Table Of ContentINTEGRATION OF THE LIFTING FORMULAS AND THE CYCLIC
HOMOLOGY OF THE ALGEBRAS OF DIFFERENTIAL OPERATORS
9
9 BORIS SHOIKHET
9
1
n
a
Abstract
J
5 We integrate the Lifting cocycles Ψ ,Ψ ,Ψ ,... ([Sh1], [Sh2]) on the Lie
2n+1 2n+3 2n+5
1
algebra Dif of holomorphic differential operators on an n-dimensional complex vec-
n
] tor space to the cocycles on the Lie algebra of holomorphic differential operators on a
A
holomorphic line bundle λ on an n-dimensional complex manifold M in the sense of
Q
Gelfand–Fuks cohomology [GF] (more precisely, we integrate the cocycles on the sheaves
. of the Lie algebras of finite matrices over the corresponding associative algebras). The
h
t main result is the following explicit form of the Feigin–Tsygan theorem [FT1]:
a
m H• (glfin(Dif );C) = ∧•(Ψ ,Ψ ,Ψ ,...).
Lie ∞ n 2n+1 2n+3 2n+5
[
2 Introduction
v
The cocycles Ψ ,Ψ ,Ψ ,... on the Lie algebra glfin(Dif ) of finite
7 2n+1 2n+3 2n+5 ∞ n
3 matrices over (polynomial, holomorphic, formal) differential operators on Cn
0 (Ψ ∈ Ci (glfin(Dif );C)), called Lifting formulas, were constructed in the author’s
9 i Lie ∞ n
works [Sh1], [Sh2].
0
8 In the present paper we study the various aspects of the notion of integral in Lie alge-
9 bra cohomology, applied to the Lie algebra glfin(Dif ), namely, we integrate the cocycles
/ ∞ n
h Ψ ,Ψ ,Ψ ,... on this Lie algebra on an n-dimensional complex manifold M.
2n+1 2n+3 2n+5
t In this way, we obtain 1-, 3-, 5-, ... cocycles on the sheaf of the Lie algebras of holo-
a
m morphic differential operators in any holomorphic line bundle λ over M (in the sense of
: Section 3).
v
Xi 0.1. It was proved in [FT1] that the cohomology algebra H•(glfin(Dif );C) is the ex-
∞ n
r terior algebra with the generators in degrees 2n+1,2n+3,2n+5,.... This result was
a
proved using the spectral sequence, connecting the Hochschild homology of an associa-
tive algebra and its cyclic homology (and the result of [T]); in particular, any explicit
formulas for the generators do not follow from this computation.
We prove that
H•(glfin(Dif );C) ≃ ∧•(Ψ ,Ψ ,Ψ ,...),
∞ n 2n+1 2n+3 2n+5
where Ψ ,Ψ ,Ψ ,... are Lifting cocycles (Theorem 4.3.6 in the case n = 1
2n+1 2n+3 2n+5
and Theorem 4.4 in the general case.) We prove this theorem using the notion of the
integral in Lie algebra cohomology, which is due to I.M.Gelfand and D.B.Fuks [GF].
First of all, let us recall the classical construction of the Virasoro 2-cocycle on the Lie
algebra Vect(S1) of smooth vector fields on the circle.
Date: September1, 1998.
1
2 BORIS SHOIKHET
We choose any (formal) coordinate system in each point x∈ S1, smoothly depending
on the point x; in this way, we obtain the map ι : Vect(S1) → W , connected with
x 1
each point x ∈ S1 (W is the Lie algebra of formal vector field on the line R1). Let Ψ
1 3
be the 3-cocycle on the Lie algebra W (in fact, dimH0(W ;C) = dimH3(W ;C) = 1,
1 1 1
dimHi(W ;C) = 0 when i6= 0,3).
1
We obtain “in any point x ∈ S1” the cocycle Ψ (x) = ι∗Ψ ∈ C3 (Vect(S1);C). In
3 x 3 Lie
fact, the cohomological class of all the cocycles Ψ (x) is the same, it does not depend
3
on the point x∈ S1 and on the choice of the (formal) coordinate system in the point x.
