Contemporary Mathematics

Extremal Kähler Metrics and Ray-Singer Analytic Torsion

Werner Müller and Katrin Wendland

Abstract. Let (X, [ω]) be a compact Kähler manifold with a fixed Kähler class [ω]. Let K_{ω} be the set of all Kähler metrics on X whose Kähler class equals [ω]. In this paper we investigate the critical points of the functional g ∈ K_{ω} ↦ Q(g) =∥ v ∥_{g} T_{0}(X, g)^{1/2}, where v is a fixed nonzero vector of the determinant line λ(X) associated to H^{∗}(X) and T_{0}(X, g) is the Ray-Singer analytic torsion. For a polarized algebraic manifold (X, L) we consider a twisted version Q_{L}(g) of this functional and assume that c_{1}(L) = [ω]. Then the critical points of Q_{L} are exactly the metrics g ∈ K_{ω} of constant scalar curvature. In particular, if c_{1}(X) = 0 or if c_{1}(X) < 0 and \frac{1}{2π} [ω] = −c_{1}(X), then K_{ω} contains a unique Kähler-Einstein metric g_{KE} and Q_{L} attains its absolute maximum at g_{KE}.

0. Introduction

Let X be a closed oriented surface. Given a Riemannian metric g on X, we denote by det Δ_{g} the zeta regularized determinant of the Laplacian Δ_{g} associated to g. Let g_{0} be a fixed metric on X and consider the conformal equivalence class Conf(g_{0}) of g_{0}. Let Conf_{0}(g_{0}) be the subset of metrics g ∈ Conf(g_{0}) with Area(X, g) = Area(X, g_{0}). In [OPS1], Osgood, Phillips and Sarnak studied the functional

(0.1)  h: g ∈ Conf_{0}(g_{0}) ↦ det Δ_{g} ∈ ℝ.

One of the main results of [OPS1] states that h has a unique maximum and this maximum is attained at the metric g̃ ∈ Conf_{0}(g_{0}) of constant Gauss curvature. This gives a new proof of the Riemann uniformization theorem.

The present paper grew out of an attempt to generalize this work of Osgood, Phillips and Sarnak to higher dimensions. There exist different possibilities for doing this. In [BCY], [CY], for example, Branson, Chang and Yang studied the analogous problem on four-manifolds. More preciseley, on a given four-manifold, the authors consider natural, conformally covariant differential operators such as the conformal Laplacian Δ + (n − 2)s_{g}/4(n − 1), and investigate extremals of the zeta function determinants of such operators in a given conformal equivalence class of metrics.

\begin{matrix}c \\ ◯ \end{matrix}0000 (copyright holder)

1991 Mathematics Subject Classification. Primary: 58G26; Secondary: 58E11, 53C55.

This paper is in final form and no version of it will be submitted for publication elsewhere.

We take a different point of view. First of all, the conformal equivalence class Conf(g_{0}) determines a unique complex structure on X so that g_{0} is hermitian with respect to this complex structure. Since dim_{ℂ} X = 1, g_{0} is a Kähler metric. Let [ω_{0}] ∈ H^{1,1}(X) be the Kähler class of g_{0} and let K_{ω_{0}} be the space of all Kähler metrics on X with Kähler class equal to [ω_{0}]. Then Conf_{0}(g_{0}) = K_{ω_{0}}. So, we may regard (0.1) as a functional on K_{ω_{0}}. The variational formula is different from the Polyakov formula, but, of course, it gives the same result as in [OPS1].

