DEFORMATIONS OF SEMISIMPLE POISSON PENCILS OF ...

DEFORMATIONS OF SEMISIMPLE POISSON PENCILS OF HYDRODYNAMIC TYPE ARE UNOBSTRUCTED GUIDO CARLET, HESSEL POSTHUMA, AND SERGEY SHADRIN Abstract. We prove that the bihamiltonian cohomology of a semisimple pencil of Poisson brackets of hydrodynamic type vanishes for almost all degrees. This implies the existence of a full dispersive deformation of a semisimple bihamiltonian structure of hydrodynamic type starting from any infinitesimal deformation.

Contents 1. Introduction 1.1. Basic setup from the theory of integrable hierarchies 1.2. Classification of Poisson pencils and the extension problem 1.3. Methods of proof and organization of the paper 1.4. Conventions 2. Theta formalism, polyvector fields and cohomology 2.1. Basic definitions ˆ 2.2. The complex (A[λ], Dλ ) 2.3. The main theorem 2.4. Vanishing of bihamiltonian cohomology 2.5. Bihamiltonian cohomology and deformations 3. Filtrations and spectral sequences 3.1. Strategy of the proof of Theorem 4 3.2. Grading and subcomplexes 3.3. The first filtration 3.4. The second filtration 3.5. The third filtration Acknowledgments References

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1. Introduction 1.1. Basic setup from the theory of integrable hierarchies. Consider a system of evolutionary PDEs with one spatial variable x and n dependent variables of the form ∂ui i j k = Aij (u)ujx + Bji (u)ujxx + Cjk ux ux + O(2 ). ∂t (Here, and in the following we use the summation convention.) The right hand side of this equation is represented as a formal power series in , where 1

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GUIDO CARLET, HESSEL POSTHUMA, AND SERGEY SHADRIN

the coefficient of k is a homogeneous differential polynomial, i.e. an homogeneous polynomial in ui,d := ∂xd ui , i = 1, . . . , n, d = 1, . . . , k, deg ui,d = d, of degree k+1 whose coefficients are smooth functions of coordinates u1 , . . . , un on some domain U ⊂ Rn . We can think of the ui (x) as (-power series of) smooth functions of x ∈ S 1 or Schwarzian functions of x ∈ R. On of the possible ways to define and study an integrable hierarchy of partial differential equations of this type uses so-called bihamiltonian structures with hydrodynamic limit. A bihamiltonian structure with hydrodynamic limit is given by: a pencil of two compatible Poisson structures on the space of local functionals of the form k {ui (x), uj (y)}a = gaij (u)∂x + Γij (u)u δ(x − y) + O(), a = 1, 2 x k,a where the leading order is given by two Poisson brackets of hydrodynamic type, and the terms of higher order in are homogeneous differential operators acting on δ(x − y), and their coefficients are homogeneous differential polynomials in the dependent variables u1 , . . . , un ; a system of Hamiltonians of the form Z Ha [u] =

dx · (ha (u) + O()) ,

a = 1, 2

with higher order terms in given by homogeneous differential polynomials in u1 , . . . , un . The evolutionary PDE above can be written as an Hamiltonian flow w.r.t. both Poisson structures ∂ui = {ui (x), Ha }a ∂t for a = 1, 2. The natural equivalence relation on these systems, and in particular on the pencils of Poisson structures with hydrodynamic limit, is given by the so-called Miura transformations, which are transformations of the dependent variables of the form ui 7→ v i (u) + O(), (1) where higher order terms in are homogeneous differential polynomials in u1 , . . . , un , and the leading term is a diffeomorphism. In this context, an important problem is to classify the pencils of Poisson structures with hydrodynamic limit, up to the equivalence given by Miura transformations. In the scalar (n = 1) case a complete solution of this classification problem has been obtained, see [13, 10, 1, 12, 4, 5]. In the general n > 0 case (see [3, 6, 7, 9, 11]) it is convenient to make the assumption that the pencil of Poisson brackets of hydrodynamic type that we are considering is semisimple. A Poisson pencil of hydrodynamic type k gaij (u)∂x + Γij (u)u a = 1, 2 x δ(x − y), k,a is semisimple if the polynomial det g1ij − λg2ij of degree n in λ has n pairwise distinct non-constant roots on U ⊂ Rn . In such case [6] one can use the roots as a set of coordinates on U , called canonical coordinates. This choice

DEFORMATIONS OF SEMISIMPLE POISSON PENCILS

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ij ensures that both metrics g1,2 are diagonal with diagonal entries respectively i i i equal to f (u), u f (u), i = 1, . . . , n for non-vanishing functions f i (u) on U . In this paper we shall consider the deformation problem of semisimple Poisson pencils of hydrodynamic type by working in canonical coordinates. The change of coordinates to canonical ones is an example of a Miura transformation of the first kind, i.e., a diffeomorphism with the terms O() in (1) equal to zero. By fixing these coordinates, we are therefore left with the problem of classifying Poisson Pencils up to Miura transformations of the second kind, that is, transformations of the dependent coordinates as in (1), with the zeroth order constant in equal to the identity. Since Miura transformations of the first and second kind obviously generate the whole Miura group, this classification problem is equivalent to the original one described above. Let us now give a precise formulation of this deformation problem in canonical coordinates.

1.2. Classification of Poisson pencils and the extension problem. Let {, }0λ = {, }02 − λ{, }01 be a semisimple Poisson pencil of hydrodynamic type [6], and let u1 , . . . , un be the associated canonical coordinates over a domain U ⊂ Rn where ui − uj 6= 0 for i 6= j. The two compatible Poisson brackets {, }01,2 are of the form k {ui (x), uj (y)}0a = gaij (u(x))δ 0 (x − y) + Γij k,a (u(x))ux (x)δ(x − y),

with a = 1, 2, i, j = 1, . . . n, where the contravariant metrics are given by g1ij = f i δij ,

g2ij = ui f i δij

(no summation over i)

j il j and Γij k,a = −ga Γlk,a , where Γlk,a are the Christoffel symbols of the metric

gaij , and f 1 (u), . . . , f n (u) are non-vanishing functions on U . A deformation of {, }0λ is given by a pencil {, }λ = {, }2 − λ{, }1 where {, }a , a = 1, 2 are compatible Poisson brackets of the form {ui (x), uj (y)}a = {ui (x), uj (y)}0a +

X k>0

Aij k,l;a

k

k+1 X

(l) Aij k,l;a (u(x))δ (x − y)

