(’1)i+s X1 ... Xs’1 [[Xs , Xs+1 ... Xi+1 UE .

202 5. DEFORMATIONS AND RECURSION OPERATORS

p

Since ˜ ∈ Ci (E) and X1 , . . . , Xi+1 ∈ CD(E), we have

p’i+s’1

˜ ∈ Cs’1

Xs+1 ... Xi+1 (E)

and by Lemma 5.12 (ii) the element [[Xs , Xs+1 . . . Xi+1 ˜]]fn belongs

p’i+s’1

to Cs’1 (E) as well. Hence, all the summands in (5.36) vanish.

Let us now de¬ne a ¬ltration in Λ— (E) — D v (E) by setting

p

F l (Λp (E) — D v (E)) = Cp’l+1 (E). (5.37)

Obviously,

F l (Λp (E) — D v (E)) ⊃ F l+1 (Λp (E) — D v (E))

and, by Proposition 5.10,

‚C F l (Λp (E) — D v (E)) ‚ F l (Λp+1 (E) — D v (E)).

Thus, ¬ltration (5.37) de¬nes a spectral sequence for the complex (5.9) which

we call the C-spectral sequence for the equation E.

Remark 5.5. As it was already mentioned before, C-spectral sequences

were introduced by A.M. Vinogradov (see [102]). As A.M. Vinogradov

noted (a private communication), the H-spectral sequence can also be viewed

as a C-spectral sequence constructed with respect to ¬bers of the bundle

π∞ : E ∞ ’ M . It is similar to the classical Leray“Serre sequence.

The term E0 of the C-spectral sequence is of the form

p,q p+q p+q

E0 = Cq+1 (E)/Cq (E), p = 0, 1, . . . , q = 0, 1, . . . , n.

To describe these modules explicitly, note that

p+q

C i Λ(E) § Λp+q’i (E) — Dv (E)

p+q

Cq (E) = h

i=p

while

p+q

p+q

C i Λ(E) § Λp+q’i (E) — Dv (E).

Cq+1 (E) = h

i=p+1

Thus

E0 = C p Λ(E) § Λq (E) — D v (E).

p,q

h

The con¬guration of the term E0 for the C-spectral sequence is given on

Fig. 5.2.

Remark 5.6. The 0-th column of the term E0 coincides with the hori-

zontal de Rham complex for the equation E with coe¬cients in the bundle of

vertical vector ¬elds. Complexes of such a type were introduced by T. Tsu-

jishita in [97].

2. SPECTRAL SEQUENCES AND GRADED EVOLUTIONARY DERIVATIONS 203

Λn — D v Λn § C 1 Λ — D v Λn § C p Λ — D v

q=n ... ...

h h h

q = n’1 Λn’1 — Dv Λn’1 § C 1 Λ — Dv Λn’1 § C p Λ — Dv

... ...

h h h

... ... ... ... ... ...

Λq — D v Λq § C 1 Λ — D v Λq § C p Λ — D v

... ... ...

h h h

... ... ... ... ... ...

Λ1 — D v Λ1 § C 1 Λ — D v Λ1 § C p Λ — D v

q=1 ... ...

h h h

Dv C 1 Λ — Dv C p Λ — Dv

q=0 ... ...

p=0 p=1 ... ... ...

Figure 5.2. The C-spectral sequence con¬guration (term E0 ).

Consider, as before, a formally integrable equation E ‚ J k (π) and the

corresponding algebra F(E) ¬ltered by its subalgebras Fi (E).

We say that an element ˜ ∈ Λp (E) — D v (E) is i-vertical if

L˜ |Fi’k’1 (E) = 0 (5.38)

and denote by Vip (E) the set of all such elements. Obviously, Vip (E) ⊃

p p

Vi+1 (E) and V0 (E) = Λp (E) — D v (E).

Proposition 5.13. For any equation E the embedding

‚C (Vip (E)) ‚ Vi’1 (E)

p+1

takes place.

Proof. Obviously, LUE (Fj (E)) ‚ Fj+1 (E) for any j ≥ ’k ’1. Consider

elements ˜ ∈ Vip (E) and φ ∈ Fi’k’2 (E). Then, by de¬nition,

L‚C (˜) (φ) = L[[UE ,˜]]fn (φ) = LUE (L˜ (φ)) ’ (’1)˜ L˜ (LUE (φ)) = 0,

which ¬nishes the proof.

