R[[x_{2}^{2}+x_{4}^{2}]]$. Hence Corollary 4.6
implies $[(x_{1}p+x_{3}q_{1})\beta_{2}]=[c_{1}f_{1}^{m-1}f_{2}\beta_{2}]$ and
$[q_{2}\beta_{2}]=[c_{2}f_{1}^{m}\beta_{2}]$ so that

$I(0,p,q)=[(c_{1}f_{1}^{m-1}f_{2}+c_{2}f_{1}^{m})\beta_{2}]\in\mathcal{D}^{2}.$

Finally, since ${\operatorname{d}}\zeta_{i}=\beta_{i}$ we get

$I(0,p,q)=[{\operatorname{d}}((c_{1}f_{1}^{m-1}f_{2}+c_{2}f_{1}^{m})\zeta_{2})]\in\mathcal{D}^{2},$

and therefore by changing $\alpha_{i}$ we obtain the statement. ∎

We prove the statement first for the case of $n=2m-1$ being odd. In the base
case, i.e. $m=0$ it is straightforward to see that $\tau={\operatorname{d}}g$
for $g$ of homogeneous degree $1$. Hence we assume

$m_{t}^{*}\alpha=t^{2m+3}\alpha\qquad\text{ and
}\qquad{\operatorname{d}}f_{1}\wedge{\operatorname{d}}f_{2}\wedge{\operatorname{d}}\alpha=0.$

From Lemma 5.2 we get that

${\operatorname{d}}\alpha=\sum_{i=1}^{2}\alpha_{i}\wedge{\operatorname{d}}f_{i}\quad\text{
with }\quad m_{t}^{*}\alpha_{i}=t^{2m+1}\alpha_{i}\quad\text{ and
}\quad{\operatorname{d}}f_{1}\wedge{\operatorname{d}}f_{2}\wedge{\operatorname{d}}\alpha_{i}=0\quad\text{
for }\ i=1,2.$

We distinguish two cases. For $m=1$ the induction hypothesis implies that

${\operatorname{d}}\alpha={\operatorname{d}}g_{1}\wedge{\operatorname{d}}f_{1}+{\operatorname{d}}g_{2}\wedge{\operatorname{d}}f_{2},$

and hence we can conclude that

$\alpha={\operatorname{d}}g_{0}+g_{1}{\operatorname{d}}f_{1}+g_{2}{\operatorname{d}}f_{2},$

For $1<m$ the induction hypothesis yields

${\operatorname{d}}\alpha={\operatorname{d}}g_{1}\wedge{\operatorname{d}}f_{1}+{\operatorname{d}}g_{2}\wedge{\operatorname{d}}f_{2}+g{\operatorname{d}}f_{1}\wedge{\operatorname{d}}f_{2},$

for some $g\in{\mathcal{R}}$. Taking ${\operatorname{d}}$ of this expression
we see that $g\mu\in\ker\delta_{\pi}$. Hence $g\in{\mathbb{R}}[[f_{1},f_{2}]]$
and we can absorb $g$ into $g_{1}$ and $g_{2}$. Therefore we obtain

$\alpha={\operatorname{d}}g_{0}+g_{1}{\operatorname{d}}f_{1}+g_{2}{\operatorname{d}}f_{2}$

for some $g_{0},g_{1},g_{2}\in{\mathcal{R}}$, which proves the Proposition for
$n$ odd.

For the even case let $n=2m$. The base case ($m=0$) can be checked by hand,
where we obtain that

$\alpha={\operatorname{d}}g_{0}+\sum_{i=1}^{2}p_{i}\zeta_{i}+g_{i}{\operatorname{d}}f_{i}$

with $g_{0}\in{\mathcal{R}}$ homogeneous of degree $2$ and
$p_{i},g_{i}\in{\mathbb{R}}$ for $i=1,2$. For $\alpha$ satisfying

$m_{t}^{*}\alpha=t^{2m+2}\alpha\qquad\text{ and
}\qquad{\operatorname{d}}f_{1}\wedge{\operatorname{d}}f_{2}\wedge{\operatorname{d}}\alpha=0$

we get from Lemma 5.2 that

${\operatorname{d}}\alpha={\operatorname{d}}\left((c_{1}f_{1}^{m}+c_{2}f_{1}^{m-1}f_{2})\zeta_{2}\right)+\sum_{j=1}^{2}({\operatorname{d}}g_{j}+\sum_{i=1}^{2}p_{i}^{j}\zeta_{i})\wedge{\operatorname{d}}f_{j}+g{\operatorname{d}}f_{1}\wedge{\operatorname{d}}f_{2}.$

Here the $p_{i}^{j}\in{\mathbb{R}}[[f_{1},f_{2}]]$ are homogeneous of degree
$2m-2$, $g_{j}\in{\mathcal{R}}$ are homogeneous of degree $2m$ for
$i,j\in\\{1,2\\}$ and $g\in{\mathcal{R}}$ is homogeneous of degree $2m-2$.
Hence we have (see (29))

(31)
$\displaystyle{\operatorname{d}}\alpha=\sum_{i,j=1}^{2}p_{i}^{j}\zeta_{i}\wedge{\operatorname{d}}f_{j}+g{\operatorname{d}}f_{1}\wedge{\operatorname{d}}f_{2}\qquad\mod\mathrm{d}V.$

######  Lemma 5.3.

We may assume that

${\operatorname{d}}\alpha=\sum_{i=1}^{2}p_{i}\zeta_{i}\wedge{\operatorname{d}}f_{1}+g{\operatorname{d}}f_{1}\wedge{\operatorname{d}}f_{2}\qquad\mod\mathrm{d}V.$

###### Proof.

Note that we have the relations

$\displaystyle\mathrm{d}f_{1}\wedge\zeta_{1}-\mathrm{d}f_{2}\wedge\zeta_{2}={\operatorname{d}}(f_{1}\zeta_{1}-f_{2}\zeta_{2})\quad\text{
and
}\quad\mathrm{d}f_{2}\wedge\zeta_{1}+\mathrm{d}f_{1}\wedge\zeta_{2}={\operatorname{d}}(f_{2}\zeta_{1}+f_{1}\zeta_{2}).$

Hence for any $p,q\in{\mathbb{R}}[[f_{1},f_{2}]]$ we have:

$\displaystyle(p+f_{2}\partial_{y}p){\operatorname{d}}f_{2}\wedge\zeta_{2}-f_{1}\partial_{y}p{\operatorname{d}}f_{2}\wedge\zeta_{1}=(p+f_{1}\partial_{x}p){\operatorname{d}}f_{1}\wedge\zeta_{1}-f_{2}\partial_{x}p{\operatorname{d}}f_{1}\wedge\zeta_{2}\qquad\mod{\operatorname{d}}V,$
$\displaystyle(q+f_{2}\partial_{y}q){\operatorname{d}}f_{2}\wedge\zeta_{1}+f_{1}\partial_{y}q{\operatorname{d}}f_{2}\wedge\zeta_{2}=-(q+f_{1}\partial_{x}q){\operatorname{d}}f_{1}\wedge\zeta_{2}-f_{2}\partial_{x}q{\operatorname{d}}f_{1}\wedge\zeta_{1}\qquad\mod{\operatorname{d}}V.$

Adding those two equations, we want to determine the image of

$\begin{pmatrix}p\\\
q\end{pmatrix}\mapsto\begin{pmatrix}q-f_{1}\partial_{y}p+f_{2}\partial_{y}q\\\
p+f_{2}\partial_{y}p+f_{1}\partial_{y}q\\\ \end{pmatrix}.$

By looking at the maximal powers of $f_{2}$ in the image and inductively
decreasing, we obtain that the map is injective, and hence surjective, thus
proving the claim. ∎

Applying ${\operatorname{d}}$ to equation (31) yields

$0={\operatorname{d}}(\sum_{i=1}^{2}p_{i}\zeta_{i})\wedge{\operatorname{d}}f_{1}+{\operatorname{d}}g\wedge{\operatorname{d}}f_{1}\wedge{\operatorname{d}}f_{2}.$

To show that this implies (25) we first prove an auxiliary Lemma.