(2)

l=0

∈ A and deg Aij k,l;a = k − l + 1. Here A denotes polynomials in u1 , . . . , un , i.e. formal power series in

the space of with differential the variables ui,s , i = 1, . . . , n, s > 0, with coefficients that are smooth functions of u1 , . . . , un . The degree is defined by setting deg ui,s = s. Two deformations are equivalent if they are related by a Miura transformation (of the second kind [11]), i.e. by a change of variables of the form X ui 7→ u ˜i = ui + k Fki , i = 1, . . . , n k>0

with Fki ∈ A and deg Fki = k. 0 An infinitesimal deformation {, }62 λ of {, }λ is given by a pair of compatible Poisson brackets of the form (2) where terms of O(3 ) are disregarded. This means that in the expansion (2) above, we only consider the the coefficients (l) Aij k,l;a (u(x)) for k up to 2, so that the highest derivative δ (x−y). appearing

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is 3. Two infinitesimal deformations are equivalent iff they are related by a Miura transformation up to O(3 ). The following theorem, which classifies the deformations of {, }0λ , was proved in [9, 6]. Theorem 1. Two deformations of {, }0λ are equivalent if and only if the corresponding infinitesimal deformations are equivalent. Given an infinitesimal deformation of {, }0λ , the functions, called central invariants, defined by   ki i ki 2 X (A1,2;2 − u A1,2;1 ) 1 i ii Aii , ci (u) := 2,3;2 − u A2,3;1 + i 2 3(f (u)) f k (u)(uk − ui ) k6=i

for i = 1, . . . , n, only depend on the single variable ui are invariant under Miura transformations. Two infinitesimal deformations of {, }0λ are equivalent if and only if they have the same central invariants. The main problem in the deformation theory of a semisimple Poisson pencil {, }0λ is the problem of extension [12]. Making a choice of central invariants c1 (u1 ), . . . , cn (un ) fixes an equivalence class of infinitesimal deformations of {, }0λ , but the question is whether there exists a full deformation {, }λ that extends an infinitesimal one to all orders in . The main result of this paper is the affirmative answer to this question. Theorem 2. Let {, }62 λ be an infinitesimal deformation of a semisimple Poisson pencil of hydrodynamic type {, }0λ . Then there exists a deformation {, }λ that extends {, }62 λ to all orders in . 1.3. Methods of proof and organization of the paper. The problem of the description of automorphisms, infinitesimal deformations, and obstructions to the extension of infinitesimal deformations for any algebraic structure can be formulated in terms of some cohomology groups associated to it. In our case, in order to prove that the deformation of a semisimple pencil of Poisson brackets is not obstructed we have to show that certain cohomology groups, called bihamiltonian cohomology, are equal to zero. There is no straightforward way to compute these cohomology groups. However, Liu and Zhang have shown that the vanishing of these cohomology groups follows from the vanishing of the cohomology of the auxiliary complex ˆ (A[λ], Dλ ), defined below, in certain degrees. It is a difficult task to compute the full cohomology of this complex, however we are able to show vanishing of such cohomology in the required degrees using a clever choice of filtrations and the structure of the associated spectral sequences. The paper is organized as follows. In Section 2 we recall the definition of the auxiliary complex and explain its relation to the problems of deformation of pencils of Poisson structures. In particular, we formulate a statement ˆ about the complex (A[λ], Dλ ) that implies Theorem 2. In Section 3 we ˆ introduce a series of filtrations on the complex (A[λ], Dλ ) that allows us to prove the key statement about its cohomology. 1.4. Conventions. Throughout the paper we use the summation convention in the sense that repeated (upper- and lower-)indices should be summed over. However, there are a few exceptions when the metrics g1ij = f i δij and


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g2ij = ui f i δij , or the tensors derived from them are involved. Such equations always involve the functions f i . To determine which indices are to be summed over, it suffices to consider the other side of the equation and the indices that appear in there. 2. Theta formalism, polyvector fields and cohomology The deformation theory of a pencil of Poisson brackets is controlled by the so-called bihamiltonian cohomology defined on the space of local polyvector fields. In order to show the vanishing of such bihamiltonian cohomology in certain degrees, from which Theorem 2 follows, we consider the cohomology ˆ of a related complex (A[λ], Dλ ), introduced by Liu and Zhang [12]. This ˆ where approach has a double advantage: first, we can work in the space A, the identifications imposed by integration are not imposed, making compuˆ tations simpler; second, we can compute the cohomology on the space A[λ] ˆ of polynomials in λ, rather than the bihamiltonian cohomology on A, and this allows us to use directly the methods of spectral sequences associated with filtrations. In this Section we review some basic definitions, mainly from [12], state our main theorem and derive its most important consequences. 2.1. Basic definitions. Consider the supercommutative associative algebra Aˆ defined as Aˆ = C ∞ (U )[[ui,1 , ui,2 , . . . ; θ0 , θ1 , θ2 , . . . ]], i

i

i

ui,s ,

where i = 1, . . . , n, s = 1, 2, . . . are formal even variables and θis , i = 1, . . . , n, s = 0, 1, 2, . . . are odd variables. An element in C ∞ (U ) is represented by a function of the coordinates ui , i = 1, . . . , n on the domain U ⊂ Rn . We define the standard gradation on Aˆ by assigning the degrees deg ui,s = deg θis = s,

s = 1, 2, . . .

and degree zero to both θi = and the elements in C ∞ (U ). The standard degree d homogeneous component of Aˆ is denoted Aˆd . The super gradation is defined by assigning degree one to θis for s > 0 and degree zero to the ˆ The super degree p homogeneous component is remaining generators of A. p ˆ denoted A . We also denote Aˆp = Aˆd ∩ Aˆp . θi0

d

ˆ The standard derivation on A, X ∂ ∂ ∂= ui,s+1 i,s + θis+1 s , ∂u ∂θi s>0

is compatible with the standard and super gradations, in particular it increases the standard degree by one and leaves invariant the super degree. ˆ Thanks to the homogeneity of ∂, the space Fˆ := Aˆ still possesses two gra∂A dations, which we keep denoting with indices p and d. The elements of Fˆ p are called local p-vectors and the projection map is denoted by an integral Z ˆ : Aˆ → F.

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The space Fˆ can be endowed with the Schouten-Nijenhuis bracket [, ] : Fˆ p × Fˆ q → Fˆ p+q−1 , which satisfies the usual graded skew-symmetry and graded Jacobi identities, see [12, 11] for more details. A Poisson bivector P is an element of Fˆ 2 that satisfies [P, P ] = 0. If P is a Poisson bivector, its adjoint action dP = [P, ·] on Fˆ by the graded Jacobi ˆ dP ). Given identity squares to zero, hence defines a differential complex (F, a Poisson bivector P , the super derivation DP on Aˆ is defined by X δP ∂ δP ∂ s s ∂ DP = +∂ , δθi ∂ui,s δui ∂θis s>0

where the variational derivatives on Aˆ w.r.t. ui and θi are defined as follows ∞

∞

X δ ∂ = (−∂)s i,s , i δu ∂u

X δ ∂ = (−∂)s s . δθi ∂θi

s=0

s=0

The super derivation DP squares to zero, and is such that the integral defines a map of differential complexes Z ˆ DP ) → (F, ˆ dP ). : (A, ˆ DP ) As pointed out in [12] this allows us to work with the complex (A, ˆ dP ). rather than with the more complicated (F, A Poisson pencil is given by two Poisson bivectors P1 , P2 which are compatible, i.e. [P1 , P2 ] = 0. For each λ, then, Pλ := P2 − λP1 is also a Poisson bivector. We denote by d1 and d2 the differentials on Fˆ corresponding to P1 and P2 , respectively. Due to compatibility, dλ := d2 − λd1 squares to zero. We denote by D1 and D2 the super derivations on Aˆ associated to P1 and P2 , respectively. Their compatibility in this case implies that Dλ := D2 − λD1 also squares to zero. In summary we can define two differential complexes associated to a Poisson pencil ˆ (A[λ], Dλ ),

ˆ (F[λ], dλ ).

