Bismut superconnections and the Chern character...

K-Theory 9: 507-528, 1995. 507 @ 1995 KluwerAcademic Publishers. Printed in the Netherlands.

Bismut Superconnections and the Chern Character for Dirac Operators on Foliated Manifolds

JAMES L. HEITSCH Department of Mathematics, University of Illinois at Chicago, Chicago, 1L 60607-7045, U.S.A. e-mail: [email protected]

(Received: August 1994)

Abstract. In this paper, we show how to define a Bismut superconnection for generalized Dirac operators defined along the leaves of a compact foliated manifold M. Using the heat operator of the curvature of the superconnection, we define a (nonnormalized) Chem character for the Dirac operator, which lies in the Haefliger cohomology of the foliation. Rescaling the metric on M by 1/a and letting a -+ 0, we obtain the analog of the classical cohomologicaI formula for the index of a family of Dirac operators. In certain special cases, we can also compute the limit as a ~ oo and show that it is the Chern character of the 'index bundle' given by the kernel of the Dirac operator. Finally, we discuss the relation of our results with the Chem character in cyclic cohomology.

Key wards: Bismut superconnection, generalized Dirac operators, foliations.

1. Bismut Superconnection for Foliations

In this section, we fol low pages 454-458 of [BV] making the necessary changes to adapt it to the foliation case.

Let M be a C °° n dimensional manifold and F a C a p dimensional foliation on M. Let go be a metric on M. This induces a metric g r on each leaf L of F . We assume that the tangent bundle T F of F is oriented and that it is spin with a fixed spin structure. Assume that p is even and denote by So = S + ® S o the bundle of spinors along the leaves of F . Given a leaf L of F , denote by V ° the Levi-Civita connection on L and on SOIL. We denote also by V ° the connection on T F over M given by the orthogonal projection of the Levi-Civita connection for go on T M . Note that V ° on T F over M restricted to a leaf L is just the V ° above.

Let £ be a complex vector bundle over M with Hermitian metric and connection. Denote by V ° the tensor product connection on So ® £JL. These data determine a smooth family D @ £ = (DL ® g) of Dirac operators, where DL @ g acts on sections o f So @ £1L. They also determine another family o f Dirac operators, D 6 g = (D (~ gx)~:eM where D @ £~ acts on sections o f ~r*(S0 @ gIL~) where 7r : Lx -+ Lz is the ho lomony cover of the leaf through x. Thus D@£ is a smooth family of Dirac operators on g, the graph of F,

5 0 8 JAMES L. HEITSCH

We denote by u the normal bundle to F in M and by v* its dual. At x 6 M, u has fiber

us : { X E TM~IX J- TF~}

and u* has fiber

= {co c T*Mxl cotTFx = 0} .

Let V/e be a Bott connection on u*. Ifwl , . . . , coq is a local framing for u*, then

q

j=l

where 0} are local one forms on M and the 0} satisfy dco{ q { = ~i=lOj A coj. That is, the composition

r(~,*) Y-% F(T*M ® v*) A F(T*M A u*)

is just a., -~ dco. V B induces a connection on Au* also denoted V B so that

F(Av*) V--~ B F ( T * M ® A u * ) & F(T*M Au*)

is also just co --> dco. (Later we shall assume that F is transversely SL, i.e. u* has a holomony invariant SLq structure which is preserved by V B.)

Now consider the bundle 1; = T F (9 u ® v* = T M (9 u* over M, and define a symmetric bilinear form g on 1; as follows. T F and u (9 u* axe orthogonal and g t T F is gotTF, gtv (9 u* is given by the canonical duality, i.e. u and u* are totally isotropic and 9 (X ,w) = co(X) for X 6 v, co E u*.

PROPOSITION 1.1. There is a unique connection V on ]2 so that

(1) V preserves u* and Vlu* : V/e (2) V preserves 9 (3) for all X , Y 6 r ( T M ) , V x Y - V y X = [X, g].

Note that, in general, V does not preserve T M but that for X, Y 6 F(TM), V x Y - VrX 6 F(TM). For the proof, see [BV] p. 455.

Recall that the dimension of v* is q, and consider the vector space V = R ~ (9 R q ® (Rq) *. Define a bilinear form Q on V as g was on '12, i.e. R p is orthogonal to R q @ (Rq) *, QIR p is the usual inner product, and QiR q (9 (Rq) * is given by the canonical duality. Let C(V, Q) be the associated Clifford algebra and set S = A(R q)* ® S0 where So is the spinor space for R p with the usual inner product. Let P0 be the representation of the Clifford algebra of R p in So. Then S is the

BISMUST SUPERCONNECTIONS AND THE CHERN CHARACTER 509

spinor space for C(V, Q) with the Clifford multiplication being defined by

p ( x ) ( ~ ® ~) = ( - 1 ) ~ ° g ~ ® ;o(X)~, p (y ) (~ ® ~) = -2i(v)~., ® ~,

; (¢ ) (~ ® s) = ¢ A ~ ® ~,

for

See [BV], p. 456 and for general facts about spinors and Clifford algebras, [LM]. The above facts allow Berligne and Vergne to give a beautiful and concise definition of the Bismut superconnection for fiber bundles which we now extend to foliations.

Consider the vector bundle S = A~'* ® So over M and the bundle of Clifford algebras C(V) over M associated to 12, g. Then the fiber over x of S, Sz is a module for the algebra C(V)x and we denote the module action by p. The connection V on V induces a connection V on S ([BV], p. 456; or more generally [LM], Ch. 4). We shall also denote by V the tensor product connection on S ® g.

A Bismut superconnection for F is the Dirac type operator on r(s ® g) defined as follows. Let X 1 , . . . , Xp be a local oriented orthonormal basis of TF, and X v + I , . . . ,Xn a local basis of u. Let Xi~,. . . ,X* be the dual basis in T F ® ~*,

~ X ~ i . e .X i = Xi f o r l ~< i ~< p, i = a~i f o r p + l ~< i ~< nwhereaJ i E u* and ~i (Xj ) = 5ij. Set

B= ~(p(X*) ® I)Vx, i=l

= ~po(X i )Vx~ + ~ v x ~ . i~l i=pq-1

(This is identical to the definition in [BV] for fiber bundles. We have only substituted a Bott connection for their connection V B on the base space/3 of the bundle.) Note that B does not depend on the choice of X ~ , . . . , X~.