It follows from this statement, that there exists a 1-form Θ on S1 with the values in
2
C2 (Vect(S1);C) such that
Lie
d Ψ (x) = δ Θ .
DR 3 Lie 2
Then Θ is a cocycle in C2 (Vect(S1);C). Indeed,
S1 2 Lie
R
δ Θ = δ Θ = d Ψ = 0.
Lie 2 Lie 2 DR 3
ZS1 ZS1 ZS1
In fact, the choice of Θ is not unique, but there exists in a sense the canonical choice
2
(see Subsec. 2.2). The cohomological class Θ does not depend on the choice of the
S1 2
coordinate systems.
(cid:2)R (cid:3)
Let M be an n-dimensional complex manifold, λ be a holomorphic line bundle on M.
Let Dif be the sheaf of the associative algebras of holomorphic differential operators
λ,M
in λ, and D• be the Dolbeault complex on M:
M
D• = 0 → C∞ →∂ Ω0,1 →∂ Ω0,2 →∂ ... .
M M M M
(cid:26) (cid:27)
We consider Γ (glfin(Dif ⊗ D• )) as a DG Lie algebra (see Remark 3.1). When
M ∞ λ,M OM M
M = Cn, the DG Lie algebra ΓCn(glfi∞n(DifCn ⊗OCn DC•n)) is quasi-isomorphic to Difn[0]
as a DG Lie algebra.
We definethecohomology ofthesheaf of Liealgebras Dif (with thebracket [a,b] =
λ,M
(a∗b−b∗a))ascohomologyoftheDGLiealgebraΓ (Dif ⊗ D• )andcohomologyof
M λ,M OM M
the sheaf glfin(Dif ) as cohomology of the DG Lie algebra Γ (glfin(Dif ⊗ D• )).
∞ λ,M M ∞ λ,M OM M
The reason is that complex of sheaves Dif ⊗ D• is quasi-isomorphic to the sheaf
λ,M OM M
Dif [0] (as sheaves of associative or Lie algebras).
λ,M
For any cocycle Ψ ∈ Ck (glfin(Dif );C) and any singular cycle σ ∈ Csing(M;C) we
Lie ∞ n l
define (in Sec. 2 and Subsec. 3.2) the integral Ψ, which is a cocycle in
σ
Ck−l(Γ (glfin(Dif R )⊗ D• )).
Lie M ∞ λ,M OM M
Thecrucialpointisthatif[ Ψ] 6= 0than[Ψ] 6= 0(here[...]standsforthecohomological
σ
class)(Theorem2.3(1)). Thereforethenotionofintegralgivesusaneffectivetooltoprove
R
the cohomological nontriviality of cocycles in C• (glfin(Dif );C).
Lie ∞ n
Let n = 1, M = CP1. In this simplest case the sheaf of holomorphic differential
operators (in any holomorphic line bundle λ) has not higher cohomology (as a sheaf),
and the DG Lie algebra Γ (glfin(Dif )⊗ D• ) is quasi-isomorphic to the Lie
CP1 ∞ λ,CP1 OCP1 CP1
algebraofglobaldifferential operatorsΓ(Difλ,CP1)[0]. WewilldenotethelastLiealgebra
by Dif . (The situation is the same for generalized flag varieties, in particular, for
λ,CP1
projective spaces and flag varieties).
INTEGRATION OF THE LIFTING FORMULAS 3
Using the method of [FT1] it is not difficult to show that HL•ie(glfi∞n(DifCP1);C) is the
exterior algebra with the unique generator in dimension 1 and two generators in each
dimension 3,5,7,....
We calculate the integral Ψ ∈ C2k−1(glfin(Dif );C) and prove, that it has
CP1 2k+1 Lie ∞ CP1
nonzero value on the (2k−1)-cycle in H (glfin(Dif );C) which is the image of some
R 2k−1 ∞ CP1
(2k −1)-cycle in H (glfin;C) under the inclusion glfin ֒→ glfin(Dif ) (A 7→ A⊗1)
2k−1 ∞ ∞ ∞ CP1
(k ≥ 1) (Subsection 4.1–4.3). Then it follows from Theorem 2.3(1) that Ψ is not
2k+1
cohomologous to zero in C• (glfin(Dif ;C) for any k ≥ 1. It follows from this result,
Lie ∞ 1
calculation of [FT1], and the Hopf algebra structure on cohomology H• (glfin(Dif );C)
Lie ∞ 1
that
H• (glfin(Dif );C) =∧•(Ψ ,Ψ ,Ψ ,...)