Remark 3. The Poisson brackets {, }1,2 introduced in (2) are elements of the space Λ2loc of local bivectors written in δ-formalism, see [7]. There is a one to one correspondence between the space of local p-vectors Λploc , written in δ-formalism and the space Fˆ p . We will not recall it here in general but rather refer the reader to [12]. For the case of a bivector, written as X {ui (x), uj (y)} = Bsij δ (s) (x − y), s>0

with Bsji ∈ A, the corresponding element in Fˆ 2 is Z X 1 θi Bsij θjs . P = 2 s>0


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ˆ 2.2. The complex (A[λ], Dλ ). Let us fix a semisimple Poisson pencil of 0 hydrodynamic type {, }λ = {, }02 − λ{, }01 , and denote by u1 , . . . , un the associated canonical coordinates on U ⊂ Rn , see §1.2. The compatible Poisson brackets {, }01,2 are represented in Fˆ12 by two bivectors Z 1 ij 0 1 ij k,1 Pa = ga θi θj + Γk,a u θi θj , a = 1, 2. 2 In canonical coordinates the Christoffel symbol of g1ij = f i δij is fi fj 1 i j i ∂ f δ + ∂ f δ − ∂ f δ Γij = ij i j k jk ik k,1 2 fj fi and P1 is given by 1 P1 = 2

Z f

i

θi θi1

fi j j,1 + j ∂i f u θi θj . f

We denote by D1 = D(f 1 , . . . , f n ) the super derivation on Aˆ corresponding to P1 . A straightforward computation gives us the following formula X ∂ D(f 1 , . . . , f n ) = ∂ s f i θi1 (3) ∂ui,s s>0 i j 1X s ∂ i j,1 0 i ∂i f j,1 0 j ∂j f i,1 0 + ∂ ∂ j f u θi + f u θj − f u θj j i 2 f f ∂ui,s s>0 i i 1X s ∂ j 0 1 j ∂j f 0 1 j ∂j f 0 1 + ∂ ∂ i f θj θj + f θi θj − f θj θi i i 2 f f ∂θis s>0 l j ∂i f l ∂i f l l,1 0 0 1X s l ∂i f ∂l f ∂ fj l θ − f uj,1 θl0 θj0 u θ + l j 2 f fl fl fj s>0

∂l f i ∂i f j j,1 0 0 f l f j ∂l f i ∂j f i i,1 0 0 u θl θj − i u θl θj fi fj f fi fi l i ∂l f i ∂l f j j,1 0 0 j ∂l f ∂j f l,1 0 0 u θ θ − f u θj θi + fl i j i f fj fi fl l i j i ∂ l ∂l f ∂l f j,1 0 0 l ∂l f ∂j f j,1 0 0 +f u θl θi + f u θl θi . i i j l f f f ∂θis f

+ fl

The super derivation corresponding to P2 is then given by D2 := D(u1 f 1 , . . . , un f n ). ˆ Our aim is to compute the cohomology of the complex (A[λ], Dλ ) with Dλ = D2 − λD1 , and D1 , D2 given above. 2.3. The main theorem. In Section 3 we prove the following vanishing ˆ theorem for the cohomology of the complex (A[λ], Dλ ). ˆ Theorem 4. The cohomology Hdp (A[λ], Dλ ) vanishes for all bi-degrees (p, d), unless d = 0, . . . , n, p = d, . . . , d + n, or d = n + 1, n + 2, p = d, . . . , d + n − 1.

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Remark 5. For n = 1 the bi-degrees for which we cannot state vanishing according to Theorem 4 are (p, d) = (0, 0), (1, 0), (1, 1), (2, 1), (2, 2), (3, 3). In [5] (and in [4] for the KdV case) we compute by a different method the ˆ Dλ ) for all bi-degrees, proving in particular that it cohomology Hdp (A[λ], vanishes also for (p, d) = (1, 0), (1, 1), (2, 2). This shows that the vanishing theorem above can be strengthened. Our result is enough, however, to prove the absence of obstructions to deformations of bihamiltonian structures, which is our main aim. Remark 6. In the proof of Theorem 4 we will distinguish two sets of indexes for which we do not prove vanishing: d = 0, . . . , n,

p = d, . . . , d + n

(Case 1),

and d = 2, . . . , n + 2, p = d, . . . , d + n − 1 (Case 2), which clearly overlap for n > 2. We distinguish the two cases since they have different sources in the complex. 2.4. Vanishing of bihamiltonian cohomology. By the following lemma of Liu and Zhang [12], the bihamiltonian cohomology of Fˆ is isomorphic ˆ to the cohomology of the complex1 (F[λ], dλ ) in almost all degrees (p, d). ˆ D1 , D2 ) or (F, ˆ d1 , d2 ). The Let (C, ∂1 , ∂2 ) be either the double complex (A, bihamiltonian cohomology of the double complex (C, ∂1 , ∂2 ) is defined by Ker ∂1 ∩ Ker ∂2 BH(C, ∂1 , ∂2 ) = . Im ∂1 ∂2 Lemma 7. The natural embedding of C in C[λ] induces an isomorphism ∼ H p (C[λ], ∂λ ) BH p (C, ∂1 , ∂2 ) = d

d

for p > 0, d > 2. As pointed out in [12], the short exact sequence of complexes ˆ ˆ ˆ 0 → (A[λ]/R[λ], Dλ ) → (A[λ], Dλ ) → (F[λ], dλ ) → 0 implies a long exact sequence in cohomology, which includes ˆ ˆ ˆ · · · → H p (A[λ]) → H p (F[λ]) → H p+1 (A[λ]) → ··· d

d

d p ˆ Hd (A[λ], Dλ ) and

ˆ for p, d > 0. It is clear that if both Hdp+1 (A[λ], Dλ ) vanish, p ˆ p ˆ then Hd (F[λ], dλ ) vanishes too. Our vanishing result for Hd (A[λ]) translates ˆ to the following statement for the cohomology of the complex (F[λ], dλ ). ˆ Corollary 8. The cohomology Hdp (F[λ], dλ ) vanishes for all bi-degrees (p, d), unless d = 0, . . . , n, p = d − 1, . . . , d + n, or d = n + 1, n + 2, p = d − 1, . . . , d + n − 1. 1A similar description in terms of a bicomplex was given in [3].


