# Variational Bihamiltonian Cohomologies and Integrable Hierarchies I:
Foundations

Si-Qi Liu, Zhe Wang, Youjin Zhang

###### Abstract.

This series of papers is devoted to the study of deformations of Virasoro
symmetries of the principal hierarchies associated to semisimple Frobenius
manifolds. The main tool we use is a generalization of the bihamiltonian
cohomology called the variational bihamiltonian cohomology. In the present
paper, we give its definitions and compute the associated cohomology groups
that will be used in our study of deformations of Virasoro symmetries. To
illustrate its application, we classify the conformal bihamiltonian structures
with semisimple hydrodynamic limits.

###### Key words and phrases:

Bihamiltonian Cohomologies, Variational Bihamiltonian Cohomologies, Conformal
Bihamiltonian Structures, Integrable Hierarchies

###### Contents

  1. 1 Introduction
  2. 2 Basic notions and Main results
  3. 3 Variational Bihamiltonian Cohomology
    1. 3.1 Variational forms on the infinite jet bundle.
    2. 3.2 Variational Hamiltonian cohomology and its triviality
    3. 3.3 Definition of the variational bihamiltonian cohomology
  4. 4 The cohomology group $\operatorname{\mathit{VBH}}^{2}_{3}$
  5. 5 Vanishing theorem of the variational bihamiltonian cohomology
    1. 5.1 Vanishing theorem and the strategy of computation.
    2. 5.2 Computation of ${}^{1}E_{1}$.
    3. 5.3 Computation of ${}^{1}E_{2}$.
  6. 6 Conformal bihamiltonian structures
    1. 6.1 Conformal Bihamiltonian Structures of hydrodynamic type
    2. 6.2 Deformed Conformal Bihamiltonian Structures
  7. 7 Conclusion

##  1\. Introduction

Since Dubrovin introduced the notion of Frobenius manifold in the beginning of
90’s of the last century [8, 9] inspired by the development of 2d topological
field theory [6, 7, 26, 36, 37], its relationship with hierarchies of
integrable PDEs has become an important research subject in the theory of
integrable systems and its applications in the study of Gromov-Witten
invariants, singularity theory and quantum field theory, see some of the
related works [1, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 28, 35, 38] and
references therein. In [15], Dubrovin and Zhang, the third author of the
present paper, proposed a project to classify hierarchies of integrable PDEs
which possess hydrodynamic limits and satisfy the following properties: each
integrable hierarchy has a bihamiltonian structure, a tau function (also
called a tau structure), and an infinite set of Virasoro symmetries that act
linearly on the tau function. The bihamiltonian structure of the integrable
hierarchy is assumed to be represented as a formal power series w.r.t. the
dispersion parameter $\varepsilon$, its coefficients are given by homogeneous
differential polynomials of the dependent variables of the integrable
hierarchy, and its dispersionless limit is a bihamiltonian structure of
hydrodynamic type. The existence of a bihamiltonian structure and of a tau
function implies that the dispersionless limit of the integrable hierarchy can
be described by a Frobenius manifold structure or a degenerate one. On the
other hand, starting from any Frobenius manifold, one can construct an
integrable hierarchy of hydrodynamic type which is called the Principal
Hierarchy of the Frobenius manifold. The flat metric and the intersection form
of the Frobenius manifold yield a bihamiltonian structure of the Principal
Hierarchy which also possesses a tau function and an infinite set of Virasoro
symmetries. The Virasoro symmetries, apart from the first two, of the
Principal Hierarchy do not act linearly on the tau function. Under the
semisimplicity condition of the Frobenius manifold, the requirement of linear
actions of the Virasoro symmetries on the tau function leads to a
dispersionful deformation, called the topological deformation, of the
Principal hierarchy. As it is shown in [15], the topological deformation of
the Principal Hierarchy is constructed via a quasi-Miura transformation
determined by the loop equation of the semisimple Frobenius manifold.

To classify dispersionful deformations of the Principal Hierarchy, Dubrovin
and Zhang introduced in [15] the notion of bihamiltonian cohomology
$\operatorname{\mathit{BH}}^{k}(M;P_{0},P_{1})$ for a bihamiltonian structure
$(P_{0},P_{1})$ of hydrodynamic type defined on the jet space of a smooth
manifold $M$. The cohomology groups
$\operatorname{\mathit{BH}}^{2}(M;P_{0},P_{1})$ and
$\operatorname{\mathit{BH}}^{3}(M;P_{0},P_{1})$ characterize the equivalence
classes of infinitesimal deformations of $(P_{0},P_{1})$ under Miura type
transformations and the obstruction of extending an infinitesimal deformations
to a full deformation respectively. Here we emphasize that the deformed
bihamiltonian structures are required to have the property that the
coefficients of the powers of the deformation parameter are given by
homogeneous differential polynomials of the dependent variables, this property
is called the polynomiality of the deformed Hamiltonian structures.

In [34] Lorenzoni studied the deformations of the bihamiltonian structure of
the dispersionless KdV hierarchy and showed, up to the fourth order
approximation of the deformation parameter, that they are parametrized by a
function of one variable. The very first computation of the bihamiltonian
cohomology groups for a semisimple bihamiltonian structure of hydrodynamic
type is given in [10, 31], from which the cohomology group $BH^{2}_{\geq
2}(M;P_{0},P_{1})$ can be obtained, and it shows that the equivalence classes
of the infinitesimal deformations of $(P_{0},P_{1})$ with $n$ dependent
variables are parametrized by a set of smooth functions
$c_{1}(u_{1}),\dots,c_{n}(u_{n})$, where $u_{1},\dots,u_{n}$ are the canonical
coordinates of the semisimple bihamiltonian structure of hydrodynamic type.
These functions are called the central invariants of the deformed
bihamiltonian structure. It is also conjectured that the cohomology group
$\operatorname{\mathit{BH}}^{3}(M;P_{0},P_{1})$ is trivial, and that any
infinitesimal deformation of $(P_{0},P_{1})$ can be extended to a full
deformation.

An important observation toward computing the cohomology group
$\operatorname{\mathit{BH}}^{3}(M;P_{1},P_{2})$ was made by the first and
third author of the present paper in [33], where they showed that one can work
in the space of differential polynomials instead of the space of the local
functionals to compute the bihamiltonian cohomology. By using this idea they
proved the afore mentioned conjecture for the bihamiltonian structure of
hydrodynamic type associated to the one dimensional Frobenius manifold (or the
dispersionless KdV hierarchy). Base on the method of [33] and on a clever
utilizing of the technique of spectral sequences and some other tools from
homological algebra, Carlet, Posthuma and Shadrin proved this conjecture for a
general semisimple bihamiltonian structure of hydrodynamic type in [5].

A deformation of the bihamiltonian structure of the Principal hierarchy of a
semisimple Frobenius manifold yields a deformation of the integrable
hierarchy. It is shown in [11] that the deformed integrable hierarchy
possesses a tau structure when the central invariants of the deformed
bihamiltonian structure are constant functions. However, it remains to be
unclear whether this deformed integrable hierarchy also possesses an infinite
set of Virasoro symmetries. In particular, it is conjectured that the
deformation of the Principal Hierarchy with all the central invariants being
equal to $\frac{1}{24}$ possesses an infinite set of Virasoro symmetries which
act linearly on the tau function of the deformed integrable hierarchy, or in
other words, it coincides with the topological deformation of the Principal
Hierarchy. An approach to prove the validity of this conjecture is as follows:
one needs to show that the quasi-Miura transformation, which transforms the
Principal Hierarchy to its topological deformation, also transforms the
bihamiltonian structure of hydrodynamic type to a deformed bihamiltonian
structure which satisfies the polynomiality property. In [2, 3] Buryak,
Posthuma and Shadrin proved the polynomiality property of the first deformed
Hamiltonian structure of the Principal Hierarchy, and in the recent preprint
[25] Iglesias and Shadrin provide a certain evidence of the validity of the
polynomiality property of the second Hamiltonian structure.

The main purpose of the present paper and of its subsequents is to show that
the deformation of the Principal Hierarchy of a semisimple Frobenius manifold
which is associated to a deformed bihamiltonian structure with constant
central invariants possesses an infinite set of Virasoro symmetries, and these
Virasoro symmetries can be lifted to the actions on the tau function of the
deformed integrable hierarchy; moreover, for the case when all the central
invariants equal to $\frac{1}{24}$, the Virasoro symmetries act linearly on
the tau function, and so this deformed integrable hierarchy is equivalent to
the topological deformation of the Principal Hierarchy. In particular, this
implies that the topological deformation of the Principal Hierarchy of a
semisimple Frobenius manifold possesses a bihamiltonian structure which
satisfies the polynomiality property. To this end, we first generalize in the
present paper the notion of bihamiltonian cohomology by considering a certain
cohomology, called the variational bihamiltonian cohomology, on the space of
variational $1$-forms of the infinite jet space of the dependent variables,
and apply it to the study of Virasoro symmetries of the deformed integrable
hierarchy in the subsequent papers.

The paper is organized as follows. In Sect. 2, we recall the basic notions of
bihamiltonian structures in terms of local functionals on the infinite jet
bundle of a super manifold, and we present the main results of this paper. In
Sect. 3, we first define the variational Hamiltonian cohomology and prove the
triviality of the cohomology groups, and then we define the variational
bihamiltonian cohomology. Sect. 4 and Sect. 5 are both devoted to the
computation of the variational bihamiltonian cohomology groups. As an
illustration of the application of this theory, we classify the conformal
bihamiltonian structures and their deformations in Sect. 6. In Sect. 7 we give
some concluding remarks.

##  2\. Basic notions and Main results

In this section, we first recall some basic definitions and constructions on
bihamiltonian structures in terms of the language of super variables. The
readers may refer to [32, 33] and the lecture note [27] for a detailed
introduction to this topic.

Let $M$ be an $n$-dimensional smooth manifold, and $\hat{M}=\Pi(T^{*}M)$ be
the super manifold of dimension $(n\mid n)$ obtained by reversing the parity
of the fiber of the cotangent bundle of $M$. Locally, one may choose a
coordinate system $(u^{1},\cdots,u^{n})$ for an open neighborhood $U\subset M$
and we will use $(\theta_{1},\cdots,\theta_{n})$ to denote the coordinate
system of the fibers. The variables $\theta_{i}$ are fermion, meaning that
they satisfy the anti-commutation relations
$\theta_{i}\theta_{j}+\theta_{j}\theta_{i}=0$. Let $J^{\infty}(\hat{M})$ be
the infinite jet bundle of $\hat{M}$ and let $\hat{\mathcal{A}}$ be the ring
of differential polynomials. In local coordinates, we can write it as

$\hat{\mathcal{A}}=C^{\infty}(U)[[u^{i,s+1},\theta^{s}_{i}\mid
i=1,\cdots,n;s\geq 0]].$

Here and henceforth we denote $u^{i,0}=u^{i}$ and
${\theta_{i}^{0}}=\theta_{i}$. The jet variables $u^{i,s},\theta_{i}^{s}$ for
$s\geq 1$ may be regarded as $s$-th order derivatives with respect to an
independent variable, which we usually denote as $x$, when regarding $u^{i}$
and $\theta_{i}$ as dependent variables. This identification is possible due
to the existence of the global vector field

(2.1) $\partial_{x}=\sum_{s\geq 0}u^{i,s+1}\frac{\partial}{\partial
u^{i,s}}+\theta_{i}^{s+1}\frac{\partial}{\partial\theta_{i}^{s}}.$

Sometimes we will use a prime to denote $\partial_{x}$, i.e.
$f^{\prime}:=\partial_{x}f$. The elements in the quotient space
$\hat{\mathcal{F}}=\hat{\mathcal{A}}/\partial_{x}$ are called local
functionals and for an element $f\in\hat{\mathcal{A}}$, we will use $\int f$
to denote its class in $\hat{\mathcal{F}}$. Let us define two degrees
$\deg_{x}$ and $\deg_{\theta}$ on the space $\hat{\mathcal{A}}$, which are
called respectively the differential gradation and the super gradation. On the
generators of $\hat{\mathcal{A}}$, these degrees are defined by

$\deg_{x}u^{i,s}=\deg_{x}\theta^{s}_{i}=s;\quad\deg_{\theta}u^{i,s}=0,\quad\deg_{\theta}\theta_{i}^{s}=1.$

We denote the space of homogeneous elements by

$\hat{\mathcal{A}}_{d}=\\{f\in\hat{\mathcal{A}}\mid\deg_{x}f=d\\},\quad\hat{\mathcal{A}}^{p}=\\{f\in\hat{\mathcal{A}}\mid\deg_{\theta}f=p\\},\quad\hat{\mathcal{A}}^{p}_{d}=\hat{\mathcal{A}}^{p}\cap\hat{\mathcal{A}}_{d}.$

It is clear that the vector field $\partial_{x}$ is homogeneous with respect
to both degrees, hence the gradations on $\hat{\mathcal{A}}$ induce a
gradation on $\hat{\mathcal{F}}$ and we will denote the space of homogeneous
elements by $\hat{\mathcal{F}}_{d}$, $\hat{\mathcal{F}}^{p}$ and
$\hat{\mathcal{F}}^{p}_{d}$ accordingly.

A nontrivial construction called the Schouten-Nijenhuis bracket equips the
space $\hat{\mathcal{F}}$ with a graded Lie algebra structure. Define the
variational derivative of an element in $\hat{\mathcal{A}}$ by

$\frac{\delta f}{\delta u^{i}}=\sum_{s\geq 0}(-\partial_{x})^{s}\frac{\partial
f}{\partial u^{i,s}},\quad\frac{\delta f}{\delta\theta_{i}^{s}}=\sum_{s\geq
0}(-\partial_{x})^{s}\frac{\partial f}{\partial\theta_{i}^{s}}.$

It can be shown that the variational derivative annihilates the image of
$\partial_{x}$ so we can define the variational derivative of a local
functional as follows:

$\frac{\delta F}{\delta u^{i}}=\int\frac{\delta f}{\delta
u^{i}},\quad\frac{\delta F}{\delta\theta_{i}}=\int\frac{\delta
f}{\delta\theta_{i}},\quad F=\int f\in\hat{\mathcal{F}}.$

Then the Schouten-Nijenhuis bracket is defined to be the following bilinear
map

$[-,\
-]:\hat{\mathcal{F}}^{p}\times\hat{\mathcal{F}}^{q}\to\hat{\mathcal{F}}^{p+q-1},\quad[P,Q]=\int\sum_{i}\frac{\delta
P}{\delta\theta_{i}}\frac{\delta Q}{\delta u^{i}}+(-1)^{p}\frac{\delta
P}{\delta u^{i}}\frac{\delta Q}{\delta\theta_{i}}.$

This bracket defines a graded Lie algebra structure on $\hat{\mathcal{F}}$,
whose sign convention is different from the usual definition, for more details
one may refer to [32].

For a local functional $P\in\hat{\mathcal{F}}^{p}$, we can associate a
derivation on $\hat{\mathcal{A}}$ defined by

(2.2) $D_{P}=\sum_{i}\sum_{s\geq 0}\partial_{x}^{s}\left(\frac{\delta
P}{\delta\theta_{i}}\right)\frac{\partial}{\partial
u^{i,s}}+(-1)^{p}\partial_{x}^{s}\left(\frac{\delta P}{\delta
u^{i}}\right)\frac{\partial}{\partial\theta^{s}_{i}}.$

We see that $D_{P}$ is a derivation of super degree $p-1$ and that
$[D_{P},\partial_{x}]=0$. The following property is important for the
construction of the cohomologies that will be given later in this and the next
section:

(2.3) $(-1)^{p-1}D_{[P,Q]}=D_{P}\comp D_{Q}-(-1)^{(p-1)(q-1)}D_{Q}\comp
D_{P};\quad P\in\hat{\mathcal{F}}^{p},\ Q\in\hat{\mathcal{F}}^{q}.$

This relation shows that the map $D$ induces a (graded) Lie algebra
homomorphism

$\hat{\mathcal{F}}\to\mathrm{Der}(\hat{\mathcal{A}}):\quad P\mapsto D_{P},$

here the natural graded Lie algebra structure on the space of derivations on
$\hat{\mathcal{A}}$, denoted by $\mathrm{Der}(\hat{\mathcal{A}})$, is given by
the graded commutators. In what follows, we will also call an element of
$\mathrm{Der}(\hat{\mathcal{A}})$ a vector field on $J^{\infty}(\hat{M})$.

Using the above notions, a Hamiltonian structure is defined to be a local
functional $P\in\hat{\mathcal{F}}^{2}$ satisfying $[P,P]=0$, and it is called
of hydrodynamic type if $P\in\hat{\mathcal{F}}^{2}_{1}$. A bihamiltonian
structure is a pair of Hamiltonian structure $(P_{0},P_{1})$ satisfying an
additional compatibility condition $[P_{0},P_{1}]=0$. Now assume that we have
a bihamiltonian structure $(P_{0},P_{1})$ of hydrodynamic type [27], locally
we represent them as follows:

$P_{a}=\frac{1}{2}\int\sum
g^{ij}_{a}(u)\theta_{i}\theta_{j}^{1}+\Gamma^{ij}_{a,k}(u)u^{k,1}\theta_{i}\theta_{j},\quad
a=1,2.$

This bihamiltonian structure is called semisimple if the roots
$\lambda^{1}(u),\cdots,\lambda^{n}(u)$ of the characteristic equation
$\det(g_{1}^{ij}-\lambda g_{0}^{ij})=0$ are distinct and not constant.

It is proved in [20] that if $(P_{0},P_{1})$ is semisimple, then the roots
$\lambda^{1}(u),\cdots,\lambda^{n}(u)$ can serve as local coordinates of the
manifold $M$ and in such a coordinate system $(P_{0},P_{1})$ can be
represented in the following forms:

$P_{0}=\frac{1}{2}\int\sum
f^{i}(\lambda)\theta_{i}\theta_{i}^{1}+A^{ij}\theta_{i}\theta_{j},\quad
P_{1}=\frac{1}{2}\int\sum
g^{i}(\lambda)\theta_{i}\theta_{i}^{1}+B^{ij}\theta_{i}\theta_{j},$

where $f^{i}$ are non-vanishing functions, $g^{i}=\lambda^{i}f^{i}$ and the
functions $A^{ij}$ and $B^{ij}$ are given by

$A^{ij}=\frac{1}{2}\left(\frac{f^{i}}{f^{j}}\frac{\partial
f^{j}}{\partial\lambda^{i}}\lambda^{j,1}-\frac{f^{j}}{f^{i}}\frac{\partial
f^{i}}{\partial\lambda^{j}}\lambda^{i,1}\right),\quad
B^{ij}=\frac{1}{2}\left(\frac{g^{i}}{f^{j}}\frac{\partial
f^{j}}{\partial\lambda^{i}}\lambda^{j,1}-\frac{g^{j}}{f^{i}}\frac{\partial
f^{i}}{\partial\lambda^{j}}\lambda^{i,1}\right).$

The coordinates $\lambda^{1},\cdots,\lambda^{n}$ are called the canonical
coordinates of $(P_{0},P_{1})$.

In what follows, when we consider semisimple bihamiltonian structures of
hydrodynamic type, it is always assumed that we choose the canonical
coordinates as local coordinates. We will use $u^{1},\cdots,u^{n}$ instead of
$\lambda^{1},\cdots,\lambda^{n}$ to denote them. In these coordinates, we
represent a bihamiltonian structure $(P_{0},P_{1})$ of hydrodynamic type as

(2.4) $P_{0}=\frac{1}{2}\int\sum
f^{i}(u)\theta_{i}\theta_{i}^{1}+A^{ij}\theta_{i}\theta_{j},\quad
P_{1}=\frac{1}{2}\int\sum
u^{i}f^{i}(u)\theta_{i}\theta_{i}^{1}+B^{ij}\theta_{i}\theta_{j}.$

Given a hydrodynamic bihamiltonian structure $(P_{0},P_{1})$, we construct the
variational bihamiltonian cohomology as follows. Consider the space of 1-forms
$\Omega$ of the infinite jet space $J^{\infty}(\hat{M})$, locally it is an
$\hat{\mathcal{A}}$-module generated by $\delta u^{i,s}$ and
$\delta\theta_{i}^{s}$ for $i=1,\cdots,n$ and $s\geq 0$:

$\Omega=\biggl{\\{}\sum_{i;s\geq 0}g_{i,s}\delta
u^{i,s}+h^{i}_{s}\delta\theta_{i}^{s}\mid
g_{i,s},h^{i}_{s}\in\hat{\mathcal{A}}\biggr{\\}}.$

Each derivation of $\hat{\mathcal{A}}$ induces an action on $\Omega$ by the
Lie derivative (see Sect. 3 for details). In particular we consider the action
of Lie derivative of $\partial_{x}$ on $\Omega$, which we still denote by
$\partial_{x}$, and we denote by $\bar{\Omega}$ the quotient space
$\Omega/{\partial_{x}\Omega}$. We can verify that the actions on
$\bar{\Omega}$ given by the Lie derivatives of $D_{P_{0}}$ and $D_{P_{1}}$,
which we denote by $\tilde{D}_{0}$ and $\tilde{D}_{1}$ respectively, equip
$\bar{\Omega}$ a structure of double complex. We grade the space
$\bar{\Omega}$ similarly as we do for the space $\hat{\mathcal{F}}$ by setting

$\deg_{x}\delta
u^{i,s}=\deg_{x}\delta\theta^{s}_{i}=s;\quad\deg_{\theta}\delta
u^{i,s}=0,\quad\deg_{\theta}\delta\theta_{i}^{s}=1.$

Then we define the variational bihamiltonian cohomology groups as follows:

$\operatorname{\mathit{VBH}}^{p}_{d}(\bar{\Omega},\tilde{D}_{0},\tilde{D}_{1})=\frac{\bar{\Omega}_{d}^{p}\cap\ker\tilde{D}_{0}\cap\ker\tilde{D}_{1}}{\bar{\Omega}_{d}^{p}\cap\operatorname{Im}\tilde{D}_{0}\tilde{D}_{1}}.$

We prove the following theorem in Sect. 4 and Sect. 5.