Lie ∞ 1 3 5 7
(in fact, the cocycles Ψ , k ≥ 1, are primitive elements with respect to the Hopf
2k+1
algebra structure on H• (glfin(Dif );C)).
Lie ∞ 1
Unfortunately, wehavenotfoundanysimplecalculation oftheintegrals usingtheDol-
beault complex as we described above. Our calculation is based on the Cˇech resolution,
and we define the notion of integral from viewpoint of the Cˇech resolution (in the case
of CPn) in Subsec. 4.1–4.2. We believe, that both definitions of the integral coincide.
Let ι: Dif ֒→ Dif be the inclusion defined by a choice of coordinate system in a
CP1 1
point of CP1. We prove that
H• (glfin(Dif );C) ≃ ∧• Ψ ; ι∗Ψ , Ψ ; ι∗Ψ , Ψ ;... .
Lie ∞ CP1 3 3 5 5 7
(cid:18)ZCP1 ZCP1 ZCP1 (cid:19)
Using the integration on CPn, we prove that
H• (glfin(Dif );C) ≃ ∧•(Ψ ,Ψ ,Ψ ,...)
Lie ∞ n 2n+1 2n+3 2n+5
for any n ≥ 1. The situation here is quite complicated, because we don’t know any way
to calculate the integrals directly. Let λ = O(λ), λ∈ Z, and we consider
Ψ2k+1 ∈ HL2kie−1(glfi∞n(Difλ,CPn);C), k ≥ n.
ZCPn
For a matrix (2k − 2n + 1)-cycle γ ∈ H (glfin;C) we consider Ψ as a
2k−2n+1 ∞ CPn 2k+1
polynomial function on λ. In fact, this is a polynomial of n-th degree. It is not easy
R
to calculate this polynomial, but we calculate its leading coefficient and prove, that it is
not equal to zero for some γ. Therefore,
Ψ2k+1 ∈ CL2kie−2n+1(glfi∞n(Difλ,CPn);C)
ZCPn
is not cohomologous to zero, and it follows from the Theorem 2.3.(1) that
Ψ ∈C2k+1(glfin(Dif );C)
2k+1 Lie ∞ n
is not cohomologous to zero for any k ≥ n. The remaining part of the proof is the same
as in the case n = 1.
Finally, let us note that the question on the cohomological nontriviality of the pull-
back j∗Ψ with respectto the inclusion j: Dif ֒→ glfin(Dif ) (D 7→ E ⊗D) remains
2k+1 n ∞ n 11
open; it solved only in the simplest case k = n ([Sh1], Sect. 2). We don’t know how to
check the nontriviality of these cocycles because it is not known any explicit formulas for
cycles in C (Dif ;C). So the case of the Lie algebra glfin(Dif ) turns out more simpler
• n ∞ n
4 BORIS SHOIKHET
than the case of the Lie algebra Dif itself, because there exist matrix cycles in the case
n
of glfin(Dif ).
∞ n
On the other hand, the values of Lifting formulas itself (not their integrals) on the
matrix cycles are equal to 0, as well as the values of the pull-back ι∗Ψ on the matrix
2k+1
cycles.
0.2. Content of the paper.
Section 1: we recall ([Sh1], [Sh2]) the construction of the Lifting cocycles
Ψ ,Ψ ,Ψ ,... on the Lie algebra glfin(Dif );
2n+1 2n+3 2n+5 ∞ n
Section 2: werecallthedefinitionof thenotion ofintegral inLiealgebracohomology
in the classical case of the Lie algebra of smooth vector fields on a real manifold; all
this material is basically standard ([GF]), the minor modifications are due to Boris
Feigin;
Section 3: we give the definition of the integral in the case of the Lie algebra of
holomorphic differential operators on a holomorphic line bundle λ on a complex
manifold M (more precisely, the sheaf of the Lie algebras, ...); when M is com-
pact, the holomorphic noncommutative residue is defined. In this case, the value
of the noncommutative residue on the identity differential operator is equal, up
to a constant depending only on M, to the Euler characteristic of the bundle λ
(Conjecture 3.3);
Section 4: we calculate the integrals in the case M = CPn, n ≥ 1, λ = O(λ). We
use the Cˇech resolution instead of the Dolbeault resolution of Section 3. We prove
here, that H• (glfin(Dif );C) = ∧•(Ψ ,Ψ ,Ψ ,...).