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Using the isomorphism of Lemma 7 we obtain the vanishing of the biˆ hamiltonian cohomology of F. ˆ d1 , d2 ) vanishes for Corollary 9. The bihamiltonian cohomology BHdp (F, all bi-degrees (p, d) with d > 2, unless d = 2, . . . , n,

p = d − 1, . . . , d + n,

or d = n + 1, n + 2,

p = d − 1, . . . , d + n − 1.

Remark 10. Notice that in particular it follows that 2 ˆ d1 , d2 ) = 0, BH>4 (F,

3 ˆ d1 , d2 ) = 0. BH>5 (F,

The vanishing of the second cohomology for d > 4 has been already proved in [9, 6], together with the results ˆ d1 , d2 ) = 0, BH22 (F,

ˆ d1 , d2 ) ∼ BH32 (F, =

n M

C ∞ (R).

i=1

The vanishing of the third cohomology for d > 5 is new, and is the most relevant for the extension problem of deformation theory. 2.5. Bihamiltonian cohomology and deformations. The deformation problem can be formulated in Fˆ as follows. Let Pλ0 = P20 − λP10 ∈ Fˆ12 [λ] be a semisimple Poisson pencil of hydrodynamic type. A deformation of Pλ0 is given by Pλ = P2 − λP1 ∈ Fˆ 2 [λ] with Pλ − P 0 ∈ Fˆ 2 and [Pλ , Pλ ] = 0. An >1

infinitesimal deformation of Pλ0 is given by Pλ62 = Pλ0 + Pλ1 + Pλ2 ,

λ

>2

Pλd−1 = P2d−1 − λP1d−1 ∈ Fˆd2 [λ]

such that 3 [Pλ62 , Pλ62 ] ∈ Fˆ>5 [λ].

A deformation Pλ of Pλ0 extends a infinitesimal deformation Pλ62 if Pλ − 3 [λ]. Pλ62 ∈ Fˆ>4 The extension problem can be stated as follows: given an infinitesimal deformation Pλ62 of Pλ0 , there exists a deformation Pλ extending Pλ62 ? As pointed out in [12] the fact that the second bihamiltonian cohomology ˆ d1 , d2 ) vanish for d > 4 and d = 2 but not for d = 3, imgroups BHd2 (F, 3 (F, ˆ d1 , d2 ) guarantees that any infinitesimal plies that the vanishing of BH>6 62 deformation Pλ can be extended to a full deformation Pλ . This result, expressed in the δ-formulation of local bivectors, is Theorem 2. 3 (F) ˆ implies that the deformations are To see that the vanishing of BH>6 unobstructed, consider that any infinitesimal deformation can be put in the form P162 = P10 , P262 = P20 + P22 by an appropriate Miura transformation. Clearly P22 is in the kernel of ˆ Let us look for a both d1 and d2 , hence identifies an element of BH32 (F). deformation of the form P1 = P10 ,

P2 = P20 + P22 + P24 + P26 + . . .

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Let us first show that exist a term P24 such that 1 d1 P24 = 0, d2 P24 + [P22 , P22 ] = 0. (5) 2 ˆ Clearly [P22 , P22 ] ∈ Fˆ63 is in Ker d1 ∩ Ker d2 , hence the vanishing of BH63 (F) implies that 1 d1 d2 Q1 = [P22 , P22 ] 2 1 4 ˆ for some Q1 ∈ F4 . Then P2 = d1 Q1 gives a solution of (5). At the next step of the deformation we want to find P26 such that d1 P26 = 0, [P22 , P24 ]

d2 P26 + [P22 , P24 ] = 0.

(6)

Fˆ83

ˆ is in Ker d1 ∩Ker d2 , hence the vanishing of BH83 (F) 1 2 4 ˆ an element Q2 ∈ F6 s.t. d1 d2 Q2 = [P2 , P2 ]. Then

As before ∈ implies that there is setting P26 = d1 Q2 gives us the required solution of (6). 3 (F) ˆ ensures that this procedure can be conSince the vanishing of BH>6 tinued indefinitely, as proved by induction in [12], the existence of a full deformation extending the infinitesimal deformation Pλ62 indeed follows. Notice that this in particular implies that any deformation can be put in the form (4), i.e., with the first Poisson tensor undeformed and the second one having only odd standard degree terms. This fact was also proved independently of the vanishing of the third bihamiltonian cohomology in [6]. 3. Filtrations and spectral sequences In this Section we give a proof of Theorem 4. It is a rather technical argument: basically, we introduce a sequence of filtrations and study the associated spectral sequences in order to show the vanishing of the cohomology in some degrees. 3.1. Strategy of the proof of Theorem 4. Before we start with the technical details, let us make some general remarks about the strategy of the ˆ proof. The theorem states that the cohomology of the complex (A[λ], Dλ ) vanishes in certain degrees (p, d). We prove this vanishing by introducing several spectral sequences associated to certain filtrations. Central to this derivation is the following simple principle: suppose we have a cochain complex (C, d) with a bounded decreasing filtration . . . ⊂ F p+1 C ⊂ F p C ⊂ . . . The associated spectral sequence is bounded and converges to E1p,q = H p+q (F p C/F p+1 C, d0 ) =⇒ H p+q (C, d), where d0 is the induced differential on the zeroth page, i.e. the associated graded complex. Suppose now that H p (Ek , dk ) = 0 for some k > 0. Since all higher pages are iterated subquotients of Ek , we see that in this case H p (C, d) = 0. In the proof we will apply this principle inductively: we introduce a filtraˆ tion on the complex (A[λ], Dλ ) which induces a spectral sequence. To show the vanishing of the cohomology of the first page E1 , we introduce another filtration on this page, which induces a spectral sequence converging to the


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second page E2 of the previous spectral sequence. In this way, we apply in total a sequence of three filtrations. For the convenience of the reader, we list them below: 1. The first filtration is associated with the degree of monomials in ˆ in the variables ui,s , s > 1, i.e. we assign degree 1 to each ui,s , A[λ] s > 1. The differential on the zeroth page of the associated spectral sequence is the part of Dλ that preserves this degree, whereas on the first page it is the part that decreases it by 1. 2. On the first page of the spectral sequence above, we consider the filtration given by the degree in θi1 , for all i = 1, . . . , n, i.e., this time deg(θi1 ) = 1, i = 1, . . . , n. On the zeroth page, the differential is given by the part of the differential in 1 (at the first page) that increases the number of θi1 by 1. 3. The complex on the zeroth page in 2 splits as direct sum of complexes Cî , i = 1, . . . , n. To compute the cohomology of the subcomplex Cî , we filter by the degree of monomials in θi1 , where this time i is fixed. On the zeroth page of the spectral sequence we finally find a complex of which we can prove the vanishing of the cohomology in the relevant degrees. Having obtained the vanishing of the cohomology in 3, we apply the principle stated above to argue that the first page of the spectral sequence in 2 vanishes in certain degree. The same argument gives the vanishing of the second page of the spectral sequence in 1, which in turn proves the vanishing of ˆ the cohomology of the original complex (A[λ], Dλ ) in certain degrees, i.e. Theorem 4. 3.2. Grading and subcomplexes. Since θis are odd variables, we have a restriction on possible gradings p and d on the space Aˆ . Indeed, the minimal possible standard degree d of a monomial in θis of super degree p = nq + r, r < n is the degree of θ10 · · · θn0 · · · θ1q−1 · · · θnq−1 θiq1 · · · θiqr equal to n(0 + · · · + (q − 1)) + rq = nq(q − 1)/2 + rq. So, the finitely generated C ∞ (U )-module Aˆpd is zero for d < nq(q − 1)/2 + rq.

