Géométrie différentielle/Differential geometry

Holomorphic torsion on Hermitian symmetric spaces

Kai Köhler

Abstract – We calculate explicitly the equivariant holomorphic Ray-Singer torsion for all equivariant Hermitian vector bundles over Hermitian symmetric spaces G/K with respect to any isometry g ∈ G. In particular, we obtain the value of the usual non-equivariant torsion. The result is shown to provide very strong support for Bismut’s conjecture of an equivariant arithmetic Grothendieck-Riemann-Roch theorem.

Torsion holomorphe pour des espaces hermitiens symétriques

Résumé – On calcule explicitement la torsion équivariante holomorphe de Ray-Singer pour tous les fibrés vectoriels hermitiens équivariants sur les espaces hermitiens symétriques compactes G/K relativement à chaque isométrie g ∈ G. En particulier on obtient la valeur de la torsion non-équivariante. Le resultat va dans le sens de la conjecture de Bismut d’un théorème de Grothendieck-Riemann-Roch arithmétique equivariant.

Version française abrégée – Soit E un fibré holomorphe hermitien sur une variété complexe compacte M. Soit □_{q} := (¯∂ + ¯∂^{∗})^{2} l’opérateur de Laplace-Kodaira agissant sur Γ(Λ^{q}T^{∗0,1}M ⊗ E). Soit g une isométrie holomorphe de M et supposons que le fibré hermitien soit invariant par l’action de g. Soit τ_{g}(M, E) la torsion équivariante comme définie dans [8]. Elle est donnée par la dérivée en zéro d’une certaine fonction zêta associée au spectre de □ et à l’action de g sur les espaces propres de □. La torsion joue un rôle crucial dans la définition d’une image directe dans la K-théorie arithmétique de Gillet-Soulé.

Considérons un groupe de Lie compact et semi-simple G. Soit G/K un espace hermitien symétrique equipé d’une métrique G-invariante ⟨·, ·⟩_{⋄}. Soit T ⊆ K un tore maximal et Ψ un système de racines d’une structure complexe invariante de G/K dans le sens de [4]. Soit ρ_{G} la demi-somme des racines positives de G. On pose (α, ρ_{G}) := 2⟨α, ρ_{G}⟩_{⋄}/∥α∥_{⋄}^{2} pour chaque poids α. Soit χ_{α} le caractère virtuel associé à α.

Choisissons une représentation irréductible V de K du poids maximal Λ. Soit E := (G×V)/K le fibré vectoriel associé. Pour exprimer la torsion de E, il faut d’abord établir quelques notations. Soit P : Z → C une fonction du type P (k) = ∑_{j=0}^{m} c_{j}k^{n_{j}}e^{ikϕ_{j}} où m, n_{j} ∈ N_{0}, c_{j} ∈ C, ϕ_{j} ∈ R. Soit ζ_{L} la fonction zêta de Lerch. On pose P^{odd}(k) := (P (k) − P (−k))/2 et

ζP := ∑_{j=0}^{m} c_{j}ζ_{L}(−n_{j}, ϕ_{j}), ζ′P := ∑_{j=0}^{m} c_{j}ζ′_{L}(−n_{j}, ϕ_{j})

et P^{∗}(p) := − ∑_{{j=0 \atop ϕ_{j} ≡0 mod2π}}^{m} c_{j} \frac{p^{n_{j}+1}}{4(n_{j} + 1)} ∑_{ℓ=1}^{n_{j}} \frac{1}{ℓ}.

Alors on obtient le résultat suivant par des méthodes similaires à [8] Theorem 1 ([9]) Le logarithme de la torsion équivariante de E sur G/K est donné par

log τ_{g} = ζ′ ∑_{α∈Ψ} χ_{ρ_{G}+Λ+kα}^{odd}(g) − ∑_{α∈Ψ} χ_{ρ_{G}+Λ−kα}(g)^{∗} ((α, ρ_{G} + Λ))

+\frac{1}{2} ∑_{α∈Ψ} ∑_{k=1}^{(α,ρ_{G}+Λ)} χ_{ρ_{G}+Λ−kα}(g) log k + \frac{1}{2} ∑_{α∈Ψ} (\frac{1}{2} − ζχ_{ρ_{G}+Λ+kα}^{odd}(g)) log \frac{∥α∥_{⋄}^{2}}{2}

pour g ∈ G.