Lie ∞ n 2n+1 2n+3 2n+5
0.3. Acknowledgements. I am very much indebted to Boris Feigin, almost all I know
about Lie algebra cohomology I learned from the numerous informal discussions with
him.
The first ideas of this paper appeared in February 1998, when I worked in the Institut
des Hautes E´tudes Scientifiques (Bures–sur–Yvette). I am grateful to the I.H.E.S. for
the hospitality and very stimulating atmosphere.
1. Lifting formulas ([Sh1], [Sh2])
1.1. It will be convenient to give the definitions in a bit more generality.
Let A be an associative algebra, we will also consider A as the Lie algebra with the
bracket [a,b] = a · b − b · a. Let Tr: A → C be a trace on the associative algebra A
(i.e. Tr[a,b] = 0 for any a,b ∈ A); we denote by Der A the Lie algebra of all the
Tr
derivations D of the associative algebra A such that Tr(Da)= 0 for any a∈ A.
Lifting formulas are the formulas for (k +1)-, (k +3)-, (k +5)- ... cocycles on the
Lie algebra A constructed by derivations D ,...,D ∈ Der A such that the following
1 k Tr
conditions (i)–(ii) hold:
(i) [D ,D ] = adQ (Q ∈ A, Q = −Q );
i j ij ij ji ij
(ii) AltD (Q )= 0
l ij
i,j,l
(Let us note that in any case AltD (Q ) lies in the center of the Lie algebra A.).
l ij
i,j,l
INTEGRATION OF THE LIFTING FORMULAS 5
In the simplest case, when Q = 0 for all i,j the formula for (k+1)-cocycle is given
ij
by the following formula:
(1) Ψ (A ,...,A ) = Alt Alt Tr(D A ·...·D A ·A ).
k+1 1 k+1 1 1 k k k+1
A1,...,Ak+1D1,...,Dk
In the general case, the r.h.s.of (1) is the “leading term” of the(k+1)-cocycle, also exist
k
terms linear, quadratic, ..., -th degree in Q .
ij
2
Let A = ΨDifn be the asshociiative algebra of the formal pseudo-differential operators
on (S1)n ([A]). One can define 2n (exterior) derivations of this algebra: ad(lnx ), ...,
1
ad(lnx ); ad(∂ ), ..., ad(ln∂ ). We use the formal symmetry between the symbols
n 1 n
∂ and x arising from the generating relation [∂ ,x ] = 1 to define ad(ln∂ ). These
i i i i i
derivations as well as the corresponding 2-cocycles firstly appear in [KK]. The trace on
ΨDif is the “noncommutative residue” ([A]): it is defined as the coefficient at the term
n
x−1...x−1∂−1...∂−1 in any coordinate system; it was proved in [A], that Tr[a,b] = 0
1 n 1 n
for any a,b ∈ A. It is easy to see ([Sh1]) that in fact ad(lnx ),ad(ln∂ ) ∈ Der ΨDif .
i i Tr n
It was proved in [Sh1] that
(2) [ad(ln∂ ),ad(ln∂ )] =
i i
1 2 (n−1)!
ad x−1∂−1+ x−2∂−2+ x−3∂−3+...+ x−n∂−n+... .
i i 2 i i 3 i i n i i
(cid:18) (cid:19)
Therefore, the set of 2n derivations
{ad(lnx ),...,ad(lnx );ad(ln∂ ),...,ad(ln∂ )}
1 n 1 n
satisfy the conditions (i), (ii) above.
