POISSON COHOMOLOGY

A NNALES DE L ’ ^INSTITUT F ^OURIER

P ING X U

Poisson cohomology of regular Poisson manifolds

Annales de l’institut Fourier, tome 42, n^o4 (1992), p. 967-988

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42, 4 (1992), 967-988

POISSON COHOMOLOGY

OF REGULAR POISSON MANIFOLDS

by Ping XU

Introduction.

Poisson cohomology plays a very important role in the study of Pois- son geometry. Finding Poisson cohomology of certain Poisson manifolds would allow us to solve some problems in deformation theory, as well as a number of other interesting problems [L] [VK1] [VK2]. For symplectic man- ifolds, Poisson cohomology is isomorphic to the usual de Rham cohomology [L]. In general, Poisson cohomology, roughly speaking, reflects two pieces of information of a Poisson manifold, the de Rham cohomology of symplectic leaves and the variation of symplectic structures along symplectic leaves.

Despite of its importance, little work has been done in computing such cohomology because of the lack of a general powerful method.

The aim of the present paper is to suggest a way of computing Poisson cohomology by means of symplectic groupoids. Using this method, we carry out a computation for a special kind of regular Poisson manifolds where the symplectic foliations are trivial fibrations. Such an idea was already suggested by Vorob'ev and Karasev in [VK2], where they compute the Poisson cohomology for certain kinds of Poisson manifolds in dimensions 1, 2 and 3.

The key point of this paper can be stated as follows. For an integrable Poisson manifold (i.e., a Poisson manifold admitting a global symplectic groupoid), the Poisson cohomology is naturally isomorphic to the de Rham

Key words : Poisson cohomology - Symplectic groupoid - De Rham cohomology.

A.M.S. Classification : 58F05.

968 PING xu

cohomology of left invariant forms on the symplectic groupoid. This fact allows one to convert the problem of computing Poisson cohomology to that of computing de Rham cohomology of certain manifolds, which is much easier to handle in general. In particular, in the case that the symplectic foliations are trivial fibrations, we show that the computation of Poisson cohomology is equivalent to the computation of de Rham cohomology of certain torus bundles.

The first section of this paper contains a general discussion on the preceding fact and its consequence for the regular Poisson manifolds in which the symplectic foliations are locally trivial fibrations.

The second section is devoted to an explicit construction of symplectic groupoids for integrable Poisson manifolds of the form P = S x Q.

In Section 3, we compute directly the de Rham cohomology of torus bundles, a result needed in Section 4.

In Section 4, we obtain the main result of this paper, namely, a general formula describing Poisson cohomology of integrable Poisson manifolds of the form P = S x Q.

As a by-product of this work, we can easily see that any volume form of an integrable Poisson manifold produces a Haar system for its symplectic groupoid. The relation between volume forms of the base Poisson manifold, Haar systems of the symplectic groupoid and the symplectic volume of the symplectic groupoid, as well as a number of possible applications to (7*- algebras of symplectic groupoids will be discussed elsewhere.

Finally, we would like to mention that a different approach to the computations of Poisson cohomology was recently carried out by I. Vaisman [V].

Acknowledgements : Most parts of this work were finished when the author was at the University of California, studying with Professor Alan Weinstein. The author would like to acknowledge his indebtedness to Pro- fessor Alan Weinstein for his invaluable guidance during its preparation.

He also wishes to thank Professor Pierre Dazord, Professor Richard Lashof, Professor Kirill Mackenzie and Jianghua Lu for many stimulating discus- sions.

1. Poisson cohomology and de Rham cohomology of invariant forms on symplectic groupoids.

Let (r -^ P, a, 13) be a Lie groupoid. By T^F we denote the tangent bundle of the foliation on F arising from the a-fibration and by T^F its dual bundle. Write ^ for the C7°°(r)-module of all smooth sections of the bundle Ty and ^ = A^; then ^ -^ ^⁺¹ with d the usual exterior derivative along the fibres becomes a cochain complex. The subset f^ ^ of f^ consisting of all forms which are invariant under the groupoid left translations is a subcomplex of (f^;, d). The cohomology groups of this subcomplex are denoted by H^ ^(T, IR). Let A —> Fo be the Lie algebroid of r with anchor map p : A —> TP. It is quite obvious that any element of (7"'(A, R) can be naturally identified with an element of f^ ^ and vice versa.

This identification commutes with the coboundary operators, and therefore establishes an isomorphism between Tr^A.IR) and H^^{Y,R) (cf. [WX]).

THEOREM 1.1. — If F is a Lie groupoid with Lie algebroid A —> FQ, then

^(r.n^^A.n).

In particular, if (F ⁼^ P, a, (3) is a symplectic groupoid, its Lie algebroid is the cotangent bundle T*P —> P with anchor p : T*P —> TP being the map naturally induced from the Poisson tensor. Therefore, C^A.IR) is naturally isomorphic to r^/^ TP), and the Lie algebroid differential d turns out [H] to be the Poisson differential [L] d^ for the multi-vector fields over P. Hence, the Lie algebroid cohomology .^(r*?,^) is isomorphic to the Poisson cohomology H^(P). Therefore by Theorem 1.1, we have

THEOREM 1.2. — Jfr ^ P is a symplectic groupoid, then H^ ^(F, IR) is isomorphic to the Poisson cohomology H^{P).