Let us de¬ne a ¬ltration in Λ— (E) — D v (E) by setting

p

F l (Λp (E) — D v (E)) = Vl’p (E). (5.39)

Obviously,

F l (Λp (E) — D v (E)) ‚ F l+1 (Λp (E) — D v (E))

and

‚C F l (Λp (E) — D v (E)) ‚ F l (Λp+1 (E) — D v (E)).

Thus, (5.39) de¬nes a spectral sequence for the complex (5.9) which we call

V-spectral. The term E0 for this spectral sequence is of the form

p,q p+q p+q

E0 = V’q (E)/V1’q (E), p = 0, 1, . . . , q = 0, ’1, . . . , ’p.

204 5. DEFORMATIONS AND RECURSION OPERATORS

p=0 p=1 ... ... ...

FV F V — Λ1 (π) F V — Λp (π)

q=0 ... ...

F V — S 1 D(M ) F V — Λp’1 (π) — S 1 D(M )

q = ’1 ... ...

... ... ... ...

F V — S p D(M )

q = ’p ...

... ...

Figure 5.3. The V-spectral sequence con¬guration for

J ∞ (π) (term E0 ).

— —

Now we shall compute the algebra HC (E) = HC (π) for the “empty equa-

tion” J ∞ (π) using the V-spectral sequence.

p,q

First, we shall represent elements of the modules E0 in a more conve-

nient way. Denote ’q by r and consider the bundle πr,r’1 : J r (π) ’ J r’1 (π)

and the subbundle πr,r’1,V : T v (J r (π)) ’ J r (π) of the tangent bundle

T (J r (π)) ’ J r (π) consisting of πr,r’1 -vertical vectors. Then we have the

induced bundle:

—

π∞,r (T v (J r (π)) ’ T v (J r (π))

—

π∞,r (πr,r’1,V ) πr,r’1,V

“ “

J ∞ (π) ’ J r (π)

and obviously,

p,’r —

= Λp’r (π) —F (π) “(π∞,r (πr,r’1,V )).

E0

—

On the other hand, the bundle π∞,r (πr,r’1,V ) can be described in the fol-

lowing way. Consider the tangent bundle „ : T (M ) ’ M , its rth symmetric

power S r („ ) : S r T (M ) ’ M and the bundle

πV — π — (S r („ )) : T v (J 0 (π)) — π — (S r T (M )) ’ J 0 (π),

where πV : T v (J 0 (π)) ’ J 0 (π) is the bundle of π-vertical vectors. Then, at

least locally,

πr,r’1,V ≈ πp,0 (πV — π — (S r („ ))).

—

It means that locally we have an isomorphism

p,’r

≈ F(π, πV ) —F (π) Λp’r (π) —C ∞ (M ) S r (D(M )).

µ : E0

Thus the term E0 of the V-spectral sequence is of the form which is presented

def

on Fig. 5.3, where F V = F(π, πV ).

2. SPECTRAL SEQUENCES AND GRADED EVOLUTIONARY DERIVATIONS 205

Let (x1 , . . . , xn ) be local coordinates in M , pj be the coordinates arising

σ

∞ (π), and ξ = ‚/‚x , . . . , ξ = ‚/‚x be the local basis in

naturally in J 1 1 n n

T (M ) corresponding to (x1 , . . . , xn ). Denote also by v j , j = 1, . . . , m, local

vector ¬elds ‚/‚uj , where uj = pj (0,...,0) are coordinates along the ¬ber of

p,’r

the bundle π. Then any element ˜ ∈ E0 is of the form

m

‚

j

θσ ∈ Λp’r (π),

j

θσ —

˜= ,

‚pj

σ

j=1 |σ|=r

while the identi¬cation µ can be represented as

m

1σ

v j — θσ —

j

„¦ = µ(˜) = ξ,

σ!

j=1 |σ|=r

σ

where σ = (σ1 , . . . , σn ), σ! = σ1 ! · · · · · σn !, ξ σ = ξ1 1 · · · · · ξnn .

σ

Let us now represent „¦ in the form

p’r

ρi § ω i — Q i ,

„¦=

i=0

where ρi ∈ F(π, πV ) — C p’r’i Λ(π), ωi ∈ Λi (π), and Qi = Qi (ξ) are homo-

h

geneous polynomials in ξ1 , . . . , ξn of the power q.