######  Theorem 2.1.

The variational bihamiltonian cohomology groups for a semisimple bihamiltonian
structure $(P_{0},P_{1})$ of hydrodynamic type have the following properties:

  1. i)

$\operatorname{\mathit{VBH}}^{2}_{3}(\bar{\Omega},\tilde{D}_{0},\tilde{D}_{1})\cong\oplus_{i=1}^{n}C^{\infty}(\mathbb{R});$

  2. ii)

If $d\geq 2$ and if both $(p,d)$ and $(p+1,d)$ are NOT in the index set $I$,
then
$\operatorname{\mathit{VBH}}^{p}_{d}(\bar{\Omega},\tilde{D}_{0},\tilde{D}_{1})=0$,
where the index set is defined by $I=I_{1}\cup I_{2}\cup I_{3}$ with

$\displaystyle I_{1}$ $\displaystyle=\\{(i,j)\mid j=0,1;\
i=j+1,\cdots,j+n+1\\};$ $\displaystyle I_{2}$ $\displaystyle=\\{(i,j)\mid
j=2,\cdots,n;\ i=j,\cdots,j+n+1\\};$ $\displaystyle I_{3}$
$\displaystyle=\\{(i,j)\mid j=n+1,n+2,n+3;\ i=j,\cdots,j+n\\}.$

  3. iii)

$\operatorname{\mathit{VBH}}^{1}_{2}(\bar{\Omega},\tilde{D}_{0},\tilde{D}_{1})=0.$

To illustrate the application of the variational bihamiltonian cohomology, in
Sect. 6 we study the conformal property of bihamiltonian structures which is
shared by the bihamiltonian structures associated with Frobenius manifolds.
Let us introduce the notations

$\displaystyle\mathrm{Der}(\hat{\mathcal{A}})^{p}:=\\{X\in\mathrm{Der}(\hat{\mathcal{A}})\mid
X(\hat{\mathcal{A}}^{k})\subset\hat{\mathcal{A}}^{k+p}\\},$
$\displaystyle\mathrm{Der}(\hat{\mathcal{A}})^{p}_{d}:=\\{X\in\mathrm{Der}(\hat{\mathcal{A}})\mid
X(\hat{\mathcal{A}}^{k}_{l})\subset\hat{\mathcal{A}}^{k+p}_{l+d}\\}.$

Then the conformal property of a bihamiltonian structure can be described as
the follows.

######  Definition 2.1.

A bihamiltonian structure $(P_{0},P_{1})$ is called conformal if there exists
a nonzero derivation $E\in\mathrm{Der}(\hat{\mathcal{A}})^{0}$ and real
numbers $\mu,\lambda_{0},\lambda_{1}$ such that:

(2.5)
$\left[E,\partial_{x}\right]=\mu\partial_{x},\quad\left[E,D_{P_{a}}\right]=\lambda_{a}D_{P_{a}},\quad
a=0,1.$

######  Example 2.1.

The bihamiltonian structure of the dispersionless KdV hierarchy is given by

$P_{0}=\frac{1}{2}\int\theta\theta^{1};\quad P_{1}=\frac{1}{2}\int
u\theta\theta^{1}.$

One can check that it is a conformal bihamiltonian structure with $E$ given by

$E=\sum_{s\geq 0}(\lambda_{1}-\lambda_{0}+s\mu)u^{(s)}\frac{\partial}{\partial
u^{(s)}}+(\lambda_{1}+(s-1)\mu)\theta^{s}\frac{\partial}{\partial\theta^{s}}.$

######  Theorem 2.2.

A semisimple bihamiltonian structure $(P_{0},P_{1})$ of hydrodynamic type is
conformal if and only if there exist real numbers $d^{1},\cdots,d^{n}$ such
that the functions $f^{1},\cdots,f^{n}$ given in (2.4) satisfy the following
identities:

(2.6) $\displaystyle\sum_{j}u^{j}\frac{\partial f^{i}}{\partial
u^{j}}=d^{i}f^{i},\quad\forall\ i;$ (2.7)
$\displaystyle(d^{i}-d^{j})\frac{\partial f^{j}}{\partial
u^{i}}=0,\quad\forall\ i,j.$

In such a case, if we require that the derivation $E$ has differential degree
0, then $E$ is an Euler type vector field given by

(2.8) $E=\sum_{i;s\geq
0}(\lambda_{1}-\lambda_{0}+s\mu)u^{i,s}\frac{\partial}{\partial
u^{i,s}}+(\lambda_{1}-(\lambda_{1}-\lambda_{0})d^{i}+(s-1)\mu)\theta_{i}^{s}\frac{\partial}{\partial\theta_{i}^{s}}.$

Given a semisimple and conformal bihamiltonian structure of hydrodynamic type,
we consider whether its deformations are still conformal. It turns out that
there is only a certain family of deformations that preserve the conformal
property, and the equivalence classes of these deformations under Miura type
transformations are parametrized by $n$ constants, as it is shown by the
following theorem.

######  Theorem 2.3.

Let $(P_{0},P_{1})$ be a semisimple and conformal bihamiltonian structure of
hydrodynamic type, then its deformation $(\tilde{P}_{0},\tilde{P}_{1})$ is
conformal if and only if the central invariants are given by:

(2.9) $c_{i}(u^{i})=C_{i}(u^{i})^{m_{i}},\quad
m_{i}=\frac{\lambda_{1}-\lambda_{0}-2\mu-(\lambda_{1}-\lambda_{0})d^{i}}{\lambda_{1}-\lambda_{0}},$

where $C_{i}$ are arbitrary constants.

##  3\. Variational Bihamiltonian Cohomology

In this section, we start with the definition of variational Hamiltonian
cohomology for a single Hamiltonian structure of hydrodynamic type and prove
its triviality. Then we define the variational bihamiltonian cohomology and
make some necessary algebraic preparations for computing the cohomology
groups.

###  3.1. Variational forms on the infinite jet bundle.

We recall the constructions of the variational forms on the infinite jet
bundle $J^{\infty}(\hat{M})$ (cf. [15] and reference therein). Let
$\mathcal{E}$ be the space of differential forms on $J^{\infty}(\hat{M})$,
locally is just the space

$\mathcal{E}=\wedge^{*}\left(\mathrm{Span}_{\hat{\mathcal{A}}}\left\\{\delta\theta_{i}^{s},\delta
u^{i,s}\mid i=1,\cdots,n,\,s\geq 0\right\\}\right).$

Here we adopt Deligne’s sign rule for these variational forms, for example

$\theta_{i}^{s}\delta\theta_{j}^{t}=-\delta\theta_{j}^{t}\theta_{i}^{s},\quad\delta\theta_{i}^{s}\delta\theta_{j}^{t}=\delta\theta_{j}^{t}\delta\theta_{i}^{s},\quad\delta
u^{i,s}\delta\theta_{j}^{t}=-\delta\theta_{j}^{t}\delta u^{i,s},\quad\delta
u^{i,s}\delta u^{j,t}=-\delta u^{j,t}\delta u^{i,s}.$

Any vector field $X\in\mathrm{Der}(\hat{\mathcal{A}})$ induces an action on
$\mathcal{E}$ by the Lie derivative, which is defined by the Cartan formula

$\mathcal{L}_{X}=\delta\comp\iota_{X}+\iota_{X}\comp\delta,$

where the de Rham differential is given by

$\delta=\sum_{i;s\geq 0}\delta u^{i,s}\frac{\partial}{\partial
u^{i,s}}+\delta\theta_{i}^{s}\frac{\partial}{\partial\theta_{i}^{s}}.$

In particular, the vector field $\partial_{x}$ defined in (2.1) induces an
action on $\mathcal{E}$, which we still denote by $\partial_{x}$. We define
the space of local functionals of forms by

$\bar{\mathcal{E}}=\mathcal{E}/{\partial_{x}\mathcal{E}},$

and represent its elements in the form $\int f$ with $f\in\mathcal{E}$. In
addition, if a vector field $X\in\mathrm{Der}(\hat{\mathcal{A}})$ commutes
with $\partial_{x}$, then its Lie derivative $\mathcal{L}_{X}$ also commutes
with $\partial_{x}$ acting on $\mathcal{E}$, hence it induces an action on
$\bar{\mathcal{E}}$ which we still denote by $\mathcal{L}_{X}$.

######  Example 3.1.

The space of 1-forms is given by

$\mathcal{E}^{1}=\biggl{\\{}\sum_{i;s\geq 0}g_{i,s}\delta
u^{i,s}+h^{i}_{s}\delta\theta_{i}^{s}\mid
g_{i,s},h^{i}_{s}\in\hat{\mathcal{A}}\biggr{\\}}.$

Let us grade $\mathcal{E}^{1}$ as we do for $\hat{\mathcal{A}}$ as follows:

$\displaystyle(\mathcal{E}^{1})_{d}=\biggl{\\{}\sum_{i;s\geq 0}g_{i,s}\delta
u^{i,s}+h^{i}_{s}\delta\theta_{i}^{s}\mid
g_{i,s},h^{i}_{s}\in\hat{\mathcal{A}}_{d-s}\biggr{\\}},$
$\displaystyle(\mathcal{E}^{1})^{p}=\biggl{\\{}\sum_{i;s\geq 0}g_{i,s}\delta
u^{i,s}+h^{i}_{s}\delta\theta_{i}^{s}\mid
g_{i,s}\in\hat{\mathcal{A}}^{p},h^{i}_{s}\in\hat{\mathcal{A}}^{p-1}\biggr{\\}}.$

For any element $\omega\in\bar{\mathcal{E}}^{1}$, we can uniquely represent it
in the form

$\omega=\int\sum_{i}g_{i}\delta u^{i}+h^{i}\delta\theta_{i},\quad
g_{i},h^{i}\in\hat{\mathcal{A}}.$

So we can identify the space $\bar{\mathcal{E}}^{1}$ with the space of
$\hat{\mathcal{A}}$-valued differential 1-forms on $\hat{M}$.

######  Example 3.2.

Let $X\in\mathrm{Der}(\hat{\mathcal{A}})^{q}$ and
$\omega\in(\mathcal{E}^{1})^{p}$ be given by

$\omega=\sum_{i;s\geq 0}g_{i,s}\delta
u^{i,s}+h^{i}_{s}\delta\theta_{i}^{s},\quad g_{i,s}\in\hat{\mathcal{A}}^{p},\
h^{i}_{s}\in\hat{\mathcal{A}}^{p-1},$

then the action of $\mathcal{L}_{X}$ on $\omega$ has the expression

$\mathcal{L}_{X}\omega=\sum_{i;s\geq 0}X(g_{i,s})\delta
u^{i,s}+(-1)^{pq}g_{i,s}\delta\left(X(u^{i,s})\right)+X(h^{i}_{s})\delta\theta_{i}^{s}+(-1)^{(p-1)q}h^{i}_{s}\delta\left(X(\theta_{i}^{s})\right).$

Let us consider the space $\mathrm{Der}(\hat{\mathcal{A}})^{\partial}$ of
derivations on $\hat{\mathcal{A}}$ which commute with $\partial_{x}$. Take an
element $X\in\mathrm{Der}(\hat{\mathcal{A}})^{\partial}$, since $[X,\
\partial_{x}]=0$ we can regard $X$ as an $\hat{\mathcal{A}}$-valued vector
field on $\hat{M}$ instead of on $J^{\infty}(\hat{M})$. Note that $\hat{M}$
admits a canonical symplectic structure

$\varpi=\sum_{i}\delta u^{i}\wedge\delta\theta_{i},$

hence the space of the $\hat{\mathcal{A}}$-valued vector fields on $\hat{M}$
is canonically identified with the space of $\hat{\mathcal{A}}$-valued
differential 1-forms on $\hat{M}$, which is the same as the space
$\bar{\mathcal{E}}^{1}$ as we explained in the Example 3.1. Let us write down
this identification explicitly. An element $X$ of
$\mathrm{Der}(\hat{\mathcal{A}})^{\partial}$ can be represented as

$X=\sum_{i;s\geq
0}\partial_{x}^{s}\left(X(u^{i})\right)\frac{\partial}{\partial
u^{i,s}}+\partial_{x}^{s}\left(X\left(\theta_{i}\right)\right)\frac{\partial}{\partial\theta^{s}_{i}}.$

Restricting it on $\hat{M}$, we get the following $\hat{\mathcal{A}}$-valued
vector field on $\hat{M}$ which we still denote by $X$:

$X=\sum_{i}X(u^{i})\frac{\partial}{\partial
u^{i}}+X(\theta_{i})\frac{\partial}{\partial\theta_{i}}.$

Using the canonical symplectic structure $\varpi$, we identify $X$ with the
1-form

$W=\iota_{X}\varpi=\sum_{i}X(u^{i})\delta\theta_{i}-X(\theta_{i})\delta
u^{i},$

which corresponds to a unique element $\omega\in\bar{\mathcal{E}}^{1}$ given
by

$\omega=\int\sum_{i}X(u^{i})\delta\theta_{i}-X(\theta_{i})\delta u^{i}.$

We note that for an element
$X\in(\mathrm{Der}(\hat{\mathcal{A}})^{\partial})^{p}_{d}$, the corresponding
element $\omega$ belongs to $(\bar{\mathcal{E}}^{1})^{p+1}_{d}$. Let us denote
this correspondence by

(3.1)
$\Phi:\mathrm{Der}(\hat{\mathcal{A}})^{\partial}\to\bar{\mathcal{E}}^{1};\quad\Phi:X\mapsto\int\iota_{\varpi}\left(X|_{\hat{M}}\right).$

Now let $P$ be a Hamiltonian structure of hydrodynamic type. By using the
identity (2.3), we see that the space
$\mathrm{Der}(\hat{\mathcal{A}})^{\partial}$ becomes a cochain complex with
the differential given by adjoint action of $D_{P}$. The question is that when
we identify elements in $\mathrm{Der}(\hat{\mathcal{A}})^{\partial}$ as
elements in $\bar{\mathcal{E}}^{1}$, then how the action of $D_{P}$ induces a
differential on $\bar{\mathcal{E}}^{1}$?

######  Lemma 3.1.

Let $P$ be a Hamiltonian structure of hydrodynamic type, then the following
identity holds true for any $X\in\mathrm{Der}(\hat{\mathcal{A}})^{\partial}$:

(3.2) $\Phi\left([D_{P},\ X]\right)=\mathcal{L}_{D_{P}}\Phi(X).$

###### Proof.

Since both the definition (3.1) and the identity (3.2) are coordinate free, we
can choose a system of local coordinates
$(v^{1},\cdots,v^{n};\phi_{1},\cdots,\phi_{n})$ on $\hat{M}$ such that the
Hamiltonian structure $P$ has the expression [12]

$P=\frac{1}{2}\int\eta^{\alpha\beta}\phi_{\alpha}\phi_{\beta}^{1},$

here $(\eta^{\alpha\beta})$ is a constant non-degenerate matrix, and summation
over repeated lower and upper Greek indices is assumed. We call such
coordinates the flat coordinates of $P$.

Now we assume that $X$ is an element of
$\mathrm{Der}(\hat{\mathcal{A}})^{\partial}$ with super degree $p$, which
means that $X(\hat{\mathcal{A}}^{q})\subset\hat{\mathcal{A}}^{p+q}$, then it
is straightforward to show that

$[D_{P},X]v^{\alpha}=D_{P}\left(X(v^{\alpha})\right)+(-1)^{p+1}\eta^{\alpha\beta}(X(\phi_{\beta}^{1})),\quad[D_{P},X]\phi_{\alpha}=D_{P}(X(\phi_{\alpha})).$

Then we arrive at

$\Phi\left([D_{P},X]\right)=\int\left(D_{P}\left(X(v^{\alpha})\right)+(-1)^{p+1}\eta^{\alpha\beta}(X(\phi_{\beta}^{1}))\right)\delta\phi_{\alpha}-D_{P}(X(\phi_{\alpha}))\delta
v^{\alpha}.$

On the other hand we have

$\displaystyle\mathcal{L}_{D_{P}}\Phi(X)$
$\displaystyle=\mathcal{L}_{D_{P}}\int
X(v^{\alpha})\delta\phi_{\alpha}-X(\phi_{\alpha})\delta v^{\alpha}$
$\displaystyle=\int
D_{P}\left(X(v^{\alpha})\right)\delta\phi_{\alpha}-D_{P}(X(\phi_{\alpha}))\delta
v^{\alpha}-(-1)^{p+1}X(\phi_{\alpha})\delta\left(\eta^{\alpha\beta}\phi_{\beta}^{1}\right)$
$\displaystyle=\int\left(D_{P}\left(X(v^{\alpha})\right)+(-1)^{p+1}\eta^{\alpha\beta}(X(\phi_{\beta}^{1}))\right)\delta\phi_{\alpha}-D_{P}(X(\phi_{\alpha}))\delta
v^{\alpha}.$

Therefore we prove the lemma. ∎

###  3.2. Variational Hamiltonian cohomology and its triviality

From now on we will use $\Omega$ to denote the space of 1-froms
$\mathcal{E}^{1}$ and $\bar{\Omega}$ to denote its quotient space
$\bar{\mathcal{E}}^{1}$. The space of homogeneous elements with differential
degree $d$ and super degree $p$ will be denoted by $\Omega^{p}_{d}$ and
$\bar{\Omega}^{p}_{d}$ respectively.

Let $P$ be a Hamiltonian structure of hydrodynamic type, we will use
$\tilde{D}_{P}$ to denote the action $\mathcal{L}_{D_{P}}$ on the space
$\Omega$ and $\bar{\Omega}$. By using the identity (3.2) we conclude that
$\tilde{D}_{P}\comp\tilde{D}_{P}=0$, so $\tilde{D}_{P}$ is a differential on
the spaces $\Omega$ and $\bar{\Omega}$.

######  Definition 3.1.

The variational Hamiltonian cohomology of $\Omega$ (and of $\bar{\Omega}$,
respectively) is defined to be the cohomology of the complex
$(\Omega^{\bullet},\tilde{D}_{P})$ (and of
$(\bar{\Omega}^{\bullet},\tilde{D}_{P}),respectively$) given by

$H^{p}_{d}(\Omega,\tilde{D}_{P})=\frac{\Omega^{p}_{d}\cap\ker\tilde{D}_{P}}{\Omega^{p}_{d}\cap\operatorname{Im}\tilde{D}_{P}};\quad
H^{p}_{d}(\bar{\Omega},\tilde{D}_{P})=\frac{\bar{\Omega}^{p}_{d}\cap\ker\tilde{D}_{P}}{\bar{\Omega}^{p}_{d}\cap\operatorname{Im}\tilde{D}_{P}}.$

By using the fundamental facts of the homological algebra we have the
following lemma.

######  Lemma 3.2.

The short exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{x}}$$\textstyle{\Omega\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\bar{\Omega}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

induces the following long exact sequence of the cohomology groups for $d\geq
1$:

$\cdots\to H^{p}_{d}(\Omega,\tilde{D}_{P})\to
H^{p}_{d}(\bar{\Omega},\tilde{D}_{P})\to
H^{p+1}_{d}(\Omega,\tilde{D}_{P})\to\cdots.$

######  Remark 3.1.

On the space $\hat{\mathcal{A}}$, the map $\partial_{x}$ has the kernel
$\mathbb{R}$, however on the space $\Omega$ the map $\partial_{x}$ is
injective.

######  Theorem 3.3 (Triviality of the variational Hamiltonian cohomology).

We have

$H^{p}_{d}(\Omega,\tilde{D}_{P})=0$

for $p>0,\ d>0$.

###### Proof.

We choose locally a system of flat coordinates
$(v^{1},\cdots,v^{n};\phi_{1},\cdots,\phi_{n})$ on $\hat{M}$ such that
$P=\frac{1}{2}\int\eta^{\alpha\beta}\phi_{\alpha}\phi_{\beta}^{1}$. Then for a
1-form

$\omega=\sum_{s\geq 0}f_{\alpha,s}\delta
v^{\alpha,s}+g^{\alpha}_{s}\delta\phi_{\alpha}^{s}\in\Omega^{p}_{d}$

we have

$\tilde{D}_{P}\omega=\sum_{s\geq 0}D_{P}(f_{\alpha,s})\delta
v^{\alpha,s}+(-1)^{p}\eta^{\alpha\beta}f_{\alpha,s}\delta\phi_{\beta}^{s+1}+D_{P}(g^{\alpha}_{s})\delta\phi_{\alpha}^{s}.$

If $\omega\in\ker\tilde{D}_{P}$, then we see that

$D_{P}(g^{\alpha}_{0})=0;\quad
D_{P}(g_{s+1}^{\alpha})+(-1)^{p}\eta^{\alpha\beta}f_{\beta,s}=0,\ s\geq 0,$

so we can write $\omega$ in the form

$\omega=\tilde{D}_{P}\left(\sum_{s\geq
0}(-1)^{p-1}\eta_{\alpha\beta}g^{\alpha}_{s+1}\delta
v^{\beta,s}\right)+g^{\alpha}_{0}\delta\phi_{\alpha}.$

For $g^{\alpha}_{0}\in\hat{\mathcal{A}}^{p-1}_{d}$, by using the triviality of
the Hamiltonian cohomology (see, for example [27, 32]) we can represent it as
$g^{\alpha}_{0}=D_{P}(h^{\alpha}_{0})$. Thus the cocycle $\omega$ must also be
a coboundary. The theorem is proved. ∎

From Lemma 3.2 we have the following corollary.

######  Corollary 3.4.

We have $H^{p}_{d}(\bar{\Omega},\tilde{D}_{P})=0$ for $p>0,\ d>0$.

###  3.3. Definition of the variational bihamiltonian cohomology

We proceed to define the variational bihamiltonian cohomology and discuss its
relations with the bihamiltonian cohomology. Let $(P_{0},P_{1})$ be a
semisimple bihamiltonian structure of hydrodynamic type and
$u^{1},\cdots,u^{n}$ be its canonical coordinates. We will use $\tilde{D}_{0}$
and $\tilde{D}_{1}$ to denote $\tilde{D}_{P_{0}}$ and $\tilde{D}_{P_{1}}$
respectively. By using the identity (3.2) we have
$\tilde{D}_{i}\tilde{D}_{j}+\tilde{D}_{j}\tilde{D}_{i}=0$ for $i,j=0,1$.

######  Definition 3.2.

The variational bihamiltonian cohomology for $(P_{0},P_{1})$ is defined to be
the following groups:

(3.3)
$\displaystyle\operatorname{\mathit{VBH}}^{p}_{d}(\Omega,\tilde{D}_{0},\tilde{D}_{1})=\frac{\Omega_{d}^{p}\cap\ker\tilde{D}_{0}\cap\ker\tilde{D}_{1}}{\Omega_{d}^{p}\cap\operatorname{Im}\tilde{D}_{0}\tilde{D}_{1}};$
(3.4)
$\displaystyle\operatorname{\mathit{VBH}}^{p}_{d}(\bar{\Omega},\tilde{D}_{0},\tilde{D}_{1})=\frac{\bar{\Omega}_{d}^{p}\cap\ker\tilde{D}_{0}\cap\ker\tilde{D}_{1}}{\bar{\Omega}_{d}^{p}\cap\operatorname{Im}\tilde{D}_{0}\tilde{D}_{1}}.$

The following lemmas are important for computing the cohomology groups.

######  Lemma 3.5.

The cohomology groups of the cochain complex
$\Omega[\lambda]=\Omega\otimes\mathbb{R}[\lambda]$ with differential
$\partial_{\lambda}=\tilde{D}_{1}-\lambda\tilde{D}_{0}$ is isomorphic to the
variational bihamiltonian cohomology, i.e.