Ce resultat correspond très bien à la conjecture d’un théorème de Grothendieck-Riemann-Roch arithmétique équivariante de Bismut [2].

The Ray-Singer analytic torsion is a positive real number associated to the spectrum of the Kodaira-Laplacian on Hermitian vector bundles over compact Hermitian manifolds [10]. It was shown by Quillen, Bismut, Gillet and Soulé that the torsion provides a metric with very beautiful properties on the determinant line bundle of direct images in K-theory over Kähler manifolds.

The main application of this construction is related to arithmetic geometry. Extending ideas of Arakelov, Gillet and Soulé constructed for arithmetic varieties X (i.e. flat regular quasi-projective schemes over Spec Z with projectiv fibre X_{Q} over the generic point) a Chow intersection ring and a K-theory by using differential geometric objects on the Kähler manifold X := X ⊗ C [11]. In particular, the K-theory consists of arithmetic vector bundles on X with Hermitian metric over X and certain classes of differential forms. Using the torsion as part of a direct image, Bismut, Lebeau, Gillet and Soulé were able to prove an arithmetic Grothendieck-Riemann-Roch theorem relating the determinant of the direct image in the K-theory to the direct image in the arithmetic Chow ring. For a generalization of these concepts to higher degrees, see Bismut-Köhler [3] and Faltings [5].

One important step in the proof of the theorem was its explicit verification for the canonical projection of the projective spaces to Spec Z by Gillet, Soulé and Zagier [6]. In particular, the Gillet-Soulé R-genus, a rather complicated characteristic class occuring in the theorem was determined this way. The discovery of the same genus in a completely different calculation of secondary characteristic classes associated to short exact sequences by Bismut gave further evidence for the theorem.

In [8], an equivariant version of the analytic torsion was introduced and calculated for rotations with isolated fixed points of complex projective spaces. The result led Bismut to conjecture an equivariant arithmetic Grothendieck-Riemann-Roch formula [2]. Redoing his calculations concerning short exact sequences, he found an equivariant characteristic class R which equals the Gillet-Soulé R-genus in the non-equivariant case and the function R^{rot} in the case of isolated fixed points. In [1], he was able to show the compatibility of his conjecture with immersions.

In this note, we give the equivariant torsion for all compact Hermitian symmetric spaces G/K with respect to the action of any g ∈ G as calculated in [9]. The result is of interest also in the non-equivariant case: The torsion was known only for very few manifolds; the projective spaces, the elliptic curves and the tori of dimension > 2 (for which it is zero for elementary reasons). Also, Wirsching [12] found a complicated algorithm for the determination of the torsion of complex Grassmannians G(p, n), which allowed him to calculate it for G(2, 4), G(2, 5) and G(2, 6). Thus, our results extend largely the known examples for the torsion.

Let M be a compact n-dimensional Kähler manifold with holomorphic tangent bundle T M. Consider a hermitian holomorphic vector bundle E on M and let ¯∂ and ¯∂^{∗} denote the associated Dolbeault operator and its formal adjoint. Let □_{q} := (¯∂ + ¯∂^{∗})^{2} be the Kodaira-Laplacian acting on Γ(Λ^{q}T^{∗0,1}M ⊗ E). We denote by Eig_{λ}(□_{q}) the eigenspace of □_{q} corresponding to an eigenvalue λ. Consider a holomorphic isometry g of M which induces a holomorphic isometry g^{∗} of E. Then the equivariant analytic torsion is defined via the zeta function

Z_{g}(s) := ∑_{q>0}(−1)^{q}q ∑_{{λ∈Spec□q \atop λ≠0}} λ^{−s} Tr g_{|Eig_{λ}(□_{q})}^{∗}

for ℜe s ≫ 0. Classically, this zeta function has a meromorphic continuation to the complex plane which is holomorphic at zero. The equivariant analytic torsion is defined as τ_{g} := exp(−Z′_{g}(0)/2). This gives for g = Id_{M} the ordinary analytic torsion τ of Ray and Singer [10].