This is our starting point for higher dimensional generalizations. We consider a compact Kähler manifold X of dimension n and we fix a Kähler class [ω] ∈ H^{1,1}(X). As above, let K_{ω} be the space of all Kähler metrics on X whose Kähler class is equal to [ω]. For a given Kähler metric g, let T_{0}(X, g) be the Ray-Singer analytic torsion associated to g [RS]. This is a certain weighted product of the regularized determinants of the Dolbeault-Laplace operators Δ_{0,q}, q = 1, ..., n. If dim_{ℂ} X = 1, then T_{0}(X, g) = c(det Δ_{g})^{1/2} for some constant c ≠ 0. Thus, we may regard the functional

τ : g ∈ K_{ω} ↦ T_{0}(X, g) ∈ ℝ

as a higher dimensional analogue of (0.1). However since, in general, the harmonic (0, q)-forms vary nontrivially for q > 0, we need to modify this functional appropriately. Let ∥ · ∥_{Q,g} be the Quillen metric on the determinant line

(0.2)  λ = ⨂_{q=0}^{n} (det H^{q}(X))^{(−1)^{(q+1)}}

[BGS3]. Fix v ∈ λ, v ≠ 0, and put Q(X, g) :=∥ v ∥_{Q,g}. Then we consider the functional

(0.3)  Q: g ∈ K_{ω} ↦ Q(X, g) ∈ ℝ.

Now recall that any variation g_{u}, u ∈ (−ε, ε), of g ∈ K_{ω} is of the form ω_{u} = ω+∂\overline{∂}φ_{u} for some φ_{u} ∈ C^{∞}(X), u ∈ (−ε, ε). Using the results of Bismut, Gillet, and Soulé [BGS3], in section 2 we compute the variation δQ/δφ for φ ∈ C^{∞}(X).

In section 3, we briefly discuss the case of a Riemann surface and the relation with [OPS1].

In section 4 we assume that X admits a metric of constant holomorphic curvature. Such manifolds may be regarded as higher-dimensional analogues of Riemann surfaces. Applying the variational formula of section 2, we show that any metric g_{KE} ∈ K_{ω} of constant holomorphic curvature is a critical point of Q. The question, if in this case g_{KE} is the only critical point of Q, remains open. Also we do not know if g_{KE} is an extremum of Q.

In section 5 we consider K3 surfaces. These are examples which do not admit any metric of constant holomorphic sectional curvature. But, on the other hand, by Yau’s theorem [Y], every Kähler class [ω] on a K3 surface contains a unique Kähler-Einstein metric g_{KE}. This metric should be a natural candidate for a critical point of Q : K_{ω} → ℝ. It follows from the variational formula of section 2 that g_{KE} is a critical point of Q if and only if the Euler-Lagrange equation

(0.4)  Δc_{2}(Ω_{KE}) = 0

holds. We do not know if there exists any K3 surface that admits a Kähler-Einstein metric satisfying (0.4). However, we can show that there are Kähler-Einstein metrics on certain K3 surfaces which do not satisfy (0.4).

In section 6 we introduce a modification of our functional which has a simpler variational formula. For this purpose we assume that X is a complex projective algebraic manifold. Then there exists a positive line bundle L over X. We choose the Kähler form ω such that [ω] = c_{1}(L). Let g ∈ K_{ω} be the metric corresponding to ω. Following Donaldson [D] and [Ti1], we introduce a certain virtual bundle

(0.5)  E = ⨁^{4κ_{2}(n+1)}(L − L^{−1})^{⊗n} ⊕ ⨁^{−κ_{1}n}(L − L^{−1})^{⊗(n+1)},

where κ_{1} and κ_{2} are integers which are defined by κ_{1} = ∫_{X} c_{1}(Ω) ∧ ω^{n−1} and κ_{2} = ∫_{X} ω^{n}, respectively. Similarly to (0.2) there is a determinant line λ(E). Moreover, any hermitian metric h on L determines a hermitian metric h^{E} on E. Let g̃ ∈ K_{ω}. With respect to the metrics (g̃, h^{E}), we form the analytic torsion T_{0}(X, E, g̃, h^{E}) of X with coefficients in E and the Quillen norm ∥ · ∥_{Q,g̃,h^{E}} on λ(E). Fix v ∈ λ(E), v ≠ 0, and set

Q_{L}(g̃, h) :=∥ v ∥_{Q,g̃,h^{E}} .

