Available at: http://www.ictp.trieste.it/~pub-off IC/98/196

United Nations Educational Scientific and Cultural Organizationand

International Atomic Energy Agency

THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

SUBMERSIONS AND EQUIVARIANT QUILLEN METRICS

Xiaonan MaHumboldt- Universitat zu Berlin, Institut far Mathematik,

unter den Linden 6, D-10099 Berlin, Germany 1

andThe Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.

Abstract

In this paper, we calculate the behaviour of the equivariant Quillen metric by submersions.We thus extend a formula of Berthomieu-Bismut to the equivariant case.

MIRAMARE - TRIESTE

November 1998

1Address after 1 December 1998. E-mail: xiaonan@mathematik.hu-berlin.de

Resume. Dans cet article, on calcule le comportement de la metrique de Quillen equivariantepar submersions. On etend ainsi une formule de Berthomieu-Bismut au cas equivariant.

Introduction

Let ξ be a Hermitian vector bundle on a compact Hermitian complex manifold X. By Hodgetheory, we can identify H(M, ξ), the cohomology of ξ, with the corresponding harmonic elementsin the Dolbeault complex Q*(M, ξ). Let hH(M,ξ) be the corresponding L2-metric on H(M,ξ).

Let λ(ξ) be the inverse of the determinant of the cohomology of ξ. Quillen defined first ametric on λ(ξ) in the case that X is a Riemann surface. So we call "Quillen metric ". This isthe product of the L2 metric on λ(ξ) by the analytic torsion of Ray-Singer of ξ. The analytictorsion of Ray-Singer [RS] is the regularized determinant of the Kodaira Laplacian on ξ. In[BGS3], Bismut, Gillet, and Soule have extended it to complex manifolds. They have estabishedthe anomaly formulas for Quillen metrics, which tell us the variation of Quillen metric on themetrics on ξ and TX by using some Bott-Chern classes.

Later, Bismut and Kohler [BK6] have extended the analytic torsion of Ray-Singer to theanalytic torsion forms T for a holomorphic submersion. In particular, the equation on ^Tgives a refinement of the Grothendiek-Riemann-Roch Theorem. They have estabished also thecorresponding anomaly formulas.

In [GS1], Gillet and Soule had conjectured an arithmetic Riemann-Roch Theorem in Arakelovgeometry. In [GS2], they have proved it for the first Chern class. The analytic torsion formsare contained in their definition of direct image.

Let i : Y —>• X be an immersion of compact complex manifolds. Let η be a holomorphicvector bundle on Y, and let (ξ, v) be a complex of holomorphic vector bundles which providesa resolution of i*rj. Then by [KM], the line A-1(r?) <g> (ξ) has a nonzero canonical sectionσ. In [BL], Bismut and Lebeau have given a formula for the Quillen norm of σ in terms ofBott-Chern currents on X and of a genus R introduced by Gillet and Soule [GS1]. Recently,in [B6], Bismut has extended this result to a relative situation. This result and the precedentworks have completed the proof of the arithmetic Riemann-Roch Theorem.

In [BerB], Bismut and Berthomieu solved a similar problem. In fact, let π : M —> B be asubmersion of compact complex manifolds. Let ξ be a holomorphic vector bundle on M. Let-R*7T*{ be the direct image of ξ. Then, by [KM], the line λ(ξ) <S> \~1(R°TT*O has a nonzerocanonical section σ. In [BerB], they have given a formula for the Quillen norm of σ in terms ofBott-Chern classes on M and the analytic torsion forms of π. Recently, Ma [Ma] has extendedthis result to a relative situation.

Another side, let G be a compact Lie group acting holomorphically on every object on X.Then Bismut [B5] defined ΛG(Ξ) the inverse of the equivariant determinant of the cohomologyof ξ on X. He also defined an equivariant Quillen metric on ΛG(Ξ) which is a central functionon G (refer also §1a)). In [B5], Bismut calculated the equivariant Quillen metric of the nonzerocanonical section of XG

l(r]) <S> G(Ξ) for a G-equivariant immersion i : Y —> X. In this way,he has generalized the result of [BL] to the equivariant case. In [B4], he also conjectured anequivariant arithmetic Riemann-Roch Theorem in Arakelov geometry. Recently, using the resultof [B5], Kohler and Roessler [KRo] have given a version of this conjecture.

In this paper, we shall extend the result of Bismut and Berthomieu to G-equivariant case.This completes the picture on G-equivariant case.

Let π : M —> B be a submersion of compact complex manifolds with fibre X. Let ξ be aholomorphic vector bundle on M. Let G be a compact Lie group acting holomorphically on Mand B, and commuting with π, whose actions lift holomorphically on ξ.

Let R*TT*( be the direct image of ξ. We assume that the Rk7r*((0 < k < dimX) are locally

free.Let σ be the canonical section of ΛG(Ξ ® AQ^-R'TT^).

Let hTM, hTB be G-invariant Kahler metrics on TM and TB. Let hTX be the metric inducedby hTM on TX. Let hξ be a G-invariant Hermitian metric on ξ. Let ω M be the Kahler form of

Let || ||A (Q^x-T-apTuE) ξ)be t h e G-equivariant Quillen metric on the line AG(£)®

attached to the metrics hTM, hξ, hTB, hH(X,ξ|X) on TM, ξ, TB, R'TT^. The purpose of this paperis to calculate the G-equivariant Quillen metric ||Σ||ΛG (nR,\-1(R»7r ξ)

For g e G, let Tdg(TM,gTM) be the Chern-Weil Todd form on Mg = {x e M,gx = x}associated to the holomorphic hermitian connection on (TM,hTM) [B5, §2(a)], which appearsin the Lefschetz formulas of Atiyah-Bott [ABo]. Other Chern-Weil forms will be denoted in asimilar way. In particular, the forms chg(ξ, hξ) on Mg are the Chern-Weil representative of theg-Chern character form of (ξ, hξ).

In this paper, by an extension of [BK6], we first construct the equivariant analytic torsionforms Tg(ωM,hξ) on Bg, such that

(0.1) — T g(ωM,h ξ) =ch

We also estabish the corresponding anomaly formulas. Remark that in [KRo], they have alsodefined the forms Tg(ωM, hξ).

Let fdg(TM, TB, hTM, hTB) e PMg/PMg,0 b e t h e B ott-Chern class, constructed in [BGS1],such that

g ( T M , TB, hTM, hTB) = Tdg(TM, hTM)-ir*(Tdg(TB, hTB))Tdg(TX, hTX).

The main result of this paper is the following extension of [BerB, Theorem 3.1]. Namely,we prove in Theorem 3.1 the formula

+ / fdg(TM,TB,h™ ,hTB)chg(ξ,hξ).JMs

We apply the methods and techniques in [BerB] and [B5], with necessary equivariant ex-tentions, to prove Theorem 3.1. The local index theory [B1] and finite progagation speed of thesolution of the hyperbolic equation [CP], [T] will also play an important role as in [BerB] and[B5].

This paper is organized as follows.In Section 1, we recall the construction of the equivariant Quillen metrics [B5]. In Section 2,

we construct the equivariant analytic torsion forms, and we prove the corrresponding anomalyformulas. In Section 3, we extend the result of [BerB] to the equivariant case. In Section 4,we state eight intermediary results which we need for the proof of Theorem 3.1, and we proveTheorem 3.1. In Sections 5-9, by combining the techniques of [BerB] and [B5], we prove theeight intermediary results.

Throughout, we use the superconnection formalism of Quillen [Q1]. The reader is referredfor more details to [B5, BGS1, BerB].

1 Equivariant Quillen metrics

This Section is organized as follows. In a), we recall the construction of the equivariantQuillen metrics of [B5, § 1]. In b), we indicate the characteristic classes which we will often use.

3

a) Equivariant Quillen metrics [B5].

Let X be a compact complex manifold of complex dimension l. Let ξ be a holomorphicvector bundle on X.

Let G be a compact Lie group. We assume that G acts on X by holomorphic diffeomor-phisms and that the action of G lifts to a linear holomorphic action on ξ.

Let E = ®f^xEi be the vector space of C°° sections of A(T*(°'1)X)cg){ = ©gmx A ^ ^ X )-y- -y-

®{ over X. Let 9 be the Dolbeault operator acting on E. Then G acts on (E, d ) by chain

homomorphisms, and we have an identification of G-spaces

(1.1) H(E,dX)~H(X,£).

Let dvX be the volume element on X associated to hTX. Let ( )A(T<°'1)X)®£ ξ b e t h e

Let hTX,hξ be G-invariant Hermitian metrics onLet dvX be the volume element on X associated

mitian product induced by hTX, hξ on A(T*(°'1)X) <g> . If s, s' G E, set

(1.2)/

A

Y'j, -y

Let 9 be the formal adjoint of d with respect to the Hermitian product (1.2). Set

(1.3) DX = ox + dx*,

K(X,ξ) =KerDX.