(3.5)
${H}^{p}_{d}(\Omega[\lambda],\partial_{\lambda})\cong\operatorname{\mathit{VBH}}^{p}_{d}(\Omega,\tilde{D}_{0},\tilde{D}_{1}),\quad
d\geq 2.$

Similarly, we have the following isomorphisms for the corresponding quotient
spaces:

(3.6)
${H}^{p}_{d}(\bar{\Omega}[\lambda],\partial_{\lambda})\cong\operatorname{\mathit{VBH}}^{p}_{d}(\bar{\Omega},\tilde{D}_{0},\tilde{D}_{1}),\quad
d\geq 2.$

###### Proof.

The lemma follows from the triviality of the variational hamiltonian
cohomology. For details, one may refer to the proof of Lemma 4.4 of [33]. ∎

######  Lemma 3.6 (Salamander lemma).

We have the following isomorphism induced by $\tilde{D}_{0}$ for $p>0,d>0$:

$\displaystyle\tilde{D}_{0}:\quad\frac{\bar{\Omega}^{p}_{d}\cap\ker\tilde{D}_{1}\tilde{D}_{0}}{\bar{\Omega}^{p}_{d}\cap(\operatorname{Im}\tilde{D}_{0}+\operatorname{Im}\tilde{D}_{1})}\xrightarrow[]{\cong}\operatorname{\mathit{VBH}}^{p+1}_{d+1}(\bar{\Omega},\tilde{D}_{0},\tilde{D}_{1}).$

###### Proof.

The lemma follows easily from the triviality of the variational Hamiltonian
cohomology. ∎

The definition of the variational bihamiltonian cohomology is comparable with
the definition of bihamiltonian cohomology. On the space $\hat{\mathcal{A}}$
there are differentials $D_{0}$ and $D_{1}$ which are defined in (2.2) by the
bihamiltonian structure $(P_{0},P_{1})$, and on the space $\hat{\mathcal{F}}$
there are the induced differential which are denoted by $d_{0}$ and $d_{1}$.
The bihamiltonian cohomology is then given by

$\displaystyle\operatorname{\mathit{BH}}^{p}_{d}(\hat{\mathcal{A}},D_{0},D_{1})=\frac{\hat{\mathcal{A}}_{d}^{p}\cap\ker
D_{0}\cap\ker
D_{1}}{\hat{\mathcal{A}}_{d}^{p}\cap\operatorname{Im}D_{0}D_{1}}\cong
H^{p}_{d}(\hat{\mathcal{A}}[\lambda],D_{1}-\lambda D_{0});$
$\displaystyle\operatorname{\mathit{BH}}^{p}_{d}(\hat{\mathcal{F}},d_{0},d_{1})=\frac{\hat{\mathcal{F}}_{d}^{p}\cap\ker
d_{0}\cap\ker
d_{1}}{\hat{\mathcal{F}}_{d}^{p}\cap\operatorname{Im}d_{0}d_{1}}\cong
H^{p}_{d}(\hat{\mathcal{F}}[\lambda],d_{1}-\lambda d_{0}).$

There is a natural cochain map

(3.7) $\delta:\quad\hat{\mathcal{F}}[\lambda]\to\bar{\Omega}[\lambda]$

between the complexes $\hat{\mathcal{A}}[\lambda]$ and $\Omega[\lambda]$ given
by the de Rham differential $\delta$. This map can be further illustrated as
follows: We first note that the space $\hat{\mathcal{F}}$ can be identified
with

$\mathrm{Der}(\hat{\mathcal{A}})^{D}:=\\{X\in\mathrm{Der}(\hat{\mathcal{A}})\mid\exists\
Y\in\hat{\mathcal{F}},X=D_{Y}\\}$

by using the map (2.2). By applying the identity 2.3, we see that the
differentials $d_{0},d_{1}$ on $\hat{\mathcal{F}}$ induce differentials
$\mathrm{ad}_{D_{0}},\mathrm{ad}_{D_{1}}$ on
$\mathrm{Der}(\hat{\mathcal{A}})^{D}$. Since it is obvious that
$\mathrm{Der}(\hat{\mathcal{A}})^{D}\subseteq\mathrm{Der}(\hat{\mathcal{A}})^{\partial}$,
the identity (3.2) shows that the map (3.7) can be viewed as a natural
embedding (with a change of signs, see the remark below):

$i:\mathrm{Der}(\hat{\mathcal{A}}[\lambda])^{D}\hookrightarrow\mathrm{Der}(\hat{\mathcal{A}}[\lambda])^{\partial}.$

Therefore we conclude that the bihamiltonian cohomology can be viewed as the
cohomology on the space $\mathrm{Der}(\hat{\mathcal{A}}[\lambda])^{D}$, while
the variational bihamiltonian cohomology is the cohomology on the space
$\mathrm{Der}(\hat{\mathcal{A}}[\lambda])^{\partial}$. In this sense, the
bihamiltonian cohomology is just a restriction of the variational
bihamiltonian cohomology onto a subcomplex.

######  Remark 3.2.

The map $\delta:\hat{\mathcal{F}}\to\bar{\Omega}$ can be viewed as a
generalization of the correspondence between the hamiltonian functions and the
hamiltonian vector field on a finite dimensional symplectic manifold. In our
case, this correspondence is twisted by a sign:

$\delta X=(-1)^{p-1}\Phi(D_{X}),\quad X\in\hat{\mathcal{F}}^{p}.$

##  4\. The cohomology group $\operatorname{\mathit{VBH}}^{2}_{3}$

In this section, we compute the bihamiltonian cohomology group
$\operatorname{\mathit{VBH}}^{2}_{3}(\bar{\Omega},\tilde{D}_{0},\tilde{D}_{1})$.
The computation of other cohomology groups will be covered in the next
section. We fix a semisimple bihamiltonian $(P_{0},P_{1})$ of hydrodynamic
type and work in the canonical coordinates such that the bihamiltonian
structure is given by (2.4). Note that if we define a derivation

$\displaystyle D(g^{1},\cdots,g^{n})=$ $\displaystyle\sum_{s\geq
0;i}\partial_{x}^{s}(g^{i}\theta_{i}^{1})\frac{\partial}{\partial u^{i,s}}$
$\displaystyle+\frac{1}{2}\sum_{s\geq
0;i,j}\partial_{x}^{s}\left(\partial_{j}g^{i}u^{j,1}\theta_{i}+g^{i}\frac{\partial_{i}g^{j}}{g^{j}}u^{j,1}\theta_{j}-g^{j}\frac{\partial_{j}g^{i}}{g^{i}}u^{i,1}\theta_{j}\right)\frac{\partial}{\partial
u^{i,s}}$ $\displaystyle+\frac{1}{2}\sum_{s\geq
0;i,j}\partial_{x}^{s}\left(\partial_{i}g^{j}\theta_{j}\theta_{j}^{1}+g^{j}\frac{\partial_{j}g^{i}}{g^{i}}\theta_{i}\theta_{j}^{1}-g^{j}\frac{\partial_{j}g^{i}}{g^{i}}\theta_{j}\theta_{i}^{1}\right)\frac{\partial}{\partial\theta_{i}^{s}}$
$\displaystyle+\frac{1}{2}\sum_{s\geq
0;i,j,k}\partial_{x}^{s}\left(\partial_{i}\left(g^{k}\frac{\partial_{k}g^{j}}{g^{j}}\right)u^{j,1}\theta_{k}\theta_{j}-\partial_{j}\left(g^{k}\frac{\partial_{k}g^{i}}{g^{i}}\right)u^{j,1}\theta_{k}\theta_{i}\right)\frac{\partial}{\partial\theta_{i}^{s}},$

for a set of functions $g^{1},\cdots,g^{n}$, then we know that

$D_{P_{0}}=D(f^{1},\dots,f^{n}),\quad
D_{P_{1}}=D(u^{1}f^{1},\dots,u^{n}f^{n}).$

Here and henceforth we will use $\partial_{i}$ to denote
$\frac{\partial}{\partial u^{i}}$.

From Lemma 3.6 it follows that in order to compute the cohomology group
$\operatorname{\mathit{VBH}}^{2}_{3}(\bar{\Omega},\tilde{D}_{0},\tilde{D}_{1})$
we only need to consider the spaces

$\mathcal{Z}:=\bar{\Omega}^{1}_{2}\cap\ker(\tilde{D}_{0}\tilde{D}_{1}),\quad\mathcal{B}:=\bar{\Omega}^{1}_{2}\cap(\operatorname{Im}\tilde{D}_{0}+\operatorname{Im}\tilde{D}_{1}).$

For an element $\omega\in\bar{\Omega}^{1}_{2}$, we can represent it uniquely
in the form

(4.1) $\omega=\int\sum_{i}X^{i}\delta u^{i}+Y^{i}\delta\theta_{i},\quad
X^{i}\in\hat{\mathcal{A}}^{1}_{2},\quad Y^{i}\in\hat{\mathcal{A}}^{0}_{2},$

where the differential polynomials can be written as

(4.2) $\displaystyle X^{i}$
$\displaystyle=\sum_{j}X^{(i)}_{j}\theta_{j}^{2}+\sum_{j,k}\left(X^{(i)}_{kj}u^{j,1}\theta_{k}^{1}+Z^{(i)}_{jk}u^{k,2}\theta_{j}\right)+\sum_{j,k,l}Z^{(i)}_{j;kl}u^{k,1}u^{l,1}\theta_{j};$
(4.3) $\displaystyle Y^{i}$
$\displaystyle=\sum_{j}Y^{(i)}_{j}u^{j,2}+\sum_{j,k}Y^{(i)}_{jk}u^{j,1}u^{k,1}.$

######  Definition 4.1.

For an element $\omega\in\bar{\Omega}^{1}_{2}$ which is represented in the
form (4.1)–(4.3), the indices $ind_{i}(\omega)$ for $i=1,\cdots,n$ with
respect to a semisimple bihamiltonian structure $(P_{0},P_{1})$ of
hydrodynamic type are defined to be the functions

$ind_{i}(\omega):=\frac{1}{f^{i}}(X^{(i)}_{i}+Y^{(i)}_{i}).$

We will see later that the indices defined above are generalizations of the
central invariants of deformations of bihamiltonian structures of hydrodynamic
type. To compute the cohomology group
$\operatorname{\mathit{VBH}}^{2}_{3}(\bar{\Omega})\cong\mathcal{Z}/\mathcal{B}$,
we need the following two lemmas.

######  Lemma 4.1.

For any given cocycle $\omega\in\mathcal{Z}$, the index $ind_{i}(\omega)$ is a
function of single variable $u^{i}$ for any $i=1,\cdots,n$.

######  Lemma 4.2.

A cocycle $\omega\in\mathcal{Z}$ is a coboundary, i.e. $\omega\in\mathcal{B}$,
if and only if

$ind_{i}(\omega)=0,\quad i=1,\cdots,n.$

######  Theorem 4.3.

The quotient space
$\mathcal{Z}/\mathcal{B}\cong\oplus_{i=1}^{n}C^{\infty}(\mathbb{R})$.

###### Proof.

Given $n$ functions of single variable
$c_{1}(u^{1}),c_{2}(u^{2}),\cdots,c_{n}(u^{n})$, we construct an element
$\tau\in\mathcal{Z}$ as follows:

$\tau=\int\delta\biggl{(}D_{1}\sum_{i}c_{i}(u^{i})u^{i,1}\log
u^{i,1}-D_{0}\sum_{i}u^{i}c_{i}(u^{i})u^{i,1}\log u^{i,1}\biggr{)}.$

One can check directly, or refer to [31], to confirm the fact that actually
$\tau$ is a 1-form with differential polynomial coefficients and that
$\tau\in\mathcal{Z}$.

By a straightforward computation we can show that

$ind_{i}(\tau)=-3c_{i}(u^{i}),\quad i=1,\cdots,n.$

Therefore, it follows from Lemma 4.1 that we can choose suitable
$c_{i}(u^{i})$ such that

$ind_{i}(\omega-\tau)=0$

for any given cocycle $\omega\in\mathcal{Z}$. Then we apply Lemma 4.2 to
conclude that the class $[\omega]$ coincides with the class $[\tau]$ in the
space $\mathcal{Z}/\mathcal{B}$. Lemma 4.2 also implies that $\tau$ is a
coboundary if and only if $c_{i}(u^{i})=0$, hence different choices of
functions $c_{1}(u^{1}),\cdots,c_{n}(u^{n})$ give different classes $[\tau]$
in $\mathcal{Z}/\mathcal{B}$. The theorem is proved. ∎

The remaining part of this section is devoted to the proof of Lemma 4.1 and
Lemma 4.2. The strategy of our proof is: We first prove the ‘only if’ part of
Lemma 4.2, then we prove Lemma 4.1. Finally we prove the ‘if’ part of Lemma
4.2.

###### Proof of Lemma 4.2 (Part 1).

We are to show that if $\omega\in\mathcal{B}$ then its indices vanish. Take
$\alpha\in\bar{\Omega}^{0}_{1}$ and write it uniquely as

$\alpha=\int\sum_{i,j}\alpha^{(i)}_{j}u^{j,1}\delta u^{i},$

then we have

(4.4) $\displaystyle\tilde{D}_{0}\alpha$
$\displaystyle=\int\sum_{i,j}D_{0}(\alpha^{(i)}_{j}u^{j,1})\delta
u^{i}+\alpha^{(i)}_{j}u^{j,1}\delta(D_{0}(u^{i}))$
$\displaystyle=\int\sum_{i,j}\left(\alpha^{(i)}_{j}f^{j}\theta_{j}^{2}+\cdots\right)\delta
u^{i}+\left(-\alpha^{(i)}_{j}f^{i}u^{j,2}+\cdots\right)\delta\theta_{i}.$

Here $\cdots$ stands for the terms that make no contribution to the index.
Then from Definition 4.1 it follows that $ind_{i}(\tilde{D}_{0}\alpha)=0$.
Similarly we have $ind_{i}(\tilde{D}_{1}\alpha)=0$. Hence the indices of a
coboundary must vanish. ∎

The above proof shows that the indices can be defined for a class
$[\omega]\in\mathcal{Z}/\mathcal{B}$. To prove Lemma 4.1, we first choose a
‘normal form’ for every class $[\omega]\in\mathcal{Z}/\mathcal{B}$ that will
simplify the computation.

######  Lemma 4.4.

For any given class $[\sigma]\in\mathcal{Z}/\mathcal{B}$, there exists a
unique $\omega\in\mathcal{Z}$ which can be represented in the form (4.1)–(4.3)
and satisfies the following conditions:

  1. (1)

$[\omega]=[\sigma]$;

  2. (2)

$X^{(i)}_{j}=0$ for $j\neq i$;

  3. (3)

$Y^{(i)}_{j}=0$ for any $i,j$;

  4. (4)

$Y^{(i)}_{ii}=0$.

Such a form $\omega$ is called the normal form of the class $[\sigma]$.

###### Proof.

We first take $\omega=\sigma$, then the first condition is satisfied. We then
adjust $\omega$ by adding elements of $\mathcal{B}$ such that other conditions
are also satisfied. Firstly, according to (4.4), we change $\omega$ to
$\tilde{\omega}=\omega+\tilde{D}_{0}\gamma$ with

$\gamma=\int\sum_{i,j}\frac{Y^{(i)}_{j}}{f^{i}}u^{j,1}\delta u^{i},$

then the third condition is satisfied. As for other conditions, we want to
find $\alpha,\beta\in\bar{\Omega}^{0}_{1}$ such that
$\tilde{\omega}+\tilde{D}_{0}\alpha+\tilde{D}_{1}\beta$ satisfies all the four
conditions.

Let us write $\alpha=\int\sum_{i}\alpha_{i}\delta u^{i}$ and
$\beta=\int\sum_{i}\beta_{i}\delta u^{i}$ for some
$\alpha_{i},\beta_{i}\in\hat{\mathcal{A}}^{0}_{1}$. We can uniquely represent
$\tilde{D}_{0}\alpha$ and $\tilde{D}_{1}\beta$ as follows:

(4.5) $\tilde{D}_{0}\alpha=\int\sum_{i}A_{i}\delta
u^{i}+W_{i}\delta\theta_{i};\quad\tilde{D}_{1}\beta=\int\sum_{i}B_{i}\delta
u^{i}+R_{i}\delta\theta_{i},$

where the differential polynomials $W_{i}$ and $R_{i}$ are given by

$\displaystyle W_{i}=$
$\displaystyle-(\alpha_{i}f^{i})^{\prime}+\frac{1}{2}\sum_{j}\left(\alpha_{i}\partial_{j}f^{i}u^{j,1}+\alpha_{j}f^{j}\frac{\partial_{j}f^{i}}{f^{i}}u^{i,1}-\alpha_{j}f^{i}\frac{\partial_{i}f^{j}}{f^{j}}u^{j,1}\right);$
$\displaystyle R_{i}=$
$\displaystyle-(\beta_{i}u^{i}f^{i})^{\prime}+\frac{1}{2}\sum_{j}\left(\beta_{i}u^{i}\partial_{j}f^{i}u^{j,1}+\beta_{j}u^{j}f^{j}\frac{\partial_{j}f^{i}}{f^{i}}u^{i,1}-\beta_{j}u^{i}f^{i}\frac{\partial_{i}f^{j}}{f^{j}}u^{j,1}\right)$
$\displaystyle+\frac{1}{2}\beta_{i}f^{i}u^{i,1}.$

Note that we should make sure that the third condition is satisfied, so if we
further write $\alpha_{i}=\sum_{j}\alpha^{(i)}_{j}u^{j,1}$ and
$\beta_{i}=\sum_{j}\beta^{(i)}_{j}u^{j,1}$, and compare the coefficients of
$u^{j,2}$ of $W_{i}$ and $R_{i}$, we arrive at
$u^{i}\beta^{(i)}_{j}+\alpha^{(i)}_{j}=0$, hence we must take
$\alpha_{i}=-u^{i}\beta_{i}$. Since the coefficients of $(u^{i,1})^{2}$ in
$W_{i}+R_{i}$ are given by $\beta_{i}^{(i)}f^{i}$, we can choose suitable
functions $\beta_{i}^{(i)}$ such that the condition (4) is satisfied. Finally
we compute the coefficients of $\theta_{j}^{2}$ in $A_{i}+B_{i}$ to obtain

$\beta^{(i)}_{j}(u^{j}-u^{i})f^{j},$

so for $j\neq i$ we can choose suitable functions $\beta^{(i)}_{j}$ such that
the condition (2) is satisfied. Thus we find a form $\omega$ which satisfy all
the four conditions, and the above computation also shows that such an
$\omega$ is unique. The lemma is proved. ∎

As a byproduct of the above computation, we have the following theorem.

######  Theorem 4.5.

We have
$\operatorname{\mathit{VBH}}^{1}_{2}(\bar{\Omega},\tilde{D}_{0},\tilde{D}_{1})=0$.

###### Proof.

Recall that by definition
$\operatorname{\mathit{VBH}}^{1}_{2}(\bar{\Omega},\tilde{D}_{0},\tilde{D}_{1})=\bar{\Omega}^{1}_{2}\cap\ker\tilde{D}_{0}\cap\ker\tilde{D}_{1}$.
Now take any cocycle $\omega$, by using the triviality of the variational
Hamiltonian cohomology we can find $\alpha,\beta\in\bar{\Omega}^{0}_{1}$ such
that

(4.6) $\omega=\tilde{D}_{0}\alpha=\tilde{D}_{1}\beta.$

Let us show that $\alpha=\beta=0$. To this end we write
$\alpha=\int\sum_{i}\alpha_{i}\delta u^{i}$ and
$\beta=\int\sum_{i}\beta_{i}\delta u^{i}$ for some
$\alpha_{i},\beta_{i}\in\hat{\mathcal{A}}^{0}_{1}$, and represent
$\tilde{D}_{0}\alpha$, $\tilde{D}_{1}\beta$, as we do in the proof of Lemma
4.4, in the form

(4.7) $\tilde{D}_{0}\alpha=\int\sum_{i}A_{i}\delta
u^{i}+W_{i}\delta\theta_{i};\quad\tilde{D}_{1}\beta=\int\sum_{i}B_{i}\delta
u^{i}+R_{i}\delta\theta_{i}.$

From the coefficients of $u^{j,2}$ in $W_{i}$ and $R_{i}$ it follows that
$\alpha_{i}=u^{i}\beta_{i}$, and from the coefficients of $(u^{i,1})^{2}$ in
$W_{i}$ and $R_{i}$ we conclude that $\beta^{(i)}_{i}=0$. Finally, from the
coefficients of $\theta^{2}_{j}$ in $A_{i}$ and $B_{i}$ for $j\neq i$ it
follows that $\beta^{(i)}_{j}=0$ when $j\neq i$. Therefore $\alpha=\beta=0$
and the theorem is proved. ∎

Now let us come back to prepare the proof of Lemma 4.1. Take a class
$[\omega]\in\mathcal{Z}/\mathcal{B}$ with $\omega$ being its normal form which
can be represented as

(4.8) $\omega=\int\sum_{i}X^{i}\delta u^{i}+Y^{i}\delta\theta_{i},\quad
X^{i}\in\hat{\mathcal{A}}^{1}_{2},\quad Y^{i}\in\hat{\mathcal{A}}^{0}_{2},$

where the differential polynomials can be written in the form

(4.9) $\displaystyle X^{i}$
$\displaystyle=X^{(i)}_{i}\theta_{i}^{2}+\sum_{j,k}\left(X^{(i)}_{kj}u^{j,1}\theta_{k}^{1}+Z^{(i)}_{jk}u^{k,2}\theta_{j}\right)+\sum_{j,k,l}Z^{(i)}_{j;kl}u^{k,1}u^{l,1}\theta_{j};$
(4.10) $\displaystyle Y^{i}$
$\displaystyle=\sum_{j,k}Y^{(i)}_{jk}u^{j,1}u^{k,1},\quad Y^{(i)}_{ii}=0.$

Further more, we require that

(4.11) $Z^{(i)}_{j;kl}=Z^{(i)}_{j;lk},\quad Y^{(i)}_{jk}=Y^{(i)}_{kj}.$

Due to the part of Lemma 4.2 that we just proved, different representatives of
a class in $\mathcal{Z}/\mathcal{B}$ have the same indices, hence to prove
Lemma 4.1 we only need to show that $ind_{i}(\omega)=X^{(i)}_{i}/f_{i}$ is a
function of $u^{i}$ for each $i=1,\dots,n$.