This functional of (g̃, h) can be turned into a functional on K_{ω} as follows. According to our choice of ω, there exists a hermitian metric h on L such that the curvature Θ_{h} of h satisfies Θ_{h} = −2πiω. Then h is determined by g up to multiplication by a positive scalar. Let

H_{ω} = \{φ ∈ C^{∞}(X) | ω + i∂\overline{∂}φ > 0\}

be the space of Kähler potentials. Given φ ∈ H_{ω}, let ω(φ) = ω + i∂\overline{∂}φ and let g(φ) be the metric with Kähler form ω(φ). It is well known that the map φ ∈ H_{ω} ↦ g(φ) ∈ K_{ω} is surjective. Furthermore, let h(φ) = e^{−2πφ}h. Then we put

Q_{L}(φ) = Q_{L}(g(φ), h(φ))^{(−1)^{n}}, φ ∈ H_{ω}.

Using the variational formula established in [BGS3], one can show that the functional Q_{L} satisfies Q(φ + c) = Q(φ) for all c ∈ ℝ and hence it can be pushed down to a functional on K_{ω}. By the same variational formula one can compute the variation of Q_{L}. Let φ_{t}, t ∈ (a, b), be a smooth path in H_{ω} and set g_{t} = g(φ_{t}). Then

(0.6)  \frac{∂Q_{L}}{∂t} (g_{t}) = c_{n} ∫_{X} φ̇_{t}(s(φ_{t}) − s_{0})ω(φ_{t})^{n},

where s(φ_{t}) is the scalar curvature of the metric g(φ_{t}), s_{0} is the normalized total scalar curvature and c_{n} > 0 is a certain constant that depends only on n and [ω]. It follows from (0.6) that the critical points of Q_{L} are exactly the metrics g ∈ K_{ω} of constant scalar curvature. Any such metric is an extremal metric in the sense of Calabi [Ca].

Furthermore, formula (0.6) agrees, up to the constant c_{n} and the sign, with the variation of the K-energy μ: K_{ω} → ℝ introduced by Mabuchi [Ma1]. The K-energy has been studied by Bando and Mabuchi [B], [BM], [Ma2], mainly in connection with Futaki’s obstruction to the existence of Kähler-Einstein metrics in the case c_{1}(X) > 0. The relation between the Quillen metric and the K-energy was first observed by Tian [Ti1, p.258]. In [Ma2], Mabuchi defined a natural Riemannian structure on K_{ω}. He proved that the sectional curvature of K_{ω} is nonpositive and that Hess(μ) is positive semidefinite everywhere. Therefore Q_{L} is also a convex functional. But this is not sufficient to determine the type of the critical point.

In section 7 we assume that K_{ω} contains a Kähler-Einstein metric. Then we can say more about the critical points of Q_{L}. The main result is the following theorem.

Theorem 0.1. Let (X, L) be a polarized projective algebraic manifold. Choose a Kähler form ω on X such that [ω] = c_{1}(L). Assume that K_{ω} contains a Kähler-Einstein metric g_{KE}. If c_{1}(X) ≤ 0, then Q_{L} has a unique maximum which is attained at g_{KE}. If c_{1}(X) > 0, then Q_{L} attains its absolute maximum on the subset K_{KE} ⊂ K_{ω} of Kähler-Einstein metrics.

To prove the first part of Theorem 0.1, we use the evolution of the metric by the complex analogue of Hamilton’s Ricci flow equation

(0.7)  \frac{∂g̃_{i\overline{j}}}{∂t} = −r̃_{i\overline{j}} + \frac{s_{0}}{2n}g̃_{i\overline{j}}, g̃_{i\overline{j}}(0) = g_{i\overline{j}},

where r̃_{ij} is the Ricci tensor of g̃ and s_{0} the normalized total scalar curvature. Cao [C] proved that (0.7) has a unique solution g̃_{i\overline{j}}(t) which exists for all time and, furthermore, if c_{1}(X) ≤ 0, then g̃(t) converges to g_{KE} as t → ∞. Now the main observation is that the K-energy decreases along the flow generated by (0.7). It was pointed out to us by Tian that one can also use equality (3.8) in [Ti1] to show that for c_{1}(X) ≤ 0, Q_{L} has a strict absolute maximum at g_{KE}.