A Poisson manifold is called integrable, by Dazord [Dl], if it is the unit space of a symplectic groupoid. For an integrable regular Poisson manifold P whose symplectic foliation is a locally trivial fibration P -n-^ Q as a different table fibration, we have a nice description for the cohomology H^ ^(r, IR). To this end, we shall introduce a vector bundle E^ —> Q by the following procedure. It is known that for each u C P, a~~^(u) is a principal bundle over the symplectic leaf through u, with the structure group being the isotropy group Iu of F at u [CDW]. We denote by H^a-1^)^) the de Rham cohomology of J^-invariant forms on a~^l(u). For any two points u

970 _{PING XU}

and v in the same symplectic leaf L, there always exists a point z e F such that a(z) = u and (3(z) = v. The left translation L^ : a^^v) —> a-^u) induces an isomorphism :

L: : H^a-^u)^) -^ H^a-\v)^).

Moreover, L^ = L^ if ^ and z^ e F are two such points, since H^ (a ¹^), IR) is fixed by the isotropy group.

Since the symplectic foliation of P is a locally trivial fibration, F is a-locally trivial. Let {U,} be an open cover of Q, such that the symplectic fibration TT : P —> Q has a local section a over each Ui. For any U^

the a-fibration a-^e^)) -°-> a(Ui) induces a canonical vector bundle SF —> €i(Ui) (^ £/,) with Hy^a-1^))^) as the fibre over each point u e U,. If U, H ^ ^ 0, the transition : Hy (a-^e^n))^) _>

^.(u)^^^"^6^^))^) is g^611 ^ the canonical identification above. Thus in this way, we obtain a canonical vector bundle S" —> Q, which is easily seen to be independent of the choice of open covers {U^}. It is not difficult to see that

^LO^^G00^^),

where C°°(Q, 2⁷¹) is the space of smooth sections of the bundle 2⁷¹. Hence, we have the following :

COROLLARY 1.1. — If P is an integrable regular Poisson manifold in which the symplectic fibration P —> Q is locally trivial as a differentiable fibration, then H^(P) is isomorphic to the space of smooth sections of the vector bundle 571.

Example 1.1. — If S is a symplectic manifold, its symplectic groupoid S x S~ z=^ S is transitive and principal. Hence, (^(O.S71) ^ S71 ^

%(S',R) and Corollary 1.1 implies that H^(S) ^ H^(S,R). This result was first proved by Lichnerowicz [L], by introducing a direct isomorphism between the two cochain complexes.

Remark. — In the preceding example, if one takes the fundamental groupoid H(S) instead of the coarse groupoid 5' x 6'- as a symplectic groupoid over S, the a-fibre is the universal covering space of S and the isotropy group becomes the fundamental group of S. Hence, the corresponding de Rham cohomology of invariant forms coincides with the usual de Rham cohomology of S. So, one obtains the same result as before.

More generally, if P = S x M, the direct product of a symplectic manifold S and a zero Poisson manifold M, then a symplectic groupoid of P is given by

S x S~ x T*M ^ S x M.

Fixing any point SQ C 5, we have a global section {so} x M for the symplectic fibration of P. Then the corresponding a-fibration over this section is isomorphic to S x T*M -a-> M, where the isotropy group over each m € M is isomorphic to T^M, and T^M acts on each a'^m) = S x T^M by translations on the second factor T^M. Hence,

s ^ Q ^ ^ ^ ^ A ™ ^

i=l

therefore,

H^(P) ^ Q%^n) ^r°°(/\ TM).

1=1

Example 1.2. — Let P = S² x IR"^, with ^ the symplectic structure on each leaf S'2 x {t}, where a; is the standard symplectic structure on the unit sphere S². First, we note that P may be viewed as so (3)* — {0}

with the linear Poisson structure. Hence, its symplectic groupoid is the transformation groupoid

S² x H-⁴- x 50(3) =^S² x R+.

It can be easily seen that the a-fibre of this groupoid is 5'0(3), and the isotropy group is the circle group S¹. Since S¹ is compact and connected, the de Rham cohomology of invariant forms on any a-fibre coincides with the de Rham cohomology of 50(3). Furthermore, since the symplectic fibration is trivial with the base space IR"^, the vector bundle 572 is trivial over IR4'. Hence,

:7n _ 0+ „ u-n

x%(50(3),R);

therefore,

^(p)- c00 (R+,7%(50(3),R)).

972 PING xu It follows immediately that

7^(P) ^ G00^4-, H) = 0°°^) (Casimirs),

H^(P) = 0 (Poisson vector fields/ Hamiltonian vector fields), H^(P) = 0 (infinitesimal deformations of the Poisson structure), H^(P) ^ G00^4-, R) = C°°(R^) (obstructions to deformation

quantization), and H^P)=0 ( A ; > 4 ) .