(7)

Moreover, the operators D1 and D2 (and, correspondingly, d1 and d2 ) are of bi-degree (p, d) equal to (1, 1). Therefore, the difference p − d is preserved by both operators, and this means that the space Aˆ can be seen as the completion of an infinite direct sum of subcomplexes indexed by difference d − p = −n, −n + 1, . . . . The inequality (7) implies that each of these subcomplexes is finite, and consequently we can compute the cohomology of each of them separately. For this general reason all the filtrations and spectral sequences introduced below will be bounded, and consequently the spectral sequences will converge in a finite number steps. 3.3. The first filtration. We now introduce the main filtration of the space ˆ in our computation. For this we consider the the degree of monomials A[λ] ˆ in the variables ui,t , i = 1, . . . , n, t > 1. More formally, we introduce in A[λ]

12


ˆ by declaring a new grading on A[λ]

g i,t ) = 1, deg(u

i = 1, . . . , n, t > 1,

g t ) = 1, deg(θ i

i = 1, . . . , n, t > 0.

On the θ-variables, this is just the ordinary super degree introduced in section 2.1. With this notation we can define

g ) > r}. ˆ := {f ∈ A[λ], ˆ F r A[λ] deg(f

Clearly, this defines a decreasing filtration

ˆ ⊂ F 1 A[λ] ˆ ⊂ F 0 A[λ] ˆ = A[λ]. ˆ · · · ⊂ F 2 A[λ] g ) = r, and write q for ˆ of degree deg(f Consider now a monomial f ∈ A[λ] i,t the degree in u , i = 1, . . . , n, t > 1 and p for the degree in θit , i = 1, . . . , n, t > 0, so that p + q = r. We see from equation (3) that the differential Dλ increases p by 1 and changes the degree q to some degree q˜ > q − 1. Therefore, the sum p + q is either preserved or increased by the differential, ˆ in other words, (F • A[λ], Dλ ) is a filtered complex. This filtration, when restricted to each subcomplex with fixed difference d − p is bounded. We now split the differential as

Dλ = ∆−1 + ∆0 + . . . ,

where where ∆−1 decreases q by 1, ∆0 preserves q, and ∆t , t > 1, increases q by t. Explicitly, we see from equation (3), that there is only one term that lowers degree q, so that we have:

∆0 =

X s>1

(−λ + ui )f i θi1+s

∂ . ∂ui,s

(8)


13

The terms in (3) that preserve the degree q are: ∂ ∂ui i s ∂f j,a 1+b ∂ i u θi + (−λ + u ) ∂ui,s b ∂uj

∆0 = (−λ + ui )f i θi1 X

+

s=a+b s,a>1;b>0

1 + 2 1 + 2 1 − 2

X

X s=a+b s,a>1;b>0

1 s ∂ (−λ + u ) ∂j f i uj,1+a θib i,s + ∂u 2 b

X

X

i

s=a+b s>1;a,b>0

s=a+b s>1;a,b>0

s i ∂i f j j,1+a b ∂ 1 (−λ + u ) f u θj i,s + j f ∂u 2 b s j ∂j f i i,1+a b ∂ 1 (−λ + u ) f u θj i,s − i b f ∂u 2

s=a+b s>1;a,b>0

X

j

s=a+b s>1;a,b>0

s i i,1+a b ∂ f u θi i,s b ∂u

X

i

s=a+b s>1;a,b>0

X

s i i,a 1+b ∂ f u θi b ∂ui,s

s=a+b s>1;a,b>0

s i i,1+a b ∂ f u θi i,s b ∂u s i i,1+a b ∂ f u θi i,s b ∂u

1 X s 1 X s i a 1+b ∂ j j a 1+b ∂ + (−λ + u ) ∂ i f θj θj + f θ i θi 2 b ∂θis 2 b ∂θis s=a+b s,a,b>0

s=a+b s,a,b>0

1 X s j ∂j f i a 1+b ∂ 1 X s i a 1+b ∂ j + (−λ + u ) f θ θ + f θi θi 2 b f i i j ∂θis 2 b ∂θis s=a+b s,a,b>0

s=a+b s,a,b>0

1 X s j ∂j f i a 1+b ∂ 1 X s i a 1+b ∂ j − (−λ + u ) f θ θ − f θi θi 2 b f i j i ∂θis 2 b ∂θis s=a+b s,a,b>0

s=a+b s,a,b>0

We now start the computation on the zeroth page of the spectral sequence ˆ comes from a grading, induced by this filtration. Since the filtration of A[λ] we obviously have M p,q M ˆ E0 := F p Aˆp+q [λ]/F p+1 Aˆp+q [λ] ∼ = A[λ], p,q

p,q

but the induced differential is given by ∆−1 . Therefore, we first compute the cohomology of ∆−1 . Introduce the space Cˆ := C ∞ (U )[[θ10 , . . . , θn0 , θ11 , . . . , θn1 ]], together with i,s s+1 ˆ Cî := C[[{u , θi , s > 1}]].

On Cî we have the de Rham differential X ∂ dî = θis+1 i,s . ∂u s>1

With this, we can now state the result of the computation of the cohomology on the zeroth page of the spectral sequence:

14


Proposition 11. The cohomology of ∆−1 is given by: n M ˆ ˆ ⊕ H(A[λ], ∆−1 ) ∼ Im dî : Cî → Cî = C[λ] i=1

Before we start the proof of this proposition, as a preliminary step, observe that the cohomology of the de Rham complex (Cî , dî ) is trivial in positive degree, i.e., Lemma 12 (“Poincaré lemma”). ( C H k (Cî , dî ) = 0

k = 0, k > 0.