From equality (5.15) it follows that in this representation the di¬erential

p,’r p,’r+1

’ E0

‚0 : E 0 in the following way

p’r n

‚Q

(’1)p’r’i ρi § dxs § ωi —

‚0 („¦) = . (5.40)

‚ξs

s=1

i=0

Thus, the di¬erential ‚0 reduces to δ-Spencer operators (see [93]) from which

p,0

it follows that all its cohomologies are trivial except for the terms E0 . But

as it is easily seen from (5.40) and from the previous constructions,

p,0 p,0 p,1

E1 = E0 /‚0 (E0 ) = F(π, πV ) —F (π) C p Λ(π).

Hence, only the 0-th row survives in the term E1 and it is of the form

0,0

‚1

0 ’ F(π, πV ) ’ ’ F(π, πV ) — C 1 Λ(π) ’ · · ·

’

p,0

‚1

’ F(π, πV ) — C Λ(π) ’ ’ F(π, πV ) — C p+1 Λ(π) ’ · · ·

p

’

Recall now that ‚1 is induced by the di¬erential ‚π and that the latter

increases the degree of horizontality for the elements from Λ— (π) — D v (π)

(Proposition 5.6). Again, we see that ‚1 is trivial. Thus, we have proved

the following

Theorem 5.14. The V-spectral sequence for the “empty” equation

E∞ = J ∞ (π) stabilizes at the term E1 , i.e., E1 = E2 = · · · = E∞ , and

C-cohomologies for this equation are of the form

p

HC (π) ≈ F(π, πV ) —F (π) C p Λ(π).

206 5. DEFORMATIONS AND RECURSION OPERATORS

Remark 5.7. When π is a vector bundle, then F(π, πV ) ≈ F(π, π) and

we have the isomorphism

p

HC (π) ≈ F(π, π) —F (π) C p Λ(π).

This result allows to generalize the notion of evolutionary derivations

and to introduce graded (or super ) evolutionary derivations. Namely, we

choose a canonical coordinate system (x, pj ) in J ∞ (π) and for any element

σ

1 , . . . , ω m ) ∈ F(π, π ) — C p Λ(π), ω j ∈ C p Λ(π), set

ω = (ω V

‚

Dσ (ω j ) — ∈ Λp (π) — D v (π).

= (5.41)

ω

‚pj

σ

j,σ

We call ω a graded evolutionary derivation with the generating form ω ∈

F(π, πV ) — C p Λ(π). Denote the set of such derivations by κp (π).

The following local facts are obvious:

(i)

(F(π)) ‚ C p Λ(π),

L ω

(ii)

p

∈ ker(‚π ),

ω

(iii) the correspondence ω ’ splits the natural projection

ω

p

p

ker(‚π ) ’ HC (π)

and thus

ker(‚π ) = im(‚π ) • κp (π).

p p’1

We shall show now that De¬nition (5.41) is independent of local coor-

dinates. The proposition below, as well as its proof, is quite similar to that

one which has been proved in [60] for “ordinary” evolutionary derivations

(see also Chapter 2).

Proposition 5.15. Any element „¦ ∈ Λ— (π) — D v (π) which satis¬es

the conditions (i) and (ii) above, i.e., for which L„¦ (F(π)) ‚ C p Λ(π) and

‚π („¦) = 0, is uniquely determined by the restriction of L„¦ onto F0 (π) =

C ∞ (J 0 (π)).

Proof. First recall that „¦ is uniquely determined by the derivation

L„¦ ∈ Dgr (Λ— ) (see Proposition 4.20). Further, since L„¦ is a graded deriva-

tion and due to the fact that

L„¦ (dθ) = (’1)„¦ d(L„¦ (θ)) (5.42)

for any θ ∈ Λ— (π) (Proposition 4.20), L„¦ is uniquely determined by its

restriction onto F(π) = Λ0 (π).

Now, from the equality ‚π („¦) = 0 it follows that

0 = [[Uπ , „¦]]fn (φ) = LUπ (L„¦ (φ)) ’ (’1)„¦ L„¦ (LUπ (φ)). (5.43)

2. SPECTRAL SEQUENCES AND GRADED EVOLUTIONARY DERIVATIONS 207

Let „¦ be such that L„¦ |F0 (π) = 0 and suppose that we have proved that

L„¦ |Fr (π) = 0. Then taking φ = pj , |σ| = r, and using equality (5.43), we

σ

obtain

n

pj i dxi = LUπ (L„¦ (pj )) = 0.