We now show that B satisfies the usual properties (suitably interpreted) for a Bismut superconnection. Note that S ® g = Au* @ So ® £, so given (local) sections w of Au* and ¢ of S ® $, w- ¢ is a (local) section of S ® g. Given any point x E M, there is a neighborhood U of x and a submersion 7r: U -+ /3 so that the leaves of FlU are just the fibers of ~r. Thus we may identify Av*tU with 7r*(AT*B). A section w of Au*IU which is the pull back of a section of AT*B is called a baselike local section of Au*, and it satisfies Vxa~ = 0 for X E TF.

PROPOSITION 1.2. Assume V B is an SLq connection. Then B satisfies

(1) B ( ~ " qS) m da~. q~ -/- (--1)deg~a¢ • BCfor~ a baselike localsection o f Au* and ¢ a local section of S ® g.

(2) B does not depend on the choice ofSLq Bott connection V B.

510 JAMES L. HEITSCH

Proof. We apply the proof of Proposition 1.14, p. 457 of [BV] to obtain (as V B is trace free)

¢)

n

= 1)vx - ¢ i=1

f~

= ~ ( p ( X ; ) ® 1)[(VBX~a)) • & + co. Vxi¢] i=1

n n

= (co . vf ,co).¢ + i=p+l i=1

since VBxco = 0 for i ~< p as such Xi C TF and co is baselike. To finish, note that

(p(X/*) @ 1)co = (--l)degwco • p (X; ) @ 1

n . B and as V B is a Bott connection, ~i=p+l&Z~Xia) : dco.

The proof that B does not depend on the choice of V B in [BV] works equally well here.

As noted above, S ® £ = h u * ® S 0 ® g a n d S 0 = S + ® S o . T h u s S 0 ® g i s Z2 graded and Au* is Z graded, so $ ® £ has a total Z2 grading and we may write S ® £ = ($+ ® g) ® ( S - ® g). It is immediate from the fact that V preserves the grading that B is an odd operator, i.e. B maps F(S + ® g) to F(S- ® g) and vice-versa.

Finally, we may use the Z grading on Au* to grade the operator B, i.e. B = B [°] + B [1] + . . . where B[q: F ( A k u * ® So ® £) --+ F ( A k + i u * ® So ® g) .

We then have

PROPOSITION 1.3. The term B [°] is given by the family D ® g of Dirac operators along the leaves of F.

Proof. This is a local result and by Proposition 1.2 we need only check it for a local section ¢ of So ® g. Then we have

P BIO3¢ = po(XO- Vx,¢

i=1

but Vxi¢ = V°~¢ for these Xi so we are done. We finish this section with two remarks. First, if F is the foliation of a fiber

bundle given by the fibers, B is a Bismut superconnection as in [B], [BV] and [D2].

BISMUST SUPERCONNECTIONS AND THE CHERN CHARACTER 51 ]

Second, since B is independent of the choice of V B, when we examine B locally (i.e. when we apply the results of [D2]), we may assume that V B is the pull back of a Levi-Civita connection from the base space B of a local submersion 7r: U -+ B defining F on U. Thus we are in a situation where we may apply the local results of [D2].

2. Asymtotics of the Curvature of B

Consider the operator B 2, the curvature of B. For w and ¢ as in Proposition 1.2, we have

:

Thus

(B2@IL = IL(CIL)

where IL is a second order differential operator on F($ @ glL). By the remark at the end of the previous section, the formulas for IL in [B], [BV] and [D2] are valid in our situation. In particular, the leading order symbol of IL is given by the metric gL. We now assume that M is compact. It follows from the construction in [D1] of the heat kernel on non compact manifolds of bounded geometry that we may form the operator e - Is acting on F(S @ £1L). In the fiber bundle case, the Schwartz kernel o fe - I s is a section of the bundle AT*B @ Horn(S0 @ $) over L, where B is the base space of the bundle. The analogous statement (i.e. replace AT*B by Au*) is not necessarily tree in our case as the holomony of t/*JL may not be trivial. Thus we now shift our attention to the graph G of F and its associated foliations.

Recall that G consists of equivalence classes of paths 7: [0, 1] --+ M such that the image of 3' is contained in a leaf of F. Two such paths are equivalent if they start and end at the same points and the holomony along them is the same. There are two natural maps s, r: ~ --+ M given by s([7] ) = 7(0), r([7] ) = 7(1). G has two natural transverse foliations ,~ and F~ whose leaves are, respectively, s-l(x), r- l(x) for x E M.

There is a canonical lift of the normal bundle u of F to a bundle u9 C T~ so that TG = TFs ® TF~ ® u9 and r.u~ = u, s.u~ = u. It is given as follows. Let [7] E G with s([7] ) = x, r([7] ) = y. Denote by exp: u ~ M the exponential map. Given X E ux and t E R sufficiently small, there is a unique curve 7t: [0, 1] -+ M so that

(i) 7t(0) = exptX, (ii) 7~ C Lexp~X,

(iii) 7~(s) E exp(~,.~(~)).

In particular 70 = 7. Thus the family ["/t] in G defines a tangent vector Jf E TG[@

It is easy to check that 8 , (J() : X and r,(3~) is the parallel translate along ~/of X to u~,.


The metric g0 on M induces a canonical metric 9o on G as follows. T ~ = TFs ® TI> ® u 9 and these bundles are mutually orthogonal. On TF~ ® u9 we define 9o to be s*90 and on TF~ we define 9o to be r*(9olTF).

We shall concentrate on the foliation F~ with leaves Lz = s-l(x) , x E M. Note that rlL~ ~ Lx is the holomony cover of L~ and it is a local isometry. Thus L~ is a manifold of bounded geometry, with bounds independent of x.

We now proceed as in Section 1 to define the Bismut superconnection B for the foliation F~. Note that the dual normal bundle u~* for Fs is .s*T*M, and we use for the Bott connection on u 2 the pull back by 8 of any torsion free, trace free connection on T*M. (e.g., the Levi-Civita connection). Note also that So for TFs is r*`so. For the auxiliary bundle we take r'g. Let w E F(AT*M) and ¢ E F(s*(AT*M) ® r*(,So ® g)). Then

B2(r%a • ¢) = r%oB2¢

so for each x E M,

where I~ is a second-order differential operator on sections of

@*(AT*M) ® r*(8o ® = s*(AT*M~) ® (r*(`so ® gIL~)).

As above, we may form the operator e -& acting on sections of 8*(AT*Mx) ® (r*(So ® gIL~)). Let PX(7 , ~) be the Schwartz kernel o f e -x~ . The action o f e -I~ commutes with the left action of AT*Mx so for each [7] E L~,

Px(7, ? ) E AT*Mx ® Hom(r*(80 ® $)[v]).

Taking the supertrace associated to the splitting ,So = ,S+® S o yields Tr~(P~(7, -~)) E AT*M~.. In fact we have

LEMMA. Tr~(Px(7 ,7) ) E / W ; . Pro@ Let U ~- R q X R p be a foliation chart in M with transversal T "" R q x {0}

and let 7r: U -+ T be the projection. For any x E U , / ~ is isometric to L~(x) (by

composition with any path in the placque of x from x to 7r (x)). Thus on s - 1 (U), the structure is equivalent to the obvious structure on s -1 (T) x R p over T x R p, with leaves Lx × {y}, x E T, y E R p. Let V T be any trace and torsion free connection on T*T and let V ° be the canonical flat connection on T*R p. Using V T ® V ° as the Bott connection in the construction of B, on s -1 (T) x R p it is easy to see that for (x, y) E T x R p, I(~,y) = Is where Ix is the curvature of the B defined

on s - l ( T ) using V r . But e -I~ takes values in AT*Tx ® Uom(r*($0 ® g)) and AT*T = A,;.

Now scale the metric g0 by 1 / a for a E R + and form the operators [x,~ and e - j~,~ with Schwartz kernel P~#(7 , ~) and supertrace Tr~(P~#(7 , "~)) E Au~. Denote by


f~ the curvature of the bundle TFs and by L the curvature of r*g. Let ./i and ch be respectively the ~] and Chern character polynomials and consider the differential form A(f~) ch (L) on G. As AT*9 = s*(AT*M) A (AT*F~), for each ['7] C G, we may consider the part of (A~(ft) ch (L))([7]) lying in s*(AT* Mz ) A (APT * F~ )[.~]

where x = s([7]), and we write it as [7](f~) ch (L)]p ~/A vol Fs where vol Fs is the volume form on Lz. Then [A (~2) ch ( L)]p,.y E s* ( A T* Mx) and we may use the proof of Theorem 3.4 of [D2] word for word to prove the following generalization of a theorem of Bismut [B] (also proven in [BV]).

THEOREM 2.1. Forall ['7] C 9 with s([7])= x,

T X~a lira r~(P (7,7)) (2~i)-(P/2)[ft(f~)ch(L)]pm a--+0

We interpret this as follows. Suppose r([7] ) = y. As TF~ = r*TF,

A(ft) ch(L)= r*(A(ftF) ch(Le))

where f~F is the curvature form of TF and Lc the curvature form of g. Clearly then

A(a) ch(L) 6 r*(AT*M) = (A6) A (AT*&)

u* = = s*(Au~). Thus, have so [A(f~)ch(L)Jp,~ is actually in (A a)b] r*(/\u~) we the following interpretation of Theorem 2.1. For each [7] E G, with s([7] ) =

= y,

lim st (7, 7)) (27ri)-(P/2)Y~([A([~F)ch(Le)(Y)])p a--+0

where the part of A(ftF)ch (Le)(g) lying in ((Au*) A (APT*F))y is denoted [A(ftF)ch(LE)(V)]p A vol/~, volF u is the volume form of Lz at y, and T~ is parallel translation along 3' from y to x of the bundle Au*.

If F is a fiber bundle foliation of M where M is compact then Theorem 2.1 is a strong local form of the index theorem for families of elliptic operators. See [B], [BV] and [D2].

3. Chern Character for Foliations

In this section, we define the Chem character for a foliation F of a compact Rie- mannian manifold M whose graph G is Hausdorff. We begin by recalling some constructions due to Haefliger [H]. Let H = {U1,.. . , Ur} be a good cover of M by foliation charts [HL]. Let Ti C U~ be the transversal and set T = uTi. We may assume that the Ti are disjoint. Let H be the holomony pseudogroup induced by / ; on T. Let ~ck(T) be the space of smooth k forms on T with compact support and


denote by Ut~(T/H) the quotient of Ft~(T) by the vector subspace generated by elements of the form a - h*a where h E H and a E f t , (T) has support contained in the range of h. Give f~c(T/H) the quotient topology of the usual C 0° topology on a~(T). The exterior derivative d: f~ck(T) ~ a~+l(T) induces a continuous differential dF: fl~(T/tI) ~ fl~+~(T/It). Note that flkc(T/H)and dF are independent of the choice of cover/J. We denote this complex by {Fl~ (Tr F) , dF} and its cohomology by Hc(Tr F).

If F is the foliation given by a fiber bundle M ~ B, ~]~(Tr F) is the usual de Rham complex of B and H2(TrF) is just the usual de Rham cohomology of B with compact supports.

As the bundle TF is oriented, there is a continuous open surjective linear map,

f : a +k(M) r)

which commutes with d. It is given as follows. Let a3 E [F+k(M) and write a~ = Wl + . . . + a~ where wi E Ft~+k(Ui). We may integrate wi along the fibers of the submersion 7ri: Ui ~ Ti to obtain ~i E f~(Ti). Define f w to be the class of Ec?~. in f~(Tr F). It is independent of the choice of the wi and of the cover/4.

Let P [ (3', ~) be the Schwartz kernel of the heat operator e -tI~ on Ly as in Section 2. For y E M, we denote also by y E Ly the constant path at y. Thus P~(y, y) makes sense. For x ELy, let 7z be any path in Ly from y to x. Composition with 7~ induces an isometry between/~x and Ly. In particular, Pf(x, x) = P~(Tx, 7.~), which we now write as Pg(x, x). Thus P[(x, x) is smooth along L v. We shall show below that it is smooth transversely. If we denote the volume form along the leaves of F by dx, and recall that Tr~(P[(x,x)) E u*, we may regard Tr~(P[(x, z)) dx as a differential form on M.

THEOREM 3.1. The element f Tr~(PY(x,x))dx in a*(Tr F)is closed. Its cohomology class in H~'(Tr F) is independent of the metric g on M.

DEFINITION 3.2. The nonnormalized Chem character ch(D ® g) of (D ® g) for the foliation F is the cohomology class of f Tr,(P~ (x, x)) dx in H~*(Tr F).

The proof of Theorem 3. t is rather lengthy and we will split it into a collection of propositions and lemmas.

Each Ui in the good cover/4 is of the form Ui ~ D p (1) x D q (1) with coordinates ( x l , . . . , xp, Yl , . . . , Yq). The transversal Ti - 0 x Dq(1) and has coordinates (Yl, . . . , Yq). The leaves of FIUi (the placques of Ui) are of the form DP(I) × {v}, c Dq(1).

We now adapt pages 110-115 of [B] to our situation. This requires some rather complicated estimates. The key ideas are the following. First, consider the space Xi = s - l (7) ) C G. Then since 6 is Hansdorff, Xi is almost a fiber bundle, i.e.

BISMUST SUPERCONNECTIONS AND THE CHERN CHARACTER 5 1 5

it has the necessary properties so that we may adapt Bismut's arguments to our case.

Second, because of the bounded geometry of the leaves of 2~ we have the following estimate (see [BE], [D1]).

(3.3) Given a nonnegative integer i, and integer multi indices ct, fl, and a real number S > 0, there is a constant C > 0 such that for all y E T, (equivalently, y E M ) , x , z E/~y and 0 < t ~< S,

l]oi+]~]+lN/OtiOxC~OzZPg(x ' z) N <~ Ct_p/2_i_lal_tNex p "-dL(x, z) 2]

Here O/Ox and O/Oz come from the coordinates on Lv obtained from the good cover b/and dr( , ) is the distance on L v.

From this we get

(3.4) Given S > 0 and 5 > O, there is C > 0 so that for all y E T, (equivalently, y E M ) , x , z E L v withdL(x,z) > ~, andO < t ~ S,

IIP?( , )11 Ce-SU4s.

Proof By (3.3), we have

lIPg(x,z)ll <~ Cot -p/2 exp[-62/4t],

which is a continuous bounded function on [0, ec). Thus there is C1 > 0 so that for all t in [0, oo)

Cot-p/2exp[-~2/4t] <~ C1Co exp[-~2/4S].

The choice of the normal bundle u defines a local parallel translation for the leaves of F and so also for the foliation Fs on Xi. See [Hu], [W]. In particular, let K be a compact subset of/~v C 32/ with y E K. Let 7o: [0, 1] -+ Ti be a parameterization of the ray in Ti from y to yC If y~ is sufficiently close to y, then for all x E K, there is a unique path 7(t) in Xi so that 7(1) E i, v0(t ), 7(0) = x and

7 '( t) E ua,-y(t). The parallel translate of x to Lu' is just 7(1)-

Given a Riemannian manifold W, a > 0 and w E W, we denote by B~'(w) the ball in W of radius a centered at w. We denote by dw(., .) the distance function on W.

Given r > O, set

can be parallel translated } R = s u p alB~U(y) to2~y, f o r a l l y ' E B ~ ( y )


i.e. B Lu (y) is the largest ball about y which can be parallel translated to Z V for y~ r close to y. Note that if F is a Riemannian foliation, then R = ~ . In the general case, 1~ ---+ ~ as r ---+ 0.

Denote by BR(y') the parallel translate of B L~ (y) to/~y,. Set

x~,R -- U BR(y') for ~' e BT'(y) c T~

and note that Xi,R is a fiber bundle over B~ ~ (y). We may use the parallel translation on Xi,R to translate the bundles S ~= and £ to BR(y) as well as all the other data involved in the construction of B and so we may view everything as taking place

on fixed bundles over BR(y) C Lv. In particular we may view Pg'(x, z) as being defined on BR(y).

Finally, let f~ be the section of u 9 over Xi corresponding to O/Oy~ on Ti. Since we use the parallel translation over rays in Ti starting at y, we have that at y

OIV Oy~ - V f~IY"

PROPOSITION 3.5 ([Prop. 2.8 of [B]). For all t > O, 0

f Tr~(P:(x,x)) dx

= - t f T r ~ [c~I~P u ] dx

in f~(Tr F) . Proof Let a be a fixed positive number. For each R, let ~R be a smooth

function on L~ so that 0 ~< ~ ~< 1, support ~R C BR(y), and ~R]BR-a(y) = 1. Further, we require that the derivatives of ~t~ are uniformly bounded for all R. We now consider all objects to be on Ly. Let I y'* denote the adjoint of I y'. For any

x, x ~ E L y and t > 0 we have

0P~' _ _ l y e , p y t { X ot ( x , ~ ' ) = - ~ f p : (x,z') = ~ , ~ ~ ,~').

Let

0 < t ' < t , y ' e B T~(y) and x , x ' e BR-2a(y).

It follows from (3.3) that

O I L 2 y

/L ~* Y~ = y ~2(u)[P~,'(x, u)1~PY_t,(u,x ') - {I~ P~, (x, u)}P:_t,(u,x')] du

fL ~ ( u ) P ~ ' ( x , u)[I~ Y' y ' = - [~ ]P~_t,(u, x ) du- y

W yt 2 yt - [Ju ~/~(u)]P~, (x, u)P~ut,(u,x' ) du

BISMUST SUPERCONNECTIONS AND THE C ~ R N CHARACTER 517

where W = BI~(y) - BR-~(y) and J~' is the positive order part of the differential operator Iv' . As in [B] we may conclude that

Py(x, x')- P:(x, x')

£ d t , L 2 ' R( )et ' = - I ~ ]Pi_t,(u,x ) d u - JLy

at' [J.Y qo2(u)]P~'(x, Y - u ) P i _ t , [ u , x ' ) d u . (3.6)

LEMIVIA 3.7. For h E Ty2~, I[h]] = 1 and ,s C R sufficiently small, consider the ray in Ti at y given by y~ = y + sh. Then

1 dr' [JY qo2(u)JPt y, (x, u)PY_t,(u,x ') dx = 0 s OJo

and

lj~ot / W ' ~olim ~ dff [JY ~2(u) ]P~ (x, u)PY_t,(u,S) du = O.

Proof Recall that as s --+ 0, R --+ ~ . By (3.3) and (3.4), for 0 < t ~ ~< t, mid the fact that the derivatives of ~R are bounded, we have that for R sufficiently large, the integrand is bounded by C e x p [ - ( R - a)2/4t] where C depends only on t. Because of the bounded geometry of the leaves of F (and so also of F~), the volume of W as a function of R grows at most exponentially. The first result now follows.

For the second, we must estimate how R depends on s. Let 5 be the Lebesgue number of/g. The cover b/induces a cover L/i of Xi whose Lebesgue number is at least 5. Let x E BR(y) C Lv. Then there is a chart U C Ltl with x E U and with distance of x to OU at least 5. In particular, there is a transversal Tx C U centered at :c with radius/> 5. Let 3/be a geodesic in Ly from y to x of length dL~(y, x). We need to estimate the amount Tx is contracted as it is parallel translated along 7 to y. The path 7 can be covered by R/~ charts from L/i since it has length dL~ (y, x) < R. The amount Tx may be contracted from one chart to the next is bounded by say A > 0, which depends only on b/. Thus the image of Tx translated to y has radius >1 5A R/6 and so s >>. 5A R/6. It now follows from the previous estimate on the integral that the second limit is zero.

This lemma allows us to ignore the second integral in (3.6) in what follows. Let

j[ 2 y~ t f ( t ' ) = 9~R(u)Pi, (x, u)[[ y ry'l,Y - ~u j . t _ t , ( u , x ) du.


Note that for

yl = y + sh,

1 / s f ( t ' ) has limit

h C TyTi, llhll = 1 and s small,

1/8[I y - IY'lPiV(x, X')

and limit

as t f - ~ 0

1/~[ff; - xf ; ' ]P[ ' (~ ,x ' ) as t' ~ t.

Also note that by (3.3), the bounded geometry of the leaves of Fs, and the remarks below, the limits are uniform over x and x ~, s small, and t in J a bounded closed interval in (0, co). Thus there is e > 0 so that for 0 << t' <<. ~, 1 / s f ( t ~) is bounded independently of x, x r, small s and t E J.

Now for 0 <<. t ~ <~ t /2, we have by (3.3) that for any multi indices a and t ,

( ol~t+tzt / Ox ,~ouÈ )PL` , ( ~, ~,)

is uniformly bounded by a super exponentially decaying function, so its L 2 norm (as a function of u) is uniformly bounded. For all x and 0 < e ~ t' ~<

t /2, 9~R( • )P~'(x, .) is uniformly bounded in the pseudo-differential operators of order zero. As I f is differentiable in y and its coefficients and their derivatives are uniformly bounded, (see Equation (3.1) of [D2]), the L 2 norm of ~R(u)( I/s)[I~ - I~']P~_t,(u, x'), as a function of u, is bounded independently of t ' , (c ~< t' <~ t /2) , x', small s and t E J. Thus on [~,t/2], 1/s f ( t ' ) is bounded

and we have f~/2 f ( t ' ) dt' <~ Cs where C is independent of x, x ~, small s and

t C J. After considering the adjoints of I~ and [Yu' to handle t /2 <~ t ~ <<. t, we may conclude as in [131 that

IIP,~'(z, x') - e l ( x , x')II = o ( t y - J l ) . (3.8)

Thus, PY(x, x') is continuous in (t, y, x, x'). In addition, (3.8) coupled with the

fact that P~,' (x, x ~) is uniformly super exponentially decaying easily implies that

as yr ~ y, ~RP~' converges to P~ in the sense of zero order pseudo-differential operators while staying uniformly bounded for 0 < ~ ~< t ~ < t.

LEMMA 3.9 (Equation (2 .47)of [B]). For 0 < t' ~ t /2 and h E TvTi, LIhN-- 1,

l i m [ ~2(u)P~+~h(x,u) [ I~+~;- I~] eYt_t,(u,x')du s-~O JLy

= f Pj(x, JL


Proof. The difference of the two integrals equals

l i r a / ~(u)[P~+Sh(X,H)-- P~,(X,U)J ['~+~;-- I'~] P~_t,(U,x')du+ s~O dg~

[ [y+sh _ O~ h] o JLy Oy ]

+ lira [ [~} t (u ) - 1]P~(x ,u)[Oi l .h] - P~ t ,(u,x')du. ~-~o iLy [Oy -

In the first limit we note that for small s, (I~ +~h - I~)/s has uniformly bounded coefficients since 0Ig/Oy exists. Thus

-

has uniformly bounded L 2 norm as a function of u. As the norm of

as a zero order pseudo-differential operator goes to zero as s -+ O, the first limit is ze ro ,

In the second term, we note that for sma/1 s, the expression in the brackets has coefficients which are uniformly bounded by a bound of the form C8 since 02I~/Oy 2 exists. By (3.3), the L 2 norm (as a function of u) of

is uniformly bounded by a bound of the form Cs. As the norm of 9~( • )P~,(x, .) as a zero order pseudo-differential operator is bounded, the second integral is bounded by a bound of the form C~ and so has limit zero.

The third limit is zero since the support of y~ (u ) - 1 is Z~ - BR-~ (y), x, x ~

are fixed elements of Ly, Ly grows at most exponentially, R -~ c~ as s ~ 0, and the integrand is super exponentially decaying.

Note that the convergence in Lemma 3.9 takes place boundedly, i.e. the integral whose limit we are taking is uniformly bounded as a function of t f. This will allow

as to interchange f~/~ and lira,_+0 below. An obvious symmetry argument shows that Lemma 3.3 also holds for t /2 <~ t' <~ t. We thus have that P~(x, x ~) is C 1 in y and

OP? . . r r , .0s " =---- ( x, x ) = - [ t dt' ]. t:~Vf ( x, v

By iteration, we have that Pg(x, x') is C ~ in y and so also in (t, x, :d, y). We remark that the above proof may be repeated for the Laplacian I of the lift

to F~ of any leafwise Dirac operator on F. Thus we have for any such jr,


COROLLARY 3.11. Suppose ~ is Hausdorff. Then the Schwartz kernel P~(~, 7) of e -tI is smooth in t, V,~ and 7. In particular, the function Qt(rl) = PrY(y, zl), where y = s( rt ), is smooth on G.

For more in this regard, see [R]. In addition, standard techniques can be used to show that

0i+l~l+lN+l'~t y ll (x,z)

satisfies an estimate similar to (3.3). Note that I is invariant by the left action of G, so PtU(~, 7) = Qt ( ( -17) • Qt(17)

is also called the kernel of e -tz. Returning to our case, it is easy to show that OPg/Oyc< is uniformly bounded.

This implies that

o fTr [Pg(x, )]dx Oy<~

t Ol~ y

which, by Tonelli's theorem, is

- fotdt' f fLyTrs [PYt (x,u)O~@al~V_-t,(u,x)] dudx

and as in [B], this equals

rosg y .

LEMMA 3.12. The above expression equals

t [OlYz Y " u)P~(u,x)] dudx.

Using the semi group property of Pt y , we thus have proven Proposition 3.5.

Proof of 3.12: Set

t aY~ - cgy~ "-


We need to show f fLyk(x,u)dudx = 0. Let {(Ui, Uj,Tijk)} be the cover of G corresponding to the good cover U (see [HL]). Then Ui, Uj E /g and 7ijk is a leafwise path from Ui to Uj. On Ti C Ui, f f L k is represented by

Ej,~ fp~ fpj k(x, u) du dx, where u E Uj is the class of the leafwise path starting at x, ending at u and parallel to 7ijk, Pi is the placque of x in Ui and Pj is the placque of u in Uj. For the chart (U~, Ui, "y), ~/a constant path, fp~ fp~ k(x, u) du dx = 0 by symmetry. For the chart (Ui, b~, ")'ijk), set Iiik = fp~ fpj k(x, u) d u dx. Now

on Tj there is a corresponding integral Ijik = fj~ fp~ h(x, u ) d u d x for the chart U -1 ( j, Ui, 7ijk)" Note that Iijk = -Ijii;. We may move the term ljik to Ti (where it

will cancel I~jk) to obtain another representative of f fL~ k(x, u) du dx. Doing this procedure a finite number of times, say over all "Yijk of less than a fixed leafwise length, we obtain a representative for f fL~ k so that on any ~/} it is given by

/P~ /L~-BR(y) l~( X, u) du dx,

where R is as large as we like. But for fixed t and t ~ with 0 < t ~ < t, .there are constants 6'1 and C2 so that

Ik(x, u)l ~< C1 exp[ -d~(x , u)/C2].

Let V and A be upper bounds for the volume and diameter of the plaques in the cover/J. Then

ffp~ fLy_Bn(y) k(x' u) dudx

<~ V j2 Ctexp[-d2L(X,u)/C2]du

C 1 V e x p [ - ( R - A)2/2C2] ffL~-BR(y)exp[-d2(x' u)/2C2] du

as x and y are in the same plaque of Ui and as dLy (y, u) > R, dLy (x, u) ~> R - A.

As Ly grows at most exponentially, the integral is bounded, and this expression can be made arbitrarily small by making R large. Thus we have the lemma.

To finish the proof that f Tr , (P~(x, x)) dx is closed, we may use the proof of Proposition 2.9 of [B], mutatis mutandis. In particular replace both f6"y and fx by f , and to obtain (2.58) of [B], apply f fL~ to (2.57) and interchange the variables x and u as in Lemma 3.12 above.

To finish the proof of Theorem 3.1 'we have

PRO PO S ITI O N 3.13. The c lass of f Trs ( P{ ( x, x ) ) dx in H ~ ( Tr F) is independent of the metric g on M (but not necessarily independent oft).

522 JAMES L, HEITSCH

Recall that c--h (D ® g) is the cohomology class of f Trs(P~(x, x)) dx.

COROLLARY 3.14. The cohomology class ch (D ® £) in/ /2(TrF) is equal to the class of

(27ri)-P/2fl(flF) ch (Lv),

where f~F is the curvature form of TF and Lc the curvature form on g as in Section 2.1.

This is immediate from Theorem 2.1, once we note that the convergence in 2.1 is uniform, [B], [BV], [D2], since the estimates on convergence are based on the local geometry of the leaf L, which is uniformly bounded over all leaves.

Proof of Proposition 3.13. Let gs, s E [0, 1] be a smooth family of metrics on M and assume that gs = go for s C [0,1],gs = 91 for s E [3,1]. Denote by g the metric on M x I given by the family g~ on M and the canonical metric on I. On M × I , we have the foliation FI whose leaves are L x {s}, where L is a leaf of F and 8 C I. Let V be a torsion and trace free connection on TM. Then V + ds. O/Os is a torsion and trace free connection on M x I . Let B8 be the Bismut superconnection on M for g~ and B that on M x I for g. A trivial computation shows that for s E [0, 41-), the curvature I of B on L x {s} is I0, the curvature

y~8 of B0 on L. Similarly for s E ( 3 1], it is /1 . If Q1 (x, x ~) is the Schwartz kernel y~8 o f e - r , then f Tr~(Q 1 (x, x)) dx is closed element of f~(Tr FI). This element is

expressible as

a(y,s) + ds A/3(y,s) (3.15)

where c~(y, 8) and/3(y, s) are in f~*(Tr F) . If p~,s (x, x') is the Schwartz kernel of e - 5 , then we have for s E [0, ¼) tO (3, 1],

f = in ~2;(Tr F) . As (3.15) is closed in ~ ( T r FI) , we have

- d F g ( y , 8)

in f~(Tr F) . It is then clear that

f0 1

I ) - = d .


4. The Riemannian Case

In this section, we justify Definition 3.2, at least for a certain class of generalized Dirac operators D ® g on Riemannian foliations, by giving an analytic interpretation of ch (/9 ® g) as the nonnormalized 'Chern character' of the 'index bundle' associated to the family.

Let M, F and D ® g be as before, and assume that F is a Riemannian foliation. Let Q~ be the Schwartz kernel of the projection onto the kernel of DOg, the lift of D ® g to/~,~. Let Q~,b) be the Schwartz kernel of the spectral projection of DOg associated to the interval (a, b), and set Q~ = Q~0,~) u - Q(-~,0)" Note that for each

y, Qg and Q v are smoothing operators. In analogy with the classical families index case we make the following assumptions.

(4.1) Qg is a smooth function of y

(4.2) There is s > 0 so that y -+ QU is a smooth map from M to the smoothing operators in the norm topology.

In particular we assume that for the operator KY = 2(D(~g)Q~ + VyQ~ + QV

(4.3) OK y/Oy~ is a smoothing operator whose k, g norm II OKv/Oyc~ l lk,e is bounded independently of y, and similarly for the adjoint I( v*

(4.4) for small s, and h E TuT with IIh[I = 1,

8 Oy k,e

where the real number Ck,g depends only on k and ~, and similarly for the adjoints.

Assumption (4.4) implies

(4.5) It i s 8 k,g

Note that these assumptions correspond to the assumption in the families case, [AS], [B], [BGV], [BV], that the kernel of DOg ]has constant rank. In this case, zero is uniformly isolated in the spectrum of DOg so (4.1) holds and one may choose s so that Q~ is the zero operator.

Let B be the super connection for Fs as in Section 3 and denote the operator QoB[1]Q0 by V. As Q0 is transversely smooth and commutes with the action of P (AT*M) on P(s* (AT*M) ® r*(S0 O £)), it is easy to see that V is a 'connection' on the 'superbundle' ker Df3g. That is fo r~ E I ' (AT*M) and ¢ C F(s*(AT*M)O <(& ® E)),


Note that the graded operator V has no zero degree term so e -vz is well defined and equals

dim M ~72j

j=O

THEOREM 4.6. Suppose F is a Riemannian foliation and that D@g satisfies 4.1-4.4. Then

c- (D ® = fTr (e-V )dx.

In fact we show more, specifically that there is a family of superconnections 1~ so that as differential forms on ~,

lim Tr.~(e -B~) = Tr~(e -v2) ~ ---+ ( X )

in the C k norm for any k (uniformly on compact subsets).

Proof of 4.6. Let g be the super-connection for Fs given by

g=B+Q .

The zero order term of 1~ is then D@g + Q~ and its spectrum has a uniform gap about 0. The curvature [ of 1~ is ir = I + K where [ is the curvature of B and K is the operator defined above. This is because VQs + Q~V = VyQ~, where the operator on the right has kernel given by applying V to the kernel of Q~. Now zf has leading order symbol the same as I , and we may form the operator e - t I for t > 0, for which the estimate (3.3) still holds.

We now outline the proof of Theorem 3.1 for I. In this case, TR --z 1 and W = 95. To obtain (3.6) for f , we need that ((K~ - K~')Jh{_t,)(u, x') is bounded

as a function of u for 0 <~ t' <<. t/2 and similarly for K* where D{(u, x') is the Schwartz kernel of e - t r . Now (4.5) implies that IlK y - / ( ¢ l J k , e is bounded, so standard estimates give the result. In particular,

.yt ^ y t

= t1((lc - - - " I ^ y

where 64 is the Dirac operator at u. The first element is bounded by assumption, the second by the super-exponential decay of/5~t, ( -, xl), and the third is bounded for k

BISMUST SUPERCONNECTIONS AND THE CHERN CHARACTER 5 2 5

large enough as the leaves Ly have bounded geometry. Thus we have equation (3.6) which now reads

^ ~ 1 P? ) -

= dff ^ y , - u J t - t ' ( u , x ' ) d u ÷ 5, (z, )[Ig zy' P. y

+ at' Pg - I ~ ]Pi-~,(u, x') du. L y

To handle the first integral we need just repeat the proof given in Section 3 without change.

To handle the second integral, we repeat the proof of Section 3, but we must justify the various steps in the proof since K is not a differential operator. We do this as follows.

1. To obtain the bound on 1/sf(t '), we need that ((1/s)(Ku~ - t(~')PtY)(z,x ') and its derivatives with respect to x are uniformly bounded for all x, x ~, small s and t E J (and similarly for K*). This follows easily by standard estimates as above.

2. To obtain (3.8), we need that ( ( 1 / s ) ( K ~ - K~')JP[_t,)(u , x') has L 2 norm (as a function of u) bounded independently of t ~, (e <~ t ~ <~ t/2), x ~, small s and t C J , (and similarly for K*). This follows from (4.4) and the fact that the L 2 norm of P~_t,(u, x') as a function of u is uniformly bounded.

3. Lemma 3.9 follows from 2. above and (4.4). The fact that the convergence takes place boundedly follows from 2. (for e ~< t t ~< t /2) and 1. above (for 0 <<. t' ~< ~), and similarly for K*.

4. To conclude that OP[/Oy~ is uniformly bounded, we need that for small s,

fotdff J[L~P~'(x'u) [ I ( y - I(y+shs PtYt'(u'x')du

is uniformly bounded. This follows from 2. above (for c <~ t ~ <~ t/2) and I. above (for 0 ~< t ~ ~ E), and from considering the adjoint K* for t/2 <~ t J <~ t.

5. For Lemma 3.12, we need that (OKY/Oyc~P~_t,)(u, x) is bounded for 0 < t ~ < t, and t ~, t fixed. This equals

~ ( u , z ) P i _ , , ( z , x ) d z .

As IJOKUOY II-k,k is bounded for all k, it follows from standard estimates and the bounded geometry of the ~,y that OK~/Oy~(u, z) is uniformly bound-

ed. As f)tvt,(z, x) is super exponentially decaying as a function of z the integral is bounded.


The rest of the proof of Theorem 3.1 proceeds as before and we have that f Try(/Sv (x, x ) ) dx is closed.

To see that c--h (D ® g) is represented by fTrs( l f '~(x, x)) dx, we proceed as in the proof of Proposition 3.13. For each s E [0, 1], let B~ be the superconnection B~ = B + sQ¢. Then the superconnection for the foliation F1 on M x I given by B = Bs + d~ • O/Os provides a form fl(y, s) so that

f Xr (P (x,x))dx

= dF fl(y, s) ds.

Now for a > 0 set 1~ = a l/Z6~B~S21 , where 6~ is the operator defined in [BGV], page 287. This is essentially the same as rescaling the metric on M by 1/a. B~

B z is a Bismut super connection for Fs and as above one can show that f Try(e- ~) represents ch (D ® g).

PROPOSITION 4.7. As differential forms on G,

lim Tr~(e -~] ) = Trs(e -v2) C~ ---+ O O

and the convergence is unffbrm on compact subsets in any C k norm.

kernel of e - - and Q(7, () is the Schwartz That is, if P~'~ (7, ~) is the Schwartz B2 V2 kernel of e- , then for all 7 E G,

l i r a = a " - - ¢ - 0 0

In particular

lira Tr,(P~'~(y,y)= Tr~(Q(y,y)). O~-'-+ ~

To prove this we need only repeat the proof of Theorem 9.19 of [BGV].

5. Comments and a Conjecture

Let T be a complete transversal of a foliation F as above. On T we may consider those differential forms which are invariant under the holomony and form their cohomology, i.e. the baselike cohomology of F, t t * ( T / F ) . If F is the foliation given by a fiber bundle with compact base space then H * ( T / F ) = H~*(Tr F) . In general, these two cohomologies are not the same, as the following example shows. Let F be the fundamental group of a surface of higher genus E. F acts on S 1 as the boundary of the Poincar6 disc D and S 1 has no nontrivial F invariant one forms.


Thus the canonical foliation of the fiat bundle D x r S 1 over E has [Il(T/F) = O. It is easy to show that integration over T defines a map from//1 (Tr F) onto R.

There is a canonical pairing of HJ(T/F) and/:/P-J (Tr F), (p = dimension of F) given as follows. Let !P be a closed baselike form on T. Then multiplication by

gives a well defined map from Q~-J(T/II) to ft~(T/It) commuting with dF. The induced map from tIS-J (Tr F) to/-/g(Tr F) depends only on the class of ~ in g j (T/F). As above, integration over T gives a well defined map fT : fIg(Tr F) -+ R, which does not depend on the choice ofT. For oz E IIJ(T/F), ~ E Hg-J(TrF) set

(fl, @ = /T'@ A c2

where #~ E #~, ~p E oz. In particular suppose c 2 is a baselike p form on T so that it defines an invariant

transverse measure for F and let oz be its class in HP(T/F). Then we have the following (which is an easy consequence of Theorem 5, p. 67 of [C2]).

THEOREM 5.1.

<c---h (/) ® g), oz> -- ~,l(27ri)-P/2ft(f~F) ch (Lg) A (p

where f~F and LE are as in Section 2.

Note that the right-hand side is (up to a constant) the topological term appearing in the foliation index theorem of Connes for measured foliations [C1]. Thus the class ch (D ® g) carries the measured index of the leafwise Dirac operator D ® g.

Now the theory of the class ch (D ® g) is in a somewhat unsatisfactory state. Corollary 3.13 gives a topological interpretation of this class, namely as the Chern character of the symbol bundle associated to the operator D ® g. Theorem 4.6 gives an analytic interpretation of it, but only in certain special cases, as the Chem character of the 'index bundle' associated to D ® g. If possible one would like to extend Theorem 4.6 to the general case.

In [CS], Connes and Skandalis have defined analytic and topological indices, ind~(D ® g) and indt(D ® g), for D ® g which live in Ko(C*(M, F)), the If theory of the C* algebra of F. They prove that these two classes are the same. We conjecture the following: there is a map

ch" I(*(C*(M, F)) --+//2(Tr F)

so that c-h(inda(D ® g)) = ~-h(D @ g) where the right hand side is as in Definition 3.2. Note that the results in [C1] imply the dual result to this conjecture, namely that the pairing with the homology H.(Tr F) (in terms of currents of order zero as distributions) passes to K*(C*(M, F)) and satisfies the required properties.


We finish with some remarks on this conjecture. Let P0((, 7) be the Schwartz kernel of the projection onto the leafwise kernel of D@ g. Then P0 represents inda(D ® g) and it is leafwise smooth. If it were transversely smooth we could define ch(ind~(D ® g)) as in Section 4 in analogy with the classical families case. In general, P0 is not transversely smooth (even in the classical case), and it appears quite difficult to extend the methods used in the classical case (see [AS], [B], [BGV]) to the foliation case.

Acknowledgement

It is a pleasure to thank Steve Hurder for several very helpful conversations.

References

[ASI Atiyah, M. E and Singer, I. M.: The index of elliptic operators: IV, Ann. of Math 93 (1971), 119-138.

[BGV] Berline, N., Getzter, E., and Vergne, M.: Heat Kernels and Dirac Operators, Springer- Verlag, Berlin, Heidelberg, 1992.

[BV] Berline, N. and Vergne, M.: A proof of Bismut local index theorem for a family of Dirac operators, Topology 26 (1987), 435-463,

[B] Bismut, L-M.: The Atiyah-Singer index theorem for families of Dirac operators: two heat equation proofs, Invent. Math. 83 (1986), 91-151.

[BE] Buttig, I. and Eichhorn, J.: The heat kernel for p-forms on manifolds of bounded geometry, Acta Sci. Math. 55 (1991), 33-51.

[C] Connes, A.: Sur la throrie non-commutative de l'intrgration, in Algkbres d'operateurs, Lecture Notes in Math. 725, Springer, New York, 1979, pp. 19-143.

[C1] Connes, A.: Cyclic Cohomology and the Transverse Fundamental Class of a Foliation, Res. Notes in Math. 123, Pitman, London, 1986.

[C2] Connes, A.: Geometrie Non-commutative, InterEditions, Paris 1990. [CS] Connes, A. and Skandalis, G.: The longitudinal index theorem for foliations, Pub. RIMS.

Kyoto Univ. 20 (1984), 1139-1183. [D1] Donnelly, H.: Asymtotic expansions for the compact quotients of properly discontinuous

group actions, Illinois J. Math. 23 (1979), 485-496. [D2] Donnelly, H.: Local index theorem for families, Michigan Math. J. 35 (1988), 11-20. [HI Haefliger, A.: Some remarks on foliations with minimal leaves, J. Differential Geom. 15

(1980), 269-284. [HL] Heitsch, J. L. and Lazarov, C.: A Lefschetz theorem for foliated manifolds, Topology 29

(1990), 127-162. [Hu] Hurder, S.: Erotic index theory for foliations, Preprint. [LM] Lawson, H. B. and Michelsohn, M.-L.: Spin Geometry, Princeton University Press, Prince-

ton, 1989. [R] Roe, J.: Finite propagation speed and Connes' foliation algebra, Math. Proc. Comb. Phil.

Soc. 102 (1987), 459-466. [W] Winkelnkemper, H. E.: The graph of a foliation, Ann. Global Anal Geom. 1 (1983), 51-75.

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