Proof. A simple proof of this fact can be given in term of an homotopy map, a procedure that we will use repeatedly in the following. For fixed i = 1, . . . , n and s > 1, let Z ∂ hi,s = s+1 dui,s , ∂θi where the integration constant is set to zero. We have hi,s dî + dî hi,s = 1 − πui,s π s+1 θi

where πp denotes the projection that sets the variable p to zero. Given a cocycle g ∈ Ci representing a cohomology class [g], the previous formula implies that the same cohomology class can be represented by the cycle πui,s πθs+1 g. Repeating this process, we can kill all variables ui,s , θis+1 with i s > 1, hence a cohomology class can be always represented by an element in C. Since Im dî always contains θis+1 with s > 1, no further simplification is possible. Proof of Proposition 11. To prove Proposition 11 we introduce two homotopy maps. Let σi be the map that acts on a rational function of λ by removing the polar part at λ = ui . For a polynomial p(λ) we have p(λ) − p(ui ) p(λ) = . σi i λ−u λ − ui Fix i = 1, . . . , n and s > 1 and let Z 1 ∂ 1 dui,s . hi,s = σi i u − λ f i ∂θis+1 ˆ which satisfies Clearly hi,s defines a map on A[λ] hi,s d0 + d0 hi,s =(1 − πui,s πθs+1 )(1 − πλ−ui )+ i   Z X f j ∂  t+1 i,s ∂   + θ du π i. j f i ∂θis+1 ∂uj,t  λ−u t>1

(9)

j

As above πp denotes the projection that sets the variable p to zero, in particular πλ−ui sets λ to ui .


15

It follows from (9) that we can kill the dependence on all the variables θis+1 with i = 1, . . . , n, s > 1, in the λ-dependent part of any cocyle. In other words a cohomology class [g] can always be represented by a cocycle ˆ and we require each monomial of the form g + g˜, where g ∈ C[λ] and g˜ ∈ A, in g˜ to have a nontrivial dependence on the variables θis+1 for s > 1. Notice that g1 is a cocycle by itself, and can no longer be simplified by quotienting by Im d0 . Let us then consider the cocycle g˜. Since g˜ does not depend on λ, it is in the kernel of d0 iff d0 g˜ = 0 and d00 g˜ = 0 where d0 = d00 − λd0 . Moreover, an element of the form d00 d0 f for f ∈ Aˆ is in the image of d0 and does not depend on λ, since d0 d0 f = d00 d0 f . Let us define, for s, t > 1 and i 6= j Z Z ∂ 1 ∂ 1 i,s duj,t . du hi,s;j,t = i u − uj f i f j ∂θis+1 ∂θjt+1 ui,s ,

We have [hi,s;j,t , d00 d0 ] = (1 − πui,s πθs+1 )(1 − πuj,t πθt+1 ) + (...)d0 + (...)d00 i

j

We did not specify the last two terms since the vanish when applied on elements in Ker d0 ∩ Ker d00 . It follows that, up to elements in Im d00 d0 , a cocycle g˜ is equivalent to the cocycle (πui,s πθs+1 + πuj,t πθt+1 − πui,s πθs+1 πuj,t πθt+1 )˜ g. i

j

i

j

This amounts to removing all monomials that contain any quadratic term of the form θis+1 θjt+1 for i 6= j, s, t > 1. Hence g˜ can be written as sum of gi ∈ Ci for i = 1, . . . , n, where gi ∈ Ker dî and each monomial in gi has a nontrivial dependence on the variables θis+1 for s > 1. Because of the vanishing of the cohomology H(Ci , dî ) we have that gi ∈ Im dî . To complete the proof we have to show that we cannot further quotient by elements in Im d0 without spoiling the form of g˜. In principle the cocycle gi ∈ imdî could be quotiented by an element of the form d0 f (λ) ∈ Ci for ˆ f (λ) ∈ A[λ]. It is easy to see that f ∈ Ci [λ], otherwise the image d0 f would also depend on some variables uj,s , θjs+1 for j 6= i. We have then X ∂f d0 f = (ui − λ)f i θis+1 i,s . ∂u s>1

But such element can be independent of λ iff it vanishes. Proposition 11 is proved. The above proof of Proposition 11 gives us more than just the computation of the cohomology of ∆−1 , namely it gives us –in principle– the full homotopy retract data: h

ˆ (A[λ], ∆−1 ) o ;

where ˆ ⊕ Bˆ := C[λ]

n M i=1

p

/

ˆ 0) (B,

i

Im dî : Cî → Cî ,

(10)

16


and i and p are cochain maps, of which i is a quasi-isomorphism, and h is a degree one map satisfying p◦i=1 i ◦ p = ∆−1 h + h∆−1 . ˆ In our case, i is just the obvious inclusion i : Bˆ ,→ A[λ], and p is is a projection operator. The explicit description of the projection operator p and the homotopy h is more complicated than i, but fortunately, we don’t need that in general when considering the differential on the first page of the spectral sequence: this differential is simply given by ∆00 = p∆0 i. In general, we can decompose ˆ = C[λ] ˆ ⊕ A[λ]

n M

ˆ Cî [λ] ⊕ C[[{mixed terms in ui,s , θis+1 , s > 1}]].

(11)

i=1

Then the projection is simply the identity on the first component and zero on the last component. On the second component it is more involved to describe as it involves the homotopy h, but on the subspace Im dî : Cî [λ] → Cî [λ] ⊂ Cî [λ], the projection is equal to evaluating a polynomial in λ at λ = ui .

3.4. The second filtration. In the previous section we have seen that on the first page of the spectral sequence, we are left with the problem of ˆ computing the cohomology of the operator ∆00 := p∆0 i on the space B. 0 However, instead of considering the full operator ∆0 , we split this operator into two summands: ∆00 = ∆000 + ∆001 , where ∆001 increases the number of factors of θi1 by 1 and ∆000 leaves it invariant. This split can be achieved by an auxilliary filtration ˆ . . . ⊂ F 2 Bˆ ⊂ F 1 Bˆ ⊂ F 0 Bˆ = B, given by ˆ with #θi1 > k}. F k Bˆ := {f ∈ B, ˆ ∆0 ) into a filtered complex and on the zeroth page of Again, this turns (B, 0 the induced spectral sequence we have to compute the cohomology of ∆001 first.