We can reply this construction in the case A= glfin(ΨDif ), here
∞ n
Tr = Tr ◦Tr .
glfi∞n(ΨDifn) ΨDifn glfi∞n
The derivations ad(lnx ), ..., ad(lnx ); ad(ln∂ ), ..., ad(ln∂ ) act in the obvious way
1 n 1 n
on the algebra glfin(ΨDif ), and the conditions (i)–(ii) hold again.
∞ n
Remark. Strictly speaking, [ad(ln∂ ),ad(lnx )] = adQ, where Q is an infinite matrix in
i i
the case of the Lie algebra glfin(Dif ) .
∞ n
By definition, the Lifting formulas for the algebra Dif are pull-backs of the Lifting
n
formulas for ΨDif with respect to the natural imbedding Dif ֒→ ΨDif , and also for
n n n
the Lifting formulas in the case of the algebra glfin(Dif ).
∞ n
We construct the Lifting formulas for glfin(Dif ) (k = 2) in Subsection 1.2 and for
∞ 1
general k in Subsection 1.3.
1.2. The case n= 1 ([Sh1], [KLR]). Let k = 2, so we have two derivations D ,D such
1 2
that [D ,D ]= adQ.
1 2
The simplest Lifting formula for 3-cocycle is given by the following formula:
(3) Ψ (A ,A ,A ) = AltTr(D A ·D A ·A )+AltTr(Q·A ·A ·A )
3 1 2 3 1 1 2 2 3 1 2 3
A,D A
6 BORIS SHOIKHET
In the general case, for i= 2,3,4,...
(4) Ψ (A ,...,A ) = AltTr(Q·A ·A ·...·A )+
2i+1 1 2i+1 1 2 2i+1
A
+AltTr(D A ·D A ·A ·...·A +D A ·A ·A ·D A ·A ·...·A +
1 1 2 2 3 2i+1 1 1 2 3 2 4 5 2i+1
A,D
+D A ·A ·A ·A ·A ·D A ·A ·...·A +...+
1 1 2 3 4 5 2 6 7 2i+1
+a ·D A ·A ·...·A ·D A ·A ·...·A )
s 1 1 2 s 2 s+1 s+2 2i+1
1
s = i+1, a = if i is even
s
where 2 .
(s = i, a = 1 if i is odd
s
Theorem ([Sh1], Sect. 1, [KLR]). The formulas for Ψ ,Ψ ,Ψ ,... given by (3), (4) de-
3 5 7
fine cocycles on the Lie algebra A.
1.3. The general case. In this Subsection we suppose that D ,...,D ∈ Der A sat-
1 k Tr
isfy conditions (i), (ii) from Subsec. 1.1.
First of all, let us write the formula for 5-cocycle in the case k = 4 (this is the simplest
Lifting formula in this case):
(5) Ψ (A ,A ,A ,A ,A ) =
5 1 2 3 4 5
= AltTr{D A ·D A ·D A ·D A ·A
1 1 2 2 3 3 4 4 5
A,D
+A ·Q ·A ·D A ·D A ·A
1 12 2 3 3 4 4 5
+D1A1·A2·Q23·A3·D4A4·A5 terms, linear in Qij
+D A ·D A ·A ·Q ·A ·A
1 1 2 2 3 34 4 5
+A ·Q ·A ·A ·Q ·A ·A } ] term, quadratic in Q
1 12 2 3 34 4 5 ij
We don’t alternate the symbols i,j in Q .
ij
Thenext step is theformula for (k+1)-cocycle for any k. We need some preparations.
We consider the interval of the length k − 2 with some marked points, such that the
distance between any two marked points is greater or equal than 2. Let us denote by I
l
k
the set of all such marked intervals with l marked points (1 ≤ l ≤ ). Denote by
2
1,...,k−1 the integral points of the interval. h i
k
Definition. Supposethatt ∈ I andi < ... < i areitsmarkedpoints(1 ≤ i , i ≤
l 1 l 1 k
2
and i −i ≥ 2 for all s = 1,...,l−1). Then h i
s+1 s
O(t) = AltTr(P ·...·P )
1,t k+1,t
A,D
where
P = A ·Q
j,t j j,j+1 when j = i for some s = 1,...,l, i.e.,
s
P = A if point j is marked
j+1,t j+1 (cid:27)
when j and j −1 are not marked and
P = D A
j,t j j j 6= k+1
P = A
k+1,t k+1
Example. If k = 6 and t = , t ∈ I then
2
O(t)= AltTr(A ·Q ·A ·D A ·D A ·D A ·D A ·A ).