Since $\omega\in\mathcal{Z}$, from the triviality of variational Hamiltonian
cohomology it follows the existence of $\alpha\in\bar{\Omega}^{1}_{2}$ such
that $\tilde{D}_{0}\omega=\tilde{D}_{1}\alpha$. Such an $\alpha$ is unique up
to the addition of an image of $\tilde{D}_{1}$, hence we can make a particular
choice of $\alpha$ in a similar way as we choose the normal form of $\omega$.
More explicitly, it is not difficult to see that there exists a unique
$\alpha$ such that

(4.12) $\alpha=\int\sum_{i}P^{i}\delta u^{i}+Q^{i}\delta\theta_{i},\quad
P^{i}\in\hat{\mathcal{A}}^{1}_{2},\quad Q^{i}\in\hat{\mathcal{A}}^{0}_{2},$

where the differential polynomials can be written as

(4.13) $\displaystyle P^{i}$
$\displaystyle=\sum_{j}P^{(i)}_{j}\theta_{j}^{2}+\sum_{j,k}\left(P^{(i)}_{kj}u^{j,1}\theta_{k}^{1}+W^{(i)}_{jk}u^{k,2}\theta_{j}\right)+\sum_{j,k,l}W^{(i)}_{j;kl}u^{k,1}u^{l,1}\theta_{j},$
(4.14) $\displaystyle Q^{i}$
$\displaystyle=\sum_{j,k}Q^{(i)}_{jk}u^{j,1}u^{k,1}$

with

(4.15) $W^{(i)}_{j;kl}=W^{(i)}_{j;lk},\quad Q^{(i)}_{jk}=Q^{(i)}_{kj}.$

###### Proof of Lemma 4.1.

Consider a class $[\omega]\in\mathcal{Z}/\mathcal{B}$ where $\omega$ is given
by the normal form (4.8)–(4.11). Let $\alpha\in\bar{\Omega}^{1}_{2}$ be the
unique element given by (4.12)–(4.15) such that
$\tilde{D}_{0}\omega=\tilde{D}_{1}\alpha$. For the bihamiltonian structure
$(P_{0},P_{1})$ given in (2.4), we denote

(4.16) $a_{ij}=\frac{1}{2}\partial_{i}f^{j},\quad
b_{ij}=\frac{1}{2}f^{i}\frac{\partial_{i}f^{j}}{f^{j}}.$

Let us first compute the differential polynomials $M^{i},N^{i},S^{i},T^{i}$
that are defined by

$\tilde{D}_{0}\omega=\int\sum_{i}M^{i}\delta
u^{i}+N^{i}\delta\theta_{i},\quad\tilde{D}_{1}\alpha=\int\sum_{i}S^{i}\delta
u^{i}+T^{i}\delta\theta_{i}.$

In what follows of the proof, we will omit the symbol of summations and use
$j,k,l$ to denote the indices that should be summed over $1,\cdots,n$. The
index $i$ is a fixed index and do not participate in the summation. It is
straightforward to obtain

$\displaystyle M^{i}$
$\displaystyle=D_{0}(X^{i})-\partial_{i}f^{j}X^{j}\theta_{j}^{1}-X^{k}\partial_{i}a_{jk}u^{j,1}\theta_{k}+\left(X^{j}a_{ij}\theta_{j}\right)^{\prime}-X^{k}\partial_{i}b_{kj}u^{j,1}\theta_{j}$
$\displaystyle+\left(X^{j}b_{ji}\theta_{i}\right)^{\prime}+X^{k}\partial_{i}b_{jk}u^{k,1}\theta_{j}-\left(X^{i}b_{ji}\theta_{j}\right)^{\prime}+Y^{k}\partial_{i}a_{kj}\theta_{j}\theta_{j}^{1}+Y^{k}\partial_{i}b_{jk}\theta_{k}\theta_{j}^{1}$
$\displaystyle-Y^{k}\partial_{i}b_{jk}\theta_{j}\theta_{k}^{1}+Y^{l}\partial_{i}\partial_{l}b_{kj}u^{j,1}\theta_{k}\theta_{j}-\left(Y^{j}\partial_{j}b_{ki}\theta_{k}\theta_{i}\right)^{\prime}$
$\displaystyle-Y^{l}\partial_{i}\partial_{j}b_{kl}u^{j,1}\theta_{k}\theta_{l}+\left(Y^{j}\partial_{i}b_{kj}\theta_{k}\theta_{j}\right)^{\prime},$
$\displaystyle S^{i}$
$\displaystyle=D_{1}(P^{i})-P^{j}\partial_{i}\left(u^{j}f^{j}\right)\theta_{j}^{1}-P^{k}\partial_{i}\left(u^{k}a_{jk}\right)u^{j,1}\theta_{k}+\left(P^{j}u^{j}a_{ij}\theta_{j}\right)^{\prime}$
$\displaystyle-P^{k}\partial_{i}\left(u^{k}b_{kj}\right)u^{j,1}\theta_{j}+\left(P^{j}u^{j}b_{ji}\theta_{i}\right)^{\prime}+P^{k}\partial_{i}\left(u^{j}b_{jk}\right)u^{k,1}\theta_{j}-\left(P^{i}u^{j}b_{ji}\theta_{j}\right)^{\prime}$
$\displaystyle-P^{j}a_{ij}u^{j,1}\theta_{j}+\left(\frac{1}{2}P^{i}f^{i}\theta_{i}\right)^{\prime}+Q^{k}\partial_{i}\left(u^{j}a_{kj}\right)\theta_{j}\theta_{j}^{1}+Q^{k}\partial_{i}\left(u^{j}b_{jk}\right)\theta_{k}\theta_{j}^{1}$
$\displaystyle-Q^{k}\partial_{i}\left(u^{j}b_{jk}\right)\theta_{j}\theta_{k}^{1}+Q^{j}a_{ij}\theta_{j}\theta_{j}^{1}+Q^{l}\partial_{i}\partial_{l}\left(u^{k}b_{kj}\right)u^{j,1}\theta_{k}\theta_{j}$
$\displaystyle-\left(Q^{j}\partial_{j}\left(u^{k}b_{ki}\right)\theta_{k}\theta_{i}\right)^{\prime}-Q^{l}\partial_{i}\partial_{j}\left(u^{k}b_{kl}\right)u^{j,1}\theta_{k}\theta_{l}+\left(Q^{j}\partial_{i}\left(u^{k}b_{kj}\right)\theta_{k}\theta_{j}\right)^{\prime},$
$\displaystyle N^{i}$
$\displaystyle=D_{0}(Y^{i})+\left(X^{i}f^{i}\right)^{\prime}-X^{i}a_{ji}u^{j,1}-X^{j}b_{ji}u^{i,1}+X^{j}b_{ij}u^{j,1}-\left(Y^{j}a_{ji}\theta_{i}\right)^{\prime}$
$\displaystyle-Y^{j}a_{ji}\theta_{i}^{1}-\left(Y_{j}b_{ij}\theta_{j}\right)^{\prime}-Y^{i}b_{ji}\theta_{j}^{1}+\left(Y^{i}b_{ji}\theta_{j}\right)^{\prime}+Y^{j}b_{ij}\theta_{j}^{1}$
$\displaystyle+Y^{j}\partial_{j}b_{ki}u^{i,1}\theta_{k}-Y^{k}\partial_{k}b_{ij}u^{j,1}\theta_{j}-Y^{i}\partial_{j}b_{ki}u^{j,1}\theta_{k}+Y^{k}\partial_{j}b_{ik}u^{j,1}\theta_{k},$
$\displaystyle T^{i}$
$\displaystyle=D_{1}(Q^{i})+\left(P^{i}u^{i}f^{i}\right)^{\prime}-P^{i}u^{i}a_{ji}u^{j,1}-P^{j}u^{j}b_{ji}u^{i,1}+P^{j}u^{i}b_{ij}u^{j,1}-\frac{1}{2}P^{i}f^{i}u^{i,1}$
$\displaystyle-\left(Q_{j}u^{i}a_{ji}\theta_{i}\right)^{\prime}-Q^{j}u^{i}a_{ji}\theta_{i}^{1}-\left(Q^{j}u^{i}b_{ij}\theta_{j}\right)^{\prime}-Q^{i}u^{j}b_{ji}\theta_{j}^{1}+\left(Q_{i}u^{j}b_{ji}\theta_{j}\right)^{\prime}$
$\displaystyle+Q^{j}u^{i}b_{ij}\theta_{j}^{1}+Q^{j}\partial_{j}\left(u^{k}b_{ki}\right)u^{i,1}\theta_{k}-Q^{k}\partial_{k}\left(u^{i}b_{ij}\right)u^{j,1}\theta_{j}$
$\displaystyle-Q^{i}\partial_{j}\left(u^{k}b_{ki}\right)u^{j,1}\theta_{k}+Q^{k}\partial_{j}\left(u^{i}b_{ik}\right)u^{j,1}\theta_{k}-\left(\frac{1}{2}Q^{i}f^{i}\theta_{i}\right)^{\prime}-\frac{1}{2}Q^{i}f^{i}\theta_{i}^{1}.$

By comparing the coefficients of $\theta_{j}^{3}$ of both sides of the
equation $N^{i}=T^{i}$ we obtain

(4.17) $P^{(i)}_{j}=0,\quad i\neq j;\quad X^{(i)}_{i}=u^{i}P^{(i)}_{i}.$

Comparing the coefficients of $u^{k,3}\theta_{j}$ on both sides of the
equation $N^{i}=T^{i}$, we conclude that

(4.18) $Z^{(i)}_{jk}=u^{i}W^{(i)}_{jk},\quad\forall\ i,j,k.$

Next, from the coefficients of $\theta_{j}^{1}\theta_{i}^{2}$ in $M^{i}$ and
$S^{i}$ for $j\neq i$, and from (4.17) it follows that

(4.19)
$\left(\partial_{j}P^{(i)}_{i}f^{j}-2P^{(i)}_{i}b_{ji}\right)(u^{i}-u^{j})=\left(X^{(i)}_{ji}-P^{(i)}_{ji}u^{i}\right)f^{i},\quad
i\neq j.$

Similarly, from the coefficients of $u^{j,2}\theta_{k}^{1}$ in $N^{i}$ and
$T^{i}$, and from the identity (4.18) it follows that

(4.20) $X^{(i)}_{kj}=u^{i}P^{(i)}_{kj},\quad\forall\ i,j,k.$

The identity (4.19) and (4.20) lead to

$\partial_{j}\frac{P^{(i)}_{i}}{f^{i}}=0,\quad i\neq j.$

Thus for each $i$ the function

$ind_{i}(\omega)=\frac{X^{(i)}_{i}}{f^{i}}=u^{i}\frac{P^{(i)}_{i}}{f^{i}}.$

depends only on $u^{i}$. The lemma is proved. ∎

Finally, let us complete the proof of Lemma 4.2. Before proceeding to the
proof, we first derive some relations satisfied by the coefficients in
$\omega$ and $\alpha$. In the rest of this section, we will continue to use
the notations which are used in the proof of Lemma 4.1.

By comparing the coefficients of $\theta_{k}\theta_{j}^{3}$,
$\theta_{j}\theta_{j}^{3}$ and $\theta_{i}\theta_{j}^{3}$ in $M^{i}$ and
$S^{i}$ and by using the relations (4.17) and (4.18) we obtain

(4.21) $Z^{(i)}_{kj}=W^{(i)}_{kj}=0,\quad i\neq j\neq k\neq i.$ (4.22)
$P^{(i)}_{i}a_{ij}=W^{(i)}_{jj}f^{j},\quad i\neq j.$ (4.23)
$P^{(i)}_{i}b_{ji}=W^{(i)}_{ij}f^{j},\quad i\neq j.$

We also compare the coefficients of $\theta_{k}^{1}\theta_{j}^{2}$,
$\theta_{i}^{1}\theta_{j}^{2}$ and $\theta_{j}^{1}\theta_{j}^{2}$ in $M^{i}$
and $S^{i}$ to arrive at

(4.24) $X^{(i)}_{kj}=P^{(i)}_{kj}u^{j},\quad i\neq j\neq k\neq i.$ (4.25)
$3P^{(i)}_{i}(u^{i}-u^{j})b_{ji}=(X^{(i)}_{ij}-P^{(i)}_{ij}u^{j})f^{j},\quad
i\neq j.$ (4.26)
$P^{(i)}_{i}(u^{i}-u^{j})a_{ij}=(X^{(i)}_{jj}-P^{(i)}_{jj}u^{j})f^{j},\quad
i\neq j.$

By comparing the coefficients of $u^{k,1}\theta_{j}^{2}$,
$u^{j,1}\theta_{j}^{2}$, $u^{i,1}\theta_{j}^{2}$, $u^{j,1}\theta_{i}^{2}$ and
$u^{i,1}\theta_{i}^{2}$ in $N^{i}$ and $T^{i}$, and by using the relation
(4.20) we arrive at the following identities:

(4.27) $Y^{(i)}_{jk}=Q^{(i)}_{jk}u^{j},\quad i\neq j\neq k\neq i.$ (4.28)
$2Y^{(i)}_{jj}f^{j}+X^{(j)}_{j}b_{ij}=2Q^{(i)}_{jj}u^{j}f^{j}+P^{(j)}_{j}u^{i}b_{ij},\quad
i\neq j.$ (4.29) $Y^{(i)}_{ji}=Q^{(i)}_{ji}u^{j},\quad i\neq j.$ (4.30)
$Y^{(i)}_{ij}=Q^{(i)}_{ij}u^{i},\quad i\neq j.$ (4.31)
$2Y^{(i)}_{ii}=2Q^{(i)}_{ii}u^{i}+\frac{1}{2}P^{(i)}_{i}f^{i}.$

###### Proof of Lemma 4.2 (Part 2).

We need to show that a cocycle with vanishing indices must be a coboundary.
Take a class $[\omega]\in\mathcal{Z}/\mathcal{B}$ with $\omega$ being its
normal form and $ind_{i}(\omega)=0$. Let $\alpha\in\bar{\Omega}^{1}_{2}$ be
the unique element given by (4.12)–(4.15) such that
$\tilde{D}_{0}\omega=\tilde{D}_{1}\alpha$. We will prove that
$\alpha=\omega=0$.

From $ind_{i}(\omega)=0$ and (4.17) we know that $X^{(i)}_{i}=P^{(i)}_{i}=0$.
Then the identities (4.18) and (4.21)–(4.23) show that the coefficients
$W^{(i)}_{jk}$ vanish when $k\neq j$ and $Z^{(i)}_{ji}=u^{i}W^{(i)}_{ji}$; the
identities (4.24)–(4.26) together with (4.20) show that the coefficients
$P^{(i)}_{jk}$ vanish when $k\neq i$ and $X^{(i)}_{ji}=u^{i}P^{(i)}_{ji}$.
Similarly, we conclude from the identities (4.11), (4.15) and (4.27)–(4.31)
that the coefficients $Q^{(i)}_{jk}=0$ when $k\neq j$ and
$Y^{(i)}_{jj}=u^{j}Q^{(i)}_{jj}$. In particular, from (4.11) we know that
$Q^{(i)}_{ii}=0$.

To simplify the notations, we use $Y^{(i)}_{j}$ and $Q^{(i)}_{j}$ to denote
$Y^{(i)}_{jj}$ and $Q^{(i)}_{jj}$. Therefore we can represent the coefficients
of $\omega$ and $\alpha$ that are given in (4.9), (4.10) and (4.13), (4.14) as
follows:

$\displaystyle X^{i}$
$\displaystyle=\sum_{j}\left(X^{(i)}_{ji}u^{i,1}\theta_{j}^{1}+Z^{(i)}_{ji}u^{i,2}\theta_{j}\right)+\sum_{j,k,l}Z^{(i)}_{l;jk}u^{k,1}u^{j,1}\theta_{l},$
$\displaystyle P^{i}$
$\displaystyle=\sum_{j}\left(P^{(i)}_{ji}u^{i,1}\theta_{j}^{1}+W^{(i)}_{ji}u^{i,2}\theta_{j}\right)+\sum_{j,k,l}W^{(i)}_{l;jk}u^{k,1}u^{j,1}\theta_{l},$
$\displaystyle Y^{i}$ $\displaystyle=\sum_{j}Y^{(i)}_{j}(u^{j,1})^{2},\quad
Q^{i}=\sum_{j}Q^{(i)}_{j}(u^{j,1})^{2},$

where the coefficients satisfy the conditions

(4.32) $\displaystyle X^{(i)}_{ji}=u^{i}P^{(i)}_{ji},\quad
Z^{(i)}_{ji}=u^{i}W^{(i)}_{ji};$ (4.33) $\displaystyle
Y^{(i)}_{j}=u^{j}Q^{(i)}_{j},\quad Y^{(i)}_{i}=Q^{(i)}_{i}=0.$

Let us first compare the coefficients of $u^{i,1}u^{i,2}\theta_{j}$ in $N^{i}$
and $T^{i}$ to arrive at

(4.34)
$2Z^{(i)}_{j;ii}=2u^{i}W^{(i)}_{j;ii}-\frac{1}{2}W^{(i)}_{ji},\quad\forall\
i,j.$

At the same time we compute the coefficients of
$u^{i,1}\theta_{j}\theta_{i}^{2}$ in $M^{i}$ and $S^{i}$, and we obtain

(4.35)
$-2Z^{(i)}_{ji}\partial_{i}f^{i}-2Z^{(i)}_{j;ii}f^{i}=-2W^{(i)}_{ji}\partial_{i}(u^{i}f^{i})-2W^{(i)}_{l;ii}u^{i}f^{i}-\frac{1}{2}W^{(i)}_{ji}f^{i},\quad\forall\
i,j.$

Then it follows from the identities (4.32), (4.34) and (4.35) that

(4.36) $Z^{(i)}_{ji}=W^{(i)}_{ji}=0,\quad\forall\ i,j.$

Next we compare the coefficients of $u^{k,1}u^{j,2}\theta_{l}$ in $N^{i}$ and
$T^{i}$ for distinct $j$, $k$ and we obtain the relations

(4.37) $Z^{(i)}_{l;jk}=u^{i}W^{(i)}_{l;jk},\quad j\neq k,\quad\forall\ i,l.$

For $j\neq i$, we compare the coefficients of
$u^{j,1}\theta_{l}\theta_{k}^{2}$ in $M^{i}$ and $S^{i}$ and we arrive at

(4.38) $Z^{(i)}_{l;jk}=u^{k}W^{(i)}_{l;jk},\quad i\neq j,\quad\forall\ k,l.$

Since $Z^{i}_{l;jk}=Z^{(i)}_{l;kj}$, $W^{i}_{l;jk}=W^{(i)}_{l;kj}$, the
identities (4.37) and (4.38) imply that the only possibly non-vanishing
coefficients are $Z^{(i)}_{l;jj}$ and $W^{(i)}_{l;jj}$; moreover, from the
identities (4.34) and (4.36) we conclude that

(4.39) $Z^{(i)}_{l;jj}=u^{j}W^{(i)}_{l;jj},\quad\forall\ i,j,l.$

We proceed to compute the coefficients of $(u^{i,1})^{2}\theta_{j}^{1}$ in
$N^{i}$ and $T^{i}$. By using (4.32) and (4.33), it is easy to deduce that

(4.40) $X^{(i)}_{ji}=P^{(i)}_{ji}=0,\quad\forall\ i,j.$

Thanks to this equation, we can also compute the coefficients of
$(u^{j,1})^{2}\theta_{j}^{1}$ of $N^{i}$ and $T^{i}$ and obtain the relations

(4.41) $f^{i}Z^{(i)}_{j;jj}=u^{i}f^{i}W^{(i)}_{j;jj}+2Q^{(i)}_{j}f^{j},\quad
i\neq j.$

On the other hand, it follows from (4.36) and the coefficients of
$u^{j,1}u^{j,2}\theta_{j}$ in $N^{i}$ and $T^{i}$ for $i\neq j$ that

(4.42) $2f^{i}Z^{(i)}_{j;jj}=2u^{i}f^{i}W^{(i)}_{j;jj}+Q^{(i)}_{j}f^{j},\quad
i\neq j.$

Thus the relations (4.41) and (4.42) together with (4.33) lead to

(4.43) $Y^{i}=Q^{i}=0,\quad\forall\ i.$

Now thanks to (4.36), (4.40) and (4.43), we are able to easily compare the
coefficients of $u^{j,1}u^{j,2}\theta_{k}$ and $u^{j,1}u^{j,2}\theta_{i}$ in
$N^{i}$ and $T^{i}$ respectively and we arrive at

(4.44) $\displaystyle Z^{(i)}_{k;jj}$ $\displaystyle=u^{i}W^{(i)}_{k;jj},\quad
i\neq j\neq k\neq i;$ (4.45) $\displaystyle Z^{(i)}_{i;jj}$
$\displaystyle=u^{i}W^{(i)}_{i;jj},\quad i\neq j,$

Therefore by combining the identities (4.39), (4.44) and (4.45) we conclude
that

$X^{i}=\sum_{j}Z^{(i)}_{j;ii}(u^{i,1})^{2}\theta_{j},\quad
P^{i}=\sum_{j}W^{(i)}_{j;ii}(u^{i,1})^{2}\theta_{j},\quad
Z^{(i)}_{j;ii}=u^{i}W^{(i)}_{j;ii}.$

and comparing the coefficients of $(u^{i,1})^{3}$ we deduce that $P^{i}=0$,
and it follows from the above relation that $X^{i}$ also vanish. The lemma is
proved.

∎

##  5\. Vanishing theorem of the variational bihamiltonian cohomology

###  5.1. Vanishing theorem and the strategy of computation.

In this section, we compute the general variational bihamiltonian cohomology
groups
$\operatorname{\mathit{VBH}}^{p}_{d}(\bar{\Omega},\tilde{D}_{0},\tilde{D}_{1})$.
The main result we are to obtain is the following theorem.

######  Theorem 5.1.

The cohomology group $H^{p}_{d}(\Omega[\lambda],\partial_{\lambda})$ vanishes
unless the bidegree $(p,d)$ belongs to the following two cases:

$\displaystyle\mathrm{Case\,1:}\quad d=0,\cdots,n;\quad p=d+1,\cdots,d+n+1,$
$\displaystyle\mathrm{Case\,2:}\quad d=2,\cdots,n+3;\quad p=d,\cdots,d+n.$

###### Proof of Theorem 2.1.

Similar to Lemma 3.2, we have the following long exact sequence:

$\cdots\rightarrow H^{p}_{d}(\Omega[\lambda],\partial_{\lambda})\rightarrow
H^{p}_{d}(\bar{\Omega}[\lambda],\partial_{\lambda})\rightarrow
H^{p+1}_{d}(\Omega[\lambda],\partial_{\lambda})\rightarrow\cdots.$

Then Theorem 2.1 follows from Theorem 5.1 together with Lemma 3.5, Theorem 4.3
and Theorem 4.5. ∎

The strategy to prove the above theorem is inspired by the work [5, 4], where
some appropriate spectral sequences are used to compute the bihamiltonian
cohomology of a bihamiltonian structure of hydrodynamic type. Our computation
in this section can be viewed as a certain generalization of that of [4],
therefore we will use the same notations as the ones used in [4] whenever
possible.