######  Lemma 5.4.

For any $g\in{\mathcal{R}}$ we have

${\mathcal{L}}_{T_{1}}g,{\mathcal{L}}_{T_{2}}g\in{\mathbb{R}}[[f_{1},f_{2}]]\quad\Rightarrow\quad{\mathcal{L}}_{T_{1}}g={\mathcal{L}}_{T_{2}}g=0.$

###### Proof.

Note that we can focus on even homogeneous degrees only. Hence we can view $g$
as an element in ring generated by

$\displaystyle
f_{1},\,f_{2},\,x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2},\,x_{1}^{2}-x_{2}^{2}-x_{3}^{2}+x_{4}^{2},\,x_{1}^{2}+x_{2}^{2}-x_{3}^{2}-x_{4}^{2},\qquad\qquad$
$\displaystyle
2(x_{1}x_{2}-x_{3}x_{4}),\,2(x_{1}x_{3}+x_{2}x_{4}),\,2(x_{1}x_{3}-x_{2}x_{4}),\,2(x_{1}x_{4}+x_{2}x_{3}),\,2(x_{1}x_{4}-x_{2}x_{3}).$

By a direct computation we can check that ${\mathcal{L}}_{T_{i}}$ maps
generators to generators. More generally, for a homogeneous monomial of degree
$n$ in the above variables we note that either the total degree $n_{f}$ in the
variables of $f_{1}$ and $f_{2}$ is preserved or the monomial is maped to
zero. Therefore it is enough to look at polynomomials in $f_{1}$ and $f_{2}$,
but on such ${\mathcal{L}}_{T_{i}}$ acts trivial. ∎

We use the Lemma to show the following.

######  Corollary 5.5.

Fix $j=1,2$. For $p_{1},p_{2}\in{\mathbb{R}}[[f_{1},f_{2}]]$ and
$g\in{\mathcal{R}}$ the condition

$\displaystyle\mathrm{d}g\wedge\mathrm{d}f_{1}\wedge\mathrm{d}f_{2}+\sum_{i=1}^{2}\mathrm{d}(p_{i}\zeta_{i})\wedge\mathrm{d}f_{j}=0$

implies $p_{1},p_{2}=0$ and $g\in{\mathbb{R}}[[f_{1},f_{2}]]$.

###### Proof.

Let $j=1$. Note that we have the relations

(32) $\displaystyle 4\iota_{T_{i}}\zeta_{1}=f_{i},\quad
4\iota_{T_{1}}\zeta_{2}=f_{2}\quad
4\iota_{T_{2}}\zeta_{2}=-f_{1}\qquad{\mathcal{L}}_{T_{i}}\zeta_{j}=0.$

Hence contracting with $T_{1}$ and $T_{2}$, respectively together with (9)
yields the equations

$\displaystyle
0=f_{1}\partial_{y}p_{1}+f_{2}\partial_{y}p_{2}+p_{2}+4\mathrm{d}g(T_{1})\quad\text{
and }\quad
0=f_{2}\partial_{y}p_{1}+p_{1}-f_{1}\partial_{y}p_{2}+4\mathrm{d}g(T_{2})$

and therefore
$\mathrm{d}g(T_{1}),\mathrm{d}g(T_{2})\in{\mathbb{R}}[[f_{1},f_{2}]]$. Hence
Lemma 5.4 implies ${\mathcal{L}}_{T_{i}}(g)=0$ and the above equations imply
$p_{1}=p_{2}=0$. The statement for $g$ then follows from (28). The proof for
$j=2$ follows along the same lines. ∎

######  Remark 5.6.

Note that the conclusion of Corollary 5.5 also holds under the assumption

$\displaystyle\mathrm{d}g\wedge\mathrm{d}f_{1}\wedge\mathrm{d}f_{2}+\sum_{i=1}^{2}p_{i}{\operatorname{d}}\zeta_{i}\wedge\mathrm{d}f_{j}=0$

The proof follows along the same lines.

###  5.2. Proof of Degree $3$

The argument consists of two steps which are proven below. Suppose that
$\gamma\in\Omega^{3}_{f}$ satisfies $\delta_{\pi}\gamma=0$ then we show the
following:

  1. Step 1:

Replacing $\gamma$ by $\gamma-q_{1}\epsilon_{1}-q_{2}\epsilon_{2}$ for some
$q_{1},q_{2}\in{\mathbb{R}}[[f_{1},f_{2}]]$ we may assume that

(33)
$\star^{-1}({\operatorname{d}}\gamma)=\iota_{\star^{-1}\gamma}{\operatorname{d}}f_{1}=\iota_{\star^{-1}\gamma}{\operatorname{d}}f_{2}=0.$

  2. Step 2:

If $\gamma$ satisfies (33) then there exists
$\alpha\in\ker\delta_{\pi}\subset\Omega^{1}_{f}$ such that

$\gamma={\operatorname{d}}f_{1}\wedge{\operatorname{d}}\alpha.$

Using (25) the last equation can be rewritten as

$\gamma={\operatorname{d}}f_{1}\wedge{\operatorname{d}}\big{(}f_{2}{\operatorname{d}}g+\sum_{i=1}^{2}p_{i}\zeta_{i}\big{)},$

for $g\in{\mathcal{R}}$ and $p_{1},p_{2}\in{\mathbb{R}}[[f_{1},f_{2}]]$, which
finishes the proof.

Proof of Step 1: Let $\gamma\in\Omega^{3}_{f}$ be such that
$m_{t}^{*}\gamma=t^{m+3}\gamma$ and $\delta_{\pi}\gamma=0$, i.e.

$\displaystyle
0=\delta_{\pi}\gamma=\iota_{\pi}({\operatorname{d}}\gamma)-{\operatorname{d}}\iota_{\star^{-1}\gamma}({\operatorname{d}}f_{1}\wedge{\operatorname{d}}f_{2}).$

Wedging the above equation with $\mathrm{d}f_{1}$ and $\mathrm{d}f_{2}$
respectively, using Proposition 3.2 and (28) implies

$\displaystyle\iota_{\star^{-1}\gamma}(\mathrm{d}f_{i})=p_{i}\in{\mathbb{R}}[[f_{1},f_{2}]],$

satisfying the equation

(34)
$\displaystyle\star^{-1}({\operatorname{d}}\gamma){\operatorname{d}}f_{1}\wedge{\operatorname{d}}f_{2}=\big{(}(\partial_{x}p_{1})(f_{1},f_{2})+(\partial_{y}p_{2})(f_{1},f_{2})\big{)}{\operatorname{d}}f_{1}\wedge{\operatorname{d}}f_{2}$

and hence in particular that
$\star^{-1}({\operatorname{d}}\gamma)\in{\mathbb{R}}[[f_{1},f_{2}]]$. Using
the relations (9)

$\displaystyle\delta(E_{1})=2,\quad\delta(E_{2})=0,\quad$

it is easy to see that $\epsilon_{i}=\star E_{i}\in\ker\delta_{\pi}$. Given
$q_{1},q_{2}\in{\mathbb{R}}[[f_{1},f_{2}]]$, the identities above also imply:

$\displaystyle\iota_{q_{1}E_{1}+q_{2}E_{2}}\mathrm{d}f_{1}=$ $\displaystyle\
(xq_{1}+yq_{2})(f_{1},f_{2}),$
$\displaystyle\iota_{q_{1}E_{1}+q_{2}E_{2}}\mathrm{d}f_{2}=$ $\displaystyle\
(yq_{1}-xq_{2})(f_{1},f_{2}).$

Hence, replacing $\gamma$ by $\gamma-q_{1}\epsilon_{1}-q_{2}\epsilon_{2}$, for
suitably chosen $q_{1}$ and $q_{2}$, we may assume that

$\displaystyle p_{1}(f_{1},f_{2})=p_{1}(f_{1})\quad\text{ and }\quad
p_{2}(f_{1},f_{2})=p_{2}(f_{1}).$

We want to show that $p_{1}=p_{2}=0$. Recall that be homogeneity of $\gamma$
and since $f_{1}$ and $f_{2}$ are homogeneous of degree $2$, we have $m=2n-1$
where $n\geq 1$ is the homogeneous degree of $p_{i}$ as a polynomial in one
variable. As such we can write

$p_{1}=c_{1}x^{n}\quad\text{and}\quad p_{2}=c_{2}x^{n},$

for some $c_{1},c_{2}\in{\mathbb{R}}$. We also write
$\gamma=\sum_{i=1}^{4}\gamma_{i}\star(\partial_{x_{i}})$. We denote by $a_{i}$
and $b_{i}$ respectively the coefficient of the
$x_{1}^{2n-1-i}x_{2}^{i}$-terms for $\gamma_{1}$ and $\gamma_{2}$, $0\leq
i\leq 2n-1$. With this notation, explicitly computing the
$x_{1}^{2n-i}x_{2}^{i}$-terms of Equation (34) yields the following
conditions:

$\displaystyle 2a_{0}=c_{1}\quad$ $\displaystyle\ \text{ and }\quad
2b_{0}=c_{2},$
$\displaystyle(-1)^{j}\binom{n}{n-j}c_{1}=2(a_{2j}+b_{2j-1})\quad$
$\displaystyle\ \text{ and
}\quad(-1)^{j}\binom{n}{n-j}c_{2}=2(a_{2j-1}+b_{2j})\quad$ $\displaystyle\
\text{ for }\ 1\leq j\leq n,$ $\displaystyle a_{2j+1}=-b_{2j}\quad$
$\displaystyle\ \text{ and }\quad a_{2j}=-b_{2j+1}\quad\text{ for }\ 0\leq
j\leq n-1,$ $\displaystyle 2b_{2n-1}=(-1)^{n-1}c_{1}\quad$ $\displaystyle\
\text{ and }\quad 2a_{2n-1}=(-1)^{n}c_{2}.$

Using the equations in lines 1-3 we obtain

$\displaystyle a_{2j+1}=-b_{2j}=c_{2}\frac{(-1)^{j}}{2}\sum_{0\leq l\leq
j}\binom{n}{l}\quad\text{ and }\quad
a_{2j}=-b_{2j+1}=c_{2}\frac{(-1)^{j}}{2}\sum_{0\leq l\leq j}\binom{n}{l}$

for $1\leq j\leq n-1$, and hence in particular that

$\displaystyle a_{2n-1}=c_{2}\frac{(-1)^{n-1}}{2}(2^{n}-1)\quad\text{ and
}\quad b_{2n-1}=c_{1}\frac{(-1)^{n}}{2}(2^{n}-1)$

which yields a contradiction for all $1\leq n$ unless $c_{1}=c_{2}=0$. This
completes the proof of Step 1.