17

Lemma 13. The differential ∆001 is given by ˜ 001 = (−λ + ui )f i θi1 ∂ ∆ ∂ui Xs+2 ∂ + f i ui,s θi1 i,s 2 ∂u s>1

−

∂j f i ∂ 1X (−λ + uj )sf j i ui,s θj1 i,s 2 f ∂u s>1

1 1X i ∂ ∂ − (−λ + uj )∂i f j θj1 θj0 0 + f (s − 1)θi1 θis s 2 2 ∂θi ∂θi s>0

−

1X 2

(−λ + uj )f j

s>0

fi

∂j ∂ (s + 1)θj1 θis s i f ∂θi

∂j f i ∂ 1 + (−λ + uj )f j i θi1 θj0 0 2 f ∂θi Proof. As explained above, we have ∆00 := p∆0 i, with ∆0 as in equation (8), and the inclusion i and projection p as described above. We first collect from (8) all the terms ∆01 that increase the number of θi1 ’s: ∂ ∂ui X X ∂ ∂f i ∂ + (−λ + ui ) j uj,s θi1 i,s + f i ui,s θi1 i,s ∂u ∂u ∂u

∆01 = (−λ + ui )f i θi1

s>1

s>1

1X ∂ 1 X i i,s 1 ∂ + (−λ + ui )s∂j f i uj,s θi1 i,s + sf u θi i,s 2 ∂u 2 ∂u s>1

+ − + + −

1X 2

∂j f i i,s 1 ∂ u θj i,s fi ∂u

(−λ + uj )∂i f j (θjs θj1 + sθj1 θjs )

s>0

∂ 1X i s 1 ∂ + f (θi θi + sθi1 θis ) s s ∂θi 2 ∂θi s>0

fi

(−λ + uj )f j

∂j ∂ (θs θ1 + sθi1 θjs ) s fi i j ∂θi

(−λ + uj )f j

∂j f i s 1 ∂ (θ θ + sθj1 θis ) s fi j i ∂θi

s>0

1X 2

(−λ + uj )sf j

s>0

1X 2

∂i ∂ uj,s θj1 i,s j f ∂u

s>1

1X 2

(−λ + ui )sf i

s>1

1X 2

s>1

fj

In terms of this operator, we have by definition that ∆001 = p∆01 i. Next, observe that by the property of being a multicomplex, we have that [∆−1 , ∆0 ] = 0. Decomposing ∆0 = ∆00 + ∆01 using the θ1 -filtration, we see that [∆−1 , ∆00 ] = 0,

[∆−1 , ∆01 ] = 0,

separately, because ∆−1 has degree 0 with respect to this filtration. The second equation shows that since i maps to the kernel of ∆−1 , the projection

18


p acting on Cî [λ] is always given by setting λ = ui in the computation of ∆00 := p∆0 i. Let us now determine which of the terms in the formula for ∆01 above are mapped to zero when composing with p. For example, the second term, ˆ when evaluated on the image (10) of i, gives either 0 on C[λ], or maps Cî to the third summand in (11). Therefore this term gives no contribution to ∆001 . The same reasoning shows that the fourth and sixth term do not contribute. The eight term vanishes identically for s = 1 and for s > 2 either maps Cî to the third summand in (11), in which case applying p gives zero, or maps Cî to Cˆj so that setting λ = uj yields zero. Therefore only the s = 0 term contributes. For the last two summands, observe that the operator (−λ + uj )θjs , s > 2 acting on Cˆj gives zero when composing with p. Collecting all the remaining terms, we obtain the expression stated in the Lemma.

3.5. The third filtration. We have arrived at the problem of computing the cohomology of the operator ∆001 given in Lemma 13 on the space Bˆ defined in (10). In this section we introduce our final filtration to estimate the possible non-vanishing (p, d)-degrees in the cohomology of ∆001 . We start with the observation that ∆001 preserves the splitting (10) of the cohomology Bˆ of ∆−1 in n + 1 summands, namely: ˆ ˆ ∆001 C[λ] ⊂ C[λ]   Im dî : Cî → Cî [λ] Im dî : Cî → Cî [λ] ⊂ ∆001  . (ui − λ) (ui − λ) This means that we can estimate its cohomology on each of these n + 1 spaces separately. ˆ are those of monomials The possible (p, d)-degrees of the elements of C[λ] 0 0 1 1 θi1 · · · θik θj1 · · · θj` , 1 6 i1 < · · · < ik 6 n, 1 6 j1 < · · · < j` 6 n. So, for the standard gradation d we have 0 6 d 6 n, and, if we fix d, then for the super gradation p we have d 6 p 6 d + n. So, without any further computation we see that these are the restrictions for the possible (p, d)-degrees of the cohomology of ∆001 . This gives us Case 1 in the statement of Theorem 4. 0 Now we want to estimate the cohomology of ∆01 in each of the spaces Im dî : Cî → Cî [λ]/(ui − λ), i = 1, . . . , n. For the rest of this Section the index i is fixed and refers to the particular we consider. space that We represent the operator ∆001 on Im dî : Cî → Cî in the following way:

∆001

=

n X k=1

θk1 Dk ,


19

where ∂ ∂uk n X ∂ ∂k f j − (uk − ui )f k j θj1 1 f ∂θj

Dk := (uk − ui )f k

j=1

+

Xs+2 s>1

2

f k uk,s

∂ ∂uk,s

n

−

1 XX k ∂k f j ∂ (u − ui )sf k j uj,s j,s 2 f ∂u s>1 j=1

∂ 1X k f (s − 1)θks s + 2 ∂θk s>2

n

−

1 XX k ∂ ∂k f j (u − ui )f k j (s + 1)θjs s 2 f ∂θj s>2 j=1

−

n 1X

2

(uk − ui )∂j f k θk0

j=1

∂ ∂θj0

∂ 1 − f k θk0 0 2 ∂θk n X 1 ∂ ∂k f j − (uk − ui )f k j θj0 0 2 f ∂θj +

1 2

j=1 n X

(uj − ui )f j

j=1

∂j f k 0 ∂ . θ f k j ∂θk0

In particular, Di := +

Xs+2 s>1 n X

1 2

2

f i ui,s

Xs−1 ∂ ∂ 1 ∂ + f i θis s − f i θi0 0 i,s ∂u 2 ∂θi 2 ∂θi

(12)

s>2

(uj − ui )f j

j=1

∂j f k 0 ∂ θ f k j ∂θi0

As a first step toward computation of the cohomology of ∆001 on the space Im dî : Cî → Cî [λ]/(ui − λ), we introduce the filtration with respect to the degree of θi1 , in the same way as we used the general θ1 -filtration before. On the zero page of the associated spectral sequence we have to compute 1 ˆ the cohomology of θi Di on Im di : Cî → Cî . The restrictions on the pos1 sible non-trivial(p, d)-degrees that we get in the cohomology of θi Di on Im dî : Cî → Cî are sufficient for the proof of Theorem 4.