For a semisimple bihamiltonian structure $(P_{0},P_{1})$ of hydrodynamic type,
we will work in its canonical coordinates $u^{1},\cdots,u^{n}$. Given
$\omega\in\Omega^{p}$, we represent it in the form

$\omega=\sum_{i;s\geq 0}g_{i,s}\delta
u^{i,s}+h^{i}_{s}\delta\theta_{i}^{s},\quad g_{i,s}\in\hat{\mathcal{A}}^{p},\
h^{i}_{s}\in\hat{\mathcal{A}}^{p-1},$

then by using the formula given in Example 3.2 we have

$\tilde{D}_{a}\omega=\sum_{i;s\geq 0}(D_{P_{a}}g_{i,s})\delta
u^{i,s}+(-1)^{p}g_{i,s}\delta(D_{P_{a}}u^{i,s})+(D_{P_{a}}h^{i}_{s})\delta\theta_{i}^{s}+(-1)^{p-1}h^{i}_{s}\delta(D_{P_{a}}\theta_{i}^{s})$

for $a=0,1$. We are going to compute the cohomology of the complex
$(\Omega[\lambda],\partial_{\lambda}=\tilde{D}_{1}-\lambda\tilde{D}_{0})$ in
what follows.

Let us define a gradation, called the $u$-degree, on $\Omega$ by setting

$\deg_{u}u^{i,s+1}=1,\ \deg_{u}\delta u^{i,s}=1,\quad i=1,\cdots,n;\quad s\geq
0,$

and the $u$-degrees of other generators are set to be 0. Let us denote the
super degree of an element $\omega$ of $\Omega$ by $\deg_{\theta}\omega$. We
filtrate the complex $\Omega[\lambda]$ by defining

$F^{k}\Omega[\lambda]=\\{\omega\in\Omega[\lambda]\mid\deg_{u}\omega+\deg_{\theta}\omega\geq
k\\},$

then we have a filtration

$\cdots F^{k+1}\Omega[\lambda]\subset F^{k}\Omega[\lambda]\subset\cdots
F^{0}\Omega[\lambda]=\Omega[\lambda].$

We also decompose the differential in the following way:

$\partial_{\lambda}=\Delta_{-1}+\Delta_{0}+\cdots,\quad\deg_{u}\Delta_{k}=k.$

Note that $\deg_{\theta}\partial_{\lambda}=1$, therefore each $\Delta_{k}$
preserves the filtration. Now let $(^{1}E_{r},d_{r}),r\geq 0$ be the spectral
sequence induced by the filtration, then by the standard construction
$(^{1}E_{0},d_{0})=(\Omega[\lambda],\Delta_{-1})$, and ${}^{1}E_{k}$ is given
by the cohomology of ${}^{1}E_{k-1}$ for $k\geq 1$. It is clear that when
restricted to $\Omega^{p}_{d}[\lambda]$, the above filtration is bounded and
therefore this guarantees the convergence of the spectral sequence. Note that
the differential $d_{r}$ with $r=0$ or $1$ is not the induced differential on
$\hat{\mathcal{F}}$ used in Sect. 3.

We will compute ${}^{1}E_{1}$ and ${}^{1}E_{2}$ in the following subsections,
to this end we need to construct some other spectral sequences as in [4]. The
following standard fact is also used in the computation.

######  Lemma 5.2.

Let $(\mathcal{C}^{\bullet},d)$ be a cochain complex in an Abelian category.
Assume that for each $p$ we have the decomposition
$\mathcal{C}^{p}=\mathcal{A}^{p}\oplus\mathcal{B}^{p}$ with
$\mathcal{A}^{\bullet}$ being a subcomplex, and $\mathcal{A}^{\bullet}$ is
acyclic. Then

(5.1) $H^{p}(\mathcal{C}^{\bullet},d)\cong
H^{p}(\mathcal{B}^{\bullet},\pi_{B}\circ d).$

Here $\pi_{B}$ is the projection from $\mathcal{C}^{\bullet}$ to
$\mathcal{B}^{\bullet}$.

###  5.2. Computation of ${}^{1}E_{1}$.

We first write down $\Delta_{-1}$ in the explicit form

(5.2) $\displaystyle\Delta_{-1}$ $\displaystyle=\sum_{s\geq
1;i}(-\lambda+u^{i})f^{i}\theta_{i}^{s+1}\frac{\partial}{\partial
u^{i,s}}+\sum_{s\geq
0;i}(-\lambda+u^{i})f^{i}\delta\theta_{i}^{s+1}\frac{\partial}{\partial\delta
u^{i,s}}$ $\displaystyle:=\sum_{i}(-\lambda+u^{i})f^{i}\delta\hat{d}_{i},$

here we introduce the de Rham-type differential

(5.3) $\delta\hat{d}_{i}=\sum_{s\geq
1}\theta_{i}^{s+1}\frac{\partial}{\partial u^{i,s}}+\sum_{s\geq
0}\delta\theta_{i}^{s+1}\frac{\partial}{\partial\delta u^{i,s}}.$

We will also use the notation

(5.4) $\hat{d}_{i}=\sum_{s\geq 1}\theta_{i}^{s+1}\frac{\partial}{\partial
u^{i,s}}.$

######  Lemma 5.3.

Each summand of the following decomposition is a cochain subcomplex with
respect to $\Delta_{-1}$:

(5.5) $\Omega[\lambda]=\hat{\mathcal{A}}[\lambda]\\{\delta\theta_{i}\mid
i=1,\cdots,n\\}\oplus\bigoplus_{i=1}^{n}\hat{\mathcal{A}}[\lambda]\\{\delta
u^{i,s},\delta\theta_{i}^{s+1}\mid s\geq 0\\}.$

Here we use the notation $R\\{g_{1},g_{2},\dots\\}$ to denote the free
$R$-module generated by $g_{1},g_{2},\dots$.

In what follows, we will use the same notations as in [5] to denote the
following subspaces:

$\displaystyle\hat{\mathcal{C}}$
$\displaystyle=C^{\infty}(u)[\theta_{1},\cdots,\theta_{n},\theta_{1}^{1},\cdots,\theta_{n}^{1}],$
$\displaystyle\hat{\mathcal{C}}_{i}$
$\displaystyle=\hat{\mathcal{C}}[[u^{i,s},\theta_{i}^{s+1}\mid s\geq 1]],$

and use $\hat{\mathcal{C}}_{i}^{nt}$ to denote the subspace of
$\hat{\mathcal{C}}_{i}$ spanned by nontrivial monomials, i.e., all monomials
that contain at least one of the variables $u^{i,s}$, $\theta_{i}^{s+1}$ for
$s\geq 1$. We also use $\hat{\mathcal{M}}$ to denote the subspace of
$\hat{\mathcal{A}}$ spanned by monomials which contain at least one of the
mixed quadratic expressions:

$u^{i,s}u^{j,t};\quad
u^{i,s}\theta_{j}^{t+1};\quad\theta_{i}^{s+1}\theta_{j}^{t+1}$

for some $i\neq j$ and $s,t\geq 1$. Then it is easy to see that there is a
decomposition

(5.6)
$\hat{\mathcal{A}}=\hat{\mathcal{C}}\oplus\left(\bigoplus_{i=1}^{n}\hat{\mathcal{C}}_{i}^{nt}\right)\oplus\hat{\mathcal{M}},$

and each summand is preserved under the action of $\hat{d}_{i}$ given by
(5.4).

######  Lemma 5.4.

We have

$H(\hat{\mathcal{A}}[\lambda]\\{\delta\theta\\},\Delta_{-1})=\hat{\mathcal{C}}[\lambda]\\{\delta\theta\\}\oplus\bigoplus_{i=1}^{n}\frac{\hat{d}_{i}(\hat{\mathcal{C}}_{i}[\lambda])}{(-\lambda+u^{i})\hat{d}_{i}(\hat{\mathcal{C}}_{i}[\lambda])}\\{\delta\theta\\}.$

Here $\\{\delta\theta\\}$ is the abbreviation of $\\{\delta\theta_{i},\
i=1,\cdots,n\\}$.

###### Proof.

It is clear from the formula (5.2) that

$H(\hat{\mathcal{A}}[\lambda]\\{\delta\theta\\},\Delta_{-1})=H(\hat{\mathcal{A}}[\lambda],\Delta_{-1})\\{\delta\theta\\},$

where the action of $\Delta_{-1}$ on $\hat{\mathcal{A}}[\lambda]$ is just
$\sum_{i}(-\lambda+u^{i})f^{i}\hat{d}_{i}$. Hence the lemma is proved by
applying the result in [5]. ∎

According to the decomposition (5.5), we still need to compute cohomology

$H(\hat{\mathcal{A}}[\lambda]\\{\delta u^{i,s},\delta\theta_{i}^{s+1}\mid
s\geq 0\\},\Delta_{-1}).$

To this end, we construct a second spectral sequence ${}^{2}E$. For a fixed
index $i$, we define the $u^{i}$-degree by setting

$\deg_{u^{i}}u^{i,s+1}=1,\quad\deg_{u^{i}}\delta u^{i,s}=1,\quad s\geq 0,$

and other generators have $u^{i}$-degree zero. Accordingly, we decompose the
differential $\Delta_{-1}$ as follows

$\Delta_{-1}=\Delta_{-1,-1}+\Delta_{-1,0},\quad\deg_{u^{i}}\Delta_{-1,k}=k,$

where

$\Delta_{-1,-1}=(-\lambda+u^{i})f^{i}\delta\hat{d}_{i},\quad\Delta_{-1,0}=\sum_{j\neq
i}(-\lambda+u^{j})f^{j}\delta\hat{d}_{j}.$

Similar to our construction of ${}^{1}E$, we filtrate
$\hat{\mathcal{A}}[\lambda]\\{\delta u^{i,s},\delta\theta_{i}^{s+1}\mid s\geq
0\\}$ with $\deg_{u^{i}}+\deg_{\theta}$, and construct the associated spectral
sequence ${}^{2}E$. Then we have

$(^{2}E_{0},d_{0})=(\hat{\mathcal{A}}[\lambda]\\{\delta
u^{i,s},\delta\theta_{i}^{s+1}\mid s\geq 0\\},\Delta_{-1,-1})$

and ${}^{2}E_{1}=H(^{2}E_{0},d_{0})$, with differential $d_{1}=\Delta_{-1,0}$.
Since on the ${}^{2}E_{2}$ page the differential becomes 0, hence this
spectral sequence becomes convergent on this page, i.e.

$H(\hat{\mathcal{A}}[\lambda]\\{\delta u^{i,s},\delta\theta_{i}^{s+1}\mid
s\geq 0\\},\Delta_{-1})\cong{{}^{2}E_{2}}.$

Let us compute ${}^{2}E_{1}=H(^{2}E_{0},d_{0})$. Take any element

(5.7) $\omega=\sum_{s\geq 0}p_{s}\delta
u^{i,s}+q_{s}\delta\theta_{i}^{s+1}\in{{}^{2}E_{0}},$

where we assume that $\omega$ is homogeneous of degree $p$ with respect to the
$\theta$-degree. Then it is easy to see that

$d_{0}\omega=(-\lambda+u^{i})f^{i}\sum_{s\geq 0}\hat{d}_{i}(p_{s})\delta
u^{i,s}+\left(\hat{d}_{i}(q_{s})+(-1)^{p}p_{s}\right)\delta\theta_{i}^{s+1}.$

So if $d_{0}\omega=0$, we have $p_{s}=(-1)^{p+1}\hat{d}_{i}(q_{s})$, and we
can write

$\omega=\delta\hat{d}_{i}\biggl{(}\sum_{s\geq 0}(-1)^{p+1}q_{s}\delta
u^{i,s}\biggr{)}.$

Let us denote

$\hat{\mathcal{B}}_{i}=\hat{\mathcal{A}}\\{\delta u^{i,s},\ s\geq 0\\},$

then the above computation shows that $\ker
d_{0}\subset\delta\hat{d}_{i}(\hat{\mathcal{B}}_{i})[\lambda]$. The inverse
inclusion $\delta\hat{d}_{i}(\hat{\mathcal{B}}_{i})[\lambda]\subset\ker d_{0}$
is clear, hence we have $\ker
d_{0}=\delta\hat{d}_{i}(\hat{\mathcal{B}}_{i})[\lambda]$.

To compute the image of the differential $d_{0}$, we take another element
$\omega\in{{}^{2}E_{0}}$ represented in the form of (5.7). Observe that

$\displaystyle d_{0}\omega$ $\displaystyle=(-\lambda+u^{i})f^{i}\sum_{s\geq
0}\hat{d}_{i}(p_{s})\delta
u^{i,s}+\left(\hat{d}_{i}(q_{s})+(-1)^{p}p_{s}\right)\delta\theta_{i}^{s+1}$
$\displaystyle=(-\lambda+u^{i})f^{i}\delta\hat{d}_{i}\sum_{s\geq
0}\left(p_{s}+(-1)^{p}\hat{d}_{i}(q_{s})\right)\delta u^{i,s},$

we see that
$\operatorname{Im}d_{0}=(-\lambda+u^{i})\delta\hat{d}_{i}(\hat{\mathcal{B}}_{i})[\lambda]$.
Therefore we obtain the following lemma.

######  Lemma 5.5.

We have

${}^{2}E_{1}=\frac{\delta\hat{d}_{i}(\hat{\mathcal{B}}_{i})[\lambda]}{(-\lambda+u^{i})\delta\hat{d}_{i}(\hat{\mathcal{B}}_{i})[\lambda]}.$

Next let us compute ${}^{2}E_{2}=H(^{2}E_{1},d_{1})$. We use the following
identification:

${}^{2}E_{1}=\frac{\delta\hat{d}_{i}(\hat{\mathcal{B}}_{i})[\lambda]}{(-\lambda+u^{i})\delta\hat{d}_{i}(\hat{\mathcal{B}}_{i})[\lambda]}\cong\delta\hat{d}_{i}(\hat{\mathcal{B}}_{i})$

by identifying $\lambda$ with $u^{i}$. Under this identification, we have

$d_{1}=\sum_{j}(u^{j}-u^{i})f^{j}\delta\hat{d}_{j}.$

Thus we can represent an element of ${}^{2}E_{1}$ as follows:

$\omega=\delta\hat{d}_{i}\sum_{s\geq 0}p_{s}\delta u^{i,s}=\sum_{s\geq
0}\hat{d}_{i}(p_{s})\delta
u^{i,s}+(-1)^{p}p_{s}\delta\theta_{i}^{s+1}\in{{}^{2}E_{1}}.$

If $\omega$ is a cocycle, i.e. $d_{1}\omega=0$, by using the fact that
$\delta\hat{d}_{i}$ commutes with $d_{1}$ we obtain

(5.8) $\sum_{j}(u^{j}-u^{i})f^{j}\hat{d}_{j}(p_{s})=0.$

Let us consider the following decomposition according to (5.6) as follows:

$p_{s}=\hat{p}_{s}+\sum_{k}p_{s}^{k}+p_{s}^{m},$

where $\hat{p}_{s}\in\hat{\mathcal{C}}$,
$p_{s}^{j}\in\hat{\mathcal{C}}_{j}^{nt}$ and $p_{s}^{m}\in\hat{\mathcal{M}}$.
Then from the equation (5.8) and the obvious fact that $\hat{d}_{j}$
annihilates $p_{s}^{k}$ unless $k=j$, we arrive at the following equations:

(5.9)
$\left(u^{k}-u^{i}\right)f^{k}\hat{d}_{k}(p_{s}^{k})=0,\quad\sum_{j}(u^{j}-u^{i})f^{j}\hat{d}_{j}(p_{s}^{m})=0.$

Therefore for $k\neq i$, we have $\hat{d}_{k}(p_{s}^{k})=0$ and by Poincaré
Lemma (Lemma 12 in [5]), this implies $p_{s}^{k}=\hat{d}_{k}(q_{s}^{k})$ for
some $q_{s}^{k}\in\hat{\mathcal{C}}_{k}^{nt}$. For the term
$p_{s}^{m}\in\hat{\mathcal{M}}$, after a rescaling by a non-zero factor, we
may rewrite the second equation of (5.9) as $\sum_{j\neq
i}\hat{d}_{j}(p_{s}^{m})=0$, then by Proposition 11 in [5] (or a simplified
version, Lemma 3.4 in [4]), we see that $p_{s}^{m}=\sum_{j\neq
i}\hat{d}_{j}(q_{s}^{m})+h_{s}$, where $q_{s}^{m}\in\hat{\mathcal{M}}$ and
$h_{s}\in\hat{\mathcal{C}}_{i}$. But such $h_{s}$ must vanish since we have
$h_{s}=p_{s}^{m}-\sum_{j\neq i}\hat{d}_{j}(q_{s}^{m})\in\hat{\mathcal{M}}$.

To conclude, we can rewrite $p_{s}$ as

$p_{s}=\hat{p}_{s}+p_{s}^{i}+\sum_{j\neq
i}\hat{d}_{i}q_{s}^{j}+d_{1}(q_{s}^{m})=\hat{p}_{s}+p_{s}^{i}+d_{1}\left(\sum_{j\neq
i}\frac{q_{s}^{j}}{(u^{j}-u^{i})f^{j}}+q_{s}^{m}\right).$

Using again the fact that $\delta\hat{d}_{i}$ and $d_{1}$ commutes, we see
that

$\displaystyle\omega$ $\displaystyle=\delta\hat{d}_{i}(\sum_{s\geq
0}p_{s}\delta u^{i,s})$ $\displaystyle=\delta\hat{d}_{i}\sum_{s\geq
0}(\hat{p}_{s}+p_{s}^{i})\delta u^{i,s}-d_{1}\delta\hat{d}_{i}\sum_{s\geq
0}\sum_{j\neq
i}\left(\frac{q_{s}^{j}}{(u^{j}-u^{i})f^{j}}+q_{s}^{m}\right)\delta u^{i,s}.$

So we see that for any $\omega\in\ker d_{1}$, there is
$\eta\in\hat{\mathcal{C}}_{i}\\{\delta u^{i,s}\mid s\geq 0\\}$ such that
$[\omega]=[\delta\hat{d}_{i}(\eta)]$ in ${}^{2}E_{2}$. Let us denote
$\hat{\mathcal{H}}_{i}=\hat{\mathcal{C}}_{i}\\{\delta u^{i,s},\ s\geq 0\\}$.
It is obvious that each element in $\delta\hat{d}_{i}(\hat{\mathcal{H}}_{i})$
is annihilated by $d_{1}$ and different elements define different classes in
${}^{2}E_{2}$, hence we arrive at

${}^{2}E_{2}\cong\delta\hat{d}_{i}(\hat{\mathcal{H}}_{i}).$

To summarize all the results above, we obtain the following theorem.

######  Theorem 5.6.

The first page ${}^{1}E_{1}$ of the spectral sequence ${}^{1}E$ can be
described as the direct sum of the following spaces:

${}^{1}E_{1}\cong\hat{\mathcal{C}}[\lambda]\\{\delta\theta\\}\oplus\bigoplus_{i=1}^{n}\frac{\hat{d}_{i}(\hat{\mathcal{C}}_{i}[\lambda])}{(-\lambda+u^{i})\hat{d}_{i}(\hat{\mathcal{C}}_{i}[\lambda])}\\{\delta\theta\\}\oplus\bigoplus_{i=1}^{n}\delta\hat{d}_{i}(\hat{\mathcal{H}}_{i}).$

###  5.3. Computation of ${}^{1}E_{2}$.

We will find suitable bidegrees $(p,d)$ such that the cohomology
${}^{1}E_{2}=H^{p}_{d}(^{1}E_{1},\Delta_{0})=0$, this means that the spectral
sequence ${}^{1}E$ collapses on the second page for these bidegrees. Then we
conclude that for the same bidegrees
$H^{p}_{d}(\Omega[\lambda],\partial_{\lambda})=0$.

We first write down explicitly the formula for $\Delta_{0}$. To avoid lengthy
expressions, we will split $\Delta_{0}$ into $\partial/\partial u^{i,s}$ part,
$\partial/\partial\delta u^{i,s}$ part, $\partial/\partial\theta_{i}^{s}$ part
and $\partial/\partial\delta\theta_{i}^{s}$ part. In the following formulae,
the index $i$ is fixed and does not participate in the summation. These
formulae are comparable to those given in [5].