1 12 2 3 3 4 4 5 5 6 6 7
A,D
INTEGRATION OF THE LIFTING FORMULAS 7
If t = , t ∈ I , then
3
O(t)= AltTr(A ·Q ·A ·A ·Q ·A ·A ·Q ·A ·A ).
1 12 2 3 34 4 5 56 6 7
A,D
We don’t alternate the symbols i,j in Q in this definition.
ij
Theorem. Let Σ = O(t). Then
l
t∈Il
P
(6) Ψ = AltTr(D A ·D A ·...·D A ·A )+Σ +Σ +...+Σ
k+1 1 1 2 2 k k k+1 1 2 [k]
A,D 2
is a (k+1)-cocycle on the Lie algebra A.
This Theorem was proven in [Sh2], Section 3.
It remains to define the cocycles Ψ ,Ψ ,Ψ ,... (for k derivations D ,...,D ).
k+3 k+5 k+7 1 k
First of all, let us suppose that Q = 0 for all i,j.
ij
Definition. The set as is the set of the sequences {a } of the length k+2s such that:
even i
(i) a ∈ {0,1} for all i= 1,...,k+2s;
i
(ii) a = 1;
1
(iii) k of the a ’s are equal to 1 and 2s of the a ’s are equal to 0.
i i
(iv) the number of 0’s between the two nearest 1’s is even; this condition also should
hold for the “tail” of the sequence {a }, as if the number’s a were placed on a circle.
i i
We define the (k+2s−1)-cochain R (A ,...,A ) for each {a } ∈ a .
a1,...,ak+2s 1 k+2s−1 i even
Roughlyspeaking,wejustshortenanysequenceofconsecutivezerostoasinglezero; after
thisprocedurethesequencewillhaveoddlength. Wechoosethefirstsuchsequenceofthe
consecutive 0’s. More precisely, let us define the sequence {a } ∈ a in the following
i even
way:
let s = min(a = 0), s = max(a = 1); then:
1 i 2 i e
i i>s1
a ,...,a =1;
1 s1−1
a ,...,a = 0;
s1 s2−2
ea = a e for s −1 ≤i ≤ k+2s−1.
i i+1 2
e e
Definition.
e
R (A ,...,A )= AltTr(P ·...·P ),
a1,...,ak+2s 1 k+2s−1 A,D 1 k+2s−1
where
P = D A for a = 1,
i j(i) i i
P = A for a = 0
i i i
e
and j(i) is defined in the following way: j(1) = 1 and j(i ) < j(i ) when i < i and
1 2 1 2
j = 1,...,n. In other words, j takes values freom 1 to k in turn.
Theorem ([Sh2], Section 1).
(7) Ψ0 = (−1)s1R (A ,...,A )
k+2s−1 a1,...,ak+2s 1 k+2s−1
{ai}X∈aeven
is a (k+2s−1)-cocycle on the Lie algebra A when all Q = 0.
ij
8 BORIS SHOIKHET
The formula (7) is the generalization of the formula (1) on the case of the higher
cocycles.
Let us consider the general case, Q 6= 0. We quantize each summand R
ij a1,...,ak+2s
separately.
Definition. (i) Denote by Circlea1,...,ak+2s the set of all circles with k+2s−1 integral
l
k
points from which l (1 ≤ l ≤ ) are marked. The distance between any two marked
2
points is ≥ 2. The points arehenuimerated by 1,...,k+2s−1. Point i may be marked
only if a = 1 and a = 1 (we suppose that a = a ).
i i+1 k+2s 1
(ii) for t ∈ Circlea1,...,ak+2s we define O(t) by the analogy with the definition above
l
after replacing the interval with the circle. We don’t alternate the symbols i,j in Q in
e e e e ij
this definition.