Let G/K be a compact hermitian symmetric space, equipped with any G-invariant metric ⟨·, ·⟩_{⋄}. We may assume G to be compact and semi-simple. Let T ⊆ K denote a fixed maximal torus. Let Θ be a system of positive roots of K (with respect to some ordering) and let Ψ denote the set of roots of an invariant complex structure in the sense of [4]. Then Θ ∪ Ψ =: Δ^{+} is a system of positive roots of G for a suitable ordering, which we fix [4, 13.7].

Let ρ_{G} denote the half sum of the positive roots of G and let W_{G} be its Weyl group. As usual, we define (α, ρ_{G}) := 2⟨α, ρ_{G}⟩_{⋄}/∥α∥_{⋄}^{2} for any weight α. For any weight b, the (virtual) character χ_{b} evaluated at t ∈ T is given via the Weyl character formula by

χ_{b}(t) = \frac{∑_{w∈W_{G}} det(w) e^{2πiwb(t)}}{∑_{w∈W_{G}} det(w) e^{2πiwρ_{G}(t)}} .

This extends to all of G by setting χ_{b} to be invariant under the adjoint action. Let V be an irreducible K-representation with highest weight Λ and let E := (G × V)/K denote the associated G-invariant holomorphic vector bundle on G/K. The metric ⟨·, ·⟩_{⋄} on 𝔤 induces a hermitian metric on E. Using similar methods as in [8], one may reduce the problem of determining Z_{g}(s) to a problem in finite-dimensional representation theory. This way one gets our key result

Theorem 2 The zeta function Z associated to the vector bundle E over G/K is given by

Z(s) = −2^{s} ∑_{{α∈Ψ \atop k>0}}⟨kα, kα + 2ρ_{G} + 2Λ⟩_{⋄}^{−s}χ_{ρ_{G}+Λ+kα}.

Let for ϕ ∈ R and s > 2

ζ_{L}(s, ϕ) = ∑_{k>0} \frac{e^{ikϕ}}{k^{s}}

denote the Lerch zeta function. Let P : Z → C be a function of the form

P (k) = ∑_{j=0}^{m} c_{j}k^{n_{j}}e^{ikϕ_{j}}

with m ∈ N_{0}, n_{j} ∈ N_{0}, c_{j} ∈ C, ϕ_{j} ∈ R for all j. Set P^{odd}(k) := (P (k) − P (−k))/2. We define analogously to [6, 2.3.4]

ζP := ∑_{j=0}^{m} c_{j}ζ_{L}(−n_{j}, ϕ_{j}),  ζ′P := ∑_{j=0}^{m} c_{j}ζ′_{L}(−n_{j}, ϕ_{j})

\overline{ζ}P := ∑_{j=0}^{m} c_{j}ζ_{L}(−n_{j}, ϕ_{j}) ∑_{ℓ=1}^{n_{j}} \frac{1}{ℓ} and P^{∗}(p) := − ∑_{{j=0 \atop ϕ_{j} ≡0 mod2π}}^{m} c_{j} \frac{p^{n_{j}+1}}{4(n_{j} + 1)} ∑_{ℓ=1}^{n_{j}} \frac{1}{ℓ}.

Then theorem 2 implies by some calculus on zeta functions

Theorem 3 The logarithm of the equivariant torsion of E on G/K is given by

− \frac{1}{2}Z′(0) = ζ′ ∑_{α∈Ψ} χ_{ρ_{G}+Λ+kα}^{odd} − ∑_{α∈Ψ} χ_{ρ_{G}+Λ−kα}^{∗} ((α, ρ_{G} + Λ))

+\frac{1}{2} ∑_{α∈Ψ} ∑_{k=1}^{(α,ρ_{G}+Λ)} χ_{ρ_{G}+Λ−kα} log k + \frac{1}{2} ∑_{α∈Ψ} (\frac{1}{2} − ζχ_{ρ_{G}+Λ+kα}^{odd}) log \frac{∥α∥_{⋄}^{2}}{2} .

One can show that the polynomial degree in k of ∑_{Ψ} χ_{ρ_{G}+Λ+kα}(g) for any g ∈ G is at most the dimension of the fixed point set of the action of g on G/K. In particular, it is less or equal #Ψ. The torsion behaves additively under direct sum of vector bundles, thus this result gives the torsion for any homogeneous vector bundle. Remark: If the decomposition of the space G/K in its irreducible components does not contain one of the spaces SO(p + 2)/SO(p) × SO(2) (p ≥ 3) or Sp(n)/U(n) (n ≥ 2), one may choose the metric ⟨·, ·⟩_{⋄} in such a way that log ∥α∥_{⋄}^{2}/2 = 0 for all α ∈ Ψ. Thus the corresponding term in theorem 3 vanishes.

We shall now compare the result with Bismut’s conjecture of an equivariant Riemann-Roch formula. Consider again a compact Kähler manifold M and a vector bundle E acted on by g and let M_{g} denote the fixed point set. Let N be the normal bundle of the imbedding M_{g} ↪ M. Let γ_{|x}^{N} (resp. γ_{|x}^{E}) denote the infinitesimal action of g at x ∈ M_{g}. Let Ω^{T M_{g}}, Ω^{N} and Ω^{E} denote the curvatures of the corresponding bundles with respect to the hermitian holomorphic connection. Define the function Td on square matrices A as Td(A):=det A/(1-exp(-A)).

Definition 1 Let Td_{g}(T M) and ch_{g}(T M) denote the following differential forms on M_{g}:

Td_{g}(T M) := Td (\frac{−Ω^{T M_{g}}}{2πi} ) det (1 − (γ^{N})^{−1} exp \frac{Ω^{N}}{2πi} )^{−1}

and

ch_{g}(T M) := Tr γ^{E} exp \frac{−Ω_{|M_{g}}^{E}}{2πi} .

Assume now for simplicity that E is the trivial line bundle. In [2], Bismut introduced the equivariant R-genus R_{g}(T M). Using this genus we may reformulate theorem 3 as follows:

Theorem 4 The logarithm of the torsion is given by the equation

2 log τ_{g}(G/K) − log vol_{⋄}(G/K) + ∑_{Ψ} (\frac{1}{2} + ζχ_{ρ_{G}+kα}^{odd}(g)) log \frac{∥α∥_{⋄}^{2}}{2}

= ∫_{(G/K)_{g}} Td_{g}(T (G/K))R_{g}(T (G/K)) − \overline{ζ} ∑_{Ψ} χ_{ρ_{G}+kα}^{odd}(g) − 2 ∑_{Ψ} χ_{ρ_{G}−kα}(g)^{∗} ((α, ρ_{G})) .

Using the R-genus, Bismut formulated a conjectural equivariant arithmetic Grothendieck-Riemann-Roch theorem [2]. Suppose that M is given by M ⊗ C for a flat regular scheme π : M → Spec Z and that E stems from an algebraic vector bundle E over M. Let ∑(−1)^{q}R^{q}π_{∗}E denote the direct image of E under π. We equip the associated complex vector space with a hermitian metric via Hodge theory. Bismut’s conjecture implies that the equivariant torsion verifies the equation

2 log τ_{g}(M, E) + ĉ_{g}^{1} (∑_{q≥0}(−1)^{q}R^{q}π_{∗}E) = π_{∗} (T̂d_{g}(T M)ĉh_{g}(E))^{(1)}

+ ∫_{(G/K)_{g}} Td_{g}(T (G/K))R_{g}(T (G/K))ch_{g}(E)  (1)