The $\partial/\partial u^{i,s}$ part of $\Delta_{0}$ reads

$\displaystyle\sum_{s\geq
1}\sum_{t=1}^{s}\binom{s}{t}u^{i,t}f^{i}\theta_{i}^{s-t+1}\frac{\partial}{\partial
u^{i,s}}+\sum_{s\geq
1;j}\sum_{t=1}^{s}\binom{s}{t}(-\lambda+u^{i})\partial_{j}f^{i}u^{j,t}\theta_{i}^{s-t+1}\frac{\partial}{\partial
u^{i,s}}$ $\displaystyle+$ $\displaystyle\frac{1}{2}\sum_{s\geq
1}\sum_{t=0}^{s}\binom{s}{t}u^{i,t+1}f^{i}\theta_{i}^{s-t}\frac{\partial}{\partial
u^{i,s}}+\frac{1}{2}\sum_{s\geq
1;j}\sum_{t=0}^{s}\binom{s}{t}(-\lambda+u^{i})\partial_{j}f^{i}u^{j,t+1}\theta_{i}^{s-t}\frac{\partial}{\partial
u^{i,s}}$ $\displaystyle+$ $\displaystyle\frac{1}{2}\sum_{s\geq
1;j}\sum_{t=0}^{s}\binom{s}{t}(-\lambda+u^{i})f^{i}\frac{\partial_{i}f^{j}}{f^{j}}u^{j,t+1}\theta_{j}^{s-t}\frac{\partial}{\partial
u^{i,s}}$ $\displaystyle-$ $\displaystyle\frac{1}{2}\sum_{s\geq
1;j}\sum_{t=0}^{s}\binom{s}{t}(-\lambda+u^{j})f^{j}\frac{\partial_{j}f^{i}}{f^{i}}u^{i,t+1}\theta_{j}^{s-t}\frac{\partial}{\partial
u^{i,s}}+(\lambda+u^{i})f^{i}\theta_{i}^{1}\frac{\partial}{\partial u^{i}}.$

The $\partial/\partial\delta u^{i,s}$ part part of $\Delta_{0}$ reads:

$\displaystyle\sum_{s\geq
0}\sum_{t=0}^{s}\binom{s}{t}f^{i}\theta_{i}^{s-t+1}\delta
u^{i,t}\frac{\partial}{\partial\delta u^{i,s}}+\sum_{s\geq
0;j}\sum_{t=0}^{s}\binom{s}{t}(-\lambda+u^{i})\partial_{j}f^{i}\theta_{i}^{s-t+1}\delta
u^{j,t}\frac{\partial}{\partial\delta u^{i,s}}$ $\displaystyle+$
$\displaystyle\sum_{s\geq
1}\sum_{t=1}^{s}\binom{s}{t}f^{i}u^{i,t}\delta\theta_{i}^{s-t+1}\frac{\partial}{\partial\delta
u^{i,s}}+\sum_{s\geq
1;j}\sum_{t=1}^{s}\binom{s}{t}(-\lambda+u^{i})\partial_{j}f^{i}u^{j,t}\delta\theta_{i}^{s-t+1}\frac{\partial}{\partial\delta
u^{i,s}}$ $\displaystyle+$ $\displaystyle\frac{1}{2}\sum_{s\geq
0}\sum_{t=0}^{s}\binom{s}{t}f^{i}\delta(u^{i,t+1}\theta_{i}^{s-t})\frac{\partial}{\partial\delta
u^{i,s}}+\frac{1}{2}\sum_{s\geq
0;j}\sum_{t=0}^{s}\binom{s}{t}(-\lambda+u^{i})\partial_{j}f^{i}\delta(u^{j,t+1}\theta_{i}^{s-t})\frac{\partial}{\partial\delta
u^{i,s}}$ $\displaystyle+$ $\displaystyle\frac{1}{2}\sum_{s\geq
1;j}\sum_{t=0}^{s}\binom{s}{t}(-\lambda+u^{i})f^{i}\frac{\partial_{i}f^{j}}{f^{j}}\delta(u^{j,t+1}\theta_{j}^{s-t})\frac{\partial}{\partial\delta
u^{i,s}}$ $\displaystyle-$ $\displaystyle\frac{1}{2}\sum_{s\geq
1;j}\sum_{t=0}^{s}\binom{s}{t}(-\lambda+u^{j})f^{j}\frac{\partial_{j}f^{i}}{f^{i}}\delta(u^{i,t+1}\theta_{j}^{s-t})\frac{\partial}{\partial\delta
u^{i,s}}.$

The $\partial/\partial\theta_{i}^{s}$ part part of $\Delta_{0}$ reads

$\displaystyle\frac{1}{2}\sum_{s\geq
0}\sum_{t=0}^{s}\binom{s}{t}f^{i}\theta_{i}^{t}\theta_{i}^{1+s-t}\frac{\partial}{\partial\theta_{i}^{s}}+\frac{1}{2}\sum_{s\geq
0;j}\sum_{t=0}^{s}\binom{s}{t}(-\lambda+u^{j})\partial_{i}f^{j}\theta_{j}^{t}\theta_{j}^{1+s-t}\frac{\partial}{\partial\theta_{i}^{s}}$
$\displaystyle+$ $\displaystyle\frac{1}{2}\sum_{s\geq
0;j}\sum_{t=0}^{s}\binom{s}{t}(-\lambda+u^{j})f^{j}\frac{\partial_{j}f^{i}}{f^{i}}\theta_{i}^{t}\theta_{j}^{1+s-t}\frac{\partial}{\partial\theta_{i}^{s}}$
$\displaystyle-$ $\displaystyle\frac{1}{2}\sum_{s\geq
0;j}\sum_{t=0}^{s}\binom{s}{t}(-\lambda+u^{j})f^{j}\frac{\partial_{j}f^{i}}{f^{i}}\theta_{j}^{t}\theta_{i}^{1+s-t}\frac{\partial}{\partial\theta_{i}^{s}}.$

The $\partial/\partial\delta\theta_{i}^{s}$ part part of $\Delta_{0}$ reads

$\displaystyle\frac{1}{2}\sum_{s\geq
0}\sum_{t=0}^{s}\binom{s}{t}f^{i}\delta(\theta_{i}^{t}\theta_{i}^{1+s-t})\frac{\partial}{\partial\delta\theta_{i}^{s}}+\frac{1}{2}\sum_{s\geq
0;j}\sum_{t=0}^{s}\binom{s}{t}(-\lambda+u^{j})\partial_{i}f^{j}\delta(\theta_{j}^{t}\theta_{j}^{1+s-t})\frac{\partial}{\partial\delta\theta_{i}^{s}}$
$\displaystyle+$ $\displaystyle\frac{1}{2}\sum_{s\geq
0;j}\sum_{t=0}^{s}\binom{s}{t}(-\lambda+u^{j})f^{j}\frac{\partial_{j}f^{i}}{f^{i}}\delta(\theta_{i}^{t}\theta_{j}^{1+s-t})\frac{\partial}{\partial\delta\theta_{i}^{s}}$
$\displaystyle-$ $\displaystyle\frac{1}{2}\sum_{s\geq
0;j}\sum_{t=0}^{s}\binom{s}{t}(-\lambda+u^{j})f^{j}\frac{\partial_{j}f^{i}}{f^{i}}\delta(\theta_{j}^{t}\theta_{i}^{1+s-t})\frac{\partial}{\partial\delta\theta_{i}^{s}}.$

To compute the cohomology $H({{}^{1}E_{1}},\Delta_{0})$, we introduce a third
spectral sequence ${}^{3}E$ by defining the $\theta^{1}$-degree
$\deg_{\theta^{1}}\theta_{i}^{1}=1$ for $i=1,\cdots,n$, and other generators
have $\deg_{\theta^{1}}$ zero. Then we filtrate ${{}^{1}E_{1}}$ via

$F^{r}({{}^{1}E_{1}})=\\{\deg_{\theta_{1}}-\deg_{\theta}\omega\leq-r\mid\omega\in{{}^{1}E_{1}}\\}.$

We also have the decomposition

$\Delta_{0}=\Delta_{0,1}+\Delta_{0,0}+\Delta_{0,-1},\quad\deg_{\theta^{1}}\Delta_{0,k}=k.$

The zeroth page $(^{3}E_{0},d_{0})$ is given by $(^{1}E_{1},\Delta_{0,1})$. We
want to determine the bidegrees $(p,d)$ such that
$H^{p}_{d}(^{3}E_{0},d_{0})=0$, then we conclude that for the same degrees
$H^{p}_{d}(^{1}E_{1},\Delta_{0})=0$. We first write down the explicit formula
for $\Delta_{0,1}$.

The $\partial/\partial u^{i,s}$ part of $\Delta_{0,1}$ reads

$\displaystyle(-\lambda+u^{i})f^{i}\theta_{i}^{1}\frac{\partial}{\partial
u^{i}}+\sum_{s\geq
1}(\frac{s}{2}+1)f^{i}\theta_{i}^{1}u^{i,s}\frac{\partial}{\partial
u^{i,s}}+\sum_{s\geq
1;j}(\frac{s}{2}+1)(-\lambda+u^{i})\partial_{j}f^{i}\theta_{i}^{1}u^{j,s}\frac{\partial}{\partial
u^{i,s}}$ $\displaystyle+$ $\displaystyle\frac{1}{2}\sum_{s\geq
1;j}s(-\lambda+u^{i})f^{i}\frac{\partial_{i}f^{j}}{f^{j}}\theta_{j}^{1}u^{j,s}\frac{\partial}{\partial
u^{i,s}}-\frac{1}{2}\sum_{s\geq
1;j}s(-\lambda+u^{j})f^{j}\frac{\partial_{j}f^{i}}{f^{i}}\theta_{j}^{1}u^{i,s}\frac{\partial}{\partial
u^{i,s}}.$

The $\partial/\partial\delta u^{i,s}$ part of $\Delta_{0,1}$ reads

$\displaystyle\sum_{s\geq 0}f^{i}\theta_{i}^{1}\delta
u^{i,s}\frac{\partial}{\partial\delta u^{i,s}}+\sum_{s\geq
0;j}(-\lambda+u^{i})\partial_{j}f^{i}\theta_{i}^{1}\delta
u^{j,s}\frac{\partial}{\partial u^{i,s}}$ $\displaystyle+$
$\displaystyle\frac{1}{2}\sum_{s\geq
0;j}s(-\lambda+u^{i})\partial_{j}f^{i}\theta_{i}^{1}\delta
u^{j,s}\frac{\partial}{\partial\delta u^{i,s}}+\frac{1}{2}\sum_{s\geq
0}sf^{i}\theta_{i}^{1}\delta u^{i,s}\frac{\partial}{\partial\delta u^{i,s}}$
$\displaystyle+$ $\displaystyle\frac{1}{2}\sum_{s\geq
0;j}s(-\lambda+u^{i})f^{i}\frac{\partial_{i}f^{j}}{f^{j}}\theta_{j}^{1}\delta
u^{j,s}\frac{\partial}{\partial\delta u^{i,s}}-\frac{1}{2}\sum_{s\geq
0;j}s(-\lambda+u^{j})f^{j}\frac{\partial_{j}f^{i}}{f^{i}}\theta_{j}^{1}\delta
u^{i,s}\frac{\partial}{\partial\delta u^{i,s}}.$

The $\partial/\partial\theta_{i}^{s}$ part part of $\Delta_{0,1}$

$\displaystyle\frac{1}{2}\sum_{s\geq
0;j}(-\lambda+u^{j})\partial_{i}f^{j}(s-1)\theta_{j}^{1}\theta_{j}^{s}\frac{\partial}{\partial\theta_{i}^{s}}+\frac{1}{2}\sum_{s\geq
0}f^{i}(s-1)\theta_{i}^{1}\theta_{i}^{s}\frac{\partial}{\partial\theta_{i}^{s}}$
$\displaystyle+$ $\displaystyle\frac{1}{2}\sum_{\begin{subarray}{c}s\geq
0;j\\\ s\neq
1\end{subarray}}(\lambda+u^{j})f^{j}\frac{\partial_{j}f^{i}}{f^{i}}(s+1)(\theta_{i}^{s}\theta_{j}^{1}-\theta_{j}^{s}\theta_{i}^{1})\frac{\partial}{\partial\theta_{i}^{s}}+\sum_{j}(-\lambda+u^{j})f^{j}\frac{\partial_{j}f^{i}}{f^{i}}\theta_{i}^{1}\theta_{j}^{1}\frac{\partial}{\partial\theta_{i}^{1}}$

The $\partial/\partial\delta\theta_{i}^{s}$ part part of $\Delta_{0}$

$\displaystyle\frac{1}{2}\sum_{s\geq
0;j}(-\lambda+u^{j})\partial_{i}f^{j}(s-1)\theta_{j}^{1}\delta\theta_{j}^{s}\frac{\partial}{\partial\delta\theta_{i}^{s}}+\frac{1}{2}\sum_{s\geq
0}f^{i}(s-1)\theta_{i}^{1}\delta\theta_{i}^{s}\frac{\partial}{\partial\theta_{i}^{s}}$
$\displaystyle+$ $\displaystyle\frac{1}{2}\sum_{\begin{subarray}{c}s\geq
0;j\\\ s\neq
1\end{subarray}}(\lambda+u^{j})f^{j}\frac{\partial_{j}f^{i}}{f^{i}}(s+1)(\delta\theta_{i}^{s}\theta_{j}^{1}-\delta\theta_{j}^{s}\theta_{i}^{1})\frac{\partial}{\partial\theta_{i}^{s}}+\sum_{j}(-\lambda+u^{j})f^{j}\frac{\partial_{j}f^{i}}{f^{i}}\delta(\theta_{i}^{1}\theta_{j}^{1})\frac{\partial}{\partial\theta_{i}^{1}}$

To simplify the above expressions, we perform a rescaling on the generators of
$\Omega$ (see [4]) as follows:

$\displaystyle\Psi\colon
u^{i,s}\mapsto(f^{i})^{\frac{s}{2}}u^{i,s};\quad\theta_{i}^{s}\mapsto(f^{i})^{\frac{s+1}{2}}\theta_{i}^{s};\
s\geq 0;$ $\displaystyle\Psi\colon\delta
u^{i,s}\mapsto(f^{i})^{\frac{s}{2}}\delta
u^{i,s};\quad\delta\theta_{i}^{s}\mapsto(f^{i})^{\frac{s+1}{2}}\delta\theta_{i}^{s};\
s\geq 0.$

Note that this is NOT induced by a change of coordinate, but just an
isomorphism of the space $\Omega$. The expression of $\Delta_{0,1}$ will be
simplified after being conjugated by $\Psi$, and since $\Psi$ leaves all the
decomposition of the complex invariant, this will not affect the computation
of cohomology groups.

The following identities are easy to verify and helpful for our computation of
the conjugated operator $\tilde{\Delta}_{0,1}=\Psi^{-1}\Delta_{0,1}\Psi$:

$\displaystyle\Psi^{-1}u^{i,s}\Psi=(f^{i})^{-\frac{s}{2}}u^{i,s};\quad\Psi^{-1}\theta_{i}^{s}\Psi=(f^{i})^{-\frac{s+1}{2}}\theta_{i}^{s};$
$\displaystyle\Psi^{-1}\delta
u^{i,s}\Psi=(f^{i})^{-\frac{s}{2}}u^{i,s};\quad\Psi^{-1}\delta\theta_{i}^{s}\Psi=(f^{i})^{-\frac{s+1}{2}}\theta_{i}^{s};$
$\displaystyle\Psi^{-1}\frac{\partial}{\partial
u^{i,s}}\Psi=(f^{i})^{\frac{s}{2}}\frac{\partial}{\partial u^{i,s}},s\geq
1;\quad\Psi^{-1}\frac{\partial}{\partial\theta_{i}^{s}}\Psi=(f^{i})^{\frac{s+1}{2}}\frac{\partial}{\partial\theta_{i}^{s}},s\geq
0;$ $\displaystyle\Psi^{-1}\frac{\partial}{\partial\delta
u^{i,s}}\Psi=(f^{i})^{\frac{s}{2}}\frac{\partial}{\partial\delta
u^{i,s}},s\geq
0;\quad\Psi^{-1}\frac{\partial}{\partial\delta\theta_{i}^{s}}\Psi=(f^{i})^{\frac{s+1}{2}}\frac{\partial}{\partial\delta\theta_{i}^{s}},s\geq
0;$ $\displaystyle\Psi^{-1}\frac{\partial}{\partial
u^{i}}\Psi=\frac{\partial}{\partial u^{i}}+\sum_{s\geq
0;j}\frac{\partial_{i}f^{j}}{f^{j}}\left(\frac{s}{2}u^{j,s}\frac{\partial}{\partial
u^{j,s}}+\frac{s+1}{2}\theta_{j}^{s}\frac{\partial}{\partial\theta_{j}^{s}}\right).$

Then by using the rotation coefficients

$\gamma_{ij}=-\frac{1}{2}\left(\frac{f^{i}}{f^{j}}\right)^{1/2}\frac{\partial_{i}f^{j}}{f^{j}}$

defined for the diagonal metric $(f^{1},\cdots,f^{n})$, we can represent
$\tilde{\Delta}_{0,1}$ in the form

$\tilde{\Delta}_{0,1}=\phi_{1}+\phi_{2}+\phi_{3},$

where

$\displaystyle\phi_{1}=$ $\displaystyle-\sum_{s\geq
1;i,j}(-\lambda+u^{i})\left(\frac{f^{i}}{f^{j}}\right)^{\frac{s+1}{2}}\left((s+2)\gamma_{ji}\theta_{i}^{1}+s\gamma_{ij}\theta_{j}^{1}\right)u^{j,s}\frac{\partial}{\partial
u^{i,s}}$ $\displaystyle-\sum_{s\geq
0;i,j}(-\lambda+u^{i})\left(\frac{f^{i}}{f^{j}}\right)^{\frac{s+1}{2}}\left((s+2)\gamma_{ji}\theta_{i}^{1}+s\gamma_{ij}\theta_{j}^{1}\right)\delta
u^{j,s}\frac{\partial}{\partial\delta u^{i,s}}$ $\displaystyle+\sum_{s\geq
2;i,j}(-\lambda+u^{j})\left(\frac{f^{i}}{f^{j}}\right)^{\frac{s}{2}}\left((1-s)\gamma_{ij}\theta_{j}^{1}-(1+s)\gamma_{ji}\theta_{i}^{1}\right)\theta_{j}^{s}\frac{\partial}{\partial\theta_{i}^{s}},$
$\displaystyle\phi_{2}=$ $\displaystyle\sum_{s\geq
2;i,j}(-\lambda+u^{j})\left(\frac{f^{i}}{f^{j}}\right)^{\frac{s}{2}}\left((1-s)\gamma_{ij}\theta_{j}^{1}-(1+s)\gamma_{ji}\theta_{i}^{1}\right)\delta\theta_{j}^{s}\frac{\partial}{\partial\delta\theta_{i}^{s}}$
$\displaystyle+\sum_{i,j}(-\lambda+u^{j})\frac{\partial_{j}f^{i}}{f^{j}}\theta_{i}^{1}\delta\theta_{j}^{1}\frac{\partial}{\partial\delta\theta_{i}^{1}},$
$\displaystyle\phi_{3}=$ $\displaystyle-\frac{1}{2}\sum_{s\geq
0;i,j}(-\lambda+u^{j})\frac{\partial_{j}f^{i}}{f^{j}}\theta_{j}^{1}\left(s\delta
u^{i,s}\frac{\partial}{\partial\delta
u^{i,s}}+(s+1)\delta\theta_{i}^{s}\frac{\partial}{\partial\delta\theta_{i}^{s}}\right)$
$\displaystyle+\sum_{i,j}(-\lambda+u^{j})(\gamma_{ij}\theta_{j}^{1}-\gamma{ji}\theta_{i}^{1})\left(\theta_{j}\frac{\partial}{\partial\theta_{i}}+\delta\theta_{j}\frac{\partial}{\partial\delta\theta_{i}}\right)$
$\displaystyle+\sum_{i}\theta_{i}^{1}\mathcal{E}_{i}+\sum_{i}(-\lambda+u^{i})\theta_{i}^{1}\frac{\partial}{\partial
u^{i}},$

here $\mathcal{E}_{i}$ is an Euler-type vector filed given by

$\mathcal{E}_{i}=\sum_{s\geq
1}\left(\frac{s}{2}+1\right)u^{i,s}\frac{\partial}{\partial
u^{i,s}}+\sum_{s\geq 0}\left(\frac{s}{2}+1\right)\delta
u^{i,s}\frac{\partial}{\partial\delta u^{i,s}}+\sum_{s\geq
0}\frac{s-1}{2}\left(\theta_{i}^{s}\frac{\partial}{\partial\theta_{i}^{s}}+\delta\theta_{i}^{s}\frac{\partial}{\partial\delta\theta_{i}^{s}}\right).$

We first simplify $\tilde{\Delta}_{0,1}$ by the following observation (which
is parallel to Lemma 3.6 of [4]).

######  Lemma 5.7.

Both $\phi_{1}$ and $\phi_{2}$ act trivially on ${}^{1}E_{1}$.

###### Proof.

We first recall from Theorem 5.6 that

(5.10)
$^{1}E_{1}\cong\hat{\mathcal{C}}[\lambda]\\{\delta\theta\\}\oplus\bigoplus_{i=1}^{n}\frac{\hat{d}_{i}(\hat{\mathcal{C}}_{i}[\lambda])}{(-\lambda+u^{i})\hat{d}_{i}(\hat{\mathcal{C}}_{i}[\lambda])}\\{\delta\theta\\}\oplus\bigoplus_{i=1}^{n}\delta\hat{d}_{i}(\hat{\mathcal{H}}_{i}).$

The vanishing of the action of $\phi_{1}$ on the cohomology

$\hat{\mathcal{C}}[\lambda]\\{\delta\theta\\}\oplus\bigoplus_{i=1}^{N}\frac{\hat{d}_{i}(\hat{\mathcal{C}}_{i}[\lambda])}{(-\lambda+u^{i})\hat{d}_{i}(\hat{\mathcal{C}}_{i}[\lambda])}\\{\delta\theta\\}$

of $\hat{\mathcal{A}}[\lambda]\\{\delta\theta\\}$ is a direct consequence of
Lemma 3.6 of [4], and the vanishing of that of $\phi_{2}$ is obvious.

Next we consider action of $\phi_{1}$ and $\phi_{2}$ on

$H(\hat{\mathcal{A}}[\lambda]\\{\delta
u^{i,s},\delta\theta_{i}^{s+1}\\},\Delta_{-1})=\delta\hat{d}_{i}(\hat{\mathcal{H}}_{i})$

for a fixed index $i$. By identifying $\lambda$ with $u^{i}$, we can represent
$\omega$ in the form

$\omega=\delta\hat{d}_{i}(\sum_{s\geq 0}p_{s}\delta u^{i,s})=\sum_{s\geq
0}\hat{d}_{i}(p_{s})\delta u^{i,s}+(-1)^{p}p_{s}\delta\theta_{i}^{s+1},\quad
p_{s}\in\hat{\mathcal{C}}_{i}.$

Under such an identification, the action of $\phi_{1}$ can be represented as

$\displaystyle\phi_{1}=$ $\displaystyle-\sum_{s\geq
1;k,j}(-u^{i}+u^{k})\left(\frac{f^{k}}{f^{j}}\right)^{\frac{s+1}{2}}\left((s+2)\gamma_{jk}\theta_{k}^{1}+s\gamma_{kj}\theta_{j}^{1}\right)u^{j,s}\frac{\partial}{\partial
u^{k,s}}$ $\displaystyle-\sum_{s\geq
0;k,j}(-u^{i}+u^{k})\left(\frac{f^{k}}{f^{j}}\right)^{\frac{s+1}{2}}\left((s+2)\gamma_{jk}\theta_{i}^{1}+s\gamma_{kj}\theta_{j}^{1}\right)\delta
u^{j,s}\frac{\partial}{\partial\delta u^{k,s}}$ $\displaystyle+\sum_{s\geq
2;k,j}(-u^{i}+u^{j})\left(\frac{f^{k}}{f^{j}}\right)^{\frac{s}{2}}\left((1-s)\gamma_{kj}\theta_{j}^{1}-(1+s)\gamma_{jk}\theta_{k}^{1}\right)\theta_{j}^{s}\frac{\partial}{\partial\theta_{k}^{s}}.$

Hence it is clear that

$\phi_{1}\omega\in\bigoplus_{j\neq i}\hat{\mathcal{C}}_{j}^{nt}\\{\delta
u^{i,s},\delta\theta_{i}^{s+1}\mid s\geq 0\\}.$

For the action of $\phi_{2}$ on $\omega$, we first regard $\omega$ as an
element of $\delta\hat{d}_{i}(\hat{\mathcal{H}}_{i})[\lambda]$, then we
observe that

$\phi_{2}\omega\in\bigoplus_{j}(-\lambda+u^{j})\hat{\mathcal{C}}_{i}[\lambda]\\{\delta\theta_{j}^{s+1}\mid
s\geq 0\\}.$

We further make the decomposition $\phi_{2}\omega=\alpha_{1}+\alpha_{2}$,
where

$\alpha_{1}\in\bigoplus_{j\neq
i}(-\lambda+u^{j})\hat{\mathcal{C}}_{i}[\lambda]\\{\delta\theta_{j}^{s+1}\mid
s\geq
0\\},\quad\alpha_{2}\in(-\lambda+u^{i})\hat{\mathcal{C}}_{i}[\lambda]\\{\delta\theta_{i}^{s+1}\mid
s\geq 0\\}.$

Finally it is easy to see that from the definition of $\phi_{3}$ that

$\phi_{3}\omega\in\hat{\mathcal{C}}_{i}[\lambda]\\{\delta
u^{i,s},\delta\theta_{i}^{s+1}\mid s\geq 0\\}.$

It follows from

$\Delta_{0,1}\Delta_{-1}+\Delta_{-1}\Delta_{0,1}=0$

that $\tilde{\Delta}_{0,1}\omega=(\phi_{1}+\phi_{2}+\phi_{3})\omega$ still
lies in the cohomology group. Since the subspaces

$\displaystyle\bigoplus_{j\neq i}\hat{\mathcal{C}}_{j}^{nt}\\{\delta
u^{i,s},\delta\theta_{i}^{s+1}\mid s\geq 0\\},$ $\displaystyle\bigoplus_{j\neq
i}(-\lambda+u^{j})\hat{\mathcal{C}}_{i}[\lambda]\\{\delta\theta_{j}^{s+1}\mid
s\geq 0\\},$ $\displaystyle\hat{\mathcal{C}}_{i}[\lambda]\\{\delta
u^{i,s},\delta\theta_{i}^{s+1}\mid s\geq 0\\}$

are disjoint and they are all invariant under the action $\Delta_{-1}$, we
conclude that $\phi_{1}\omega$, $\alpha_{1}$ and $\alpha_{2}+\phi_{3}\omega$
lie in the cohomology group.