(iii)
Σa1,...,ak+2s = O(t)
l
t∈CircleXa1,...,ak+2s
l
Theorem ([Sh2], Section 3). The formula
(8) Ψ = Ψ0 + (−1)s1 Σa1,...,ak+2s
k+2s−1 k+2s−1 l
{ai}X∈aeven Xl≥1
defines a (k + 2s − 1)-cocycle on the Lie algebra A, where Ψ0 is defined by the
k+2s−1
formula (7).
2. Integration in the Lie algebra cohomology [GF]
2.1. C∞-case. Let M be an n-dimensional C∞-manifold, let W be the Lie algebra of
n
C∞-vector fields on Rn and let Vect(M) be the Lie algebra of C∞-vector fields on M.
The integration is a procedure, corresponding to any cocycle ξ ∈ Ck (W ;C) and any
Lie n
singular cycle σ ∈ Csing(M,C) the cocycle ξ ∈ Ck−l(Vect(M);C) such that:
l σ Lie
(i) ξ =0 if [ξ] = 0;
σ R
(ii) ξ = 0 if [σ] = 0
(cid:2)R σ (cid:3)
(here [...] denotes the cohomological class).
(cid:2)R (cid:3)
Let x ∈ M be any point of M, and let ϕ: U → M be any coordinate system in the
point x (U is a neighbourhood of 0 in Rn and ϕ(0) = x). The map ϕ induces the map
ϕ : Vect(M) → W , and the corresponding map of the cochain complexes:
Vect n
ϕ∗ : C• (W ;C) → C• (Vect(M);C).
Vect Lie n Lie
Lemma. Let ξ ∈ C• (W ;C) be a cocycle. Then the cohomological class [ϕ∗ (ξ)] does
Lie n Vect
not depend on the choice of the point x∈ M and of the coordinate system ϕ: U → M.
Proof. The ad-action of the Lie algebra Vect(M) moves points of M and induces the
trivial action in cohomology.
Let ξ ∈ Ck (W ;C) be a cocycle. We choose a coordinate system ϕ : U → M
Lie n x
(ϕ (0) = x) in all the points x∈ M, smoothly depending on the point x; by definition,
x
ξ(x) = ϕ∗ (ξ) ∈ Ck (Vect(M);C).
x,Vect Lie
INTEGRATION OF THE LIFTING FORMULAS 9
According to Lemma, all the cocycles ξ(x) (x ∈ M) are cohomologous to each other.
Therefore, there exists an element Θ ∈ Ω1 ⊗Ck−1(Vect(M);C) such that d ξ(x) =
1 M Lie DR
δ Θ (here d denotes the de Rham differential and δ denotes the differential in the
Lie 1 DR Lie
cochain complex).
Furthermore, we will find elements Θ ∈ Ωi ⊗Ck−i(Vect(M);C) such that
i M Lie
d ξ(x) = δ Θ
DR Lie 1
d Θ = δ Θ
(9) DR 1 Lie 2 .
.................
d Θ = δ Θ
DR n−1 Lie n
It is obvious that if solution of (9) exist, it is not unique.
Proposition. For any singular cycle σ ∈ Csing(M;C) and for any solution {Θ } of the
l i
system (9) the integral of l-form
Θ ∈ Ck−l(Vect(M);C)
l Lie
Zσ
is a cocycle.
Proof.
byStokes formula
δ Θ = δ Θ = d Θ = 0.
Lie l Lie l DR l−1
Zσ Zσ Zσ
It is obvious (see Subsec. 2.3) that the condition (ii) holds for any solution of the
system (9). We have in mind (but we don’t make the exact statement) that solution
of (9) is unique if we require the execution of the condition (i).
2.2. The solution of (9). In this Subsection we construct the canonical solution of the
system (9).
Let x∈ M and we have choose a coordinate system in any point, smoothly depending
onthepoint. Thenanytangentvectorv ∈ T M inthepointxdeterminesaninfinitesimal
x
1-parametric group of diffeomorphisms on Rn, i.e. the element t ∈ W .
v n
In the cochain complex of any Lie algebra g the following identity holds:
(10) δ ◦ι ±ι ◦δ = ad(t)
Lie t t Lie
(t ∈ g and ι : C• (g;C)→ C•−1(g;C) is the substitution of t for the first argument).
t Lie Lie
Let t be any C∞-vector field on M such that t | (u) = ϕ (t ). It follows directly
v v ϕ x,Vect v
from (10) that
e e
(11) δ ◦ι ±ι ◦δ =ad(t )
Lie tv tv Lie v
in the cochain complex C• (Vect(M);C).