Proposition 14. For the cohomology of the differential θi1 Di on the space dî (Cî ) ⊂ Bˆ we have: H p dî (Cî ), θi1 Di = 0, d

20


unless d = 2, 3, . . . , n + 2, p = d, d + 1, . . . , d + n − 1. Proof. In order to compute the cohomology of θi1 Di on Im dî : Cî → Cî , we represent this space as a direct sum of subcomplexes indexed by all possible non-trivial monomials in ui,>1 and θi>2 . Let us denote the set of all nontrivial monomials in these variables by M . Then we have: Lemma 15. The isomorphism M Im dî : Cî → Cî := Cˆ · dî (m) m∈M

is preserved by the differential θi1 Di , and decomposes the cochain complex (Im(dî ), θi1 Di ) into a direct sum of sub-complexes indexed by all monomials in the variables ui,>1 and θi>2 . Proof. First, a short computation shows that [Di , dî ] = f i dî . Next, consider a monomial m ∈ M in the variables ui,>1 and θi>2 . Then it follows from the identity above that θi1 Di (C dî (m)) = (θi1 Di (C)) · dî (m)) + C · θi1 Di (dî (m)) = (θ1 Di (C)) · dî (m)) + C · θ1 dî (Di + f i )(m). i

i

We see from equation (12) that (θi1 Di (C)) ⊂ C and that (Di + f i ) acts on m by multiplication with a scalar (depending on m). This shows that θi1 Di preserves the subspace Cˆ · dî (m) and proves the Lemma. Because of this Lemma, we can consider an infinite direct sum of complexes, each of which is isomorphic, as a bi-graded vector space, to C with shifted (p, d) gradation. The gradation is shifted by (pm + 1, dm + 1), where (pm , dm ) in the gradation of m ∈ M (and, therefore, (pm + 1, dm + 1) is the gradation of dî (m)). Let us discuss the action of θi1 Di . Observe that this operator is linear over the ring of functions in u1 , . . . , un and θj1 , j 6= i. So, we omit the coefficients from this ring in the computations below, assuming that there can be an arbitrary coefficient that would be preserved. We denote by C(m) the eigenvalue of the operator Xs−1 Xs+2 ∂ ∂ ui,s i,s + θis s 2 ∂u 2 ∂θi s>1

s>2

on dî (m). Observe that C(m) is always a positive half-integer, and the minimal value of C(m) is equal to 1/2 for m = ui,1 . Consider monomials θj01 · · · θj0` such that 1 6 j1 < . . . < j` 6 n, and jk 6= i for all k = 1, . . . , `. We have: θi1 Di : θj01 · · · θj0` dî (m) 7→ f i C(m)θi1 θj01 · · · θj0` dî (m) So, since C(m) 6= 0, we see the subspace spanned by the elements θj01 · · · θj0` dî (m) and θi1 θj01 · · · θj0` dî (m) forms an acyclic subcomplex of Cˆ · dî (m).


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Now we consider monomials θi0 θj01 · · · θj0` such that 1 6 j1 < · < j` 6 n, and jk 6= i for all k = 1, . . . , `. Modulo the acyclic subcomplex that we introduced in the previous paragraph, we have: 1 1 0 0 0 ˆ i θi Di : θi θj1 · · · θj` di (m) 7→ f C(m) − θ1 θ0 θ0 · · · θj0` dî (m). 2 i i j1 Note that C(m) − 12 is not equal to zero if m 6= ui,1 . Therefore, if m 6= ui,1 , then the quotient of Cˆ · dî (m) modulo an acyclic subcomplex is an acyclic subcomplex, and so Cˆ · dî (m) is acyclic. The only possible case when we can have non-trivial cohomology is the case of m = ui,1 , that is, the case of the complex Cˆ · θi2 . In this case, after taking the quotient modulo the acyclic subcomplex, the cohomology of θi1 Di is represented by a product of θi0 θi2 by an arbitrary function in u1 , . . . , un , θ10 , . . . , θn0 (θi0 is omitted), and θ10 , . . . , θn0 . This means that we have nontrivial cohomology only for gradation d = 2, 3, . . . , 2 + n, and once we fixed d, the possible values of gradation p are d, d + 1, . . . , d + n − 1. This proves the Proposition. Proof of Theorem 4. Recall from Remark 6 that we have to prove the vanishing of the bihamiltonian cohomology, except for the two cases 1 and 2 specified in that remark. The computations in this subsection show that the cohomology of ∆001 vanishes, unless d = 0, 1, . . . , n; p = d, d + 1, . . . , d + n ˆ (Case 1, coming from the sub-complex C[λ]), or d = 2, 3, . . . , n + 2, p = ˆ Going d, d + 1, . . . , d + n − 1 (Case 2, coming from the sub-complex dî (C)). back via the second to the first spectral sequence, we conclude the vanˆ ishing of the cohomology groups of the complex (A[λ], Dλ ) in the same (p, d)-degrees: this is exactly the statement of Theorem 4. Acknowledgments The authors would like to thank P. Lorenzoni for useful discussions. This work was supported by the Netherlands Organization for Scientific Research. References [1] Arsie, Alessandro; Lorenzoni, Paolo. On bihamiltonian deformations of exact pencils of hydrodynamic type. J. Phys. A 44 (2011), no. 22, 225205, 31 pp. [2] Arsie, Alessandro; Lorenzoni, Paolo. Inherited structures in deformations of Poisson pencils. J. Geom. Phys. 62 (2012), no. 5, 1114–1134. [3] Barakat, Aliaa. On the moduli space of deformations of bihamiltonian hierarchies of hydrodynamic type. Adv. Math. 219 (2008), no. 2, 604–632. [4] Carlet, Guido; Posthuma, Hessel; Shadrin, Sergey. Bihamiltonian cohomology of KdV brackets. arXiv:1406.5595 [5] Carlet, Guido; Posthuma, Hessel; Shadrin, Sergey. The bihamiltonian cohomology of a scalar Poisson pencil. Preprint 2014. [6] Dubrovin, Boris; Liu, Si-Qi; Zhang, Youjin. On Hamiltonian perturbations of hyperbolic systems of conservation laws. I. Quasi-triviality of bi-Hamiltonian perturbations. Comm. Pure Appl. Math. 59 (2006), no. 4, 559–615. [7] Dubrovin, Boris; Zhang, Youjin. Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov - Witten invariants. arXiv:math/0108160. [8] Getzler, Ezra. A Darboux theorem for Hamiltonian operators in the formal calculus of variations. Duke Math. J. 111 (2002), no. 3, 535–560.

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[9] Liu, Si-Qi; Zhang, Youjin. Deformations of semisimple bihamiltonian structures of hydrodynamic type. J. Geom. Phys. 54 (2005), no. 4, 427–453. [10] Liu, Si-Qi; Zhang, Youjin. On quasi-triviality and integrability of a class of scalar evolutionary PDEs. J. Geom. Phys. 57 (2006), no. 1, 101–119. [11] Liu, Si-Qi; Zhang, Youjin. Jacobi structures of evolutionary partial differential equations. Adv. Math. 227 (2011), no. 1, 73–130. [12] Liu, Si-Qi; Zhang, Youjin. Bihamiltonian cohomologies and integrable hierarchies I: A special case. Comm. Math. Phys. 324 (2013), no. 3, 897–935. [13] Lorenzoni, Paolo. Deformations of bihamiltonian structures of hydrodynamic type. J. Geom. Phys. 44 (2002), no. 2-3, 331–375. Korteweg-de Vries Instituut voor Wiskunde, Universiteit van Amsterdam, Postbus 94248, 1090GE Amsterdam, Nederland E-mail address: [email protected], [email protected], [email protected]