By Hodge theory,

(1.4) K(X,ξ

Clearly, for g e G, g commute to DX, so (1.4) is an identification of G-spaces.

Clearly K(X,ξ) inherits a G-invariant metric from ( ). Let hH(X,ξ) be the correspondingmetric on H(X,ξ).

Let F be a finite dimension G-vector space. Let hF be a G-invariant metric on F. Then we

have the isotypical decomposition

F = ewedRomG(W, F) <8) W.

and this decomposition is orthogonal with respect to hF. Let

(1.5) det(F,G) = © W e g

For W <EG, set

(1.6) λW(ξ) = (det(HomG(W,H(X,ξ)

Put

(1.7)

In the sequel, ΛG(Ξ) will be called the inverse of the equivariant determinant of the cohomology

of ξ. Then ΛG(Ξ) is a direct sum of complex lines.

Let | | λW (ξ) be the metric on λW(ξ) induced by hH(X,ξ) . Set

The symbol | |ΛW(Ξ) w m ' 3 e c a l l e d the (equivariant) L2 metric on ΛG(Ξ).

Take g eG. Set

(1.9) Xg = {x eX,gx = x}.

Then X g is a compact complex totally geodesic submanifold of X.Let P be the orthogonal projection operator from E on K(X, ξ) with respect to the Hermitian

product (1.2). Set P1- = 1 — P. Let N be the number operator of E, i.e. N acts by multiplicationby i on Ei. Then by standard heat equation methods, we know that as t —>• 0, for any g e G,

fceN,

(1.10) T tf

Definition 1.1. For s G C,Re(s) > dimX, set

(1.11) θX(g)(s) = - T r ^ A T p * ' 2 ) - ^ ] .

By (1.10), θX(s) extends to a meromorphic function of s G C which is holomorphic at s = 0.

Definition 1.2. For J G G , set

f)0x (g)(1-12) log(|| L W V

The symbol || ||ΛG(Ξ) will be called a Quillen metric on the equivariant determinant ΛG(Ξ).

b) Some characteristic classes.

Let X be a complex manifold. Let hTX be a Hermitian metric on TX. Let L be a holomorphicvector bundle over X. Let hL be a Hermitian metric on L.

Let V L be the holomorphic Hermitian connection on (L,hL). Let RL be its curvature.Let g be a holomorphic section of End(L). We assume that g is an isometry of L. Then g is

parallel with respect to VL.

Let 1,eiθ1, • • ,eiθq(0 < θj < 2π) be the locally constant distinct eigenvalues of g actingon L on X. Let Lθ°, Lθ1, • • •, Lθq (Θ0 = 0) be the corresponding eigenbundles. Then L splitsholomorphically as an orthogonal sum

(1.13) L = Le°(B---(BLe«.

Let hL θ 0 • • hL q be the Hermitian metrics induced by hL. Then VL induces the holomorphicHermitian connections VL '°, • • •, V L ' 9 on (Le°,hLe°), • • • {Le",hL<>q). Let Re°, • • •, Re" be theircurvatures.

If A is (q,q) matrix, set

A x(1.14) Td(A) = det(

\-e~Ah

e(A) = det(A), ch(A) = Tr[exp(A)].

The genera associated to Td and e are called the Todd genus and the Euler genus.

Definition 1.3. Set

— N L 9 ° T H — NL<>j

"•(" '» = T d < z » r > n ? - ( Z * )

r) Le°

Tdi(i»L)

chg(L,hL) =

Then the forms in (1.15) are closed forms on X, which lie in PX, and their cohomology classdoes not depend on the g invariant metric hL. We denote these cohomology classes by Tdg(L),Td'g(L),---,chg(L).

2 Equivariant analytic torsion forms and anomaly formulas

This Section is organized as follows. In a), we describe the Kahler fibrations. In b), we con-struct the Levi-Civita superconnection in the sense of [B1]. In c), we indicate results above theequivariant superconnection forms. In d), we construct the equivariant analytic torsion forms.In e), we prove the anomaly formulas, along the lines of [B5], [BK6].

a) Kahler fibrations.

Let π : M —>• B be a holomorphic submersion with compact fibre X . Let TM, TB bethe holomorphic tangent bundles to M,B. Let TX be the holomorphic relative tangent bundleTM/B. Let JTX be the complex structure on the real tangent bundle TRX. Let hTX be aHermitian metric on TX.

Let THM be a vector subbundle of TM, such that

(2.1) TM = THM®TX.

We now define the Kahler fibration as in [BGS2, Definition 1.4].

Definition 2.1. The triple (π, hTX,THM) is said to define a Kahler fibration if there existsa smooth real 2-form Ω of complex type (1,1), which has the following properties :

a) Ω is closed.

b) TRHM and T R X are orthogonal with respect to ω,c) If X, Y G TRX, then ω(X, Y) = (X, JTXY)hTX.

Now we recall a simple result of [BGS2, Theorems 1.5 and 1.7].

Theorem 2.2. Let ω be a real smooth 2-form on M of complex type (1,1), which has thefollowing two properties :

a) ω is closed.b) The bilinear map I , F e T R X —ω (JTXX, Y) defines a Hermitian product hTX on

TX.

For x G M, set

(2.2) TxH(M) = {Y <E TxM; for any X G TxX,ω(X,Y) = 0}.

Then THM is a subbundle of TM such that TM = THM © TX. Also (π,hTX,THM) is aKdhler fibration, and ω is an associated (1,1)- form.

A smooth real (l,l)-form J on M is associated to the Kdhler fibration (π,hTX,THM) ifand only if there is a real smooth closed (1,1)-form η on B such that

(2.3) UJ1 — ω = 7T*r].

b) The Bismut superconnection of a Kahler fibration.

Let ωM be a real (1,1) form on M taken as in Theorem 2.2.

Let ξ be a complex bundle on M. Let hξ be a Hermitian metric on ξ. Let V T X ,V^ be

the holomorphic Hermitian connections on (TX, hTX), (ξ,hξ). Let RTX, Lξ be the curvatures of

VTX, V«. Let vA(T* (° ' 1 ) x) be the connection induced by VTX on A(T*(°-1)X). Let

be the connection on A(T*(0>1)X) ® ξ,

Definition 2.3. For 0 < p <dim X, b G B, let Ebp be the vector space of C°° sections of

b. Set

(2.4) Eb = ®f™xEl E+ = (BpevenEp

b, E~ = ( b

As in [B1, §1f)], [BGS2, §1d)], we can regard the Eb's as the fibres of a smooth Z-graded infinitedimensional vector bundle over the base B. Smooth sections of E over B will be identified withsmooth sections of A(T*(0>1)X) <g> ξ over M.

Let ( ) be the Hermitian product on E associated to hTX,hξ defined in (1.2).

If U G TRB, let UH be the lift of U in TRHM, so that TT*C/H = U.

Definition 2.4. HUE TRB, if s is a smooth section of E over B, set

(2.5) V§s = £

By [B1, §1f)], V E is a connection on the infinite dimension vector bundle E. Let V E andV E be the holomorphic and antiholomorphic parts of V E .

For b G B, let d h be the Dolbeault operator acting on Eb, and let d h* be its formal adjoint

with respect to the Hermitian product (1.2).

The bundle A(T*(°'1)X)cg){ is a c(TRZ)-Clifford module. In fact, if U G TX, let U' G T<°'^X

correspond to U by the metric hTX. If U, V G TX, set

(2.6) c(U) = V2U'A, C(V) = -y/2iv.

Let PTX be the projection TM ~ T H M © TX -^ TX.

If U, V are smooth vector fields on B, set

(2.7) T ( U H , VH) = -PTX[UH, VH].

Then T is a tensor. By [BGS2], we know that as a 2-form, T is of complex type (1,1).

Let f1, • • •, fm be a base of TRB, and let f1 , • •, fm be the dual base of T&B.

Definition 2.5. Set

(2-8) hf^^

Then c(T) is a section of (A(T^JB)§End(A(T*(°'1)X) cg){))odd. We also define c(T(1,0)),

by formulas similar to (2.8), so that

(2.9) c(T) = c(T(1,0)) + c(T(0 , 1)).

Definition 2.6. For u > 0, let B u be the Bismut superconnection constructed in [B1, §3],

[BGS2, §2a)],

(2.10)

Let NV be the number operator defining the Z-grading on A(T*^°'l>X) <g> ξ and on E. NV

acts by multiplication by p on AP(T<°'^X) <g> ξ. If U, V G TRB, set

(2.11) ωHH(U, V) = ωM(UH,VH).

Definition 2.7. For u > 0, set

iωHH

(2.12) Nu = NV + .

c) Equivariant superconnection forms and double transgression formulas.