2. Symplectic groupoids of regular Poisson manifolds.

Given a Poisson manifold in which the characteristic foliation is a trivial fibration TT : P = S x Q —> Q, the Poisson structure is described by a smooth map from Q to Z2(S), the space of all closed two forms on S, denoted by {^y\y 6 Q}.

According to Dazord [Dl], P is integrable if and only if the map V 1—)> [^y] ^Tom Q ^° H2(S^H) is a submersion onto an affine subspace of T^^IR) whose underlying vector space is generated by elements of H2^,!). In other words, there exist [Ci], • • • , [Ck] e H2(S,I) and a submersion (^i(^/), • • • ^ g k ( y ) ) '• Q —^ ^^k such that

hJ-M+^zQ/)^].

1=1

Symplectic groupoids over such Poisson manifolds have been investigated by Dazord [Dl] [D2]. For completeness, in the following, we shall give an explicit construction of the Symplectic groupoids directly. Let us first recall that a RIL ("realisation isotrope de Libermann") is a complete symplectic realization which is a surjective submersion with connected and isotropic fibres [D2]. According to Dazord [Dl], in order to construct a symplectic groupoid over the Poisson manifold P = S x Q, it suffices to find a RIL for the Poisson manifold P * P~ = S x S~ x Q. To this end we need the following :

PROPOSITION 2.1. — Let P = S x Q be a Poisson manifold with Poisson structure {^y\y G Q} such that

k

^y = ^o + ^9i(y)Ci + dsOy, 1=1

where [C\ ] , • • ' , [C^] G 7:f2(Sf,Z) are linearly independent in H2^^), {9\{y) i ' ' ' •> 92{y)) '• Q —> ^k ls a submersion, {Oy\y C Q} is a family of one forms on S and ds is the exterior derivative with respect to S.

Let E -p-> S be a Tk -principal bundle with connection (0i, • • • ,^/c) and

( k \

curvature (Ci, • • • , C^). Let F = E x r*Q/ ^ Hdgi and

\ i=i )

/ k \ k / k \

^=p* ^o+^^(^/)Q +^*W)+^7r*(^(^))A^- ^ A + ^ ^ A 7 z ,

\ 1=1 / i=i \ 1=1 7 / k \

where 0y is considered as a one-form on P of type (1, 0), ( T*Q/ ^ Rdgi j

\ z=i / is the quotient space ofT*Q under the IR^ -action denned by :

/ k \

( t i , - - - , ^ ) ^ ^ ) - [y^+^t,dg,(y) , V ( ^ O e r * Q ,

\ 1=1 / / k \

TT : ( T*Q/\^Hdgi ) —> Q is the natural projection, (71, • • • , 7 ^ ) is a

\ i=i )

connection of this principal bundle, d\ is the standard symplectic 2-form on T*Q and (f) = p x TT. Then (F, ^) -^ P is a RIL.

/ k \

Here, ( dX + Y^ dgi A 7^ ) is considered as a two-form on the quotient

\ 1=1 7 / k \

space T*0/ V^ Rdgi . This is justified by the following lemma.

V 2=1 7

k

LEMMA 2.1. — dX + ^^ dgi A 7^ is the pull back of a two-form on / k \1=1

the quotient space ( T*Q/Y^IRd^ ) , and therefore can be viewed as a

\ 2=1 7

( kk \ \ two-form on T*Q/ ^ Hdgi .

\ i=i 7

974 _{PING XU}

Proof. — By p, we denote the natural projection T*Q —> Q. Let Xi, • • • , Xk be a family of vector fields on T*Q generating the Reaction described above, and <f>[, • • • , ^ their flows. Then,

^(y, 0 == (y, ^ + tdgi(y)), for any (y, ^) € T*Q.

For any given v e T(y^)(T*<3), it follows from the definition of A and the relation p o <f>\ = p that

((^rw = \Wv)

=^+tdg,(y))(TpT^v)

=^+tdgi(y))(Tpv)

=X(v)+t(p*dgi)(v).

Then. (<^)*A = \+tp*dgi. Therefore, L^.A = p*(^. However, it is easy to see that (^.A)(y,0 = $(TpX,) = 0 for all (y^) e r*Q. since TpX, = 0.

Thus ix.rfA = Lx,\ - dix,\ = p*dgi. Hence,

I k \ ix. \d\+^p*dg^^

\ J=i /

k k

= ix,d\ + ^ ix, (p*dg_,) A ^; - ^ ^*ri^ A ^,7,

3=1 j=l

k k

= P*dgi + ^ i(TpX,)(dgj) A ^ - ^ ^*ri^- A ^.

j=i j-=i

=0.

k

Therefore, [dX + ^ dp, A 7,^) is indeed the pull back of a certain two-form 1=1

k

on (r^Q/^Rdgi).

i=l

In fact, locally around any fixed point in Q, we may choose a coordi- nate system U - ^ x C^ with coordinates ( y , , . . . , y,, ^,... ^,), such

/ fc \

that p,Q/) = y^ (z = 1 , . . . , fc). Then, T*Q/ ^ Rd^ ^ ffT x H^-^ with V ^=1 /

coordinates (y^ . • . , yr.pk^i, • • ' ,pr), and the connection 7, can be taken

k r

to be dpi. Hence, d\ + ^ , dgi A 7^ = ^1 dj^ A dyi^ which is obviously a

1=1 Z=/C+1

/ fc \

two-form on the quotient space ( T*Q/ ^ ^ IRc?^ ) . Q.E.D.