„¦

L„¦ dpj ’

(’1) σ σ

σ+1

i=1

In other words,

n n

pj i L„¦ (pj i ) dxi

L„¦ dxi =

σ+1 σ+1

i=1 i=1

= L„¦ (dpj ) = (’1)„¦ d(L„¦ (pj ) = 0.

σ σ

Since L„¦ (pj i ) ∈ C — Λ(π), we conclude that L„¦ (pj i ) = 0, i.e., we have

σ+1 σ+1

L„¦ |Fr+1 (π) = 0.

Remark 5.8. The element Uπ = j,σ dpj ’ i pj i dxi —‚/‚pj itself

σ σ

σ+1

1 , . . . , ω m ),

is an example of an evolutionary derivation: Uπ = ω , ω = (ω

where ω j = duj ’ i pj i dxi .

1

Since

F(π, πV ) — C — Λ(π) = F(π, πV ) — C i Λ(π)

i≥0

—

is identi¬ed with the module HC (π), it carries the structure of a graded Lie

algebra. The corresponding operation in F(π, πV ) — C — Λ(π) is denoted by

{·, ·} and is called the graded Jacobi bracket. Thus, for any elements ω ∈

F(π, πV ) — C p Λ(π) and θ ∈ F(π, πV ) — C q Λ(π) we have {ω, θ} ∈ F(π, πV ) —

C p+q Λ(π) and

{ω, θ} + (’1)pq {θ, ω} = 0,

(’1)(p+r)q {ω, {θ, ρ}} = 0,

where ρ ∈ F(π, πV ) — C r Λ(π) and , as before, denotes the sum of cyclic

permutations.

To express the graded Jacobi bracket in more e¬cient terms we prove

the following

Proposition 5.16. The space κ— (π) = i≥0 κi (π) of super evolutionary

derivations is a graded Lie subalgebra in Λ— (π) — D v (π), i.e., for any two

generating forms ω, θ ∈ F(π, πV ) — C — Λ(π) the bracket [[ ω , θ ]]fn is again

an evolutionary derivation and

fn

[[ ω, θ ]] = {ω,θ} . (5.44)

fn

Proof. First note that it is obvious that [[ ω, θ ]] lies in ker(‚π ).

208 5. DEFORMATIONS AND RECURSION OPERATORS

Consider a vector ¬eld X ∈ CD(π). Then, since X =X = 0,

ω θ

from equality (4.45) on p. 175 it follows that

fn fn fn

= (’1)ω [[X, ’ (’1)(ω+1)θ [[X,

X [[ ω, θ ]] ω ]] θ ]] ω.

θ

Let X = Di , where Di is the total derivative along xi in the chosen coordi-

nate system. Then we have

‚ ‚

[[Di , ω ]]fn = Dσ (ω j ) — Di , j + Dσ+1i (ω j ) — j = 0.

‚pσ ‚pσ

j,σ

Since any X ∈ CD(π) is a linear combination of the ¬elds Di , one has

fn

CD(π) [[ ω, θ ]] = 0,

fn

∈ C — Λ(π) — D v (π). Hence, Proposition 5.15 implies that the

i.e., [[ ω , θ ]]

fn

bracket [[ ω , θ ]] is an evolutionary derivation.

From (5.44) and from Proposition 5.16 it follows that if (ω 1 , . . . , ω m )

and (θ 1 , . . . , θm ) are local representations of ω and θ respectively then

m

j θi

{ω, θ}i = i

) ’ (’1)ω j

ω j (θ θ i (ω ), (5.45)

j=1

where i = 1, . . . , m.

For example, if ω = LUπ (f ) = df ’ i Di (f ) dxi and θ = LUπ (g), where

f, g ∈ “(π), then

‚g i ‚f i

i j j

{ω, θ} = LUπ (Dσ (f )) § LUπ + LUπ (Dσ (g )) § LUπ ,

‚pj ‚pj

σ σ

j,σ

where i = 1, . . . , m. In particular,

i j

{ωσ , ω„ } = 0, (5.46)

j

i

where ωσ , ω„ are the Cartan forms (see (1.27) on p. 18).