At first, we assume that the direct image i?*7r*£ of ξ by π is locally free. For b G B,let H(Xb,ξ|Xb) be the cohomology of the sheaf of holomorphic sections of ξ|Xb. Then theH(Xb,ξ|Xb)'s are the fibres of a Z-graded holomorphic vector bundle H(X,ξ|X) on B, and-R*7r*£ = H(X, ξ|X). So we will write indifferently R'TT*( or H(X, ξ|X) .

By (1.4), the K(X b,ξ|Xb) are the fibres of a smooth bundle K(X, ξ|X) over B. By [BGS3,Theorem 3.5], the isomorphism of the fibre (1.4) induces a smooth isomorphism of Z-graduedvector bundles on B

(2.13) H(X, ξ | X ) ~ K(X, ξ | X ) .

Then K(X, ξ|X) inherits a Hermitian product from (E, ( )). Let hH(X,ξ|X) be the correspond-

ing smooth metric on H(X,ξ|X). Let PH(X,ξ|X) be the orthogonal projection operator from

E on H(X,ξ|X) ~ K(X, ξ|X). Let Vi?(-X'^x-) be the holomorphic Hermitian connection on

Let G be a compact Lie group. We assume that G acts holomorphically on M,B,ξ, andthat ξ, M are G-equivariant (vector) bundles over M,B. Let ωM,hξ be G-invariant.

Then R'TT^ is also a G-equivariant vector bundle over B, and hH(X,ξ|X) is also G-invariant.

Definition 2.8. Let PB be the vector space of real smooth forms on B, which are sums offorms of type (p,p). Let PB,° be the vector space of the forms α G PB such that there existsmooth forms β, γ on B for which α = d[3 + d^.

Let Φ be the homomorphism of Aeven(T^B) into itself: α -• (2nr)- d e g a/ 2<x

Theorem 2.9. For u > 0, the forms $Trs[gexp(-B%)} and $Trs[gNuexp(-B%)} lie in PBg.

The forms &Trs[gexp(—Bu2)] are closed and that their cohomology class is constant. Moreover

(2-14) ^Tra[geM-Bl)] = - ~

PROOF: Since g commutes with Nu,Bu, etc, by proceeding as in [BGS2, Theorem 2.9], we

have Theorem 2.9. •

For g G G, we have also a holomorphic submersion π : Mg —> Bg with compact fibre Xg.

Put

C-h9 = 1x9 ^ T d f l ( T X , hTX)chg(ξ, hξ),C0,g = fXg(-Td'g(TX, hTX) + dimXTd g(TX, h

Set

(2.16) c

Theorem 2.10. As u -• 0

(2.17) <£>Trs[gexp(-B*)]= f Tdg(TX, hTX)chg(ξ, hξ) +O

There are forms C'- G PB§ {j > — 1) suc/i £/ia£ /or A; G N, as « —>• 0

(2.18)

Also

(2-19) ^J1'9 T P " 1 ' 9 , 'C0 ) f l = C0,g in

4̂s « —> +oo

(2.20)

PROOF: By combining the technique of [BGS2, Theorem 2.2, 2.16] and [B7, Theorem 4.9-4.11], we have the equations (2.17),(2.18),(2.19).

Equation (2.20) was stated in [BK6, Theorem 3.4] if g = 1. By proceeding as in [BeGeV,Theorem 9.23], we also have (2.20). •

d) Higher analytic torsion forms.

For s G C,Res > 1, set

1 1 * ; ^o

Using (2.18), we see that ζ1(s) extends to a holomorphic function of s G C near s = 0.

9

For s G C,Res < ^ set2 ,

f+oo

Then ζ2(s) is a holomorphic function of s.

Definition 2.11. Set

(2.21) T g ( ω

M , h ξ ) = ^ ( C i +

Then Tg(ωM,hξ) is a smooth form on Bg. Using (2.18),(2.20), we get

(2.22) - / + 0 0 ($Tra\gNueM-Bl)]-<h'g(H(X,Z\x),hH(x'W))%

Theorem 2.12. The form T g (ω M ,hξ) lies in PBg, Morever

(2.23) ^-Tg(uM,ht) = chg(H(X,ξ|X),hH(X,ξ|X)) - J Tdg(TX,hTX)chg(ξ,hξ).

PROOF: As we saw before, the forms ΦTrs[gNu e x p ( - ^ ) ] lie in PBg. So the form Tg(ωM, hξ) G

PBg. Using Theorem 2.10 and equation (2.14), the proof of our Theorem 2.12 proceeds as the

proof of [ BGS2, Theorem 2.20]. •

e) Anomaly formulas for the analytic torsion forms.

Now let (w/M, h't) be another couple of objects similar to ( ω M , hξ). We denote with a ' theobjects associated to (w/M,h'^).

By [BGS1, § 1(f)], thereare uniquely defined Bott-Chern classes Td f l(TX, gTX,g'TX),

) G

= Tdg(TX,g'TX) - Tdg(TX,gTX),

Let C be a smooth section of T^X®End(A(T*(0>1)X) ®{). Let e1, • • •, e2n be an orthonormalion

c ( e . ) ) 2 = ^ (

base of T^X. We use the notation

2n V T X 2n

10

Theorem 2.13. The following identity holds

(2.24) - / [fdg(TX,hTX,h)chg(C,^)

+Tdg(TX,tiTX)chg(t,ht, /*'«)] in PBg/PBg,°.

In particular, the class ofTg(ω,hξ) in PBg /PBg,° only depends on (hTX,hξ).

PROOF: Assume first that hξ = h'^. Let c G [0,1] —ω cM be a smooth family of G-invariant(1,1)-forms on M verifying the assumptions of Theorem 2.2 such that ω0M = ωM, ω1M = u'M.

Then all the objects considered in Section 2 a)-d) now depend on the parameter c. Most ofthe time, we will omit the subscript c. The upperdot • is often used instead of J^.

Set

Q = - ~l *

Let e1, • • •, en be an orthonormal base of T R Z with respect to h^x. Let f1, • • •, f2m be a baseof T R B , and that f1, • • •, f2m is the corresponding dual base of T^B. Set

. . Mu= -^(e,,^)^)^)(2.26)4 . HH

\ . (ei, JTXei).

By the arguments of [BGS2, Theorem 2.11], we know there is p G N,/Xj e PBg,(j > —p) suchthat as u —>• 0, we have the asymptotic expansion

(2.27) ^ p

By proceeding as in [BK6, § 2,3], we easily find an analogue of [BK6, Theorem 3.16],

(2.28) T .(ω, hξ) =

In (2.28), the 0l'(O) are universal formulas of g,ωcM,hξ as in [BK6].Let da, da be two odd Grassmann variables which anticommute with the other odd elements

in A(T^B) or c(T R X). Set

2 d ^ d[B M] + dd{(2.29) Lu = -B2

U - dau—^ - da[Bu, -Mu] + dada{- — (uMu)).

If α G C(da, da), let [α]dada G C be the coefficient of da da in the expansion of α. By an analogueformula of [BK6, Theorem 3.17], we know that the class of —/xo in PBg /PBg,° coincides withthe class of the constant term in the asymptotic expansion of ΦTrs[g exp(Lu)]dada.

Let V^ be the connection on Λ(da ©da)§A(T^ JB)§A(T*(°' 1)X)§{ on fibre X.

Vu = vA(r-(Oll)*>®« + I (SQejt fS

da [u ,u i ω , . , u(2.30) - - y j - c ( . ) - - ( e f c , . ) d a u

11

Let KX be the scalar curvature of (X, hTX). Set

(2.31) L* = Lξ +

By [BK6, Theorem 3.18], we get

Lu = f(VU) 2 " V.( . (e,, jTXej)y^ Vjg{i) ( e j, J ^

(2.32) + ^ u, (ej, JTXej) - uK8 - | C (e,) C ( e j )L'«(e,, ej)

Let Pu(x,x',b)(b G B,x,x' G Xb) be the smooth kernel associated to exp(Lu) with respect to

• T h e n

(2.33) ΦTr s[ gexp(Lu)] = ̂ ^ ^ ^ ( g - ^ , x, b)]

By standard estimates on heat kernels, for b G B, the problem of calculating the limit of (2.33)

when u —> 0 can be localized to an open neighbourhood Uε of Xbg on Xb. Using normal geodesic

coordinates to Xbg in Xb, we will identify Uε to an ε-neighbourhood of Xg in

Let k(x, z)(x G X g , z G NXg/X,R, |z| < ε ) be defined by

(2.34) dvX = k(x,z)dvXg(x)dvNXg/X(z).

Then

k(x,0) = 1.

Cleary

(2.35) = l e X 9 S\z\<^-^NX9/X v2dimNx3/x<S>Trs[gPu(g-Hx, ^z), (x, ^z))}, r dvXg(x)dvNXg/X(z)

k(x, y/UZ) ^ . j dimX

Of course, since we have used normal geodesic coordinates to Xg in X, if (x, z) G

(2.36) g-\x,z) = (x,g-lz).

Take x0 G Xbg, by using the finite propagation speed as in [B6, § 11b)] , we may insteadassume that Xb by (TX)x0 with 0 G (TX)x0 representing X0 and that the extended fibrationover Cm coincides with the given fibration over B(0, ε).

Take y G Cm, set Y = y + y. We trivialize Λ(da ©da)®A(T^5)§A(T*(°' 1)X)§{ by paralleltransport along the curve t —> tY with respect to V^.

Let ρ(Y) be a C°° function over C m which is equal to 1 if \Y\ 4 ε, and equal to 0 if \Y\ 2 ε.

Let H x 0 be the vector espace of smooth sections of (A(da©da)§A(T^JB)§A(T*(°'1)X)§{) ;roover (TRX)X0. For u > 0, let L1u be the operator

-vATX

(2-37) 4 = (1 - p

2 ( F ) ) ( ^ ^ ) - ρ2(Y)Lu.

12

For u > 0, s G Hx0, set

(2.38)

Let ebase of N

7-2 _ r -

be an orthonormal base of (TRXG)X0, and let e2i'+i, • • •, e2n be an orthonormal

Xg/X,R,x0.

Let L3

u be the operator obtained from Lu2 by replacing the Clifford variables c(ej)(1 < j < 21')

by the operators ^0- — ^iei-

Let Pl

a(z,z') (z,z' G (TRX)X0, \Z'\ < f, i = 1,2,3) be the smooth kernel associated to

exp(-LJj) with respect to J^™* • I f α G C(e j , ie j)(i<j<2i')' l e t [ α ] m a x e C be the coefficient

of e1 A • • • A e21 in the expansion of α. Then as in [B5, Proposition 11.12], if z G NXg/X,R

(2.39)dada

Let i ? j ^ , Lξ|Xg, • • • be the restrictions of RTX,L^,--- over X g . Let V e i be the ordinary

differentiation operator on (TRX)X0 in the direction ei. By [ABoP, Proposition 3.7], and

(2.32), as u -• 0,

(2.40)

ej + 2

—daa2 —

\ — — i • \

Y, e j — daa1 + dada{- u (Y, ej))

dadaej, J TX /?

and a1,a2 are 1-forms of A(T^X © T R 5 ) .Let

(2.41)

dada

\ i . '

Y,ej + dada(2iω. (Y,ej)

.(ej,JTXej)+L| \Xg

of of dvTX (zf)

Let PQ (Z> z ) be the heat kernel of exp(—LQ ) over ( T R X ) X 0 with respect to (2TXΠ)DIMX0(ZX

By proceeding as in [B5, §11g)- §11i)], we have :

There exist γ > 0 , c > 0 , C > 0 , r G N such that for u G]0, 1], z, z' G ( T R X ) X 0 , we have

(2.42), z' z'\)r z'\2),

,z s cuγ(1+ z \)

From (2.34), (2.39)-(2.42), we get

w < |

(2.43){-I

'(n-U

Clairly for U, V G TX,

(2.44)LTX

dc

13

* \ (V + \

So

(2.45) I* = -\ (V, + 1

+L|ξXg -

By proceeding as in [B4, (3.16)-(3.21)],

/NXg/X,R

(2.46)dc J

By using [BGS1, Remark 1.28 and Corollary 1.30] and proceeding as in [BK6, §3h)], we finishthe proof of Theorem 2.13 in the case where hξ = h'^.

To prove (2.24) in the full generality, one only needs to consider the case where ωM = u'M.Then by using Theorem 2.12 and by proceeding as in [BGS1, §1f)], i.e. by replacing B byB x P 1 , one easily obtains (2.24) in this special case. •

3 The equivariant Quillen norm of the canonical section σ

This Section is organized as follows. In a), we describe the canonical section σ. In b), weannounce a formula for the equivariant Quillen norm of σ.

In this Section, we make the same assumptions as in Section 2c), and we use the same no-tation as in Sections 1,2.

a) The canonical section σ.

Let M, B be compact complex manifolds of complex dimension n and m. Let π : M —>• B bea holomorphic submersion with fibre X. Let ξ be a holomorphic vector bundle on M. Let G bea compact Lie group. We assume that ξ, M are G-equivariant holomorphic bundles over M,B.

We assume that the sheaves Rkir*t;(0 < k < dimX) are locally free.

If given W G G, Xw,^w are complex lines, if λ = (BWeQ\\y, [*• = ®WeQlJ-w, set

(3.1) A"1 = ®WeQXw> A <g>//= e W e g A w <g>//w.

Now we use the notation of Section 1. Set

(3.2) \G(Rkir*O = det(H(B,

By proceeding as in [BerB, §1b)] and [B5, §3b)], for W eG, the line λW(ξ (8) X^(R'TTS has

canonical nonzero section σW. Set

(3.3) σ = e w e d a w e λG(ξ) ® X^\R'7T^)-

14

b) A formula for the Quillen norm of canonical section σ.

Let hT M , hTB be G-invariant Kahler metrics on TM and TB. Let hTX be the metric inducedby h T M on TX. Let hξ be a G-invariant Hermitian metric on ξ.

On Mg, we have the exact sequence of holomorphic Hermitian vector bundles

(3.4) 0 -> TX -> T M -> TT*T5 -> 0.

By a construction of [BGS1,§1f)], there is a uniquely defined class of forms Tdg(TM,TB,

hTM,hTB) G P M G / P M G , ° , such that

,a ^ gfd f l(TM, TB, hTM, hTB) = Tdg(TM, hTM)

— 7T (lClg(J i ) , ft JJ lClg(J A , ft J.

Let ΩM be the Kahler form of h T M . Let || ||λG (t)®\-'i-(R».K ξ) be the G-equivariant Quillen

metric on the line λG(ξ) ® Ag^E'Tr*^) attached to the metrics hTM ,hξ,hTB ,hH(X,ξ|X) on

Now we state the main result of this paper, which extends [BerB, Theorem 3.1].Theorem 3.1. For g G G, the following identity holds,

) = -f Tdg(TB,hTB)Tg(ωM,hξ)

fdg(TM,TB,h™,hTB)chg(£,ht)./Mg

PROOF: The proof of Theorem 3.1 will be given in Sections 4-9. •

Remark 3.2. By Theorem 2.13, to prove Theorem 3.1 for any Kahler metric hTM, hTB, we

only need to etablish (3.6) for one given metrics hTM, hTB. So by replacing hTM by hTM+7T*hTB,

we may and we will assume that h™ is a Kahler metric on TM and

(3.7) hTM = h™+7T*hTB.

4 A proof of Theorem 3.1

This Section is organized as follows. In a), we introduce a 1-form on R*j_ x R*j_ as in [BerB,§ 3a)]. In b), we state eight intermediary results which we need for the proof of the Theorem3.1 whose proofs are delayed to Sections 5-9. In c), we prove Theorem 3.1.

In this Section, we make the same assumption as in Section 3. Also, we assume that hTM isgiven by formula (3.7). In the sequel, g G G is fixed once and for all.

a) A fundamental closed 1-form.

Recall that NV denotes the number operator of A(T*(°'l>X). Let NH be the number operatorof A(T*(0>1)l3). By (2.2), we have the identification of smooth vector bundles over M

( TM ~ TX © THM,

[ } THM ~ ir*TB.

This identification determines an identification of Z-graded bundles of algebra

(4.2) A(T<°^M) = A ( T * ( ° ' 1 ) J B ) § ( ° 1 )

15

So the operator NV and NH acts naturally on A(T*(°>VM). Of course, N = NV + NH defines

the total grading of A(T<°<VM) ® ξ and Q(M,£).

Definition 4.1. For T > 0, let hTMT be the Kahler metric on TM

(4.3) hT

MT = ^h™ + 7r*hTB.

Let ( ) T be the Hermitian product (1.2) on Q(M,£) attached to the metrics hTMT, hξ. LetDTM be the corresponding operator constructed in (1.3) acting on Q(M, ξ). Let *T be the Hodgeoperator associated to the metric hTMT. Then *T acts on A(T^M) <g>£.

T h e o r e m 4.2. Let α u ,T be the 1-form on ~R*+ x ~R*+

(4.4) aM)T = ^ T r , [5iV exp(-u 2 ^' 2 )] + dTTr, [g *^

Then ΑU,T is closed.

PROOF: Clearly g is an even operator which commutes with the operators d ,dT , *T, NV,NH. By using [BerB, (4.27), (4.28), (4.30)], the proof of Theorem 4.2 is identical to the proofof [BerB, Theorem 4.3]. •

Take ζ, A, T, 0 < ζ < 1 < A < +oo, 1 < T0 < +oo. Let Γ = ΓΑ,A,T0 be the oriented contourin R* x R*

The contour Γ is made of four oriented pieces Γ1, • • •, Γ4 indicated above. For 1 < k < 4, set

(4.5) Ik0 = [ α.

Theorem 4.3. The following identity holds,

(4.6) Σ41Ik° = 0.

PROOF: This follows from Theorem 4.2.

b) Eight intermediate results.

Let d be the formal a

metrics hTB,hH(X,ξ|X). Set

(4.7)

ID. D

Let d be the formal adjoint of the operator d acting on Q(B, _R*7r*{), with respect to the

= KerD B .

16

By Hodge theory,

(4.8) H'(B, E*7r*0 ~ F.

Let Q be the orthogonal projection from Q,(B, R'TT^) on F with respect to the Hermitian prod-uct (1.2) attached to the metrics hTB,hH(X,ξ|X). Set Q± = 1-Q.

Let a e]0,1] be such that the operator DB,2 has no eigenvalues in ]0,2a].Definition 4.4. For T > 0, set

(4.9) ET =

Let PT be the orthogonal projection operator from Q(M, ξ) on E T with respect to ( ) T .

Let ET[0,a] (resp. E T]0,a]) be the direct sum of the eigenspaces of DTM,2 associated to eigenvalues\ rr, i / \ in i\ T i r,M,2,[0,a] / 1-.M,2,l0,al\ , , , , . , . r ,-.M,2 , j-^[0,al

A e [0, a] (resp. λ e]0,a]). Let DT' >l ' '(resp. DT' '' ' ') be the restriction of DT' to ET[

0 ,

(resp. ET ) . Let PT (resp. PT ) be the orthogonal projection operator from Q(M, ξ) on

ET[0,a] (resp. ET]0,a]) with respect to ( ) T . Set p]«'+°°[ = 1 - PT[0,a].

For 0 < k < n, g e G, set

(4.10) χ g ( ξ

Then by the Lefchetz fixed point formula of Atiyah-Bott [ABo],

(4.11) χg(ξ) = I Td g(TM)ch g(ξ),

X9(Rkir*0 = I Tdg(TY)chg(Rk7rS-JBs

We now state eight intermediary results contained in Theorems 4.5 - 4.12 which play anessential role in the proof of Theorem 3.1. The proof of Theorems 4.5 -4.12 are deferred toSections 5-9.

Theorem 4.5. For any u > 0,

(4.12) lim TrJ#JVexp(-u 2I}Jf 2)j = Trs\gN exp(-u2DB'2)].

For any u > 0, there exists C > 0 such that for T > 1,

(4.13) Tr s[gNV exp(—u2DT

M,2)] — Sj=odimX(—1) jχg(R j

For any ε > 0, There exists C > 0 such that for u> ε, T > 1,

(4.14) Tr[#exp(-i

Theorem 4.6. For any u > 0,

(4.15) lim Tr

There exist c > 0, C > 0 such that for u > 1,T > 1,

(4.16) Tr[^iVexp(-u 2

JD^' 2)p] a '+ o o [] < cexp(-CV).

17

l- rn. [ r-1M,2,[0,a]

lim Tr \gDT

l 2

Theorem 4.7. The following identity holds,

(4.17)

For T > 1 large enough, for 0 < i < dim M,

(4.18) 1

= 0.

Let (Er,dr) (r > 2) be the Leray spectral sequence associated to Π,Ξ. By [Ma, Theo-

rem 14.1], the Dolbeault complex (Q(M,(),d ) filtered as in [BerB, §1a)] calculate the Leray

spectral sequence. Then as in [BerB, Section 4], for r > 2, Er is equipped with a met-

ric hEr associated to hTM ,hTB ,hξ. For r > 2, let r| |ΛG(Ξ) be the corresponding metric on1

For r > 1, let N|Er, NH|Er, NV|Er be the restriction of N, NH, NV to ^ .

Theorem 4.8. The following identity holds,

Tr log(T)

(4.19) = log-MO

For T > 1, let | |ΛG(Ξ),T be the L 2 metric on the line ΛG(Ξ) associated to the metricson TM, ξ.

Theorem 4.9. The following identity holds,

lim -̂ log 2( - di log(T)}

(4.20) = log

For u > 0, let Bu be the Bismut superconnection on Q,(X,£\x) constructed in Definition 2.6

which is attached to hTM,hξ on TM,ξ. Let 7VM be the operator defined in (2.12) associated to

the metric Ji™.

Theorem 4.10. For any T > 1,

(4.21) 0 Tr s [g *~l

/e ^(*T

TB)ΦTrs \gNT2 e x p

Let uM,uM,uB be the Kahler forms associated to h™,h™,hTB. Let V? M be the holo-morphic Hermitian connection on (TM,hTMT), and let RTTM be its curvature.

T h e o r e m 4 . 1 1 . There exist C > 0 such that for ε e ]0 , 1],ε < T < 1,

(4.22)

f —Td T/εT/ε) b=0

18

Theorem 4.12. There exist δ e]0,1],C > 0 such that for ε e]0,1],T > 1,

(4-23) \fts[g*T)eM*Th

Besides, at a formal level, Theorems 4.5- 4.9 can be obtained formally from [BerB,Theorem4.8-4.12] by introducing in the right place the operator g. This will permit us to transfer formallythe discussion in [BerB, Section 4] to our situation.

c) Proof of Theorem 3.1.

By Theorem 2.12,

(4.24) chfl(E-7T»O = / Tdg(TX)chg(ξ)•JX9

We also have the obvious equality

(4.25) Td'g(TM) = n* (rdg(TB))Tdg(TX) + IT* (rdg(TB))Tdg(TX).

By Theorem 4.3, Theorems 4.5-4.12, and proceeding as in [BerB, § 4c),d)], using (4.24),

(4.25), we get (3.6). •

5 A proof of Theorems 4.5, 4.6 and 4.7

The proof of Theorems 4.5, 4.6 and 4.7 is essentially the same as the proof of [BerB, Theorem4.8, 4.9 and 4.10] given in [BerB, § 5], where the corresponding results were established whenG is trivial.

Now we use the notation of [BerB, §5].

At first, for each U e TB, (gU)H = gUH, so the operator CT in [BerB, (5.7)] commutewith the action of G.

Let ( )oo be the Hermitian product on E00 associated to the metrics n*hTB © hTX, hξ onTM,ξ defined by (1.2).

Let EI}T,EQ,E^T(/J, > 0) be the vector spaces defined in [BerB, Definition 5.12]. Then for

any T > 0, the linear isometric embedding JT of EitOO in E1,T defined in [BerB, Definition 5.16]

is G-equivariant. Let E^ be the orthogonal space to E10,T in E00 with respect to ( ) o o . It follows

from the previous considerations that for any T > 0, the orthogonal splitting E00 = E10,T © E10

,T

of E00 considered in [BerB, (5.29)] is G-invariant, i.e. G acts on E10,T and E10,

T,

Therefore the matrix of the unitary operator g with respect to the splitting E00 = E®T(BE®'^

can be written in the form

L ° G1,T

and moreover

(5.2) g0,TJT = JTG.

The proof of Theorems 4.5 , 4.6 and 4.7 then proceeds as in [BerB, § 5 c)-g)].

19

6 A proof of Theorems 4.8-4.9

In this Section, we give a proof of Theorems 4.8 and 4.9. These generalize [BerB, §6], wherethe corresponding results were proved in the case where G is trivial.

We use the notation of [BerB,§ 6].

a) Proof of Theorem 4.8.

At first we can verify the formulas of [BerB,Theorem 6.1-6.5] are G-equivariant.By using [B6, Theorem 1.4], and by proceeding as in [BerB, §6(d)], we obtain (4.16).

This completes the proof of Theorem 4.8. •

b) Proof of Theorem 4.9.

For WeG, let

, W

q = HomG(W, H(X, ξ)) <8> W,1 ) E w = HomG(W, E^) <g> W.

Then

(6.2) H(X,ξ)=

By proceeding as in [BerB, §6(e)] for HWq, λW(ξ), w e deduce that as T —> +oo

,W] log(T)λW(ξ)

(6.3) -• log

Then Theorem 4.9 follows from (6.3). •

7 A proof of Theorem 4.10

This Section is organized as follows. In a), we show that the proof of (4.18) can be localizednear TT~1(B9). In b), given b0 G Bg, we replace M by (TRB)B 0 JQ,O, and rescaling on certainClifford variables. In c), we prove (4.18).

Recall that in this Section, we will calculate the asymptotics as ε —> 0 of certain supertracesinvolving eD^,£ for a fixed T > 1 .

a) The proof is local on TT~1(B9).

Let dvM (resp. dvB, resp. dvX) be the volume form on M (resp. B, resp. on the fibre X)associated to the metric ir*hTB © h T X on TM ~ ir*TB ®TX (resp. hTB on TB, resp. hTX onTX).

Let αB, α M be the injective radius of B, M. In the sequel, we assume that given 0 < α < α0 <\ inf{αB, αM} is chosen small enough so that ifyeB, dB(g~ly, y) < α, then dB(y, Bg) < α0/4,and ifxeM, dM(g~1x, x) < α , then dM(x, Mg) < α 0 /4 . IfxeB, let BB(x, α) be the openball of center x and radius α in B.

20

f ( t ) = 1 for |t| < α/20 for |t| > α.

Let f be a smooth even function defined on R with values in [0,1], such that

(7.1)

Set

(7.2)

Definition 7.1 . For u G]0, 1], a G C, set

(7.3)

Clearly

(7.4)

/•+ooFu(a)= / exp(itaV2)exp(

j-oo

r+oo ,-V dtexp(itaV2) exp( )g(ut)-

2 2π

Fu(a) + Gu(a) = exp(-a2).

The functions Fu(a),Gu(a) are even holomorphic functions. So there exist holomorphicfunctions Fu(a), Gu{a) such that

(7.5) u(a) = Fu(a2), Gu(a) = Gu(a2).

The restrictions of Fu, Gu, Fu, Gu to R lie in S(R).From (7.4), we deduce that

(7.6)

Proposition 7.2. For δ > 0, there exist c > 0, C > 0 such that for 0 < ε < δ, T > 1,

n2

(7.7) < cexp(-£'

PROOF. The proof of our Theorem is essentially the same as the proof of [BerB, Proposition8.3].

For T > 1 fixed, we use (7.7) with ε = T and T replace by T/ε, we find

(7.8) ^s\g*r),T/ε ~2>'

Let Fe(eD^/e)(x,x') (x,xr e M) be the smooth kernel associated to Fε(εDT/εM) with respect

to the volume (2πdv)dim M • Using (7.3) and finite propagation speed [CP,§7.8], [T,§ 4.4], it is clear

that for ε e]0,1], T > 1, x, x' G M, if dB(x, x') > α, then

and moreover, given x G M, Fε(εDT/εM)(x, •) only depends on the restriction of D^/e to TT~1BB(TTX, α).For u G T R B , V G TRX, let c(U),c(V) denote the corresponding Clifford multiplication

operators acting on TT*A(T<°^B),A(T<°^X) associated to hTB,hTX defined as in (2.6). Set

(7.9)* £

21

Then by (7.9), we get

(7-10) Tr, [g *~l

/e ^-{*T/e)Fe{eD^/e)\ = Tr, [g *~)e ^ ( * r / e ) F e « r ) ] •

Let Fe(A'eT)(x,x') (x,x' G M) be the smooth kernel associated to the operator Fe(A'eT)

with respect to , d , ^ M.

For α > 0, let Uα0(Bg) be the set of b G B such that dB(b, Bg) <α0. We identify Uα0(Bg) to{(b, Y);b G Bg, Y G NBg/B,R, |Y| < α 0 } by using geodesic coordinates normal to Bg in B, then

/* r ft ~i / / 7 ? 7i //•

(7 ~\~\\ Tr \ n ±~l (± , \T? (A1 \( n~l'r ^r\\ M

JB9jY€NBg/BtKJx s T/

'IT

d

By (7.11), we see that the proof of Theorem 4.10 is local near ir~l(Bg).

b) Rescaling of the variable Y and of the Clifford variables.

Taking b0 G Bg, we identify BB(b0,α0) with B(0,α0) C (TB)b0 = Cm by using normalcoordinates.

Take y G Cm, |y| < α0, set Y = y + y. We identify TB|Y to TB{0} by parallel transportalong the curve t —* tY with respect to the connection VTB. We lift horizontally the pathst G R+ -• tY into paths t G R+ -^ x t G M with x t G X t y , ^ e THM. If x 0 G Xb0, weidentify TXxt,ξxt to T X x 0 , ξ x 0 by parallel transport along the curve t —>• xt with respect to theconnections VTX, V^. These trivializations induce corresponding trivializations of A(T*(°'l>B),

Let Qb0 = Q(Xbo,£\Xb ) be the vector space of smooth sections of (A(T*(0>1)X) ®£)\xb on

Xb0. Then Qfco is naturally equipped with a Hermitian product ( ) attached to h TX|Xb°, h ξ|Xb0.There is also a smooth Z-graded vector bundle K C flb0

o v e r (TB)b0 ^ R 2 m which coin-cide with KerDX on B(0,2α0), with KerDbX0 over TRB\B(0, 3α0) and such that if KL is theorthogonal bundle to K in Q^,

(7.12) K

Let Pb be the orthogonal projection operator from Qbo on Kb. Set P^- = 1 — Pb.

Let if : R —>• [0,1] be a smooth function such that

, 7 i r i (t) = 1 for | t | < a 01 ' = 0 for | t | > 2 a 0 .

Let ATB be the standard Laplacian on ( T R B ) B 0 with respect to the metric hTB|b0. Let i ^ be

the vector space of smooth sections of 7r*A(T*(°'1)_B)6o®(A(T*(°'1)X)®{)|Xf) over (TRB)b0 xXb0.

Let L\ T be the operator

/ — F2ATB

(7.14) Ll>T = ^{\Y\)A'lT + (1 - ^ 2 ( |F |)) (

For ε > 0, s G Hb0, set

(7.15) Sεs(Y,x) =

22

Put

(7.16)

Let Op b e t h e set of differential o p e r a t o r s ac t ing on s m o o t h sections of ( Λ ( T*(0 > 1)

c(TRB)®Op.

over R 2 m x Xb0. Then we find that

Let f1, • • •, /2m' be an orthonormal basis of (TRBG)B0, let / w + i ; • • , f 2m be an orthonormalbasis of NBg/B,R,b0.

Definition 7.3. For ε > 0, set

(7.17) ce(/,-) = ^ P A - ^ ^ . , 1 < i < 2m'.

Let Lε,T3,Mε,T3 be obtained from L2ε,T,

c(fj)(1 < j < 2m') by the operators cε(fj).

Let P; T ((F,a;),(F', a ; ' )) , Fe{UE

by replacing the Clifford variables

,x),{Y>,x') e (TRB)b0 xXbo) be the

smooth kernels associated to exp(—Liε,T), F e ( L * T ) calculated with respect to ( (2TBΠ))BDIM ° • Using

finite propagation speed, we see that if (Y,x) e (TRB)b0 x Xb0, |Y| < α0/4, then

(7.18) F (A'e>T) (g-\Y, x), (Y, x)) = F£{Ll

e>T) (g~l{Y, x), (Y, x)).

We observe that for any k G N, c > 0, there is C > 0, C" > 0 such that for ε > 0,

(7.19) sup |a F £(a 2) - exp(-a 2 ) < c e x p ( ^ ) .

Using (7.19), and proceeding as in Proposition 7.2, we find for |Y| <

(7.20)1

ε,T) -exp(-LlT))((Y,x),(Y',x>))\ < cexp(^) .^

By (7.18), (7.19), we can replace Fε(Lε,T1) by exp(-L^ T ) in (7.11).

We know that Pε,T3((Y, x), (Y1, x1)) lies in (End(A(T£.B») ®c(NB9/B>n)) §c(T R X 6 o ) §End({).' ^ ' b0

Then M^TP^T(g~1(Y,x), (Y,x)) can be expanded in the form

P (n~ (V o^ (V TW — V! , /**! A • • • A f i P A 7 J- • • • i £

(7.21) E i l - i p ^ " ^ ( 5 - 1 ( y ; a ; ) , (r,a;)) e

Set

(7.22)

Proposition 7.4. IfYe NBg/B,R,b0, the following identity holds

(7.23)

23

PROOF. Since g acts like the identity on A((T*(°'1)JB»), g G c(NB9/B}R)bo § c(TRXb0)

®End({). Therefore the rescaling of the Clifford variable in (7.17) has no effect on g. Iden-

tity (7.23) is now a trivial consequence of [Ge]. •

c) Proof of Theorem 4.10.

Recall that for u > 0, the Bismut superconnection Bu associated to hTM and hξ was con-structed in Section 2b). Also we observe that Bu is unchanged if hTM is changed into h™.

Let RTB be the curvature of VTB. Also V/a denote the ordinary differentiation operator on(TRB)b0 in the direction fα.

Then as in [BerB, (7.30), (7.35)], we have as ε -• 0

(7.24) L3

£T^LlT.

-nHH

and

(7.25) e-̂ L;

By [BerB, (7.38)], we get , as ε -• 0

(7.26) M Ε , T -• M 0 3

-(BTBYf) Y+ 1Tr(RTB)+r, \ U Y7ja/hTB + r, i r l - " )Zi / Zi

= -(Nv-dimX)

By [B4, (3.16)-(3.21)], [BerB, § 7d)], we have

Trs [9[MlTPlT{g-\Y, x), (Y, x))]maxN Bg/B,R,b0

dvNBg/B(Y)dvXb(x)

(2π) d i m M

(7.27) = (-ifimB9^{Tdg(TB,hTB)^Trs[g(NT2 - di

T h e o r e m 7 . 5 . F o r T > 1 fixed, t h e r e e x i s t c > 0, C > 0, r G N s u c h t h a t f o r ε G ] 0 , 1 ] ,

(7.28) - x) (Y' x'\) A / | ) r e x p ( - C | F - F / | 2 ) .

To prove Theorem 7.5, we establish at first a uniform estimate on the kernel Pε,T3.

Theorem 7.6. There is C > 0 such that for m G N, there exist c > 0, r G N suchany ε G]0, 1], (Y,x), {Y>,x>) G ( T R B ) B 0 x Xb0,

(7.29) sup

a ' |

dYadY la 7PlT{(Y,x),{Y',x'))

' | ) r exp(-C|F - Y'

PROOF of Theorem 7.6. Set

(7.30)

Let E° be the vector space of square integrable sections of (A(T^B9)®

(A(T*(°-1)X) <8> ) | X b over (TRB)b0 x Xb 0. For 0 < q < 2m = 2 d i m B g , let

for

be the vector

24

space of square integrable sections of (A«(T^5») § c(NB9/B}R))bo § (A(T<°'^X)iS)O\xbo- Then

E° = ©^™0_B°. Similarly, if p G R, Ep and Eqp denote the corresponding pth Sobolev spaces.

If s G Eq0, set

Let ( ε, 0 be the Hermitian product attached to | |ε,0. If s G E1, put

(7 32) s 2 = |s|2 + S 2 m |Vf s 2 + XIV s 2

Using the technique in [BerB, §9d)], special [BerB,(9.51)] (in our situation, T is fixed), thebounds in (7.29) with C = 0 are easily obtained . To get the required C > 0, we proceed as inthe proof of [B5, Theorem 11.14]. •

PROOF of Theorem 7.5. Using Theorem 7.6 , and proceeding as in [B5, §11 i)], [BL, §11

q)], we have Theorem 7.5 . •

For b G B\ Y G NBg/B,R,b, |Y| < α0, let k(b, Y) be defined by

(7.33) dvB(b,Y) = k(b,Y)dvBg(b)dvNBg/B(Y).

Using Theorem 7.5, (7.18), (7.20), (7.23), (7.27), we get over Bg

(7.34) k(b,Y)dvNBg/B(Y)dvX(x)/(2π)dimM

T2 — dimX) exp(—BT2

By (7.7), (7.34), the proof of Theorem 4.10 is complete. •

8 A proof of Theorem 4.11

This Section is organized as follows. In a), we reformulate Theorem 4.11. In b), we indicatethat the proof is localized near ir~l(B9) by Proposition 7.2. In c), we prove the estimate (8.1).

We make the same assumption and we use the same notation as in Sections 4 and 7.

a) A reformulation of Theorem 4.11.

Theorem 8.1. There exists C > 0 such that for 0 < u < 1,T > 1,

— (* D M'2)] — 2 ~™

b=0

Cu2

~ ~T~'

Remark 8.2. Theorem 8.1 implies Theorem 4.11. In fact, for 0 < ε < 1,ε < T < 1 we

use (8.1), with u = T and T replaced by j , then we find that the right-hand side of (8.1) is

dominated by

^- = CεT < Cε.

25

So we have proved (4.19).

b) Localization of the problem near TT~1(B9).

By Proposition 7.2 and the argument in Section 7b), the proof of (8.1) can be localized near

Thus , we are entitled to choose b0 G Bg as in Section 7b), to replace M by Cm x Xb0 and to

trivialize the vector bundles as indicated in Section 7b). Then we will prove (8.1) in this situation.

c) Proof of Theorem 8.1.

By (7.9)

Therefore

. . r i d , . , u2 M2xl f -1 d , . , 9 , / 9 . I(8.3) Tr skf * T —— (*r)exp( o-DT ' ) = Tr skf * T ——(*r)exp(—«M 1 / T 1 ) .

L c/T T^ J L c/T 1/-L>1 jWe will use the notation of Section 7 with ε replaced by T1, and T by 1. By (7.24), we see

that as T -• +oo

Let P e ' T ((F,x), (Y',x')) be the smooth kernel associated to the operator exp(—u2Liε,T)

calculated with respect to ^ d h n M b° . For Y G NBg/B,R,b0, x G Xb0, set

(8.5) Qε,u(Y,x) = T

By (7.23), for Y G NBg/B,R,b0, x G Xb0, we have

(8.6) Trs[g*^ ^ *TPl/TXu{g-\Y,x),(Y,x))\

By (8.6) and the argument of Section 7b), to calculate the asymptotics of (8.3) as u —>• 0

uniformly in T > 1, we have to find the asymptotics as u —>• 0 of

-^NB9/B,

Let dx(x,x') be the distance function on (X,hTXb0). Then d((Y,x), (Y',x')) = (\Y-Y'\2 +

dx{x,x')2)1/2 be a distance function on (TRB)b0 x Xb0.

Proposition 8.3. There exist c,C > 0,p,r G N such that for any (Y,x),(Y',xr) G

(TRB)b0 xXbo, £6[O,1],U6]O,1],

(8.8)

exp - C|

26

PROOF. By proceeding as in the proof of Theorem 7.6, the bounds in (8.8) with C = 0 are

easily obtained . To get the required C > 0, we proceed as in the proof of [B5, Theorem 11.14].

Let u e R ^ k(u) be a smooth even function such that

k(u) = 0 for |u| < 1/2,

1 for |u| > 1.(8.9)

For q G R*+,a G C, set

(8.10) Kq(a) = 2 cos(tV2a)exp(-t2/2)k(-)-^=.Jo Q V2vr

Clearly, Kq(a) is an even holomorphic function of a, therefore, there is a holomorphic function

a G C —K Kq(a) such that

(8.11) Kq(a) = Kq(a2).

Using finite propagation speed for the solution of hyperbolic equations for cos(sLε,-J [ C P ,

§7.8], [T,§4.4], we find there is a fixed constant d > 0 such that

if d((Y,x),(Y',x'))>Jq.

By using the proof of Theorem 7.6, and [B5, Theorem 11.14], there is a C > 0 such that there

exist c > 0,p, r G N for which given q G N, (Y,x), (F',^') G (T R B) b 0 x Xb0, ε G [0,1], u G]0, 1],

then

(8.13)

From (8.13), we have (8.8).

|Y '\Y

By (8.8), to calculate the asymptotics of (8.7) as u —> 0, we can localize near {0} x

We identify Uα0({0} x Xb

g

0) to {(Y,x,X),Y G ,X G X| < α0} by0

geodesic coordinates normal to {0} x bg in TB x X.

For Y G TRB,xe Xg,X G N X g / X , R b , 0 | < ao/4, let fc'(F,a;,X) be defined by(8.14) dvX(Y,x,X) = k'(Y,x,X)dvNxg/x(X)dvX9(x).

Xg/X

Using (8.4), we find there exist smooth functions a'T _n(x), • • • ,a'T0(x) (x G Mg) such that as

u —> 0, for x G Xbg0

XeNxg/x,]X]<ao/4 Q1/T,u((0, Y), (x, X ) ) ^ ^ , x, X)'YeNBg/B,\Y\<a0/4

dvNXg/X(X)dvNBg/B(Y)

(8.15)

By (7.11),(7.26),(8.4)-(8.8),(8.15), we know that there exist aT,j depending continuously on

T G [1, +oo] such that for any u G]0, 1],T G [1, +oo]

(8.16)-1 9

rs\9*T 1( - i = - dim MaT,ju2jCU

Set

(8.17)

27

By [B5, (2.44), (2.63)], for T > 1, as u -> 0

(8.18) Tvs[g*^-^(*T)eM-u2D^'2)] = u 2 2 bo,g + O(u2).

By (8.16), (8.18), we get (8.1). •

9 A proof of Theorem 4.12

This Section is organized as follows. In a), as in [ BerB, §9], we reduce the problem to alocal problem near Bg. In b), we summarize very briefly the content of [ BerB, § 9 c)]. In c),we establish key estimates on the kernel of F£(L^T). In d), we prove Theorem 4.12.

a) Finite propagation speed and localization.