V 2=1 )

Sketch of proof of Proposition 2.1. — It is obvious that uj is a closed 2- form. We shall use a local argument below to show that uj is non-degenerate.

Take U ^ IR^ x IR7'"^ with coordinates (^/i, • • • , y ^ ^ y ^ i , • • • , ^/r) to be a local system as in the proof of Lemma 2.1, and let V be any open submanifold of S, such that E\^ is trivial and (7i, • • • , Ck are exact on V. We assume that E\^> ^ V x T^ with coordinates (9, a;i, • • • , a;fc), such that 0^ = ai + cte^ and d = dai with a^ € ^(V),^ = I , - - , A:). Then, r[vx^ is isomorphic to V x T^ x R7' x ^"^ with coordinates (9, a;i, • • • , Xk.yi, • • • , ^Pfe+i, • • • ,Pr), and under these coordinates

k k k r

uj = UJQ + ^ ^^i + c?(^) + ^ ^ A a^ + ^ d^ A d^ + ^ ^ A dp^.

i=l i=l i=l i=k-\-l

k

By using the fact that c<;o + Y^ ^^i + ^(^) is non-degenerate when being restricted on each symplectic leaf y ^constant, one can easily show that uj1=1 is non-degenerate.

r\

The tangent space of each ^-fibre is spanned by ——, (z = 1, • • • , k) 9xi

and ——, (2 = k + 1, • • • ,r), and therefore is obviously isotropic. For any Qpi

function f{q^y) e 0°°^ x U)^ it is not difficult to see that

^9f 9 ^ 9f 9

( ) ^^{= / - l}r ^ ^^-^ ^ ^ '

where Xf is the Hamiltonian vector field of / on the Poisson submanifold V x U C P. It is trivial to see that for all /, g € C'°°(V x U),

{^f,4>*g}=i^(X^f,X^g)

^(X/,Xg)

=^{f,g},

since Xf and Xg are always tangent to V x U. Therefore, 0 is a Poisson morphism, or in other words, a symplectic realization.

976 PING xu

r\ r\

The coefficients in the front of —— and —— in Equation (1) depend dxi opi

neither on the coordinates Xi nor on pi, so X^f is complete provided Xf is complete. In other words, (p is a complete symplectic realization. Q.E.D.

An immediate consequence of Proposition 2.1 is the following theo- rem, in which an explicit construction of a symplectic groupoid over the Poisson manifold P = S x Q is described (see also [Dl] [D2]). We will see in the next section that this construction plays a key role in our later computations.

THEOREM. 2.1. — Suppose that the Poisson structure {ijjy} on P = S x Q is given by

k

[^y}=[^}+^9z(y)[Ci]

i=l

as in Proposition 2.1. Let E -p-> S x S be a T1^-principal bundle with / k \

curvature (Ci, -d),. • • , (C^, -C^), F = E x T*Q/ ^ Rdg, ) and F ^

_ \ z=i 7

S x S x Q be the RIL of S x S~ x Q as constructed in Proposition 2.1;

then r —> P with a = OQ o 0 and f3 = /3o o <p is a symplectic groupoid over P, where OQ and f3o : S x S~ x Q —> S x Q (= P) are given by OQ = (pri,id) and f3o = (pr^id), respectively.

3. De Rham cohomology of torus bundles.

In order to find the Poisson cohomology for Poisson manifolds dis- cussed in the last section, we need to compute the de Rham cohomology of principal torus bundles. For circle bundles, the de Rham cohomology follows easily from the Gysin sequence [BT]. However, for higher k, the spectral sequence of a T^-principal bundle becomes very complicated, which makes it difficult to find the de Rham cohomology directly by using the spectral sequence. Here, instead, we compute the de Rham cohomology directly by geometric methods.

Given a T^-principal bundle E -7L-> M, the relative cohomology [BT]

is defined as the cohomology of the complex d : ^(TI-) —> f^+^Tr), where ^(Tr) = ^(M) C ^n-l(^) and d(^,0} = (^,7r*cc; - d0), for all

u: C ^(M) and 0 C f^"1^). The short exact sequence

0 ——> ^n-l(£;) -^ ^(TT) -^ ^(M) ——. 0

yields a long exact sequence

——— ^n(TT) ^ ^"(M) ^ ^(^ ^ ^^(TT) ^ JT+^M) ——— .