From our computation of ${}^{1}E_{1}$ given in 5.2, it follows that terms in
the subspace

$\bigoplus_{j\neq i}\hat{\mathcal{C}}_{j}^{nt}\\{\delta
u^{i,s},\delta\theta_{i}^{s+1}\mid s\geq 0\\}$

vanish in ${}^{1}E_{1}$ and hence action of $\phi_{1}$ vanishes. Similarly,
from the computation of the spectral sequence ${}^{2}E$ given in 5.2, it
follows that the terms in the subspace

$\bigoplus_{j\neq
i}(-\lambda+u^{j})\hat{\mathcal{C}}_{i}[\lambda]\\{\delta\theta_{j}^{s+1}\mid
s\geq 0\\}$

also vanish, hence $\alpha_{1}=0$. For the same reason, the multiples of
$(-\lambda+u^{i})$ in the subspace $\hat{\mathcal{C}}_{i}[\lambda]\\{\delta
u^{i,s},\delta\theta_{i}^{s+1}\mid s\geq 0\\}$ are trivial in the cohomology
as well, hence in particular $\alpha_{2}=0$ which implies the vanishing of
$\phi_{2}$. The lemma is proved. ∎

This above lemma shows that $\tilde{\Delta}_{0,1}=\phi_{3}$ and therefore each
summand in the decomposition (5.10) of ${}^{1}E_{1}$ is preserved by
$\tilde{\Delta}_{0,1}$. In what follows, we will show that for some bidegrees
$(p,d)$ the action of $\tilde{\Delta}_{0,1}$ is acyclic on each summand when
restricted to the elements with super degree $p$ and differential degree $d$.

######  Lemma 5.8.

We have
$H^{p}_{d}(\hat{\mathcal{C}}[\lambda]\\{\delta\theta\\},\tilde{\Delta}_{0,1})=0$
unless

$d=0,\cdots,n;\quad p=d+1,\cdots,d+n+1.$

###### Proof.

Indeed, possible bidegrees $(p,d)$ of elements of in
$\hat{\mathcal{C}}[\lambda]\\{\delta\theta\\}$ are precisely those excluded in
the lemma. The lemma is proved. ∎

To compute the cohomology of $\tilde{\Delta}_{0,1}$ on the space

$\frac{\hat{d}_{i}(\hat{\mathcal{C}}_{i}[\lambda])}{(-\lambda+u^{i})\hat{d}_{i}(\hat{\mathcal{C}}_{i}[\lambda])}\\{\delta\theta\\},$

let us first identify this space with
$\hat{d}_{i}(\hat{\mathcal{C}}_{i})\\{\delta\theta\\}$ by sending $\lambda$ to
$u^{i}$. After this identification, the action of $\tilde{\Delta}_{0,1}$ reads
(we keep the same notation)

$\displaystyle\tilde{\Delta}_{0,1}=$
$\displaystyle\sum_{j}\theta_{j}^{1}\mathcal{E}_{j}-\frac{1}{2}\sum_{j,k}(-u^{i}+u^{j})\frac{\partial_{j}f^{k}}{f^{j}}\theta_{j}^{1}\delta\theta_{k}\frac{\partial}{\partial\delta\theta_{k}}+\sum_{j}(-u^{i}+u^{j})f^{j}\theta_{j}^{1}\frac{\partial}{\partial
u^{j}}$
$\displaystyle+\sum_{j,k}(-u^{i}+u^{j})(\gamma_{kj}\theta_{j}^{1}-\gamma_{jk}\theta_{k}^{1})\left(\theta_{j}\frac{\partial}{\partial\theta_{k}}+\delta\theta_{j}\frac{\partial}{\partial\delta\theta_{k}}\right).$

######  Lemma 5.9.

We have
$H^{p}_{d}(\hat{d}_{i}(\hat{\mathcal{C}_{i}})\\{\delta\theta\\},\tilde{\Delta}_{0,1})=0$
unless

(5.11) $d=2,\cdots,n+3;\quad p=d,\cdots,d+n.$

###### Proof.

To compute the cohomology, we introduce a forth spectral sequence ${}^{4}E$
given by a filtration of
$\hat{d}_{i}(\hat{\mathcal{C}_{i}})\\{\delta\theta\\}$ using the
$\theta_{i}^{1}$-degree, which is defined by
$\deg_{\theta_{i}^{1}}\theta_{i}^{1}=1$ and by setting the degrees of other
generators to be zero. By decomposing the differential $\tilde{\Delta}_{0,1}$
with respect to the $\theta_{i}^{1}$-degree, we conclude that the zeroth page
of this spectral sequence is given by
${}^{4}E_{0}=\hat{d}_{i}(\hat{\mathcal{C}_{i}})\\{\delta\theta\\}$ with the
differential $\mathcal{D}_{i}$ given by

$\mathcal{D}_{i}=\theta_{i}^{1}\mathcal{E}_{i}+\sum_{j}(u^{i}-u^{j})\gamma_{ji}\theta_{i}^{1}\left(\theta_{j}\frac{\partial}{\partial\theta_{i}}+\delta\theta_{j}\frac{\partial}{\partial\delta\theta_{i}}\right).$

The idea to compute $H(^{4}E_{0},\mathcal{D}_{i})$ is as follows. We first
make the decomposition

$d_{i}(\hat{\mathcal{C}_{i}})\\{\delta\theta\\}=d_{i}(\hat{\mathcal{C}_{i}})\\{\delta\theta_{i}\\}\oplus\bigoplus_{j\neq
i}d_{i}(\hat{\mathcal{C}_{i}})\\{\delta\theta_{j}\\}.$

Note that $\bigoplus_{j\neq
i}d_{i}(\hat{\mathcal{C}_{i}})\\{\delta\theta_{j}\\}$ is an invariant subspace
of $\mathcal{D}_{i}$ while
$d_{i}(\hat{\mathcal{C}_{i}})\\{\delta\theta_{i}\\}$ is not. Nevertheless, if
we can find proper bidegrees $(p,d)$ such that $\mathcal{D}_{i}$ is acyclic on
$\bigoplus_{j\neq
i}d_{i}(\hat{\mathcal{C}_{i}})\\{\delta\theta_{j}\\}\cap\Omega^{p}_{d}$, then
from Lemma 5.2 we know that the cohomology of
$d_{i}(\hat{\mathcal{C}_{i}})\\{\delta\theta\\}\cap\Omega^{p}_{d}$ is given by
the cohomology of the space
$d_{i}(\hat{\mathcal{C}_{i}})\\{\delta\theta_{i}\\}\cap\Omega^{p}_{d}$ with
the differential given by the projection of $\mathcal{D}_{i}$.

Take any monomial $\mathfrak{m}$ in $u^{i,s},\theta_{i}^{s+1}$ for $s\geq 1$
and any monomial $g\in\hat{\mathcal{C}}$. For $j\neq i$, we have

$\mathcal{D}_{i}(g\hat{d}_{i}(\mathfrak{m})\delta\theta_{j})=\theta_{i}^{1}\biggl{(}\sum_{j}(u^{i}-u^{j})\gamma_{ji}\theta_{j}\frac{\partial
g}{\partial\theta_{i}}+(w_{i}(g)+w_{i}(\mathfrak{m})-1)g\biggr{)}\hat{d}_{i}(\mathfrak{m})\delta\theta_{j},$

here for a monomial $\omega\in\Omega$, the rational number $w_{i}(\omega)$ is
defined by ${\mathcal{E}}_{i}\omega=w_{i}(\omega)\omega$.

So for any fixed $\mathfrak{m}$ and $j\neq i$, the subspace
$\hat{\mathcal{C}}\hat{d}_{i}(\mathfrak{m})\\{\delta\theta_{j}\\}$ is
invariant under the action of $\mathcal{D}_{i}$ , and we can concentrate first
on computing the cohomology of this subspace. The following argument is
basically the same as that of Lemma 3.10 of [4], nonetheless we still write it
down for the convenience of readers. Let us decompose $\hat{\mathcal{C}}$ in
the form

$\hat{\mathcal{C}}=\hat{\mathcal{C}}^{i}_{0}\oplus\theta_{i}\hat{\mathcal{C}}_{0}^{i},$

where $\hat{\mathcal{C}}^{i}_{0}$ is the subspace of $\hat{\mathcal{C}}$
spanned by monomials that do not contain $\theta_{i}$. For
$g\in\hat{\mathcal{C}}^{i}_{0}$, the action of $\mathcal{D}_{i}$ is given by

(5.12)
$\mathcal{D}_{i}(g\hat{d}_{i}(\mathfrak{m})\delta\theta_{j})=\theta_{i}^{1}(w_{i}(\mathfrak{m})-1)g\hat{d}_{i}(\mathfrak{m})\delta\theta_{j},$

therefore the subspace
$\hat{\mathcal{C}}^{i}_{0}\hat{d}_{i}(\mathfrak{m})\\{\delta\theta_{j}\\}$ is
acyclic. Indeed, since $w_{i}(\mathfrak{m})\geq\frac{3}{2}$ if
$\hat{d}_{i}(\mathfrak{m})\neq 0$, so a nonzero cocycle of $\mathcal{D}_{i}$
must contain $\theta_{i}^{1}$. Assume we have a cocycly $\theta_{i}^{1}h$ for
some $h$, then it follows from (5.12) the existence of a suitable constant $c$
such that $\theta_{i}^{1}h=c\mathcal{D}_{i}(h)$.

Due to Lemma 5.2, in order to compute the cohomology of
$\hat{\mathcal{C}}\hat{d}_{i}(\mathfrak{m})\\{\delta\theta_{j}\\}$ we only
need to compute the one for the subspace
$\theta_{i}\hat{\mathcal{C}}_{0}^{i}\\{\delta\theta_{j}\\}$. The projection of
$\mathcal{D}_{i}$ on
$\theta_{i}\hat{\mathcal{C}}_{0}^{i}\hat{d}_{i}(\mathfrak{m})$ is just a
multiplication by $\theta_{i}^{1}(w_{i}(\mathfrak{m})-\frac{3}{2})$, which is
acyclic if $w_{i}(\mathfrak{m})\neq\frac{3}{2}$. So the nontrivial cocycles
are given by elements of
$\theta_{i}\hat{\mathcal{C}}_{0}^{i}\hat{d}_{i}(u^{i,1})\delta\theta_{j}=\hat{\mathcal{C}}_{0}^{i}\theta_{i}\theta_{i}^{2}\delta\theta_{j}$,
which have the following possible bidegrees:

$d=2,\cdots,2+n;\quad p=d+1,\cdots,d+n.$

Thus unless a bidegree $(p,d)$ takes the values given above, the subcomplex

$\bigoplus_{j\neq
i}d_{i}(\hat{\mathcal{C}_{i}})\\{\delta\theta_{j}\\}\cap\Omega^{p}_{d}$

is acyclic.

Thus to compute the cohomology of ${{}^{4}E_{0}}$, we only need to consider
the cohomology of the space
$d_{i}(\hat{\mathcal{C}_{i}})\\{\delta\theta_{i}\\}$ due to Lemma 5.2. The
differential is the projection of $\mathcal{D}_{i}$ which can be represented
as

$\theta_{i}^{1}\mathcal{E}_{i}+\sum_{j}(u^{i}-u^{j})\gamma_{ji}\theta_{i}^{1}\theta_{j}\frac{\partial}{\partial\theta_{i}}.$

Now by repeating the argument above we see that the nontrivial cocycles are of
the form $\hat{\mathcal{C}}^{i}_{0}\theta_{i}^{2}\delta\theta_{i}$ or
$\hat{\mathcal{C}}^{i}_{0}\theta_{i}\theta_{i}^{3}\delta\theta_{i}$. By
counting the possible bidegrees of these elements, we complete the proof of
the lemma. ∎

######  Remark 5.1.

By a more careful analysis, we can prove that actually

$H^{2n+3}_{n+3}(\hat{d}_{i}(\hat{\mathcal{C}_{i}})\\{\delta\theta\\},\tilde{\Delta}_{0,1})=0.$

But this is not important for our consideration of the deformation problem.

Finally we are to compute the cohomology of $\tilde{\Delta}_{0,1}$ on the
space $\delta\hat{d}_{i}(\mathcal{H}_{i})$. Recall that on this space we have
to identify $\lambda$ with $u^{i}$, and therefore the differential reads

$\displaystyle\tilde{\Delta}_{0,1}=$ $\displaystyle-\frac{1}{2}\sum_{s\geq
0;k,j}(-u^{i}+u^{j})\frac{\partial_{j}f^{k}}{f^{j}}\theta_{j}^{1}\left(s\delta
u^{k,s}\frac{\partial}{\partial\delta
u^{k,s}}+(s+1)\delta\theta_{k}^{s}\frac{\partial}{\partial\delta\theta_{k}^{s}}\right)$
$\displaystyle+\sum_{k,j}(-u^{i}+u^{j})(\gamma_{kj}\theta_{j}^{1}-\gamma_{jk}\theta_{k}^{1})\theta_{j}\frac{\partial}{\partial\theta_{k}}+\sum_{j}\theta_{j}^{1}\mathcal{E}_{j}+\sum_{j}(-u^{i}+u^{j})\theta_{j}^{1}\frac{\partial}{\partial
u^{j}}.$

######  Lemma 5.10.

We have $H^{p}_{d}(\delta\hat{d}_{i}(\mathcal{H}_{i}),\tilde{\Delta}_{0,1})=0$
unless

$d=3,\cdots,n+3;\quad p=d,\cdots d+n-1.$

###### Proof.

The idea is very similar to the proof of Lemma 5.9. We introduce another
spectral sequence by filtrating $\delta\hat{d}_{i}(\mathcal{H}_{i})$ using
$\deg_{\theta_{i}^{1}}$. The differential on the zeroth page reads

$\mathcal{D}_{i}=\theta_{i}^{1}\mathcal{E}_{i}+\sum_{j}(u^{i}-u^{j})\gamma_{ji}\theta_{i}^{1}\theta_{j}\frac{\partial}{\partial\theta_{i}}.$

Let us denote
$\psi=\sum_{j}(u^{i}-u^{j})\gamma_{ji}\theta_{i}^{1}\theta_{j}\frac{\partial}{\partial\theta_{i}}$.
For any monomial $\mathfrak{m}$ in $u^{i,s},\theta_{i}^{s+1}$ with $s\geq 1$,
and any monomial $g\in\hat{\mathcal{C}}$, we have

$\displaystyle\mathcal{D}_{i}\delta\hat{d}_{i}(g\mathfrak{m}\delta u^{i,s})=$
$\displaystyle\mathcal{D}_{i}(g\hat{d}_{i}(\mathfrak{m})\delta
u^{i,s}+(-1)^{p}g\mathfrak{m}\delta\theta_{i}^{s+1})$ $\displaystyle=$
$\displaystyle\theta_{i}^{1}\left(w_{i}(g)+w_{i}(\mathfrak{m})-1+\frac{s}{2}+1\right)g\hat{d}_{i}(\mathfrak{m})\delta
u^{i,s}$
$\displaystyle+(-1)^{p}\theta_{i}^{1}\left(w_{i}(g)+w_{i}(\mathfrak{m})+\frac{s}{2}\right)g\mathfrak{m}\delta\theta_{i}^{s+1}$
$\displaystyle+\theta_{i}^{1}\left(\psi(g)\hat{d}_{i}(\mathfrak{m})\delta
u^{i,s}+(-1)^{p}\psi(g)\mathfrak{m}\delta\theta_{i}^{s+1}\right)$
$\displaystyle=$
$\displaystyle-\delta\hat{d}_{i}\left(\theta_{i}^{1}\psi(g)\mathfrak{m}\delta
u^{i,s}+\theta_{i}^{1}\left(w_{i}(g)+w_{i}(\mathfrak{m})+\frac{s}{2}\right)g\mathfrak{m}\delta
u^{i,s}\right).$

So the subspace $\hat{\mathcal{C}}\delta\hat{d}_{i}(\mathfrak{m}\delta
u^{i,s})$ is invariant under the action of $\mathcal{D}_{i}$. Similarly, we
make the decomposition
$\hat{\mathcal{C}}=\hat{\mathcal{C}}^{i}_{0}\oplus\theta_{i}\hat{\mathcal{C}}^{i}_{0}$.
For $g\in\hat{\mathcal{C}}_{i}^{0}$, it is easy to see that

$\mathcal{D}_{i}\delta\hat{d}_{i}(g\mathfrak{m}\delta
u^{i,s})=-\delta\hat{d}_{i}\left(\theta_{i}^{1}\left(w_{i}(\mathfrak{m})+\frac{s}{2}\right)g\mathfrak{m}\delta
u^{i,s}\right).$

On the other hand, for a monomial in $u^{i,s},\theta_{i}^{s+1}$ with $s\geq
1$, we must have that $w_{i}(\mathfrak{m})\geq\frac{1}{2}$, thus the subspace
$\hat{\mathcal{C}}^{i}_{0}\delta\hat{d}_{i}(\mathfrak{m}\delta u^{i,s})$ is
acyclic. By applying Lemma 5.2 again, we know that we only need to consider
the cohomology of
$\theta_{i}\hat{\mathcal{C}}^{i}_{0}\delta\hat{d}_{i}(\mathfrak{m}\delta
u^{i,s})$ with the differential being the projection of $\mathcal{D}_{i}$
given by the multiplication of

$-\theta_{i}^{1}\left(w_{i}(\mathfrak{m})+\frac{s}{2}-\frac{1}{2}\right).$

So the only possible nontrivial cocycle is given by the case when $s=0$ and
$w_{i}(\mathfrak{m})=\frac{1}{2}$, i.e. when the monomial $\mathfrak{m}$ is of
the form

$\theta_{i}\hat{\mathcal{C}}^{i}_{0}\delta\hat{d}_{i}(\theta_{i}^{2}\delta
u^{i})=\hat{\mathcal{C}}^{i}_{0}\theta_{i}\theta_{i}^{2}\delta\theta_{i}^{1}.$

By counting the possible bidegrees of such elements, we complete the proof of
the lemma. ∎

Let us summarize all the results obtained above. We first construct a spectral
sequence ${}^{1}E$ to compute the cohomology group
$H^{p}_{d}(\Omega[\lambda],\partial_{\lambda})$. Then we transform the
computation to finding suitable bidegrees $(p,d)$ such that the second page
${}^{1}E_{2}=0$ when restricted to $\Omega^{p}_{d}$. To this end we introduce
a third spectral sequence ${}^{3}E$, and we conclude that the first page
${}^{3}E_{1}$ vanishes for suitable bidegrees in Lemmas 5.8–Lemma 5.10, thus
${}^{3}E$ converges to ${}^{1}E_{2}$ and so ${}^{1}E_{2}$ vanishes.
Consequently ${}^{1}E$ converges to
$H^{p}_{d}(\Omega[\lambda],\partial_{\lambda})$. In this way, we prove Theorem
5.1.

The following proposition is an illustration of applications of the
variational bihamiltonian cohomology.

######  Proposition 5.11.

Let $(P_{0},P_{1})$ be a semisimple bihamiltonian structure of hydrodynamic
type, and let $X\in\mathrm{Der}(\hat{\mathcal{A}})_{1}^{0}$ commute with
$\partial_{x}$, $D_{P_{0}}$ and $D_{P_{1}}$. Then for any deformation
$(\tilde{P}_{0},\tilde{P}_{1})$ of $(P_{0},P_{1})$, there exists a unique
$\tilde{X}\in\mathrm{Der}(\hat{\mathcal{A}})^{0}_{\geq 1}$ with leading term
given by $X$ such that $\tilde{X}$ commutes with $\partial_{x}$,
$D_{\tilde{P}_{0}}$ and $D_{\tilde{P}_{1}}$.

###### Proof.

Let us decompose $\tilde{X}$ and $\tilde{P}_{a}$ according to the differential
degree as follows:

$\displaystyle\tilde{X}$ $\displaystyle=X^{[0]}+\sum_{k\geq 1}X^{[k]},\quad
X^{[k]}\in\mathrm{Der}(\hat{\mathcal{A}})_{k+1}^{0},\ X^{[0]}=X;$
$\displaystyle\tilde{P}_{a}$ $\displaystyle=P_{a}+\sum_{k\geq
1}P_{a}^{[k]},\quad P_{a}^{[k]}\in\hat{\mathcal{F}}^{2}_{k+1},\quad a=0,1.$

The condition that $\tilde{X}$ commutes with $D_{\tilde{P}_{a}}$ is equivalent
to the following equations:

(5.13)
$\left[D_{P_{a}},X^{[k]}\right]+\sum_{l=1}^{k}\left[D_{\tilde{P}_{a}^{[l]}},X^{[k-l]}\right]=0,\quad
k\geq 1,\ a=0,1.$

To prove the uniqueness of $\tilde{X}$, we only need to show that if $X=0$
then $\tilde{X}=0$. Indeed, if $X=0$, then we have the following equation for
$X^{[1]}$:

$\left[D_{P_{a}},X^{[1]}\right]=0,$

which is equivalent to $\tilde{D}_{a}\mathcal{X}^{[1]}=0$. Here
$\mathcal{X}^{[k]}\in\bar{\Omega}^{1}_{k+1}$ is the 1-form corresponds to
$X^{[k]}$. Recall that

$\operatorname{\mathit{VBH}}^{1}_{d}(\bar{\Omega},\tilde{D}_{0},\tilde{D}_{1})=\bar{\Omega}^{1}_{d}\cap\ker\tilde{D}_{0}\cap\ker\tilde{D}_{1},$

then by the vanishing of
$\operatorname{\mathit{VBH}}^{1}_{2}(\bar{\Omega},\tilde{D}_{0},\tilde{D}_{1})$
given by Theorem 4.5, we conclude that $X^{[1]}=0$. In a similar way, we can
prove recursively that $X^{[k]}=0$ and the uniqueness is proved.