Proof of Step 2: Assume $\gamma$ satisfies (33). Observe that for any
$\alpha\in\Omega^{1}_{f}$ we have

(35) $\displaystyle\star\circ\sharp\
(\iota_{\star^{-1}\gamma}(\alpha))=\alpha\wedge\gamma$

where $\sharp:\Omega^{0}_{f}\to\mathfrak{X}^{0}_{f}$ is the canonical
identification. Hence Corollary 3.11 implies that

$\displaystyle\gamma=\mathrm{d}f_{1}\wedge\beta$

for some
$\beta\in\mathcal{D}^{2}({\operatorname{d}}f_{1},{\operatorname{d}}f_{2})$
with $m_{t}^{*}\beta=t^{m+1}\beta$. Thus, it remains to show that
$\beta={\operatorname{d}}\alpha$ with $\alpha\in\ker\delta_{\pi}$. From
Proposition 4.3 we know that

$\displaystyle\beta=I(c,p,q)+\sum_{i=1}^{2}\mathrm{d}f_{i}\wedge\alpha_{i}$

for some
$c\in{\mathbb{R}},p\in{\mathbb{R}}[[x_{2}]],q\in{\mathbb{R}}[[x_{2},x_{4}]]^{2}$
and $\alpha_{i}\in\Omega^{1}({\mathcal{R}})$. Moreover, since we are
interested in $\gamma$ we may assume that $\alpha_{1}=0$, and rename
$\alpha_{2}$ to $\alpha$. Then Equation (33) implies:

(36)
$\displaystyle\mathrm{d}f_{1}\wedge\mathrm{d}I(c,p,q)=\mathrm{d}f_{1}\wedge\mathrm{d}f_{2}\wedge\mathrm{d}\alpha.$

Hence it suffices to show that

$\beta=\sum_{i=1}^{2}\mathrm{d}f_{i}\wedge\alpha_{i}\mod{\operatorname{d}}V.$

That is, we may assume $I(c,p,q)=0$. If $\beta$ has constant coefficients,
i.e. $m=1$, then this follows immediately from the fact that
$\mathrm{d}\zeta_{i}=\beta_{i}$. Hence let’s assume $\beta$ is homogeneous of
degree $m>1$. To see the statement, we recall that the coefficients of
${\operatorname{d}}f_{1}\wedge{\operatorname{d}}f_{2}$ are (by definition) in
the ideal ${\mathcal{J}}$ defined in (14), so that (36) is zero
$\operatorname{mod}{\mathcal{J}}$. This means the left hand side of (36)
satisfies the hypothesis of Lemma 3.18 taking $g\cdot\mu=0$, which in turn
implies that $x_{2}p+x_{4}q_{1},q_{2}\in R[[x_{2}^{2}+x_{4}^{2}]]$. Hence
$m=2n+1$ for $1\leq n$, otherwise we are done. It then follows from Corollary
4.6 that $[(x_{1}p+x_{3}q_{1})\beta_{2}]=[c_{1}f_{1}^{n-1}f_{2}\beta_{2}]$ and
$[q_{2}\beta_{2}]=[c_{2}f_{1}^{n}\beta_{2}]$ so that

$I(0,p,q)=[(c_{1}f_{1}^{n-1}f_{2}+c_{2}f_{1}^{n})\beta_{2}].$

Finally, since ${\operatorname{d}}\zeta_{i}=\beta_{i}$, and
${\operatorname{d}}(c_{1}f_{1}^{n-1}f_{2}+c_{2}f_{1}^{n})=0\
\operatorname{mod}{\mathcal{J}}$ we obtain

$I(0,p,q)=[{\operatorname{d}}((c_{1}f_{1}^{n-1}f_{2}+c_{2}f_{1}^{n})\zeta_{2})],$

as desired. Hence the proof is complete.

###  5.3. Proof of Degree $2$

Let $\beta\in\Omega^{2}_{f}$ such that

$\displaystyle\delta_{\pi}(\beta)=$
$\displaystyle\iota_{\star^{-1}\mathrm{d}\beta}(\mathrm{d}f_{1}\wedge\mathrm{d}f_{2})-\mathrm{d}\iota_{\pi}(\beta)=0.$

Applying ${\operatorname{d}}$ to the equation implies that
${\operatorname{d}}\beta\in\ker\delta_{\pi}$. By a direct computation we can
check that

$q\zeta_{1}\wedge\zeta_{2}\in\ker\delta_{\pi}$

for any $q\in{\mathbb{R}}[[f_{1},f_{2}]]$. Moreover, the relations

(37)
$\displaystyle{\operatorname{d}}f_{1}\wedge\zeta_{1}\wedge\zeta_{2}=f_{1}\epsilon_{2}-f_{2}\epsilon_{1},\quad{\operatorname{d}}f_{2}\wedge\zeta_{1}\wedge\zeta_{2}=f_{2}\epsilon_{2}+f_{1}\epsilon_{1}\quad\text{
and }\quad\mathrm{d}(\zeta_{1}\wedge\zeta_{2})=2\epsilon_{2}$

imply that we have

(38)
$\displaystyle{\operatorname{d}}(q\zeta_{1}\wedge\zeta_{2})=(2q+f_{1}\partial_{x}q+f_{2}\partial_{y}q)\epsilon_{2}+(f_{1}\partial_{y}q-f_{2}\partial_{x}q)\epsilon_{1}.$

Therefore, by replacing $\beta$ by $\beta-q\zeta_{1}\wedge\zeta_{2}$ and using
(27) we can assume that

${\operatorname{d}}\beta=\mathrm{d}f_{1}\wedge\mathrm{d}f_{2}\wedge\mathrm{d}g+q\epsilon_{1}+\mathrm{d}f_{1}\wedge\mathrm{d}\big{(}p_{i}\zeta_{i}\big{)}$

for some $q,p_{i}\in{\mathbb{R}}[[f_{1},f_{2}]]$ and $g\in{\mathcal{R}}$.
Taking the differential ${\operatorname{d}}$ of this equation yields

$0={\operatorname{d}}q\wedge\epsilon_{1}+2q\mu=(f_{1}\partial_{x}q+f_{2}\partial_{y}q+2q)\mu$

and hence $q=0$ and we can conclude that

$\beta=g\mathrm{d}f_{1}\wedge\mathrm{d}f_{2}+\mathrm{d}f_{1}\wedge
p_{i}\zeta_{i}+{\operatorname{d}}\alpha$

for $p_{i}\in{\mathbb{R}}[[f_{1},f_{2}]]$, $g\in{\mathcal{R}}$ and
$\alpha\in\Omega^{1}_{f}$. In other words, the map

$\displaystyle{\operatorname{d}}:\\{\beta=g\mathrm{d}f_{1}\wedge\mathrm{d}f_{2}+\mathrm{d}f_{1}\wedge
p_{i}\zeta_{i}+{\operatorname{d}}\alpha\\}\cap\ker\delta_{\pi}\to\\{\gamma\in\ker\delta_{\pi}|\
\gamma\text{ satisfies }\eqref{eq: transverse degree 1}\\}$

is surjective. Therefore, it is enough to study its kernel, i.e. we may assume
$\beta={\operatorname{d}}\alpha$ such that

$\delta_{\pi}(\beta)=\iota_{\star^{-1}\mathrm{d}\beta}(\mathrm{d}f_{1}\wedge\mathrm{d}f_{2})-\mathrm{d}\iota_{\pi}(\beta)=-\mathrm{d}\iota_{\pi}({\operatorname{d}}\alpha)=0.$

By Proposition 3.2 this implies that $\alpha\in\ker\delta_{\pi}$ and hence
(25) implies the statement.

##  6\. Poincare series

In this section we use the Hilbert-Poincare series to determine the dimension
of the Poisson homology groups. We achieve this by constructing short exact
sequences for the kernels of $\delta_{\pi}$ as described in Proposition 5.1.
We obtain the following result:

######  Proposition 6.1.

For the Poisson homology spaces we have the following Hilbert-Poincare series:

$\displaystyle HP_{H_{0}}(t)=$ $\displaystyle\
\frac{4t^{2}+4t+1}{(1-t^{2})^{2}},$ $\displaystyle HP_{H_{1}}(t)=$
$\displaystyle\ 4t\frac{t^{3}+t^{2}+2t+1}{(1-t^{2})^{2}},$ $\displaystyle
HP_{H_{2}}(t)=$ $\displaystyle\ 2t^{2}\frac{4t^{2}+2t+1}{(1-t^{2})^{2}}.$

In degree $1$ we have the following statement.

######  Lemma 6.2.

The following sequence is exact:

$\displaystyle
0\to{\mathbb{R}}[[f_{1},f_{2}]]\xrightarrow{\theta_{1}}{\mathcal{R}}(-2)^{2}\oplus{\mathcal{R}}\oplus{\mathbb{R}}[[f_{1},f_{2}]](-2)^{2}\xrightarrow{\beta}\ker\delta_{\pi,1}\to
0,$

where the maps are given by

$\displaystyle\theta_{1}:u\mapsto\left((-\partial_{f_{1}}u,-\partial_{f_{2}}u),u,0\right)\quad\text{
and
}\quad\sigma_{1}:((g_{1},g_{2}),g_{0},(p_{1},p_{2}))\mapsto{\operatorname{d}}g_{0}+\sum_{i}p_{i}(f_{1},f_{2})\zeta_{i}+g_{i}{\operatorname{d}}f_{i}.$

###### Proof.

Injectivity of $\theta_{1}$ follows from the definition, while surjectivity of
$\sigma_{1}$ follows from (25). Let

$\displaystyle\alpha:=\sigma_{1}(g_{1},g_{2},g_{0},p_{1},p_{2})=0.$

Wedging with $\mathrm{d}f_{1}$, applying $\mathrm{d}$ to both sides and using
Corollary 5.5 implies that $p_{1}=p_{2}=0$ and
$g_{2}\in{\mathbb{R}}[[f_{1},f_{2}]]$. Wedging with
$\mathrm{d}f_{1}\wedge\mathrm{d}f_{2}$ implies that
$g_{0}\in{\mathbb{R}}[[f_{1},f_{2}]]$ by Proposition 3.2 and (28). Hence the
statement follows by contraction with $E_{1}$ and $E_{2}$ by (9). ∎

In degree $2$ we obtain the following short exact sequences for the kernel.

######  Lemma 6.3.

The following sequence is exact:

$\displaystyle
0\to{\mathbb{R}}[[f_{1},f_{2}]](-2)^{2}\xrightarrow{\theta_{2}}{\mathcal{R}}(-2)^{2}\oplus{\mathcal{R}}(-4)\oplus{\mathbb{R}}[[f_{1},f_{2}]](-4)^{3}\oplus{\mathbb{R}}[[f_{1},f_{2}]](-2)^{2}\xrightarrow{\sigma_{2}}\ker\delta_{\pi,2}\to
0,$

where the maps are given by

$\displaystyle\
\theta_{2}:(u_{1},u_{2})\mapsto\left((u_{1},u_{2}),\partial_{f_{2}}u_{1}-\partial_{f_{1}}u_{2},0,0\right)\quad\text{
and }$ $\displaystyle\sigma_{2}:$ $\displaystyle\
((g_{1},g_{2}),g,(p,p_{1},p_{2}),(q_{1},q_{2}))\mapsto
p\zeta_{1}\wedge\zeta_{2}+g\mathrm{d}f_{1}\wedge\mathrm{d}f_{2}+\sum_{i=1}^{2}\mathrm{d}f_{1}\wedge
p_{i}\zeta_{i}+\mathrm{d}(q_{i}\zeta_{i})+\mathrm{d}g_{i}\wedge\mathrm{d}f_{i}.$

###### Proof.

Note that $\theta_{2}$ is injective, $\sigma_{2}$ is surjective by the
description of $\ker\delta_{\pi,2}$ in (26) and we have

$\displaystyle\operatorname{Im}\theta_{2}\subset\ker\sigma_{2}.$

Let us determine the kernel of $\sigma_{2}$. To do so, we set

$\displaystyle\beta:=\sigma_{2}((g_{1},g_{2}),g,(p,p_{1},p_{2}),(q_{1},q_{2})).$

We first note that due to (38), (9) and (12) we have

$\displaystyle\mathrm{d}\beta\wedge(f_{2}\mathrm{d}f_{1}-f_{1}\mathrm{d}f_{2})=2(f_{1}^{2}+f_{2}^{2})(2p+f_{1}\partial_{x}p+f_{2}\partial_{y}p)\mu,$

implying that $p=0$ if $\beta=0$. Wedging with $\mathrm{d}f_{1}$ and applying
Corollary 5.5 implies that $q_{1}=q_{2}=0$ and
$g_{2}\in{\mathbb{R}}[[f_{1},f_{2}]]$. Applying $\mathrm{d}$ and using again
Corollary 5.5 implies the statement. ∎

In degree $3$ we obtain the following short exact sequence for the kernel of
the differential.

######  Lemma 6.4.

The following sequence is exact:

$\displaystyle
0\to{\mathbb{R}}[[f_{1},f_{2}]](-4)\xrightarrow{\theta_{3}}{\mathcal{R}}(-4)\oplus{\mathbb{R}}[[f_{1},f_{2}]](-4)^{4}\xrightarrow{\sigma_{3}}\ker\delta_{\pi,3}\to
0,$

where the maps are given by:

$\displaystyle\theta_{3}:u\mapsto(u,0),\quad\text{ and
}\quad\sigma_{3}:(g,(q_{1},q_{2},p_{1},p_{2}))\mapsto\mathrm{d}f_{1}\wedge\mathrm{d}f_{2}\wedge\mathrm{d}g+\sum_{i=1}^{2}q_{i}\epsilon_{i}+\mathrm{d}f_{1}\wedge\mathrm{d}(p_{i}\zeta_{i}).$

###### Proof.

The map $\theta_{3}$ is injective by definition and surjectivity of
$\sigma_{3}$ follows from (27). To show that

$\displaystyle\operatorname{Im}\theta_{3}=\ker\sigma_{3}$

we define

$\displaystyle\gamma:=\sigma_{3}(g,(q_{1},q_{2},p_{1},p_{2})).$

Note that by (9) and (12) we have the relations

$\displaystyle(f_{1}\mathrm{d}f_{1}+f_{2}\mathrm{d}f_{2})\wedge\gamma=2(f_{1}^{2}+f_{2}^{2})q_{1}\mu,\quad\text{
and
}\quad(f_{2}\mathrm{d}f_{1}-f_{1}\mathrm{d}f_{2})\wedge\gamma=2(f_{1}^{2}+f_{2}^{2})q_{2}\mu.$

Hence $\gamma=0$ implies that $q_{1}=q_{2}=0$. Therefore Corollary 5.5 implies
the statement. ∎

Now we are ready to proof Proposition 6.1.

###### Proof of Proposition 6.1.

We use the exact sequence of degree preserving maps

(39) $\displaystyle
0\to\ker\delta_{\pi,i+1}\hookrightarrow\Omega^{i+1}_{f}\xrightarrow{\delta_{\pi,i+1}}\ker\delta_{\pi,i}\to
H_{i}\to 0.$

Let us first compute the Hilbert-Poincare series of the kernels of
$\delta_{\pi,i}$. Therefore we note that

$\displaystyle HP_{{\mathcal{R}}(-n)}=\frac{t^{n}}{(1-t)^{4}}\quad\text{ and
}\quad HP_{{\mathbb{R}}[[f_{1},f_{2}]](-n)}=\frac{t^{n}}{(1-t^{2})^{2}}$

for all $n\in\mathbb{N}_{0}$. For any two graded vector spaces $U$ and $V$ we
have that

$\displaystyle HP_{U\oplus V}=HP_{U}+HP_{V},$

and for any exact sequence of graded vector spaces

$\displaystyle 0\to V_{0}\to\dots\to V_{k}\to 0$

with degree preserving maps in between, that

$\displaystyle HP_{V_{k}}=\sum_{i=0}^{k-1}(-1)^{k-1-i}HP_{V_{i}}.$

From Lemma 6.4 we obtain

$\displaystyle
HP_{\ker\delta_{\pi,3}}(t)=\frac{t^{4}}{(1-t^{2})^{2}}+\frac{t^{4}}{(1-t)^{4}}.$

Similarly, Lemma 6.3 and Lemma 6.2 imply

$\displaystyle
HP_{\ker\delta_{\pi,2}}(t)=\frac{3t^{4}}{(1-t^{2})^{2}}+\frac{t^{4}+2t^{2}}{(1-t)^{4}}\quad\text{
and }\quad
HP_{\ker\delta_{\pi,1}}(t)=\frac{2t^{2}-1}{(1-t^{2})^{2}}+\frac{2t^{2}+1}{(1-t)^{4}}.$