(We identify the first arithmetic Chow group ĈH^{1}(Spec Z) with R). Here ĉ_{g}^{1}, T̂d_{g} and ĉh_{g} denote certain equivariant arithmetic characteristic classes which are only defined in a non-equivariant situation up to now (see [11]). Bismut [1] has proven that this formula is compatible with the behaviour of the equivariant torsion under immersions and changes of the occuring metrics. In the non-equivariant case, equation (1) has been proven by Gillet, Soulé, Bismut and Lebeau [7]. In our case, the ĉ_{g}^{1} term in (1) should be independent of g. By the definition of ĉ^{1}, it should equal minus the logarithm of the norm of the element 1 ∈ H^{0}(G/K), thus − log vol_{⋄}(G/K). Hence, theorem 4 fits very well with Bismut’s conjecture.

We consider now the case g = Id. For this action, the equivariant torsion equals the original Ray-Singer torsion. The values of the characters χ_{ρ_{G}+kα} at zero are given by the Weyl dimension formula

χ_{ρ_{G}+kα}(0) = dim V_{ρ_{G}+kα} = ∏_{β∈Δ^{+}} (1 + k \frac{⟨β, α⟩}{⟨β, ρ_{G}⟩}) .

In particular, the first term in theorem 3 is given by ζ′ applied to the odd part of the polynomial

∑_{α∈Ψ} χ_{ρ_{G}+kα}(0) = ∑_{α∈Ψ} ∏_{β∈Δ^{+}} (1 + k \frac{⟨β, α⟩}{⟨β, ρ_{G}⟩}) .

At a first sight, this looks like a polynomial of degree #Δ^{+}, but it has in fact degree ≤ #Ψ, thus all higher degree terms cancel. By combining theorem 4 with the arithmetic Riemann-Roch theorem, we get the following formula:

Theorem 5 The direct image of the arithmetic Todd class is given by

(π_{∗}T̂d(T M))^{(1)} = ∑_{Ψ} (\frac{1}{2} + ζ (dim V_{ρ_{G}+kα})^{odd} ) log \frac{∥α∥_{⋄}^{2}}{2}
+ \overline{ζ} ∑_{Ψ} (dim V_{ρ_{G}+kα})^{odd} + 2 ∑_{Ψ} (dim V_{ρ_{G}+kα})^{∗} ((α, ρ_{G})) .

References

[1] J.-M. Bismut. Equivariant immersions and Quillen metrics. preprint 56, Orsay, 1993.

[2] J.-M. Bismut. Equivariant short exact sequences of vector bundles and their analytic torsion forms. Comp. Math., 93:291–354, 1994.

[3] J.-M. Bismut and K. Köhler. Higher analytic torsion forms for direct images and anomaly formulas. J. alg. geom., 1:647–684, 1992.

[4] A. Borel and F. Hirzebruch. Characteristic classes and homogeneous spaces, I. Am. J. Math., 80:458–538, 1958.

[5] G. Faltings. Lectures on the arithmetic Riemann-Roch theorem. Princeton university press, 1992.

[6] H. Gillet and C. Soulé. Analytic torsion and the arithmetic Todd genus. Topology, 30:21–54, 1991. With an Appendix by D. Zagier.

[7] H. Gillet and C. Soulé. An arithmetic Riemann-Roch theorem. Invent. math., 110:473–543, 1992.

[8] K. Köhler. Equivariant analytic torsion on P^{n}C. Math. Ann., 297:553–565, 1993.

[9] K. Köhler. Holomorphic torsion on Hermitian symmetric spaces. preprint 1, MPI, 1994,to appear in J. Reine Angew. Math.

[10] D. B. Ray and I. M. Singer. Analytic torsion for complex manifolds. Ann. Math., 98:154–177, 1973.

[11] C. Soulé, D. Abramovich, J.-F. Burnol, and J. Kramer. Lectures on Arakelov Geometry. Cambridge University Press, 1992.

[12] C. Wirsching. Eigenwerte des Laplace-Operators für Graßmann-Varietäten. PhD thesis, Universität München, 1992.

Max-Planck-Institut für Mathematik, Gottfried-Claren-Str. 26, D-53225 Bonn e-mail: Koehler@MPIM-Bonn.MPG.De