Hence, we have

LEMMA 3.1. — The following exact sequence holds for any n G N : 0 —, jr^M)/^ —> ^(E) —> L⁷² —^ 0,

where K71 = %*(^n(7^)) and L71 is the icerneJ ofz* : T^+^TT) —^ ^rn+l(M).

On the cochain level, z* is given by %*(cj, ^) = uj.

Remark. — Since T^k is a compact group, both de Rham cohomology and relative de Rham cohomology of the bundle E can be computed by using the de Rham subcomplex consisting of all invariant differential forms.

In the rest of this section, we will work on this invariant subcomplex without any further mention.

In order to compute ^(I^), it suffices to compute K71 and L71

according to Lemma 3.1. To this end, let us choose a connection on E given by k invariant one-forms ai, • • • , Ok G ^(E) and denote the corresponding curvature by 0:1, • • • , cj/c e f^(M). Given any (^, 0) e ^(Tr), we may always assume that

k

0 = TT*(/) 4-y^ ^ 0^1 A a^ A • • • A a^ A 7r*0^...^,

r==l Kzi<i2---<ir<^

with ^i,...,^ € ^"^(M) and 0 <E Q^'^M). Since we are working with relative cohomology, we can assume that (j) = 0. By an elementary computation, we have

TT*^ - d0

( \ k

= TT* ( uj - ^ a;, A (f)i j + ^ ^ o^ A • • • A a^

\ i / r=li-i<---<ir

ATT* ((-l)r-l^,..•,^.+^(-l)t ^ ^A^,..^^^,..^) .

1=0 i-^<---<it<l<it+i<---<Zr

978

Hence, (u,6) e Z^TI-) with

PING XU

e=^L ]C ^i A ^ A — A c ^ A T r * ^ . . . ^

r==l Kzi<i2---<ir<A;

if and only if

(2) ^=^^/\^

i

and (3)

W^-.^D-^ E ^,,.,^,,,=o,

*-0 zi<---<it<Kit+i<---<ir

for all r such that 1 < r < k. On the other hand, (^0) e Bⁿ(7^) if and only if uj = d^ and

k

6 = TT^ + ^ ^ ^ A a,, A ... A a^ A TT*^,.,^

r=l l<ii<i2---<ir<A;

where 7, 0 e ^-^(M) and

^,..,.=(-l)^{^}-^l^,.,,+E(-l)^t ^ ^A^,,,,,,^,,,

, *-o ii<---<^<^<^^i<...<^

for certain ^,...^ C ^-^^M).

K^0)] ^^n ^ and only if (^) e Z^+^TT) and [^ = 0. Let us denote by L^ (M) the group consisting of all elements ©[<^,...,J C

^ H^^M), such that for all 1 ^ r < I - 1, there exists

Kii<---<il^k

^i,...,z, C f^-^M) satisfying Equation (3) and ^[^ A 0,] = 0. Then, it is not difficult to see that

^ ^ ^(M) e • . . e L^^M) e... e Lr^W.

Similarly, we have

(4) ( k

^ ^ [ y ^ ^ ^ ) ^ ^ ) 4)iefln ^^ ^ there exists ^,...,^ e ^-^(M) [ [ ^[ satisfying Equation (3) for all r, 1 < r < k. I

For instance, we have by definition,

L^M) = { (9[0,] e ^^(M) ^>,] A [0,] = ol ,

I i i=l i=l )

which can be easily seen to depend only on the cohomology classes [o^], (z = 1, • • • , k). Indeed, this fact is true for all such groups L^' (M).

LEMMA 3.2. — L^^M) and K⁷¹ depend only on the Chern class ([a;i],—,[a;fc]).

Proof. — We will prove this lemma only in the case that k = 2. The general case follows from the same argument. By the remark above, the lemma already holds for L^' (Af). So, it suffices to show this for L^' (M) and Kn'. By definition,

( 3 <^i, <^2 C ^-^(M) such that

L^{M) = { [^1,2] € ^""(M) d(^i + ^2 A (^1,2 = 0, d^2 - ^i A ^1,2 = 0 and [uj\ A (^i + 0^2 A (^2] = 0

and

K^ = < [0:1 A 0i + cj2 A 02] ^{01,02 € Q.n~'2}(M) s.t. d0i +^2 A 0i^2 = 0 and d02 — 0:1 A 0i 2 = 0, for some 0i 2 ^ Zn~3(M).

Assume that uj[ = cji + d0\ and c<;2 = ^2 + ^2 with 0i,02 ^ ^^^(M), and suppose that (^1,^2 ^ ^1(M) satisfy d(f)\ + 0^2 A (^1^2 = 0 and d(f>^ — (jj\ A 0i^2 = 0. Set

^1 = ^1 — ^2 A 01,2

and

02 = ^2 + ^1 A 01,2;

then it can be checked directly that

d(f)[ + Cc;2 A 01,2 = d(f)\ + C<;2 A 01,2 = 0.