The proof of the existence is very similar to the one given in Sect. 6.2 and
we omit the details here. For a sketch of the proof, we may first assume
$P_{a}^{[1]}=0$ and hence we can choose $X^{[1]}=0$, then by using
$\operatorname{\mathit{VBH}}^{2}_{\geq
4}(\bar{\Omega},\tilde{D}_{0},\tilde{D}_{1})=0$, we can recursively solve the
equations (5.13) to obtain $X^{[\geq 2]}$. The proposition is proved. ∎

######  Remark 5.2.

This result is a generalization of the fact that a bihamiltonian vector filed
is uniquely determined by its leading term. This generalized version can be
applied to the case when the flows are not in the space
$\mathrm{Der}(\hat{\mathcal{A}})^{D}$. Typical examples of such kind of flows
are given by Virasoro symmetries.

##  6\. Conformal bihamiltonian structures

###  6.1. Conformal Bihamiltonian Structures of hydrodynamic type

This section is devoted to the proof of the Theorem 2.2. To this end, we
consider a semisimple bihamiltonian structure $(P_{0},P_{1})$ of hydrodynamic
type which is represented as in (2.4).

Assume that the bihamiltonian structure $(P_{0},P_{1})$ is conformal, we are
going to find a derivation $E$ such that it satisfied the equation (2.5). We
first make the following decomposition

$E=E^{[0]}+E^{[\geq 1]},\quad
E^{[0]}\in\mathrm{Der}(\hat{\mathcal{A}})^{0}_{0},\,E^{[\geq
1]}\in\mathrm{Der}(\hat{\mathcal{A}})^{0}_{\geq 1}.$

According to the equation (2.5), it is easy to see that $E^{[\geq 1]}$ is a
vector field that commutes with $D_{P_{0}}$ and $D_{P_{1}}$ and thus it is an
element of $\operatorname{\mathit{VBH}}^{1}_{\geq
1}(\bar{\Omega},\tilde{D}_{0},\tilde{D}_{1})$. In what follows we assume that
$E\in\mathrm{Der}(\hat{\mathcal{A}})^{0}_{0}$.

Denote $E(u^{i})=F^{i}$ and $E(\theta_{i})=\sum_{j}G^{j}_{i}\theta_{j}$, where
$F^{i}$ and $G^{j}_{i}$ are some smooth functions of $u^{1},\dots,u^{n}$. We
are to solve these functions from the equations given in (2.5).

Firstly, from the equation $[E,\partial_{x}]=\mu\partial_{x}$ it follows that

$E(u^{i,s})=\partial_{x}^{s}(F^{i})+s\mu u^{i,s},\quad
E(\theta^{s}_{i})=\sum_{j}\partial_{x}^{s}(G^{j}_{i}\theta_{j})+s\mu\theta_{i}^{s}.$

By comparing the $\theta_{j}^{1}$ coefficients $(j\neq i)$ of both sides of
the equation

(6.1) $[E,D_{P_{a}}]u^{i}=\lambda_{a}D_{P_{a}}(u^{i})$

for $a=0,1$ we obtain the following equations:

$\displaystyle f^{i}G^{j}_{i}-(\partial_{j}F^{i})f^{j}=0,\quad j\neq i$
$\displaystyle u^{i}f^{i}G^{j}_{i}-(\partial_{j}F^{i})u^{j}f^{j}=0,\quad j\neq
i.$

From these equations it follows that $G^{j}_{i}=0$ for $j\neq i$. We also
compare the $\theta_{i}^{1}$ coefficients of both sides of the equation (6.1)
to obtain

$\displaystyle\sum_{j}\partial_{j}f^{i}F^{j}+f^{i}(G^{i}_{i}+\mu)-\partial_{i}F^{i}f^{i}=\lambda_{0}f^{i},$
$\displaystyle\sum_{j}\partial_{j}(u^{i}f^{i})F^{j}+u^{i}f^{i}(G^{i}_{i}+\mu)-\partial_{i}F^{i}u^{i}f^{i}=\lambda_{1}u^{i}f^{i}.$

We solve these equations to arrive at

$F^{i}=(\lambda_{1}-\lambda_{0})u^{i},\quad
G^{i}_{i}=\lambda_{1}-\mu-\frac{1}{f^{i}}\sum_{j}(\lambda_{1}-\lambda_{0})u^{j}\partial_{j}f^{i}.$

Together with the fact that $G^{j}_{i}=0$ for $j\neq i$, we conclude that

$E(u^{i})=(\lambda_{1}-\lambda_{0})u^{i},\quad
E(\theta_{i})=\left(\lambda_{1}-\mu-\frac{1}{f^{i}}E(f^{i})\right)\theta_{i}.$

For simplicity, we will denote $E(\theta_{i})=G^{i}\theta_{i}$.

Next we compare the $\theta_{i}$ coefficients of the equation

(6.2) $[E,D_{P_{0}}]u^{i}=\lambda_{0}D_{P_{0}}(u^{i})$

to arrive at the following equation:

$\displaystyle\sum_{j,k}$
$\displaystyle\frac{f^{i}\partial_{j}\partial_{k}f^{i}-\partial_{j}f^{i}\partial_{k}f^{i}}{f^{i}}(\lambda_{0}-\lambda_{1})u^{k}u^{j,1}+\sum_{j}\partial_{j}f^{i}(\lambda_{0}-\lambda_{1})u^{j,1}$
$\displaystyle+\frac{1}{2}\sum_{j,k}\partial_{k}\partial_{j}f^{i}(\lambda_{1}-\lambda_{0})u^{k}u^{j,1}+\frac{1}{2}\sum_{j}\partial_{j}f^{i}u^{j,1}G^{i}=\frac{1}{2}\sum_{j}(\lambda_{0}-\mu)\partial_{j}f^{i}u^{j,1}.$

By further comparing the $u^{j,1}$ coefficients for both sides of the above
equation, one can show that the above equation is actually equivalent to

(6.3)
$E\left(\frac{\partial_{j}f^{i}}{f^{i}}\right)=(\lambda_{0}-\lambda_{1})\left(\frac{\partial_{j}f^{i}}{f^{i}}\right),\quad\forall
i,j.$

For $j\neq i$, compare the coefficients of $u^{i,1}\theta_{j}$ and
$u^{j,1}\theta_{j}$ on both sides of the equation (6.2), one obtains the
following equations:

(6.4) $\displaystyle
E\left(f^{i}\frac{\partial_{i}f^{j}}{f^{j}}\right)+f^{i}\frac{\partial_{i}f^{j}}{f^{j}}G^{j}=(\lambda_{0}-\mu)f^{i}\frac{\partial_{i}f^{j}}{f^{j}},\quad
j\neq i.$ (6.5) $\displaystyle
E\left(f^{j}\frac{\partial_{j}f^{i}}{f^{i}}\right)+f^{j}\frac{\partial_{j}f^{i}}{f^{i}}G^{j}=(\lambda_{0}-\mu)f^{j}\frac{\partial_{j}f^{i}}{f^{i}},\quad
j\neq i.$

Finally let us compare the coefficients on both sides of the equation

$[E,D_{P_{0}}]\theta_{i}=\lambda_{0}D_{P_{0}}(\theta_{i}).$

For $j\neq i$, we consider first the coefficients of
$\theta_{i}\theta_{j}^{1}$ and obtain that

(6.6)
$E\left(f^{j}\frac{\partial_{j}f^{i}}{f^{i}}\right)+f^{j}\frac{\partial_{j}f^{i}}{f^{i}}G^{j}=(\lambda_{0}-\mu)f^{j}\frac{\partial_{j}f^{i}}{f^{i}}-\partial_{j}G^{i}f^{j},\quad
j\neq i.$

Therefore it is easy to conclude that $\partial_{j}G^{i}=0$ for $j\neq i$ by
comparing (6.5) and (6.6). We also compare the coefficients of
$\theta_{i}\theta_{i}^{1}$, after a straightforward computation, we arrive at
an equation

(6.7)
$E\left(\frac{\partial_{i}f^{i}}{f^{i}}\right)=(\lambda_{0}-\lambda_{1})\left(\frac{\partial_{i}f^{i}}{f^{i}}\right)-\partial_{i}G^{i}.$

Hence we conclude that $\partial_{i}G^{i}=0$ by comparing (6.7) with (6.3),
and $G^{i}$ must be a constant. This means that there exists real numbers
$\alpha^{i}$ such that $E(f^{i})=\alpha_{i}f^{i}$, which gives the condition
(2.6) with $d^{i}$ determined by $\alpha^{i}=(\lambda_{1}-\lambda_{0})d^{i}$.
Substitute the expression for $G^{j}$ into (6.4), we obtain the condition
(2.7), and hence the ‘only if’ part of the Theorem 2.2 is proved.

The ‘if’ part of the Theorem 2.2 can be checked easily by a straightforward
computation and hence the theorem is proved.

###  6.2. Deformed Conformal Bihamiltonian Structures

In this section, we will use the theory of variational bihamiltonian
cohomology to prove Theorem 2.3. Let $(P_{0},P_{1})$ be a conformal semisimple
bihamiltonian structure of hydrodynamic type as described in Theorem 2.2 and
$(\tilde{P}_{0},\tilde{P}_{1})$ be any of its deformation. We are going to
find a deformation $\tilde{E}\in\mathrm{Der}(\hat{\mathcal{A}})^{0}$, such
that

(6.8)
$\left[\tilde{E},\partial_{x}\right]=\mu\partial_{x};\quad\left[\tilde{E},D_{\tilde{P}_{a}}\right]=\lambda_{a}D_{\tilde{P}_{a}},\quad
a=0,1.$

We first decompose $\tilde{P}_{a}$ and $\tilde{E}$ as follows:

$\tilde{P}_{a}=\sum_{k\geq 0}\tilde{P}_{a}^{[k]},\
\tilde{P}_{a}^{[k]}\in\hat{\mathcal{F}}^{2}_{k+1};\quad\tilde{E}=\sum_{k\geq
0}E^{[k]},\ E^{[k]}\in\mathrm{Der}(\hat{\mathcal{A}})^{0}_{k}.$

We also make the same decomposition for both sides of the equation
$[\tilde{E},D_{\tilde{P}_{a}}]=\lambda_{a}D_{\tilde{P}_{a}}$ to obtain

(6.9)
$\sum_{i=0}^{l}\left[E^{[i]},D_{P_{a}^{[l-i]}}\right]=\lambda_{a}D_{P_{a}^{[l]}},\quad
a=0,1;\ l\geq 1.$

According to the result of bihamiltonian cohomology, we can assume (by doing a
Miura transformation if necessary) that $P_{a}^{[1]}=0$, hence we can choose
$E^{[1]}=0$. Let us proceed to find the deformations $E^{[\geq 2]}$. Let us
denote

$W_{a}^{[l]}=\sum_{i=0}^{l-1}\left[E^{[i]},D_{P_{a}^{[l-i]}}\right]-\lambda_{a}D_{P_{a}^{[l]}},$

then we can recursively determine the deformation by solving the equation

$\left[D_{P_{a}^{[0]}},E^{[l]}\right]=W_{a}^{[l]}.$

######  Lemma 6.1.

Assume that we have found $E^{[i]}$ for all $i<k$ such that the equations in
(6.9) are satisfied for $l<k$. Then we have the following identity:

$\left[W_{b}^{[k]},D_{P_{a}^{[0]}}\right]+\left[W_{a}^{[k]},D_{P_{b}^{[0]}}\right]=0,\quad
a,b=0,1.$

###### Proof.

The proof is a straightforward computation by using the equations

$\left[E^{[i]},D_{P_{a}^{[0]}}\right]=W_{a}^{[i]},\,i<k;\quad\left[D_{\tilde{P}_{a}},D_{\tilde{P}_{b}}\right]=0,$

and the (graded) Jacobi identities. ∎

In order to apply the theory of variational bihamiltonian cohomology, we need
the following fact.

######  Lemma 6.2.

We have $\left[E^{[l]},\partial_{x}\right]=0$ and
$\left[W_{a}^{[l]},\partial_{x}\right]=0$ for $l\geq 1$.

###### Proof.

This follows directly from the assumption
$[\tilde{E},\partial_{x}]=\mu\partial_{x}$. ∎

As a consequence of the above lemma, we can regard the vector fields $E^{[l]}$
and $W_{a}^{[l]}$ as elements of the space $\bar{\Omega}$. We denote the
corresponding 1-forms by $\mathcal{E}^{[l]}$ and $\mathcal{W}_{a}^{[l]}$
respectively. More precisely, we have

$\mathcal{E}^{[l]}\in\bar{\Omega}^{1}_{l},\quad\mathcal{W}_{a}^{[l]}\in\bar{\Omega}^{2}_{l+1}.$

Now the equations we are to solve can be rewritten in the form

(6.10) $\tilde{D}_{a}\mathcal{E}^{[l]}=\mathcal{W}_{a}^{[l]},\quad a=0,1,\
l\geq 2,$

and Lemma 6.1 gives the following conditions

$\tilde{D}_{a}\mathcal{W}_{a}^{[l]}=0,\quad a=0,1.$

Thus by using the triviality of the variational Hamiltonian cohomology we can
find an element $\gamma^{[l]}\in\bar{\Omega}^{1}_{l}$ such that

$\tilde{D}_{0}\gamma^{[l]}=\mathcal{W}_{0}^{[l]}.$

Then the solution of $\mathcal{E}^{[l]}$ can be represented by

$\mathcal{E}^{[l]}=\gamma^{[l]}+\tilde{D}_{0}\alpha^{[l]},\quad\alpha^{[l]}\in\Omega^{0}_{l-1}.$

The 1-form $\alpha^{[l]}$ should be determined by the equation

$\tilde{D}_{1}\mathcal{E}^{[l]}=\tilde{D}_{1}\left(\gamma^{[l]}+\tilde{D}_{0}\alpha^{[l]}\right)=\mathcal{W}_{1}^{[l]}.$

By using Lemma 6.1 again, we see that
$\mathcal{W}_{1}^{[l]}-\tilde{D}_{1}\gamma^{[l]}$ lies in
$\ker\tilde{D}_{0}\cap\ker\tilde{D}_{1}$. Therefore from the fact that
$\operatorname{\mathit{VBH}}^{2}_{\geq
4}(\bar{\Omega},\tilde{D}_{0},\tilde{D}_{1})=0$ it follows that we can always
solve the above equation to obtain $\alpha^{[l]}$ for $l\geq 3$.

Now let us try to find $\mathcal{E}^{[2]}$ by solving the equations in (6.10)
for $l=2$. We will work in the canonical coordinates $u^{1},\dots,u^{n}$ of
$(P_{0},P_{1})$. According to the results in [31], we can choose a Miura type
transformation such that $\tilde{P}_{0}=P_{0}$, $P_{1}^{[1]}=0$ and the
derivation $D_{P_{1}^{[2]}}$ is given by the 1-form
$\tilde{D}_{0}\mathcal{T}$, where $\mathcal{T}$ is given by

$\mathcal{T}=\int\delta\left(D_{P_{1}}\sum_{i}c_{i}(u^{i})u^{i,1}\log
u^{i,1}-D_{P_{0}}\sum_{i}u^{i}c_{i}(u^{i})u^{i,1}\log u^{i,1}\right).$

Here the functions $c_{i}(u^{i})$ are the central invariants of the deformed
bihamiltonian structure $(\tilde{P}_{0},\tilde{P}_{1})$ . Since
$P_{0}^{[2]}=0$, we can choose $\gamma^{[2]}=0$. Then we have

$W_{1}^{[2]}=\left[E^{[0]},D_{P_{1}^{[2]}}\right]-\lambda_{1}D_{P_{1}^{[2]}},$

here $E^{[0]}=E$ is described as in Theorem 2.2. Let us use $T$ to denote the
derivation given by the 1-form $\mathcal{T}$, then we have

$\displaystyle W_{1}^{[2]}$
$\displaystyle=\left[E^{[0]},\left[D_{P_{0}},T\right]\right]-\lambda_{1}\left[D_{P_{0}},T\right]$
$\displaystyle=-\left[D_{P_{0}},\left[T,E^{[0]}\right]\right]-\left[T,\left[E^{[0]},D_{P_{0}}\right]\right]-\lambda_{1}\left[D_{P_{0}},T\right]$
$\displaystyle=-\left[D_{P_{0}},\left[T,E^{[0]}\right]\right]+\lambda_{0}\left[D_{P_{0}},T\right]-\lambda_{1}\left[D_{P_{0}},T\right]$
$\displaystyle=\left[D_{P_{0}},(\lambda_{0}-\lambda_{1})T-\left[T,E^{[0]}\right]\right].$

Denote by $\beta\in\bar{\Omega}^{1}_{2}$ the 1-form corresponding to the
derivation $(\lambda_{0}-\lambda_{1})T-\left[T,E^{[0]}\right]$, then we see
that

$\tilde{D}_{1}\mathcal{E}^{[2]}=\tilde{D}_{1}\tilde{D}_{0}\alpha^{[2]}=\mathcal{W}_{1}^{[2]}=\tilde{D}_{0}\beta.$

To solve this equation for $\mathcal{E}^{[2]}$, we need to check that
$[\tilde{D}_{0}\beta]\in\operatorname{\mathit{VBH}}^{2}_{3}(\bar{\Omega},\tilde{D}_{0},\tilde{D}_{1})$
is trivial. According to Lemma 4.2 we only need to check that the indices
$ind_{i}(\beta)$ for $i=1,\cdots,n$ vanish.

We first note that

$ind_{i}(\mathcal{T})=-3c_{i}(u^{i}).$

In another word, if we represent $\mathcal{T}$ in the form

$\mathcal{T}=\int\sum_{i}X^{i}\delta u^{i}+Y^{i}\delta\theta_{i},\quad
X^{i}\in\hat{\mathcal{A}}^{1}_{2},\quad Y^{i}\in\hat{\mathcal{A}}^{0}_{2},$

where $X^{i}$ and $Y^{i}$ are given by

$\displaystyle X^{i}$
$\displaystyle=\sum_{j}X^{(i)}_{j}\theta_{j}^{2}+\sum_{j,k}\left(X^{(i)}_{kj}u^{j,1}\theta_{k}^{1}+Z^{(i)}_{jk}u^{k,2}\theta_{j}\right)+\sum_{j,k,l}Z^{(i)}_{j;kl}u^{k,1}u^{l,1}\theta_{j},$
$\displaystyle Y^{i}$
$\displaystyle=\sum_{j}Y^{(i)}_{j}u^{j,2}+\sum_{j,k}Y^{(i)}_{jk}u^{j,1}u^{k,1},$

then we must have $X^{(i)}_{i}+Y^{(i)}_{i}=-3c^{i}(u^{i})f^{i}$. From the
explicit formula (2.8) for $E^{[0]}$ it follows that

$\displaystyle\left[T,E^{[0]}\right]u^{i}=$
$\displaystyle(\lambda_{1}-\lambda_{0})X^{(i)}_{i}u^{i,2}-E^{[0]}\left(X^{(i)}_{i}\right)u^{i,2}$
$\displaystyle-X^{(i)}_{i}(\lambda_{1}-\lambda_{0}+2\mu)u^{i,2}+\cdots$
$\displaystyle=$ $\displaystyle-2\mu
X^{(i)}_{i}u^{i,2}-E^{[0]}\left(X^{(i)}_{i}\right)u^{i,2}+\cdots;$
$\displaystyle\left[T,E^{[0]}\right]\theta_{i}=$
$\displaystyle-\left(\lambda_{1}-(\lambda_{1}-\lambda_{0})d^{i}-\mu\right)Y^{(i)}_{i}\theta_{i}^{2}+E^{[0]}\left(Y^{(i)}_{i}\right)\theta_{i}^{2}$
$\displaystyle+\left(\lambda_{1}-(\lambda_{1}-\lambda_{0})d^{i}+\mu\right)Y^{(i)}_{i}\theta_{i}^{2}+\cdots$
$\displaystyle=$ $\displaystyle 2\mu
Y^{(i)}_{i}\theta_{i}^{2}+E^{[0]}\left(Y^{(i)}_{i}\right)\theta_{i}^{2}+\cdots.$

Here we omit all the terms that do not contribute to the computation of
indices. Then by the definition of the index we conclude that

$ind_{i}(\beta)=3\left(\lambda_{1}-\lambda_{0}-2\mu-(\lambda_{1}-\lambda_{0})d^{i}\right)c_{i}(u^{i})-3E^{[0]}\left(c_{i}(u^{i})\right).$

The equation $ind_{i}(\beta)=0$ is actually an ODE for $c_{i}(u^{i})$, which
can be easily solved to give the solution

$c_{i}(u^{i})=C_{i}(u^{i})^{m_{i}},\quad
m_{i}=\frac{\lambda_{1}-\lambda_{0}-2\mu-(\lambda_{1}-\lambda_{0})d^{i}}{\lambda_{1}-\lambda_{0}},$

where $C_{i}$ are arbitrary constants. Thus we prove the Theorem 2.3.

##  7\. Conclusion

In this paper, we propose a generalization of the bihamiltonian cohomology,
called the variational bihamiltonian cohomology, to deal with more general
bihamiltonian flows. The eventual goal for developing the theory of
variational bihamiltonian cohomology is to prove the following conjectures.

######  Conjecture 7.1.

For any deformation of the bihamiltonian structure of the Principal Hierarchy
associated to a semisimple Frobenius manifold with constant central
invariants, the Virasoro symmetries $\frac{\partial}{\partial s_{m}}$ can be
deformed to be symmetries of the deformed integrable hierarchy.

The above conjecture is verified for the case $m=-1$ in [11]. We will study
this conjecture in detail and give a proof of it in the second paper [30] of
this series of papers. The proof is based on the variational bihamiltonian
cohomology theory established in this paper and the construction of the super
tau-covers for any tau-symmetric bihamiltonian integrable hierarchies which
are generalizations of the results given in [29].

Among all the possible deformations of the bihamiltonian structure with
constant central invariants, there is a particular deformation with all the
central invariants being $\frac{1}{24}$. Such a deformation is called the
topological deformation, and it is conjectured that:

######  Conjecture 7.2.

When the central invariants of the deformation of the bihamiltonian structure
of the Principal Hierarchy are all equal to $\frac{1}{24}$, the corresponding
deformation of Virasoro symmetries can be represented by linear actions on the
tau function.

Apart from the above-mentioned goal, there are some other interesting problems
concerning the variational bihamiltonian cohomology itself. For example, the
natural map

$\delta:\hat{\mathcal{F}}\to\bar{\Omega}$

induces an isomorphism from the bihamiltonian cohomology to the variational
bihamiltonian cohomology for most of the bidegrees $(p,d)$, in particular, for
the bidegress where both the bihamiltonian cohomology and the variational
bihamiltonian cohomology vanish and for the bidegree $(p,d)=(2,3)$. Then it is
natural to ask if this map is indeed a quasi-isomorphism of the complex, and
if so can we obtain a homotopy inverse? Also, we can generalize the notion of
variational bihamiltonian cohomology to include all the possible differential
forms on $J^{\infty}(\hat{M})$, then we have: