Equivariant analytic torsion on P n C

Transcription

1 Euivariant analytic torsion on P n C Kai KÖOHLER Universitée de Paris-Sud Bâat ORSAY Abstract for the Zentralblatt der Mathematik: The subject of the paper is to calculate an euivariant version of the complex Ray-Singer torsion for all bundles on the P C and for the trivial line bundle on P n C, for isometries which have isolated fixed points The result can for all n be expressed with a special function, which is very similar to the series defining the Gillet-Soul R -genus 99 Mathematics Subject Classification: 58G26, 4J20, 53C30

2 Euivariant analytic torsion on P n C Kai KÖOHLER Abstract: We calculate an euivariant version of the complex Ray-Singer torsion for all bundles on the P C and for the trivial line bundle on P n C, for isometries with isolated fixed points The result gives for all n a part of the Gillet-Soul R -function Keywords: Determinants and determinant line bundles, Arakelov geometry, Homogeneous manifolds 990 AMS-Subject classification: 58G26, 4J20, 53C30 Running title: Euivariant analytic torsion on P n C Adress: Kai K hler Math matiues B t 425 F-9405 ORSAY France Koehler@matupsmatupsfr 2

3 Introduction The analytic torsion was constructed by Ray and Singer [RS] as an analytic analogue to the Reidemeister torsion Bismut, Gillet and Soul [BGS] proved as an extension of a result of Quillen important properties of the torsion in connection with vector bundles on fibrations: Let π : M B be a proper holomorphic map of compact complex manifolds and let ξ be a hermitian holomorphic vector bundle on M Let Rπ ξ be the right-derived direct image of ξ Then the analytic torsion of the fibres of π induces a metric on the Knudsen-Mumford determinant λ KM := (det Rπ ξ which is a holomorphic line bundle on B The curvature of this Quillen metric as well as its behaviour under changes of the metrics on M and ξ was expressed in [BGS] explicitly by means of secondary Bott-Chern classes In particular this gives a refinement of the Riemann-Roch theorem for families On the other hand let i : Y X be an embedding of compact complex manifolds Let η be a hermitian holomorphic vector bundle on Y and let ξ be a resolution of η by a complex of vector bundles on X Bismut and Lebeau [BL] calculated the relation between the Quillen metrics of η and ξ With the help of this result, Gillet and Soul [GS2] were able to prove a Riemann-Roch theorem in Arakelov geometry for the first Chern class of the direct image (see [S] for the theorem and some background information This theorem was later proved by Faltings [F] for higher degrees The proof of the Riemann-Roch theorem uses a calculation of Gillet, Soul and Zagier [GS] of the torsion for the trivial line bundle on the complex projective spaces P n C This led Gillet and Soul to conjecture this theorem, which was the initial motivation for [BL] In particular this rather difficult calculation gives in particular the Gillet-Soul R -genus, which appears explicitly in the theorem This is the additive genus associated to the series 3

4 R(x = l odd ( 2ζ ( l + ζ( l l j= x l j l!, where ζ is the Riemann zeta function To obtain this series, one has to caculate the torsion of P n C for every n Let us consider now a holomorphic isometry g of a hermitian vector bundle E over a compact K hler manifold M One can define in a natural way an euivariant version of the torsion This euivariant torsion appeared already in Ray s [R] calculation of the real analytic torsion for lens spaces In this paper we present the calculation of the euivariant analytic torsion for all holomorphic bundles on P C and for the trivial line bundle on P n C, where the projective spaces are euipped with the Fubini-Study metric We consider only rotations with isolated fixpoints For a rotation by angles π Q, we obtain a closed expression involving the gamma function For arbitrary angles a function R rot, which is similar to the Gillet-Soul R -function, appears as an infinite series This is relatively easy to calculate because the defining ζ -function Z has no singularities in contrast to the situation in [GS] The similarity of R rot and R might help to find an euivariant Riemann-Roch formula in Arakelov geometry, where the two functions correspond to the extremal cases: isolated fixed points or identity map In fact, Bismut [B3] found further evidence for such a formula: He constructed analytic torsion forms associated to a short exact seuence of hermitian holomorphic vector bundles euipped with a holomorphic unitary endomorphism g In his result, a series R(ϕ, x appears with the properties R(0, x = R(x, R(ϕ, 0 = R rot (ϕ As the appearance of the R -genus in [B2] gave evidence for the existence of the Riemann-Roch theorem, he now conjectures an euivariant Riemann-Roch formula The function R rot and s > 0, ζ rot (ϕ, s be the Dirichlet series can be obtained as follows: Let for 0 < ϕ < 2π 4

5 Then ζ rot ζ rot (ϕ, s := k sin kϕ k s can be seen as the imaginary part of a Lerch zeta function We set R rot (ϕ := s ζ rot (ϕ, 0 The following is obtained by classical results: Proposition R rot is eual to If ϕ = 2π p R rot (ϕ = C + log ϕ ϕ l l odd with p, N, 0 < p <, then ζ ( l( l+ 2 ϕ l l! R rot (ϕ = 2 log cot ϕ 2 + l= log Γ ( j sin jϕ In the last chapter we give some other functional properties of R rot Let E := O(k O(k n be a holomorphic vector bundle on P C, euipped with the standard metric (ie the curvature of O( is the Fubini-Study K hler form By a theorem of Grothendieck, each holomorphic vector bundle on P C is of this form Then we find Theorem 2 The euivariant analytic torsion τ(e, ϕ with respect to a rotation by an angle ϕ ]0, 2π[ is given by 2 log τ(e, ϕ = 2R rot (ϕ sin ϕ 2 n cos(k j + ϕ n 2 + j= j= k j + m= sin(2m k j + ϕ 2 sin ϕ 2 log j We see in particular that the euivariant torsion τ gives already for the trivial line bundle O on P C the function Let now Φ := log τ(o, ϕ = cot ϕ 2 ( i l odd ( iϕ 0 0 iϕ n+ ζ ( l (iϕl l! C + log ϕ ϕ be an element of the (canonical maximal Cartan subalgebras of su(n+, hence an infinitesimal rotation on P n C = SU(n + / S(U( U(n Assume that all the ϕ j we have 5 are distinct Then

6 Theorem 3 bundle O on P n C is given by n+ 2 log τ(o, e Φ = ( n The euivariant torsion τ(o, e Φ for the trivial line j,k= j k n+ 2iR rot (ϕ j ϕ k (e i(ϕ k ϕ l log n! l= l k I Definition of the torsion Let M be a K hler manifold of complex dimension n with holomorphic tangent bundle T M and K hler form ω M, ξ a hermitian vector bundle on M and the Dolbeault operator acting on sections of Λ T (0, M ξ We define a hermitian product on the vector space of smooth sections of Λ T (0, M ξ by (η, η := (η(x, η ω n (x (2π n n! M as in [GS] Consider the adjoint operator relative to this product and the Kodaira-Laplace operator := ( + 2 : Γ(Λ T (0, M ξ Γ(Λ T (0, M ξ Let g be a holomorphic isometry of M Assume that the bundle and its hermitian metric are holomorphically invariant under the induced action of g Let Eig λ ( be the eigenspace of corresponding to the eigenvalue λ and g the of g induced action on Γ(Λ T (0, M ξ Consider the ζ -function Z(g, s := >O ( + λ s Trg Eig λ ( λ Spec λ 0 for s 0 The euivariant torsion of M relative to the action of g is then defined as an exponential of the derivative at zero Z (g, 0 of the holomorphic continuation of Z(g,, τ(g := e 2 Z (g,0 The eigenvalues and eigenspaces for the Kodaira Laplacian for the trivial line bundle on P n C were determined explicitly by Ikeda and Taniguchi [IT] If one regards P n C as SU(n + /S(U( U(n, the eigenspaces 6

7 can be described by sums of irreducible representations of SU(n + We are using their method and results in our proof; see also Malliavin and Malliavin [MM] II The Laplacian on O(k-bundles over P C Let P C be the one-dimensional complex projective space euipped with the usual Fubini-Study metric That means, P C is isometric to the 2-sphere with radius /2 Take G := SU(2 and K := S(U( U( with the corresponding Lie algebras g and k We euip G with the metric g 2 R (X, Y 2trXY, which is minus one half of the Killing form Then we may represent P C as the homogeneous space G/K with the induced metric Let Λ be the weight of g which acts on the Cartan subalgebras k by diag(iϕ, iϕ ϕ 2π and let ρ K k : k C ( iϕ 0 e ikϕ 0 iϕ be the of kλ, k Z, induced representation of K This gives an action of K on the right of G C as follows: (g, x h = (gh, ρ K k (h x for g G, x C and h K Then the holomorphic line bundle O(k is the homogeneous vector bundle O(k = G C := (G C/K ρ K k It is well known that O(2 = T P C = T (0, P C By a theorem of Grothendieck [G], each holomorphic vector bundle E on P C is a direct sum E = O(k O(k n, k,, k n Z, so it suffices to calculate the torsion for O(k Obviously, Z (, 0 behaves additively under direct sum of vector bundles 7

8 We euip O(k with the induced metric If is the uniue holomorphic hermitian connection on the bundle of forms with coefficients in O(k, ΛT (0, P C O(k, and (e, e 2 a real orthonormal frame in the real tangent bundle T R P C, we define the horizontal (or Bochner Laplacian as := 2 ( en 2 2 e n e n We know that the curvature tensor of O( is simply 2i times the K hler form of P C By applying Licherowicz s formula (cf Bismut [B, Prop 2], we find that the Kodaira Laplacian acting on T (0, P C O(k is given by 0, = 2 + k 2 + To find a better expression for, we consider the Casimir Operators of G and K For a given compact Lie algebra with Killing form B and orthonormal basis {X,, X n } with respect to B, its Casimir operator is defined as Cas := i X i X i Cas is independent of the choice of the basis Let Cas G be the Casimir operator of G, acting on C (G by derivation, and Cas K operator of K, acting on C via the representation ρ K k 2 easily verified (cf for example [BGV, Prop 56] that the Casimir Then it is 2 = Cas G + Cas K on sections of T (0, P C O(k = G ρ K k 2 C The factor 2 appears because we take half of the negative Killing form as metric on G For X k we have ρ K k 2 (X = i(k + 2, so hence ρ K k 2(Cas K = (k + 2 2, Lemma 4 0, = 4 Cas G k 2 (k 2 + 8

9 III Construction of the defining ζ -function Let (ρ G l, EG l be the irreducible representation G End(EG l with highest weight lλ, l N Then we have ρ G l (Cas G = l(l + 2 Id E G l To determine the eigenspaces of 0,, we use as Ikeda and Taniguchi the following Frobenius law of Bott [Bo]: Proposition 5 For finite dimensional representations (ρ K, E K and (ρ G, E G of K and G, we have the canonical isomorphism of vector spaces by Hom G (E G, Γ(G ρ K E K = Hom K (E G, E K Now we know that the characters χ G l of ρ G l and χ K k of ρk k χ G l ( e iϕ 0 0 e iϕ = sin(l + ϕ sin ϕ (cf Br cker, tom Dieck [BD, Ch 5, p 267], and ( χ K e iϕ 0 k 0 e iϕ = e ikϕ, hence we find the decomposition χ G l = n l n even n l n odd χ K n when l even χ K n when l odd Now we can see by Proposition 5 that (ρ G l, EG l subspace of Γ(G ρ K n C iff n l and n l(mod 2: are given occurs as irreducible Lemma 6 Γ(T (0, P C O(k contains the L 2 -dense subspace E k+2 +2l G l 0 The density of this subspace follows from the Peter-Weyl theorem (cf [Bo] By Lemma 4, the eigenvalues of 0, for O(k are given by { l(l + k + on E G k+2l for l when k l(l k on E G k 2+2l for l 0 when k < So we finally obtain the 9

10 ( e Lemma 7 Let g := iϕ 0 0 e iϕ G, ϕ ]0, π[, be an element of the maximal torus K (which corresponds to the rotation of S 2 by the angle 2ϕ Then the ζ -function Z k (g, of the O(k-bundle on P C is for s > 2 given by Z k (g, s = E = l l 0 G k+2 +2l ker 0, sin(2l + k + ϕ sin ϕ χ G k+2 +2l ( 0, E G k+2 +2l l s (l + k + s In particular, Z k (g, s = Z k 2 (g, s This is in fact an immediate conseuence of the Poincar duality s IV The derivative at zero of the Lerch zeta function Define for 0 < ϕ < 2π, Re s > 0 the zeta function ζ rot (ϕ, s by ζ rot sin lϕ (ϕ, s := l s ζ rot continuous holomorphically to the whole complex plane Let ϕ = 2π p, p, N, 0 < p < be a rational angle and ζ(, the Hurwitz zeta function We obtain ζ rot (ϕ, s = = j= l=0 j= l= sin(l + jϕ (l + j s = sin jϕ s ζ(s, j j= sin jϕ s l=0 By using the euations (see for example [WW, Chap XIII] we find Because of ζ(0, x = 2 x and ζ(s, x = log Γ(x s s=0 2π s ζ rot (ϕ, 0 = j= sin jϕ = 0 and j= ( l + j s sin jϕ ( log Γ( j 2π log ( 2 j j= j sin jϕ = 2 cot ϕ 2 this is eual to s ζ rot (ϕ, 0 = 2 log cot ϕ 2 + sin jϕ log Γ( j j= 0

11 V The derivative at zero for arbitrary angles We are using Kummer s Fourier series for the logarithm of the Γ- function ( cos 2πnx log Γ(x = 2 log 2π+ C + log 2πn + sin 2πnx (0 < x < 2n nπ n With the orthogonal relations sin 2πjp cos 2πjn = 0, j= j= sin 2πjp sin 2πjn and the Fourier series of the identity function it follows that s ζ rot (ϕ, 0 = 2 x log = log 2 = 2 ( δ p n( mod δ p n( mod n [ C + log 2π p pπ = C + log ϕ ϕ log nπ sin 2πnx (0 < x <, ( C + log (2π n+p (n + pπ + n + ( C + log(2πn + ϕ 2πn + ϕ n C + log (2π n p (n pπ C + log(2πn ϕ 2πn ϕ We have the identities (see [WW] or Bismut and Soul [B2, Appendix] ( n + x = π cot πx n x x = 2 ζ(l + x l, n l and n ( log n n + x log n = 2x log n n x n 2 n n so we obtain ( log( + x n n + x s ζ rot (ϕ, 0 = C + log ϕ + ϕ π = C + log ϕ ϕ odd ( x 2l = 2 ζ n (l + x l l 0 log( x n = 2 n x n n l odd l odd l odd = 2 l odd ( ζ (l + ζ(l + + ζ ( l( l+ 2 x l n ζ(l + l j= ϕ l l! l odd l j= l j= j j x l, ] j C log 2π ζ(l + ( ϕ l 2π

12 This gives the Proposition by continuity VI The torsion on P C Recall now the zeta function Z k of Lemma 7 with ϕ 0 By a Taylor expansion of the denominator with respect to k+ l, we find for s 0 s Z k(g, s = sin(2l + k + ϕ ( log l log(l + k + sin ϕ l s + (l + k + s l s (l + k + s l = l l> k+ = l = hence for s = 0 s Z k(g, 0 = sin(2l + k + ϕ sin ϕ sin(2l k + ϕ sin ϕ 2 cos k + ϕ sin 2lϕ sin ϕ 2 cos k + ϕ sin ϕ log l l 2s ( + log l l 2s ( k + s l k + s l log l k+ sin(2l k + ϕ l 2s + log l sin ϕ l 2s + O(s l= k+ s ζ rot sin(2l k + ϕ (2ϕ, 2s + log l sin ϕ l 2s + O(s, l= k+ 2 cos k + ϕ R rot sin(2l k + ϕ (2ϕ + sin ϕ sin ϕ l= log l Remark that this computation breaks down for ϕ = 0 because of the singularity of the Riemann ζ -function The isomorphism g corresponds to a rotation of the sphere by an angle 2ϕ, so we obtain Theorem 2 VII The zeta function on P n C Now we regard as in [IT] the complex projective space P n C as the homogeneous space SU(n + /S(U( U(n Let {( iϕ 0 } n+ h := ϕ j = 0 0 iϕ n+ be the canonical maximal Cartan subalgebra of the Lie algebra su(n + Let Λ j, j n, be the fundamental weight Λ j : diag(iϕ,, iϕ n+ 2 j ϕ k 2π

13 In the following, Λ(k, 0, denotes the irreducible SU(n+ -representation with highest weight given by (k Λ +Λ +kλ n for all k, n 0 Ikeda and Taniguchi found that the spaces Λ(k, 0, 0 ( = 0 k 0 Λ(k, 0, k k + Λ(k, 0, + (0 < < n Λ(k, 0, n ( = n k n can be regarded as L 2 -dense subspaces of Γ(Λ T (0, P n C, where the Laplacian acts on Λ(k, 0, by multiplication with k(k + n + We denote by χ(k, 0, the character to the representation Λ(k, 0, Hence we find for our zeta function n ( Z(, s = ( + χ(k, 0, k s (k + n + s + = k k + χ(k, 0, + k s (k + n s +( n+ n χ(k, 0, n k s (k + s k n n = =( + χ(k, 0, k s (k + n + s k The telescope effect in the summation is not caused by accident, but by the natural splitting of each eigenspace Eig λ ( into Eig λ ( ker and Eig λ ( ker, which are isomorphic The character χ Λ of an irreducible SU(n + -module with highest weight Λ = m Λ + m 2 (Λ 2 Λ + + m n (Λ n Λ n, m m n m n+ = 0, can classically be calculated by Weyl s character formula One finds with e j := e iϕ j ( iϕ 0 χ Λ = det(em l+n+ l j n+ j,l= det(e n+ l j n+ j,l= 0 iϕ n+ In our case one gets after a rotation of the first rows e n e n n+ e n e n n+ e n+ ( e n+ ( n+ χ(k, 0, = exceptional e n+ +k e n+ +k n+ -th row e n+ (+ e n+ (+ : ( + n+ e e n+ e k e k n+ 3

14 We see immediately χ(k (n +, 0, = χ( k, 0, and χ(k, 0, = 0 for k { n,, } \ {0, n } VIII The torsion on P n C Remark that χ(k, 0, as a function in k can be regarded as a linear combination of exponentials exp ik(ϕ j ϕ l with j, l n + So the function k log k χ(k, 0, k2s is a linear combination of Lerch ζ -functions Hence it follows, if all the ϕ j are distinct, for s 0 n Z (, s = =( +( k n = =( +( k + n = =( + k χ(k, 0, logk k s (k + n + s k n+ ( χ(k, 0, χ( k, 0, log k k 2s log(n + χ( n, 0, (n + 2s χ( k, 0, log k k s (k n + s + O(s ( χ(k, 0, χ( k, 0, log k k 2s log n! + O(s, because of χ( n, 0, = ( The Laplace expansion theorem for determinants shows n n+ ( + χ(k, 0, = = j= Hence we obtain some Vandermonde determinants: e k j e n e n n+ e n e n n+ : e e n+ e k e k n+ 4

15 ( +( χ(k, 0, χ( k, 0, = n+ j,l= j l ( (ej k ( el k e n e n l e n n+ e n e n n+ ( n+l e l e j : e ḙ l e n+ ( (ej k ( el k n+ ( el e l e j e k n+ = ( n j,l= k= k l (the indicates that the l-th column is missing By using ( ej k ( el k = 2i sin k(ϕj ϕ l e l e j and the definition of R rot (ϕ, we find Theorem 3 IX Remarks about the function R rot The function R rot has a rather simple definition and hence a lot of special properties Here we only give a few of them Theorem 8 The following identities hold ( R rot (ϕ = R rot (2π ϕ (0 < ϕ < 2π, (2 2R rot (2ϕ = R rot (ϕ + R rot (π + ϕ + log 2 cot ϕ (0 < ϕ < π, 3R rot (3ϕ = R rot (ϕ + R rot( 2π (3 3 + ϕ R rot( 2π 3 ϕ log 3 cot 3ϕ (0 < ϕ < 2π 2 3, (4 R rot sinh ϕx (π + ϕ = log x dx ( π < ϕ < π sinh πx 0 Proof: is trivial by the definition of R rot 2 follows from 2 s ζ rot (2ϕ, s = ζ rot (ϕ, s + ζ rot (π + ϕ, s We see by the formulas of IV that ζ rot (ϕ, 0 = 2 cot ϕ 2 The result follows then by derivation In the same way, one gets 3 from 3 s ζ rot (3ϕ, s = ζ rot (ϕ, s + ζ rot( 2π 3 + ϕ ζ rot( 2π 3 ϕ To see the integral formula 4 we are using the Fourier series π 2 sinh ϕx sinh πx = ( l l x 2 sin lϕ ( ϕ < π + l2 5

16 and the definite integral 0 x s dx x 2 + l 2 = l s π 2l +s cos sπ 2 We have for s < ζ rot ( l sin lϕ (π + ϕ, s = = 2 π = cos πs 2 The desired result follows 0 cos πs 2 s sinh ϕx x sinh πx dx ( s < 0 ( l lx s dx x 2 + l 2 sin lϕ Acknowledgement: I would like to thank Prof J-M Bismut for sharing the idea of this problem Also I would like to thank Prof C Deninger for helpful comments This article is a part of the author s thesis References [B] [B2] [B3] [BD] [BGS] [BGV] [BL] J-M Bismut: D ly s asymptotic Morse ineualities: a heat euation proof, J Funct Anal 72(987, J-M Bismut: Koszul complexes, harmonic oscillators and the Todd class, with an appendix by J-M Bismut and C Soul, JAMS 3 (990, J-M Bismut: Euivariant short exact seuences of vector bundles and their analytic torsion forms, to appear T Br cker, T tom Dieck: Representations of Compact Lie Groups, Graduate Texts Math 98 (985, Springer-Verlag J-M Bismut, H Gillet, C Soul :Analytic torsion and holomorphic determinant bundles I, II, III, Comm in Math Physics 5 (988, 49 78, 79 26, N Berline, E Getzler, M Vergne: Heat Kernels and Dirac Operators, Grundlehren math Wiss 298 (992, Springer-Verlag J-M Bismut, G Lebeau: Complex immersions and Quillen metrics, to appear in Publ Math IHES 6

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