Proposition 9.1. There exists C > 0, such that for 0 < ε < 1, T > 1

(9.1) |Trs [g *~}e &{*T/e)Ge

PROOF. By an analogue of the McKean Singer formula [MKS], we find that

(9.2) Trs[gNVHε(DB)] = E^x(-l) jJxg(R jTT*OHs(0).

Using (9.2) and proceeding as in [BerB, Proposition 9.1], we have (9.1). •

By (7.6) and (9.1), to establish Theorem 4.12, we only need to establish the following result.Theorem 9.2. If α > 0 is small enough, there exist δ > 0, C > 0, such that for 0 < ε < 1,

T > 1

(9-3) T±a\g*-}e-^-(*T/e)Fe

\<

PROOF. The remainder of the Section is devoted to the proof of Theorem 9.2.

Using (7.1), we deduce that

(9-4) Tr s [g *~l

/£ -^{*T/e)Fe{eD^/e)] = Tr s [g *~l

/£ -^

Let F£(AfT)(x,x')(x,x' G M) be the smooth kernel associated to F£(AfT) with respect to

dvM/(2π) d i m M . Using finite propagation speed , it is clear that if x € M, Fe(A'2T)(x,-) only

depends on the restriction of Aε, T to TT~1(BB(TTX, α)).

As in Section 7, the proof of (9.3) is local near ir~l(B9).

b) The matrix structure of the operator L\T as T —>• +oo.

We use the same trivialization and notation as in Section 7.Also by using (7.18), (7.23), for Y e (NBg/B)b0, we get

(9.5)

28

Let F° be the vector space of square integrable sections of A(T^Bg)) ® c(NBg/B,R) (§>S~lK

over (TRB)b0. Then Fε° is a Hilbert subspace of E°. Let i^0'"1 be its orthogonal in E°. Let pε

be the orthogonal projection operator from E° on Fε0.For a fixed ε > 0, the analysis of the matrix structure of L\ T as T —> +00 is the same as

in [BerB, § 9 c)]. Of course, the rescaling on the Clifford variables which depends on ε > 0, isdifferent, but this does not introduce any extra difficulty.

Then [BerB, Theorem 9.3] still holds for essentially the same reasons as in [BerB].

c) Uniform bounds on the kernel of F£(L^T).

We now establish an extension of [BerB,Theorem 9.6].Theorem 9.3. There exist C > 0, r e N, for which if m1 e N, there exists C > 0 such that

if\a\,\a'\ < ml, ε e]0,l], T > 1, (Y,x),(Y',x>) e (TKB)boxXbo,

B\<*\+W\ ~(9-6) dYadYlalFe{L%T){{Y, x), (Y1, x')) < c(l + | F | + \Y'\Y exp(-C|F - Y>\2

PROOF. Recall ( ε,0 be the Hermitian product on E° defined by (7.31). If s G E1, put

(9-7) \s\lTA = T2\PtYs\% + \PeYs\l0 + sMV / a 4 2

> 0 + T2nvezPtYs\%.

The bounds in (9.6) with C = 0 are easily obtained by proceeding as in [BerB,Theorem9.6]. To get the required C > 0, we proceed as in the proof of [B5, Theorem 11.14 and 13.14J.B

d) Proof of Theorem 9.2.

Let Ξε be the analogue of the elliptic second order differential operator considered in [BerB,Definition 9.7]. The minor difference with [BerB] is that here only the Clifford variables c(fl)(1 < l < 2dimBg) are rescaled, while in [BerB], the Clifford variables c(fl)(1 < l < 2dimB)were rescaled. Because our Clifford rescaling introduces fewer diverging terms than in [BerB,§9], the analogue of [BerB, Theorem 9.8] still holds.

Proceeding as in [BerB, §9f), e)] and [B5, §13 j)], we obtain Theorem 9.2. •

Acknowledgements. I learned subjects from Professor Jean-Michel Bismut. I'm very muchindebted to him for very helpful discussions, suggestions and for the time he spent in teachingme. Without his help, this paper may never have appeared in this form. I would also likethank the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy, especiallyProfessor M.S. Narasimhan for hospitality.

29

References

[ABo] Atiyah M.F., Bott R. A Lefchetz fixed point formula for elliptic complexes, I. Ann. ofMath, 86 (1968), 374-407.

[ABoP] Atiyah M.F., Bott R. and Patodi.V.K., On the heat equation and the Index Theorem,Invent.Math. 19 (1973), 279-330.

[BeGeV] Berline N., Getzler E. and Vergne M., Heat kernels and the Dirac operator, Grundl.Math. Wiss. 298, Springer, Berlin-Heidelberg-New York 1992.

[BerB] Berthomieu A., Bismut J.-M., Quillen metric and higher analytic torsion forms. J.Reine Angew. Math 457, 1994,85-184.

[B1] Bismut J.-M.,The index Theorem for families of Dirac operators: two heat equationproofs, Invent.Math. ,83 (1986), 91-151.

[B2] Bismut J.-M., Superconnection currents and complex immersions, Invent. Math. 99(1990), 59-113.

[B3] Bismut J.-M., Koszul complexes, harmonic oscillators and the Todd class, J.A.M.S. 3(1990), 159-256.

[B4] Bismut J.-M., Equivariant short exact sequences of vector bundles and their analytictorsion forms. Comp.Math.,93 (1994), 291-354.

[B5] Bismut J.-M., Equivariant immersions and Quillen metrics, J. Diff. Geom. 41 (1995).53-159.

[B6] Bismut J.-M., Families of immersions, and higher analytic torsion, Asterisque 244,1997.

[B7] Bismut J.-M., The Atiyah-Singer Index Theorems: a probabilistic approch, II. TheLefschetz fixed point formulas, J.Funct. Anal. 57 (1984), 329-348.

[BCh] Bismut J.-M., Cheeger J., η-invariants and their adiabatic limits, J.A.M.S., 2 (1989),33-70.

[BGS1] Bismut J.-M., Gillet H., Soule C., Analytic torsion and holomorphic determinantbundles.I, Comm.Math. Phys. 115 (1988), 49-78.

[BGS2] Bismut J.-M., Gillet H., Soule C., Analytic torsion and holomorphic determinantbundles.II, Comm.Math. Phys. 115 (1988), 79-126.

[BGS3] Bismut J.-M., Gillet H., Soule C., Analytic torsion and holomorphic determinantbundles.III, Comm.Math. Phys. 115 (1988), 301-351.

[BK6] Bismut J.-M., Kohler K., Higher analytic torsion forms for direct images and anomalyformulas. J. of. Alg. Geom. 1 (1992), 647-684.

[BL] Bismut J.-M. and Lebeau G., Complex immersions and Quillen metrics. Publ. Math.IHES., Vol. 74, 1991, 1-297.

[CP] Charazain J., Piriou A., Introduction a la theorie des equations aux derivees partielles,Paris: Gauthier-villars 1981.

[D] Dai X., Adiabatic limits, non multiplicativity of signature and Leray spectal sequence,J.A.M.S. 4(1991), 265-321.

[DM] Dai X., Melrose R.B., Adiabatic limit of the analytic torsion. Preprint.[Ge] Getzler E., A short proof of the Atiyah-Singer Index Theorem, Topology, 25 (1986),

111-117.

[GrH] Griffiths P., Harris J., Principles of Algebraic Geometry, New-York,Wiley 1978.[GS1] Gillet H., Soule C., Analytic torsion and the arithmetic Todd genus, Topology 30

(1991), 21-54.[GS2] Gillet H., Soule C., An arithmetic Riemann-Roch Theorem, Invent.Math., 110(1992),

473-543.

[KM] Knudsen P.F., Mumford D., The projectivity of the moduli space of stable curves, I,Preliminaires on "det" and "div", Math. Scand. 39 (1976), 19-55.

30

[KRo] Kohler K., Roessler D., Un theoreme du point fixe de Lefschetz en geometrie d'Arakerov,C.R.A.S. Paris, t326, serie I (1998), 719-722.

[Ma] Ma Xiaonan., Formes de torsion analytique et familles de submersions, These, Orsay1998.

[Ma1] Ma Xiaonan., Formes de torsion analytique et familles de submersions, C.R.A.S. Paris,324, serie I (1997), 205-210.

[MKS] McKean H., Singer I.M., Curvature and the eigenvalues of the Laplacian, J. Diff.Geom. 1(1967), 43-69.

[Q] Quillen D., Determinants of Cauchy-Riemann operators over a Riemann surface, Funct.Anel. appl. 14 (1985), 31-34.

[RS] Ray D.B., Singer I.M., Analytic torsion for complex manifolds, Ann. of Math. 98(1973), 154-177.

[T] Taylor M., Pseudodifferential operators, Princeton Univ Press, Princeton 1981.

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u

A

To T