Additionally, note that by Example 3.8 we have

$\displaystyle
HP_{\ker\delta_{\pi,0}}(t)=HP_{\Omega^{0}_{f}}(t)=\frac{1}{(1-t)^{4}},$
$\displaystyle\ \qquad HP_{\Omega^{1}_{f}}(t)=\frac{4t}{(1-t)^{4}},$
$\displaystyle HP_{\Omega^{2}_{f}}(t)=\frac{6t^{2}}{(1-t)^{4}},$
$\displaystyle\ \qquad HP_{\Omega^{3}_{f}}(t)=\frac{4t^{3}}{(1-t)^{4}}.$

Hence (39) for $i=2$ implies that

$\displaystyle HP_{H_{2}}(t)=$ $\displaystyle\
HP_{\ker\delta_{\pi,2}}(t)+HP_{\ker\delta_{\pi,3}}(t)-HP_{\Omega^{3}_{f}}(t)$
$\displaystyle=$ $\displaystyle\
\frac{3t^{4}}{(1-t^{2})^{2}}+\frac{t^{4}+2t^{2}}{(1-t)^{4}}+\frac{t^{4}}{(1-t^{2})^{2}}+\frac{t^{4}}{(1-t)^{4}}-\frac{4t^{3}}{(1-t)^{4}}$
$\displaystyle=$ $\displaystyle\
\frac{4t^{4}}{(1-t^{2})^{2}}+2t^{2}\frac{(1-t)^{2}}{(1-t)^{4}}\ =\
2t^{2}\frac{4t^{2}+4t+1}{(1-t^{2})^{2}}.$

For $i=1$ we get

$\displaystyle HP_{H_{1}}(t)=$ $\displaystyle\
HP_{\ker\delta_{\pi,1}}(t)+HP_{\ker\delta_{\pi,2}}(t)-HP_{\Omega^{2}_{f}}(t)$
$\displaystyle=$ $\displaystyle\
\frac{2t^{2}-1}{(1-t^{2})^{2}}+\frac{2t^{2}+1}{(1-t)^{4}}+\frac{3t^{4}}{(1-t^{2})^{2}}+\frac{t^{4}+2t^{2}}{(1-t)^{4}}-\frac{6t^{2}}{(1-t)^{4}}$
$\displaystyle=$ $\displaystyle\
\frac{3t^{4}+2t^{2}-1}{(1-t^{2})^{2}}+\frac{(1-t)^{2}(1+t)^{2}}{(1-t)^{4}}\ =\
4t\frac{t^{3}+t^{2}+2t+1}{(1-t^{2})^{2}}.$

Finally, for $i=0$ we compute

$\displaystyle HP_{H_{0}}(t)=$ $\displaystyle\
HP_{\ker\delta_{\pi,0}}(t)+HP_{\ker\delta_{\pi,1}}(t)-HP_{\Omega^{1}_{f}}(t)$
$\displaystyle=$ $\displaystyle\
\frac{1}{(1-t)^{4}}+\frac{2t^{2}-1}{(1-t^{2})^{2}}+\frac{2t^{2}+1}{(1-t)^{4}}-\frac{4t}{(1-t)^{4}}$
$\displaystyle=$ $\displaystyle\
\frac{2t^{2}-1}{(1-t^{2})^{2}}+2\frac{(1-t)^{2}}{(1-t)^{4}}\ =\
\frac{4t^{2}+4t+1}{(1-t^{2})^{2}}.$

∎

##  7\. Proof of the main theorem

In this section we prove Theorem 1.2. In each degree the strategy of the proof
is the same: by Proposition 6.1 the given set of generators has the right
dimension. Hence it is enough to show that none of the elements generated by
the given module are in the image of the Poisson differential.

###  7.1. Degree 0

For the proof in degree $0$ we need the following auxiliary statement.

######  Lemma 7.1.

Let $g\in{\mathbb{R}}[[f_{1},f_{2}]]$,
$(c,p,(q_{1},q_{2}))\in{\mathbb{R}}\oplus{\mathbb{R}}[[x_{2}]]\oplus{\mathbb{R}}[[x_{2},x_{4}]]^{2}$
and $\alpha\in\Omega^{1}_{f}$ such that:

$\displaystyle g\cdot\mu=$
$\displaystyle\,\mathrm{d}f_{1}\wedge\mathrm{d}I(c,p,(q_{1},q_{2}))+\mathrm{d}f_{1}\wedge\mathrm{d}f_{2}\wedge\mathrm{d}\alpha.$

Then $g=0$ and the same statement holds if we exchange the roles of
$\mathrm{d}f_{1}$ and $\mathrm{d}f_{2}$.

###### Proof.

We prove the statement for $g$ being homogeneous of degree $2n$ by induction
on $n$.

$n=0,1$ : If $g\in{\mathbb{R}}$ then the statement holds by a degree count of
the coefficients. If $g$ is homogeneous of degree $1$ then the statement
follows from Lemma 3.18.

Induction step: Assume that $g\in{\mathbb{R}}[[f_{1},f_{2}]]$ is homogeneous
of degree $2n$. Then Lemma 3.18 implies

$x_{2}p+x_{4}q_{1},q_{2}\in{\mathbb{R}}\cdot(x_{2}^{2}+x_{4}^{2})^{n-1},\qquad\text{
and }\qquad g\in(f_{1}^{2}+f_{2}^{2}){\mathbb{R}}[[f_{1},f_{2}]].$

Hence, by Corollary 4.6 we have:

$\displaystyle
I(c,p,(q_{1},q_{2}))=(c_{1}f_{1}^{n-1}+c_{2}f_{1}^{n-2}f_{2})\beta_{2}+\sum_{i=1}^{2}\mathrm{d}f_{i}\wedge\alpha_{i},$

for some $c_{i}\in{\mathbb{R}}$ and $\alpha_{i}\in\Omega^{1}_{f}$, with
$i=1,2$. Therefore, writing $g$ as $(f_{1}^{2}+f_{2}^{2})\widetilde{g}$ and
replacing $\alpha$ by $\alpha-\alpha_{2}$, again denoted by $\alpha$, the
original equation becomes

$\displaystyle(f_{1}^{2}+f_{2}^{2})\widetilde{g}\cdot\mu=$
$\displaystyle\,\mathrm{d}f_{1}\wedge\mathrm{d}f_{2}\wedge\mathrm{d}\alpha.$

Using (9), (12) and (37) we note that

$\displaystyle
2g\mathrm{d}f_{1}\wedge\mathrm{d}f_{2}\wedge\zeta_{1}\wedge\zeta_{2}=(f_{1}^{2}+f_{2}^{2})g\cdot\mu.$

Hence combining the two equations above, we obtain by Proposition 4.3 that

$\displaystyle
2\widetilde{g}\zeta_{1}\wedge\zeta_{2}-\mathrm{d}\alpha=I(0,\widetilde{p},(\widetilde{q}_{1},\widetilde{q}_{2}))+\sum_{i=1}^{2}\alpha_{i}\wedge\mathrm{d}f_{i}$

for some
$(\widetilde{p},(\widetilde{q}_{1},\widetilde{q}_{2}))\in{\mathbb{R}}[[x_{2}]]\times{\mathbb{R}}[[x_{2},x_{4}]]^{2}$
and $\alpha_{i}\in\Omega^{1}_{f}$. Note that the homogeneous degrees of
$\widetilde{p},\widetilde{q}_{1},\widetilde{q}_{2}$ and $\alpha_{i}$ are two
smaller than those of $p,q_{1},q_{2}$ and $\alpha$, respectively. Now taking
the de-Rham differential and wedging with $\mathrm{d}f_{1}$ implies by (9),
(12) and (37) that

$\displaystyle
2((f_{1}^{2}+f_{2})^{2}\partial_{y}\widetilde{g}+f_{2}\widetilde{g})\cdot\mu=\mathrm{d}f_{1}\wedge\mathrm{d}I(0,\widetilde{p},(\widetilde{q}_{1},\widetilde{q}_{2}))+\mathrm{d}f_{1}\wedge\mathrm{d}f_{2}\wedge\mathrm{d}\alpha_{2}.$

Therefore, the induction hypothesis implies that

$\displaystyle(f_{1}^{2}+f_{2})^{2}\partial_{y}\widetilde{g}+f_{2}\widetilde{g}=0,$

and hence $\widetilde{g}=0$ concluding the induction. The proof for
$\mathrm{d}f_{2}$ follows along the same lines. ∎

Now we are ready to prove Theorem 1.2 for degree $0$. We distinguish between
even and odd homogeneous degrees.

The odd degree: The zeroth Poisson homology group is realized by the free
${\mathbb{R}}[[x_{2}^{2},x_{4}^{2}]]$-module generated by the linear
monomials. This follows from the description of a boundary by Proposition 3.2
and Lemma 3.17.

The even degree: By a similar argument as in the odd degree we obtain that the
coefficients in even degree are contained in the ideal ${\mathcal{I}}$ if and
only if they are of the form

$\displaystyle(f_{1}^{2}+f_{2}^{2})p$

where $p\in{\mathbb{R}}[[f_{1},f_{2}]]$. Hence we only need to check if there
exists an $\alpha\in\Omega^{1}({\mathcal{R}})$ such that

$\displaystyle(f_{1}^{2}+f_{2}^{2})p\mu=\mathrm{d}f_{1}\wedge\mathrm{d}f_{2}\wedge\mathrm{d}\alpha$

We show that such an $\alpha$ can only exist if $p=0$. This follows from Lemma
7.1.

###  7.2. Degree 1

We first prove another auxiliary lemma.

######  Lemma 7.2.

Let $g\in{\mathbb{R}}[[f_{1},f_{2}]]$ and
$a_{i}\in{\mathbb{R}}[[x_{2}^{2},x_{4}]]$ and $\gamma\in\Omega^{3}_{f}$ such
that

$0=\left(g-\star^{-1}({\operatorname{d}}\gamma)\right)\mathrm{d}f_{1}\wedge\mathrm{d}f_{2}+\mathrm{d}\left(\left(\sum_{i=1}^{4}a_{i}x_{i}-\iota_{\star^{-1}\gamma}({\operatorname{d}}f_{2})\right)\mathrm{d}f_{1}+\iota_{\star^{-1}\gamma}({\operatorname{d}}f_{1})\mathrm{d}f_{2}\right).$

Then $g=a_{1}=a_{2}=a_{3}=a_{4}=0$ and $\gamma\in\ker\delta_{\pi}$.

###### Proof.

Note that by wedging with $\mathrm{d}f_{1}$, Proposition 3.2 and (28) imply

$\iota_{\star^{-1}\gamma}({\operatorname{d}}f_{1})\in{\mathbb{R}}[[f_{1},f_{2}]].$

Using (35), (9) and Corollary 3.11 we obtain that

$\displaystyle\gamma=r_{1}\epsilon_{1}+r_{2}\epsilon_{2}+\mathrm{d}f_{1}\wedge\beta$

for unique $r_{1}\in{\mathbb{R}}[[f_{1},f_{2}]]$,
$r_{2}\in{\mathbb{R}}[[f_{2}]]$ and some $\beta\in\Omega^{2}_{f}$. Wedging
with $\mathrm{d}f_{2}$ and using (9), (35), Proposition 3.2 and (28) implies

$\displaystyle\left(\sum_{i=1}^{4}a_{i}x_{i}\right)\mu+\beta\wedge\mathrm{d}f_{1}\wedge\mathrm{d}f_{2}\in{\mathbb{R}}[[f_{1},f_{2}]]\mu.$

Hence Lemma 3.17 yields

$\displaystyle a_{1}=a_{2}=a_{3}=a_{4}=0\qquad\text{ and
}\qquad\beta\wedge\mathrm{d}f_{1}\wedge\mathrm{d}f_{2}\in(f_{1}^{2}+f_{2}^{2}){\mathbb{R}}[[f_{1},f_{2}]]\mu.$

This implies that

$\beta=r\zeta_{1}\wedge\zeta_{2}+\widetilde{\beta}+{\operatorname{d}}f_{1}\wedge\alpha_{1}+{\operatorname{d}}f_{2}\wedge\alpha_{2},$

for some $r\in{\mathbb{R}}[[f_{1},f_{2}]]$,
$\alpha_{1},\alpha_{2}\in\Omega^{1}_{f}$ and $\widetilde{\beta}$ representing
a non-trivial class in
$\mathcal{D}^{2}({\operatorname{d}}f_{1},{\operatorname{d}}f_{2})$. Note that

$\displaystyle
r_{1}\epsilon_{1}+r_{2}\epsilon_{2}+r{\operatorname{d}}f_{1}\wedge\zeta_{1}\wedge\zeta_{2}\in\ker\delta_{\pi}.$

Hence it is enough to show

$\displaystyle\gamma:=\mathrm{d}f_{1}\wedge(\widetilde{\beta}+{\operatorname{d}}f_{2}\wedge\alpha_{2})\in\ker\delta_{\pi}.$

In this case, the original equation is equivalent to

$\displaystyle
g\cdot\mu=\mathrm{d}f_{1}\wedge\mathrm{d}\widetilde{\beta}+\mathrm{d}f_{1}\wedge\mathrm{d}f_{2}\wedge\mathrm{d}\alpha_{2}.$

Using Lemma 7.1 we obtain $g=0$, which concludes the proof. ∎

To prove the statement in degree one, it is, by a dimension count and
Proposition 6.1, enough to show that for any non-trivial choice of
coefficients $a_{j},b_{j}\in{\mathbb{R}}[[x_{2}^{2},x_{4}]]$ and
$q_{i},p_{i}\in{\mathbb{R}}[[f_{1},f_{2}]]$, the $1$-cycle

$\displaystyle\alpha:=\mathrm{d}(\sum_{j=1}^{4}a_{j}x_{j})+\sum_{j=1}^{4}b_{j}x_{j}\mathrm{d}f_{1}+\sum_{i=1}^{2}q_{i}\mathrm{d}f_{i}+p_{i}\zeta_{i}$

is not a boundary. That is, for all $\beta\in\Omega^{2}_{f}$ we have

$\widetilde{\alpha}:=\alpha-\delta_{\pi}(\beta)=0\quad\Rightarrow\quad
a_{j}=b_{j}=q_{i}=p_{i}=0.$

If we take the differential of $\widetilde{\alpha}=0$ we obtain:

$\displaystyle
0=\mathrm{d}\widetilde{\alpha}={\operatorname{d}}\left(\sum_{i=1}^{2}q_{i}\mathrm{d}f_{i}+p_{i}\zeta_{i}\right)+\mathrm{d}\left(\sum_{j=1}^{4}b_{j}x_{j}+\iota_{\star^{-1}{\operatorname{d}}\beta}({\operatorname{d}}f_{2})\right)\wedge\mathrm{d}f_{1}-{\operatorname{d}}\iota_{\star^{-1}{\operatorname{d}}\beta}({\operatorname{d}}f_{1})\wedge\mathrm{d}f_{2}.$

Wedging with ${\operatorname{d}}f_{1}$, Corollary 5.5 implies that $p_{i}=0$.
Applying Lemma 7.2 for $\gamma=-{\operatorname{d}}\beta$ implies that
$b_{j}=0$ for $j=1,2,3,4$, $\partial_{x}q_{2}=\partial_{y}q_{1}$ and
${\operatorname{d}}\beta\in\ker\delta_{\pi}$. The arguments from Section 5.3
give

$\displaystyle\beta=r\zeta_{1}\wedge\zeta_{2}+g\mathrm{d}f_{1}\wedge\mathrm{d}f_{2}+\mathrm{d}f_{1}\wedge
p_{i}\zeta_{i}+{\operatorname{d}}\overline{\alpha},$

for $r,p_{i}\in{\mathbb{R}}[[f_{1},f_{2}]]$, $g\in{\mathcal{R}}$ and
$\overline{\alpha}\in\Omega^{1}_{f}$. Note all terms except
${\operatorname{d}}\overline{\alpha}$ are in $\ker\delta_{\pi}$. Therefore, we
have

$\displaystyle
0=\widetilde{\alpha}=q_{1}{\operatorname{d}}f_{1}+q_{2}{\operatorname{d}}f_{2}+{\operatorname{d}}\left(\iota_{\star^{-1}{\operatorname{d}}\overline{\alpha}}({\operatorname{d}}f_{1}\wedge{\operatorname{d}}f_{2})+\sum_{j=1}^{4}a_{j}x_{j}\right).$

Wedging the expression with
${\operatorname{d}}f_{1}\wedge{\operatorname{d}}f_{2}$ yields

${\operatorname{d}}\left(\iota_{\star^{-1}{\operatorname{d}}\overline{\alpha}}({\operatorname{d}}f_{1}\wedge{\operatorname{d}}f_{2})+\sum_{j=1}^{4}a_{j}x_{j}\right)\wedge{\operatorname{d}}f_{1}\wedge{\operatorname{d}}f_{2}=0.$

By Proposition 3.2, (28) and (35) this is equivalent to

${\operatorname{d}}\overline{\alpha}\wedge{\operatorname{d}}f_{1}\wedge{\operatorname{d}}f_{2}+\left(\sum_{j=1}^{4}a_{j}x_{j}\right)\mu\in{\mathbb{R}}[[f_{1},f_{2}]]\mu.$

Then, Lemma 3.17 implies $a_{j}=0$ for $j=1,2,3,4$ and by Lemma 7.1 we have

$\star\circ\sharp\left(\iota_{\star^{-1}{\operatorname{d}}\overline{\alpha}}({\operatorname{d}}f_{1}\wedge{\operatorname{d}}f_{2})\right)={\operatorname{d}}\overline{\alpha}\wedge{\operatorname{d}}f_{1}\wedge{\operatorname{d}}f_{2}=0.$

Hence $\widetilde{\alpha}=0$ implies $q_{1}=q_{2}=0$ which completes the
proof.

###  7.3. Degree 2

We proceed similar as in the previous cases. Consider

$\displaystyle\beta=$ $\displaystyle\
p\zeta_{1}\wedge\zeta_{2}+q{\operatorname{d}}f_{1}\wedge{\operatorname{d}}f_{2}+\sum_{i=1}^{2}p_{i}{\operatorname{d}}(f_{1}\zeta_{i})+q_{i}{\operatorname{d}}\zeta_{i}+{\operatorname{d}}\left(\sum_{j=1}^{4}a_{j}x_{j}\right)\wedge\mathrm{d}f_{1},$

where $p,q,p_{i},q_{i}\in{\mathbb{R}}[[f_{1},f_{2}]]$ and
$a_{j}\in{\mathbb{R}}[[x_{2}^{2},x_{4}]]$. We want to show that

$\displaystyle
0=\widetilde{\beta}:=\beta-\delta_{\pi}(\gamma)\quad\Rightarrow\quad
p=q=p_{i}=q_{i}=a_{j}=0$

Replicating the argument for Lemma 6.3, replacing Corollary 5.5 with Remark
5.6 we obtain

$p=p_{1}=p_{2}=q_{1}=q_{2}=0\quad\text{ and
}\quad\iota_{\star^{-1}\gamma}({\operatorname{d}}f_{1}),\star^{-1}{\operatorname{d}}\gamma\in{\mathbb{R}}[[f_{1},f_{2}]].$

As such we are left with

$0=\widetilde{\beta}=q{\operatorname{d}}f_{1}\wedge{\operatorname{d}}f_{2}-\iota_{\pi}({\operatorname{d}}\gamma)+{\operatorname{d}}\left(\iota_{\star^{-1}\gamma}({\operatorname{d}}f_{2})+\sum_{j=1}^{4}a_{j}x_{j}\right)\wedge{\operatorname{d}}f_{1}-{\operatorname{d}}\iota_{\star^{-1}\gamma}({\operatorname{d}}f_{1})\wedge{\operatorname{d}}f_{2}.$

Hence Lemma 7.2 implies $q=a_{j}=0$. Counting the elements and comparing them
to Proposition 6.1 implies that we have a representative for every cohomology
class.

###  7.4. Degree 3

By Proposition 3.2, elements in the image of the differential have the form

$\displaystyle\mathrm{d}f_{1}\wedge\mathrm{d}f_{2}\wedge\mathrm{d}g,$

for $g\in{\mathcal{R}}$. Hence we can argue as in the proof of Lemma 6.4.

##  8\. Proof of Corollary 1.4

A Poisson structure coming from a deformation of the volume form is equals
$g\pi$ for some $g\in{\mathcal{R}}$ with positive constant term. Using (3)
this corresponds to the $2$-form
$g{\operatorname{d}}f_{1}\wedge{\operatorname{d}}f_{2}$.

From the definition of the differential we see that

$g{\operatorname{d}}f_{1}\wedge{\operatorname{d}}f_{2}\ \in\ker\delta_{\pi}$

By Proposition 3.2 and Theorem 1.2 we can write

$g{\operatorname{d}}f_{1}\wedge{\operatorname{d}}f_{2}=\beta+\iota_{\pi}({\operatorname{d}}\gamma)-{\operatorname{d}}\iota_{\star^{-1}\gamma}({\operatorname{d}}f_{1}\wedge{\operatorname{d}}f_{2})$

for some $\beta=\beta(a_{i},p,p_{j},q,q_{j})$ representing a class in the
second Poisson homology according to Theorem 1.2 and
$\gamma\in\Omega^{3}_{f}$. Following the first part of the argument from
subsection 7.3 we can conclude that $p=p_{i}=q_{i}=0$. Using the first part of
the proof of Lemma 7.2 we can conclude that $a_{i}=0$ and

$\displaystyle\gamma=r_{1}\epsilon_{1}+r_{2}\epsilon_{2}+\mathrm{d}f_{1}\wedge(r\zeta_{1}\wedge\zeta_{2}+\widetilde{\beta}+{\operatorname{d}}f_{2}\wedge\alpha)$

for unique $r_{1}\in{\mathbb{R}}[[f_{1},f_{2}]]$,
$r_{2}\in{\mathbb{R}}[[f_{2}]]$, some $r\in{\mathbb{R}}[[f_{1},f_{2}]]$,
$\alpha\in\Omega^{1}_{f}$ and where $\widetilde{\beta}$ represents a non-
trivial class in
$\mathcal{D}^{2}({\operatorname{d}}f_{1},{\operatorname{d}}f_{2})$. Note that
this implies in particular, that

$g{\operatorname{d}}f_{1}\wedge{\operatorname{d}}f_{2}=q{\operatorname{d}}f_{1}\wedge{\operatorname{d}}f_{2}+\delta_{\pi}(\gamma)$

for some $q\in{\mathbb{R}}[[f_{1},f_{2}]]$ and some $\gamma\in\Omega^{3}_{f}$
such that

$\iota_{\star^{-1}\gamma}({\operatorname{d}}f_{i})=0.$

In other words, we can find a vector field $X\in\mathfrak{X}_{f,2}^{1}$ (see
(11)) and $q\in{\mathbb{R}}[[f_{1},f_{2}]]$ satisfying

$g\pi=q\pi+{\operatorname{d}}_{\pi}(X).$

This allows us to prove the following which implies Corollary 1.4.

######  Claim 8.1.

Given $g\in{\mathcal{R}}$ with positive constant term, there exists a formal
diffeomorphism $\phi$ satisfying

$\phi^{*}(g\pi)=p\pi$

for some $p\in{\mathbb{R}}[[f_{1},f_{2}]]$ with positive constant term.

To see this, first observe that the flow $\phi_{t}$ of $X$ satisfies:

(40)
$\phi_{1}^{*}(g\pi)-g\pi=\int_{0}^{1}\frac{{\operatorname{d}}}{{\operatorname{d}}t}\phi_{t}^{*}(g\pi){\operatorname{d}}t=[X,g\pi]=-g{\operatorname{d}}_{\pi}X+({\mathcal{L}}_{X}g)\pi.$

Let us write $g$ as $g=\sum_{i=0}^{\infty}g_{i}$, where $g_{i}$ is homogeneous
of degree $i$ and $g_{0}>0$. By induction on $i$ we show that if
$g_{j}\in{\mathbb{R}}[[f_{1},f_{2}]]$ for all $j<i$ then we can change $g\pi$
such that also $g_{i}\in{\mathbb{R}}[[f_{1},f_{2}]]$.

We start with $i=1$. From cohomology statement we can find $\widetilde{X}_{1}$
homogeneous of degree $2$ such that

(41) $g_{1}\pi={\operatorname{d}}_{\pi}\widetilde{X}_{1}$

Taking $X_{1}:=\frac{1}{g}\widetilde{X}_{1}$ we obtain from (40) that:

$\phi_{1}^{*}(g\pi)-g\pi=-g{\operatorname{d}}_{\pi}X_{1}+\mathcal{O}(|x|^{4})=-{\operatorname{d}}_{\pi}\widetilde{X}_{1}+\mathcal{O}(|x|^{4})$

Hence we may assume $g_{1}=0$. In even degrees (41) will be of the form

$g_{2j}\pi=q_{2j}\pi+{\operatorname{d}}_{\pi}\widetilde{X}_{2j}$

for some $q_{2j}\in{\mathbb{R}}[[f_{1},f_{2}]]$ contributing to $q$. Repeating
the argument for increasing $i\in\mathbb{N}$ we map $g\pi$ to $q\pi$.

##  9\. Proof of Corollary 1.5

In degree zero, the result follows from Theorem 1.2 as ${\operatorname{d}}$
maps representatives to representatives.

In degree four, we note that for the image of ${\operatorname{d}}$ we have by
(8) that

$\displaystyle{\operatorname{d}}\left(\sum_{i=1}^{2}p_{i}\zeta_{2}\wedge{\operatorname{d}}\zeta_{i}+q_{i}{\operatorname{d}}f_{1}\wedge{\operatorname{d}}\zeta_{i}\right)=$
$\displaystyle\,2p_{2}\mu+(\partial_{x}p_{1}+\partial_{y}p_{2}){\operatorname{d}}f_{1}\wedge\zeta_{2}\wedge{\operatorname{d}}\zeta_{1}+(\partial_{x}p_{2}-\partial_{y}p_{1}){\operatorname{d}}f_{1}\wedge\zeta_{2}\wedge{\operatorname{d}}\zeta_{2}$
$\displaystyle=$
$\displaystyle\,(2p_{2}+f_{2}(\partial_{x}p_{1}+\partial_{y}p_{2})+f_{1}(\partial_{x}p_{2}-\partial_{y}p_{1}))\mu$

which implies that ${\operatorname{d}}$ is surjective in homology.

In degree three, note that from the discussion in degree four, the kernel of
${\operatorname{d}}$ is parameterized by

$q_{1},q_{2},p_{1}\in{\mathbb{R}}[[f_{1},f_{2}]]$

since the differential operator $2+x\partial_{x}+y\partial_{y}$ is invertible
on ${\mathbb{R}}[[x,y]]$. To determine the image of ${\operatorname{d}}$ let
$\beta=\beta(a_{i},p,p_{j},q,q_{j})$ be a representative of a class in the
second Poisson homology. We can write $\beta$ as

$\beta:=p\zeta_{1}\wedge\zeta_{2}+q{\operatorname{d}}f_{1}\wedge{\operatorname{d}}f_{2}+\sum_{i=1}^{2}p_{i}{\operatorname{d}}f_{1}\wedge\zeta_{i}+q_{i}{\operatorname{d}}\zeta_{i}+\sum_{i=1}^{4}{\operatorname{d}}(a_{i}x_{i})\wedge{\operatorname{d}}f_{1}$

for $p,p_{i},q,q_{i}\in{\mathbb{R}}[[f_{1},f_{2}]]$ and
$a_{i}\in{\mathbb{R}}[[x_{2}^{2},x_{4}]]$. Applying ${\operatorname{d}}$ and
using (7) and (8) yields

$\displaystyle{\operatorname{d}}\beta=\left(\partial_{x}p{\operatorname{d}}f_{1}+\partial_{y}p{\operatorname{d}}f_{2}\right)\wedge\zeta_{1}\wedge\zeta_{2}+2p\zeta_{2}\wedge{\operatorname{d}}\zeta_{1}+\sum_{i=1}^{2}\partial_{y}p_{i}{\operatorname{d}}f_{2}\wedge{\operatorname{d}}f_{1}\wedge\zeta_{i}$
$\displaystyle+(\partial_{x}q_{1}+\partial_{y}q_{2}-p_{1}){\operatorname{d}}f_{1}\wedge{\operatorname{d}}\zeta_{1}+(\partial_{x}q_{2}-\partial_{y}q_{1}-p_{2}){\operatorname{d}}f_{1}\wedge{\operatorname{d}}\zeta_{2}\quad$

Let
$\widetilde{q}_{1},\widetilde{q}_{2},\widetilde{p}_{1}\in{\mathbb{R}}[[f_{1},f_{2}]]$
and

$\widetilde{p}_{2}:=-(2+x\partial_{x}+y\partial_{y})^{-1}(y\partial_{x}-x\partial_{y})\widetilde{p}_{1}.$

Let $\gamma=\gamma(\widetilde{p}_{i},\widetilde{q}_{i})$ be the corresponding
representative of a class in $\ker{\operatorname{d}}$. We want to determine
for which $p,p_{i},q,q_{i},a_{i}$ as above and $g\in{\mathcal{R}}$ we have

$0=\gamma+{\operatorname{d}}\beta+{\operatorname{d}}g\wedge{\operatorname{d}}f_{1}\wedge{\operatorname{d}}f_{2}.$

Wedging with ${\operatorname{d}}f_{1}$ and ${\operatorname{d}}f_{2}$
respectively, yields

$\displaystyle(f_{1}^{2}+f_{2}^{2})\partial_{y}p+2f_{2}p+f_{1}\widetilde{p}_{2}+f_{2}\widetilde{p}_{1}=0=(f_{1}^{2}+f_{2}^{2})\partial_{x}p+2f_{2}p-f_{2}\widetilde{p}_{2}+f_{1}\widetilde{p}_{1}$

and hence we obtain

$-\widetilde{p}_{1}=2p+f_{1}\partial_{x}p+f_{2}\partial_{y}p.$

Therefore, we may assume $\widetilde{p_{1}}=\widetilde{p_{2}}=p=0$.
Contracting the remaining equation with $T_{1}$ and $T_{2}$ respectively,
yields using (32) that

$\displaystyle
0=\partial_{x}q_{2}-\partial_{y}q_{1}-p_{2}-f_{1}\partial_{y}p_{1}-f_{2}\partial_{y}p_{2}+\widetilde{q}_{1}+4{\operatorname{d}}g(T_{1})$
$\displaystyle
0=\partial_{x}q_{1}+\partial_{y}q_{2}-p_{1}-f_{2}\partial_{y}p_{1}+f_{1}\partial_{y}p_{2}+\widetilde{q}_{2}+4{\operatorname{d}}g(T_{2})$

By Lemma 5.4 this implies
${\operatorname{d}}g(T_{1})={\operatorname{d}}g(T_{2})=0$. Hence
$\widetilde{q}_{1},\widetilde{q}_{2}$ can be realized by choices of $p_{2}$
and $p_{1}$, respectively.

In degree two, let’s denote a representative of the second Poisson homology by
$\beta=\beta(\widetilde{a}_{i},\widetilde{q},\widetilde{q}_{j},\widetilde{p},\widetilde{p}_{j})$
as above. Then by the above discussion, $\beta\in\ker{\operatorname{d}}$ iff
$\widetilde{p}=0$ and $\widetilde{p}_{i}$ such that

$\widetilde{p}_{2}=(1+y\partial_{y})^{-1}\left(\partial_{x}\widetilde{q}_{2}-\partial_{y}\widetilde{q}_{1}-f_{1}\partial_{y}\widetilde{p}_{1}\right)\quad\text{
and
}\quad\widetilde{p}_{1}=(1+y\partial_{y})^{-1}\left(\partial_{x}\widetilde{q}_{1}+\partial_{y}\widetilde{q}_{2}+f_{1}\partial_{y}\widetilde{p}_{2}\right).$

We denote by $\alpha=\alpha(a_{i},b_{i},q_{j},p_{j})$ a representative of the
first Poisson homology and $\gamma\in\Omega^{3}_{f}$. We want to find $\alpha$
and $\gamma$ such that

$\displaystyle
0=\beta+{\operatorname{d}}\alpha+\iota_{\pi}({\operatorname{d}}\gamma)-{\operatorname{d}}\iota_{\star^{-1}\gamma}({\operatorname{d}}f_{1}\wedge{\operatorname{d}}f_{2})$

where ${\operatorname{d}}\alpha$ is given by

$\displaystyle{\operatorname{d}}\alpha=\sum_{j=1}^{2}{\operatorname{d}}(p_{j}\zeta_{j})+\left(\partial_{x}q_{2}-\partial_{y}q_{1})\right){\operatorname{d}}f_{1}\wedge{\operatorname{d}}f_{2}+\sum_{i=1}^{4}{\operatorname{d}}\left(b_{i}x_{i}\right)\wedge{\operatorname{d}}f_{1}.$

Wedging the equation above with ${\operatorname{d}}f_{1}$ and contracting with
$T_{1}$ and $T_{2}$ yields by (8) and (32) that

$\displaystyle 0=$
$\displaystyle\,f_{1}\partial_{y}p_{1}+f_{2}\partial_{y}p_{2}+p_{2}+\widetilde{q}_{2}+d(\iota_{\star^{-1}\gamma}({\operatorname{d}}f_{1}))(T_{1})$
$\displaystyle 0=$
$\displaystyle\,f_{2}\partial_{y}p_{1}-f_{1}\partial_{y}p_{2}+p_{1}+\widetilde{q}_{1}+d(\iota_{\star^{-1}\gamma}({\operatorname{d}}f_{1}))(T_{2})$

Using again Lemma 5.4 implies
$d(\iota_{\star^{-1}\gamma}({\operatorname{d}}f_{1}))(T_{1})=d(\iota_{\star^{-1}\gamma}({\operatorname{d}}f_{1}))(T_{2})=0$.
Hence $\widetilde{q}_{1}$ and $\widetilde{q}_{2}$ can be realized by choices
of $p_{1}$ and $p_{2}$, respectively. That is, we may assume
$\widetilde{q}_{i}=\widetilde{p}_{i}=p_{i}=0$. From Lemma 7.2 implies that
$\widetilde{a}_{i}=b_{i}$ and $\partial_{x}q_{2}=\partial_{y}q_{1}$.

Finally, to determine the kernel of ${\operatorname{d}}$ in degree one, we
note that from the discussion in degree two we immediately get
$p_{1}=p_{2}=b_{i}=0$ and $\partial_{x}q_{2}=\partial_{y}q_{1}$. Using Theorem
1.2 we therefore obtain the statement.

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Lauran Toussaint,

Vrije Universiteit Amsterdam, 1081 HV Amsterdam, The Netherlands

E-mail address<EMAIL_ADDRESS>
Florian Zeiser,

University of Illinois at Urbana-Champaign, 61801 Urbana, United States

E-mail address<EMAIL_ADDRESS>