Lie e e e
Lemma.
(12) (d ξ(x))v = adt (ξ(x)).
DR v
Remark. The r.h.s. of (12) does not depend on prolongation t of vector field t .
e v v
Proof. The derivation in the direction of the vector v corresponds to the infinitesimal
e
1-parametric group of diffeomorphisms and ad(t ) corresponds to the Ad-action of the
v
diffeomorphism.
e
10 BORIS SHOIKHET
It follows from (11), (12) that
(d ξ(x))v = δ ◦ι ξ(x)
DR Lie tv
or, in other words, if we set
e
(13) Θ (v)(A ,...,A ) = ξ(x)(t ,A ,...,A )
1 1 k−1 v 1 k−1
then (d ξ(x))v = δ Θ (v).
DR Lie 1
e
It is obvious that Θ does not depend on the prolongation t of the vector field t .
1 v v
Furthermore, let us set
e
(14) Θ (v ,...,v )(A ,...,A ) = ξ(x)(t ,...,t ,A ,...,A ).
i 1 i 1 k−i v1 vi 1 k−i
Theorem. The elements {Θ } defined by the formula (14), Θ ∈ Ωi ⊗
i e e i M
Ck−i(Vect(M);C), gives us a solution of the system (9).
Lie
Proof. We have proved the statement in the case i = 1. We prove in the case i = 2, the
general case in analogous.
By the Cartan formula, we have:
(15) d (Θ )(v (x),v (x)) = Θ ([v ,v ](x))−v (Θ (v (x))+v (Θ (v (x)).
DR 1 1 2 1 1 2 1 1 2 2 1 1
In the formula (15) v and v are vector fields such that v | = v (x).
1 2 i x i
According to Lemma, we have:
(16) v (Θ (v ))(x) = ad(t )(Θ (v (x)),
1 1 2 v1 1 2
(17) v (Θ (v ))(x) = ad(t )(Θ (v (x))
2 1 1 v2 1 1
e
if we choose the vector fields v and v such that [v ,v ] = 0. Note also that
1 2 1 2
e
[t ,t ] = 0 for any two vectors v (x),v (x). Therefore
v1(x) v2(x) 1 2
(18) d (Θ )(v ,v ) = −[t ,ξ(x)(t ,A ,...,A )]+[t ,ξ(x)(t ,A ,...,A )].
e eDR 1 1 2 v1 v2 1 k−1 v2 v1 1 k−1
Now it follows from the cocycle condition for ξ(x) that the r.h.s. of (18) is equal to
e e e
δ Θ (v ,v ).
Lie 2 1 2
2.3. Conditions (i) and (ii).
Theorem. (1) If [ξ] = 0 then ξ = 0 for any manifold M and σ ∈Csing(M;C).
σ •
(2) If [σ] = 0 then ξ = 0 for any ξ.
σ (cid:2)R (cid:3)
Proof. (1): Let ξ =δL(cid:2)iReη. (cid:3)Then
Θ (v ,...,v )(A ,...,A ) =(δ η)(t ,...,t ,A ,...,A ).
i 1 i 1 k−i Lie v1 vi 1 k−i
We have:
e e
(δ η)(t ,...,t ,A ,...,A ) =δ (η(t ,...,t ,A ,...,A )
Lie v1 vi 1 k−i Lie v1 vi 1 k−i
modulo the following expressions:
e e e e
(i) ad(t )η(t ,...,t ,...,t ,A ,...,A );
vj v1 vj vi 1 k−i
(ii) η([t ,t ],t ,...,t ,...,t ,...,t ,A ,...,A ).
evα veβ v1 be vα e vβ vi 1 k−i
By formula (12), the summands of the type (i) are equal to (d η)(v ) and therefore
DR j
their pulle-baceks one the sinbegular cbyecle σ aree equal to 0.
On the other hand, [t ,t ] = 0 for any v and v , because the corresponding
vα vβ α β
1-parametric groups commute with each other.
e e