Similarly, d02 ~^[ A0i,2 = 0. However, by a straightforward computation, we have

L^[ A0'i +0;2 A 02 = CJi A 0 i +C<;2 A 0 2 + d(9\ A 01 +02 A 02 — 0 1 A 02 A 01,2).

Hence,

[uj[ A 0'i + ^ A 02J == [<^i A 0i + Ci;2 A 02].

980 PING XU

So, L⁷^' (M) and Kⁿ depend only on the cohomology classes ofc<;i and uj^.

Q.E.D.

We summarize the above results in the following :

THEOREM 3.1. — Suppose that E is a T^k-principal bundle over M with the first Chern class ([c^i], • • • , [<^k]) ', then

^(E) ^ L^'^M) e • • • © ^^(M) e • • • © L^-^^M) e TT^M),

where ^(M) = ^(M)/^, Lm'k{M) (K71 resp.) is the subgroup of Q) H^rn(M) (^(M) resp.) defined preceding Lemma 3.2 and

Kii<---<ii<k

I^'^M) is assumed to be 0 ifm< 0.

In particular, if E is a circle bundle with Chern class [c<;], i.e., when k = 1, Theorem 3.1 implies that

^(E) ^ L^~~^{M) C Hn{M)/Kn, where

L^'\M) = { [<f>] € H^W | M A [4>] = 0}

and

^ = { M A [ct>] | [cf>} € H^^M) } .

This result coincides with the well-known Gysin sequence of circle bun- dles [BT]. Although Theorem 3.1 gives an explicit formula for ^(E), the subgroups ^^m' (M) there are somehow elusive. Nevertheless for low di- mensional cohomology, we can describe them more explicitly.

COROLLARY 3.1. — Assume that [c<;i], • • • , [cjk] are linearly indepen- dent in H²(M, H ) ; then

1. L°^(M) = 0, for all k,r € N ; 2. H\E) ^H\M);

3. H^E) ^ L^(M) © ^^2(M)/(span{[^]}) ;

4. moreover, ifH^l(M) == 0 (for example, when M is simply connected), H3{E) ^ L^(M) C ^(M), and H^(E) ^ ^(M) C L^(M) C

f k 1

H\M)/K\ where K^ = ^ ^[^] A [^]| [^] € ^(M) V ; and

l z = = l J

5. if H^^M) is spanned by the wedge products of H^^M) with [c<^], (z =

k

1, • • • ,^), i.e, H^M) = ^(H^M) A {[a;,]}); then

i=l

L^(M) ={Aeso(k,H^<2(M)) = so(k,R) (g) H²(M)\ A A o ; = 0 } , where so(k, H2(M)) is the space of all kxk skew-symmetric matrices with values in H2^) and uj = ([c^i], • • • , [cc^;])^

Proof. — (1) follows immediately from the definition of L^^M) and the fact that [c<;i], • • • , [c^/c] are linearly independent in H2(M^ IR).

(2) Obviously, K1 = 0. Moreover L^(M) = 0 by (1), so H\E) ^ L^(M) © H\M)/K¹ ^ H\M).

(3) According to Equation (4), [0] e K2 C H2(M) if and only if

k

0 = V ^ A (j)i for some ^ , ( % = l - - ' , k ) ^ such that d^z = 0. Thus,

i=l

k

^ = a, (z = l , - - - , ^ ) are constants and ^ = V c ^ . Therefore, JC2 = 1=1

span{[^]} and

H2^) ^ Ly{M) C L^'^M) C ^(M)/^2

= L^(M) C ^2(M)/(span{[^]}).

(4) By Theorem 3.1,

H^^E) ^ L^(M) C ^^(M) C ^^(M) © H^M)/^.

H^M) = 0 implies that L^(M) = 0. Moreover by (1), we have L^(M) = 0. We will show below that K3 = 0.

k

By definition, [0] C K³ if and only if 0 = Y ^ z A (^ for some 1=1

^i 5 • • • 5 ^fe ^ ^(M), so that the following equations hold for some (pij €

^ ° ( M ) , ( z , j = l , - - - , / c ) :

(5) c^- + ^ ^ A (f)ij - ^ ^ A (^, = 0

i<j j<i

(6) d^j = 0.

982 PING xu

Equation (6) implies that <^j = c^- are constants. Then, according to Equation (5), we have

Y^Cij[uji}-^Cji[^i] = 0

i<j j<i

for all j = 1, • • • , k. It follows from the linear independence of [c^i], • • • , [cjk]

that Cij = 0. Hence, Equation (5) reduces to d(f)j = 0 , (j = 1, • • • , k). Since H^^M) == 0, (f)j must be exact; therefore 0 is exact. Hence, K3 = 0.

Similarly, we can show that H^(E) ^ ^(M)^^)^⁴^)/^⁴ and K⁴ = [ ^[^] A [^]| [^] e H\M) \.

i=l

(5) By definition, {[<^,J} C (]) ^(M) belongs to L^(M) if

K^i<»2^A;

and only if for all 1 ^ i < k, there is ^ 6 ^²(M) such that

^ ⁺ ^ ^ ^A ^^ - ^ ^ ^A <^ ⁼ ° ^anci

Ki i<l k

^[o;,A^] =0.

i=l

In fact, the second condition can be implied by the first one, as in the following. Suppose that ^ € ^(M), (z = l . - ' - A * ) satisfy the

k

first condition; Y ^ z A ^ must be closed. Hence, one can find [^] e i==i

A; A;

^(M), (z = 1, • • • k) such that Y^[^ A (f)i} = Y^[^z] A [^] by assumption.

1=1 i=i

Therefore, the above two conditions are satisfied if ^ is replaced by <^ — ^ for all 1 < i < k. This shows that {[^i,zj} € ^^(M) if and if it is a solution of the equations :

]^H A [(t)^} - Y^[^i] A [<^] = 0 , i = 1, • • • k.

Ki i<l

That is,

L^(M) ={Ae5o(k,H²(M)) =so(k,H)(S)H²{M)\ A/\UJ=O}.

Q.E.D.

4. Main theorem.

With the results in previous sections, we can give the main theorem of this paper. First of all, we need two lemmas, which can be easily proved by a direct verification.

To any vector bundle V -^—>• Q, there corresponds a vector bundle, denoted by £¹⁷¹, with H^(7^~^l(q),R) (the invariant de Rham cohomology of 7^~l(q) under the translations of ^~^{q) itself) as the fibre at each point q € Q. We have

LEMMA 4.1. — E71 is isomorphic to f^V*, where V* is the dual bundle ofV.

By ^Q-s we denote the foliation on Q induced from the fibration ( ^ • • • ^ ) : 0 — ^ .

/ k \

LEMMA 4.2. — The dual bundle of T*Q/^Rd^ -^ Q is

\ i=l )

isomorphic to the vector bundle TFq —> Q.

THEOREM 4.1. — Suppose that P = S x Q is a Poisson manifold with the Poisson structure {^y\y € Q}, such that

k

[^]=k)]+^Q/)[QL 1=1

where [Ci], • • • , [Ck] C H2^, Z) are linearly independent in H<2(S, R) and (^i? • • • i 9 k ) '- Q —^ ^k ^ a submersion. Then for the Poisson cohomology of P we have :

n n—i

H^P)^Q)H^[£}^roc(f\(TyQ))^

i=0

where H^S] == C^^Q) (g) H^S) and £ is any principal bundle over S with the first Chern class ([C'i], • • • , [C^]). In other words, H^(P) can be described by the following formula :

n n—i

H^(P) ^ (3(4"^ [^©^'r1'''!5]®- • •<SL{~l'k[S}(BHi[S})®TOO(/\(T^Q)),

»=o

984 where

PING xu

L^m-^k[S]=^^oo(Q,L^m-^k(S))^C^OO (0)^^(3)^

H^S] = r°°(Q, H^S)) ^ C°°{Q) 0 7T(5),

r'm.k (

and L^ {S), H^S) are as defined in Section 3, with ([Ci],.. • , [C/,]) as the first Chern class.

Proof. — For any fixed point s € 6', {s} x Q is a global section of the symplectic fibration P —> Q. According to Theorem 2.1,

a-\{s} xQ)= ^(a^^s} x Q))

=^~\{s}xSxQ)

/ k \

=p-\{s}xS)x [T-Q/^Rdg, .

\ 1=1 /

So the fibre bundle Q-¹^} x Q) ^ Q becomes p~\{s} x S) x

(

bundle is a direct product of the trivial bundle p^^s} x S) x Q —> Q

k

with the vector bundle T*Q/^Rd^ -^ Q. Moreover, since f3 = A) o ^ 1=1

and 0 = p x TT, the principal bundle a^^s^y) -^ S is isomorphic to / k \

p-\{s} x S) x T^Q/^Hdg,{y) ^ {s} x S x {y}^

\ i=l )

( ^k \

where TT maps all the points in T^Q/ ^ Rdgi{y) into one point y . The

\ 1=1 )

k

structure group 1^^ is isomorphic to TkxT^Q/ ^ Rdgi(y) (^ TkxRr-k), where the action of T^ on p-1 ({5} x S) is induced from that on E by viewing

k

P ({s} x S) as an invariant subspace of E, and T;Q/ ^ Rdg^y) acts on

itself by translations. l-

Hence, by Theorem 3.1, we have H^(a-\s^y)^)

n / k \

^^H^(p-\{s} x S))(^H^\ T^Q/^Rdg,(y)

i=0 \ i=l )

-(])Jr(f)^/\r^,

1=0

where S is the principal bundle p'¹^} x S) -^p-^ {s} x S, having curvature (—Ci, • • • , —Cfc). Combining this fact with Lemma 4.1 and Lemma 4.2, we deduce that the vector bundle 5^ introduced preceding Corollary 1.1 is

n n—i

isomorphic to ^H\£) x (/\ T^q) —> Q, or isomorphic to 1=0

^(L^(5)eL^

'

(^)e-•eL^

'

(5)e^

(5))x(/\^^^—Q.

1=0

Hence, the conclusion follows immediately from Corollary l.l.Q.E.D.

In particular, combining Theorem 4.1 with Corollary 3.1, we obtain the following formulas expressing Poisson cohomology groups of low dimen- sions (n==l, 2, 3, and 4), the first three of which exactly coincide with the results obtained by Vorob'ev and Karasev in [VK2].

COROLLARY 4.1. — Under the assumption of Theorem 4.1,

1. ^(P^r^A

^))^

^];

2. ^(P)^^

(A

(^^Q))e^

(A

(^^))07T

[5]eL^

[5]e^

[5];

and

3. moreover, if we assume that ^(S) == 0, then

H^(P) ^ ^

(A

(T^))e^

(A

(^J•Q))0^

[5]eL^

[5]©^

[5],and

4. H^(P) ^^

(A

(^^))e^

(A

(T^))0^

[5]e^

(A

(^^))0 (L

^] e H^S}) e Ly[s} © L^[5] e B^s}.

On the other hand, as k = 1, the formulas in Theorem 4.1 in fact become very simple.

COROLLARY 4.2. — Suppose that {^y\y € Q} is the Poisson structure on Poisson manifold P = S x Q and [ujy\ = [d;o] + g(y)[C}, where

986 _{PING XU}

[C] e H2(S, T) is not zero in H2^, R) and g : Q —>R is a submersion.

Then

n—z

HW ^ Q^r

'

^] eB^s}) 0r°°(/\(r^)),

i=0

where L^^S] = C°°(Q) 0 L^\S), H^S] = C°°(Q) ^ (7T(^)/^), L^\S) = {[0] e ^(5)1 [C] A [^] = 0}, 7P = {[C] A [0]| [^] e JP-2^)}

and TQ is the foliation on Q induced from the fibration g : Q —> R.

Remark. — It is possible to generalize the formula in Theorem 4.1 to those Poisson manifolds in which the symplectic foliations are locally trivial fibrations. If P is such a Poisson manifold with symplectic fibration

p —^ Q^ to this fibration, we can associate a canonical flat vector bundle for each n e N , denoted by 271 —> Q, with ^(Tr-1^),^) being the fibre at each q e Q. The flat connection there can be defined as follows.

Given any two points x,y e Q and any path -y(t) in Q connecting x and y , let <^ : TT-¹^) —> Tr-^^)) be a family of diffeomorphisms such that

^° = id. We define the parallel translation between ^(Tr-¹^),^) and H71-^-1^)^) to be the isomorphism (<^1)*. It can be justified by the following argument that this connection is well-defined. Suppose that ^ is another family of such diffeomorphisms; we let 9^ = (<!>*)-1 o ^^ which are automorphisms of Tr-^). Thus, 61 is homotopic to 6°, which is the identity. So (61)* = id; therefore, (<^)* = (^1)*. It also can be easily checked that 571 is a flat bundle. In particular, if the base space Q is simply connected, 271 becomes a trivial bundle without holonomy.

The Poisson structure on P defines a global section uj of the vector bundle 22 —> Q. We define a subbundle P of TQ by saying that v e TqQ belongs to V if and only if V^ = 0. Since V is a flat connection, i.e., V[x,y] = [Vx,Vy] for all X,Y C X{Q), P is a distribution. Given any fixed q e Q, let ^ denote the morphism from TqQ to ^(TT-^^.R) defined by ^q{v) = V^o; for all v e TqQ. It is obvious that TqQ/Vq ^ Im^q as a vector space. According to Dazord [Dl], if P is integrable,

^q must be of constant rank while q varies in Q. In fact, the image of ^q in H2(7^~l(q),n) is spanned by the first Chern classes of the canonically associated principal bundle over the symplectic leaf Tr"1^) of the symplectic groupoid. Therefore, V is of constant rank, and hence defines a foliation on Q, denoted by J='Q. Obviously, this foliation on Q is the generalization of TQ already defined when P = S x Q.

Suppose that the rank ^>q = k for all q € Q, and there exist k parallel sections [C\\,' • • , [Ck\ for the bundle S2 —> Q, such that they span the space Im^q for all q e Q. Then, the Poisson cohomology of P can still be described by the same formula in Theorem 4.1, where 1/^(5') should be considered as a subbundle of E⁷⁷¹ —> Q in an evident way, and ^^[S] as the space of all the sections. jFT[5'] should be understood similarly. This is true, because a symplectic groupoid of P can be obtained by patching together the symplectic groupoids over the Poisson submanifolds of P in which the symplectic foliations are trivial, as constructed in Theorem 2.1.

However, we do not know a general formula.

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[Wl] A. WEINSTEIN, The local structure of Poisson manifolds, J. Diff. Geom., 18 (1983), 523-557.

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Manuscrit recu Ie 3 janvier 1991.

Ping XU,

Department of Mathematics University of California Berkeley, CA 94720 (USA).

Current address :

Department of Mathematics University of Pennsylvania

Philadelphia, PA 19104-6395 (USA).