The case c_{1}(X) > 0 is more complicated. First of all, there are obstructions to the existence of Kähler-Einstein metrics [Fu], [Ti2]. Moreover, the subspace K_{E} ⊂ K_{ω} of Kähler-Einstein metrics may have positive dimension. If K_{E} ≠ ∅, it was proved in [BM] and [B] that μ takes its minimum on K_{E}. This implies the second statement of the theorem.

Theorem 0.1 has some implication for the spectral determination of Kähler-Einstein metrics. As above, let g be a Kähler metric on X such that [ω_{g}] = c_{1}(L) and pick a hermitian metric h on L such that Θ_{h} = −2πiω_{g}. Recall that h is uniquely determined by g up to multiplication by a positive constant. Let q ∈ {0, ..., n} and k ∈ ℕ. The Dolbeault-Laplace operator in Λ^{0,q}(X, L^{⊗k}), associated to (g, h), remains unchanged if we multiply h by λ ∈ ℝ^{+}. Therefore, it is uniquely determined by g and will be denoted by Δ_{0,q,k}^{g}. Let Spec(Δ_{0,q,k}) denote the spectrum of Δ_{0,q,k}^{g}. By the above, the spaces H^{0,q}(X, L^{⊗k}) of L^{⊗k}- valued harmonic (0, q)-forms also depend only on g. Although the inner product in H^{0,q}(X, L^{⊗k}) associated to (g, h) depends on h, the induced L^{2}-norm on λ(E) is invariant under multiplication of h by positive scalars and therefore depends only on g. We denote it by ∥ · ∥_{g}.

Corollary 0.2. Let (X, L) be a polarized projective algebraic manifold and suppose that c_{1}(X) ≤ 0. Choose a Kähler form ω on X such that [ω] = c_{1}(L). Let g_{KE} be the unique Kähler-Einstein metric in K_{ω}. Let g ∈ K_{ω} and suppose that the following holds:

1) Spec(Δ_{0,q,k}^{g}) = Spec(Δ_{0,q,k}^{g_{KE}}) for all q = 0, ..., n and k = −(n+1), ..., n+1.

2) ∥ · ∥_{g}=∥ · ∥_{g_{KE}}.

Then g = g_{KE}.

It seems likely that the corollary can be improved. One would expect that g_{KE} is already uniquely determined by the spectra of the Dolbeault-Laplace operators.

In the final section 8 we briefly discuss the relation to moduli spaces. These are slight modifications of results due to Fujiki and Schumacher [FS]. First we consider a metrically polarized family of compact Hodge manifolds (π : 𝔛 → S, ω̃, L). Using L, we introduce the family version of the virtual bundle (0.5) which we also denote by E. Associated to E is the determinant line bundle λ(E) on S equipped with the Quillen metric h^{λ(E)}. If the family (π : 𝔛 → S, ω̃, L) is effective, then by Fujiki and Schumacher [FS], one can define the generalized Weil-Petersson metric ĥ_{W P} on S and it follows that the first Chern form c_{1}(λ(E), h^{λ(E)}) of the determinant line bundle satisfies

c_{1}(λ(E), h^{λ(E)}) = a_{n}ω̂_{W P}

where a_{n} = −2^{n+1}κ_{2}(n + 1)!. This result extends to the moduli space 𝔐_{H,e} of extremal Hodge manifolds [FS]. Acknowledgement. The authors would like to thank Kai Köhler for some very helpful comments and remarks.

1. Preliminaries

Let (X, g) be a compact Kähler manifold of complex dimension n. We denote by ω = ω_{g} the Kähler form of g. Then the volume form of g is given by dv_{g} = ω_{g}^{n}/n!. We always choose the holomorphic connection on X. The Riemann and Ricci curvature tensors will be denoted by R and r, respectively, and the scalar curvature by s_{g}. In terms of the Ricci tensor r, the Ricci form ρ is defined by

ρ(φ, ψ) = r(Jφ, ψ),

where J denotes the complex structure of X. For a Kähler manifold, iρ is the curvature of the canonical line bundle K = (Λ^{n}T^{1,0}X)^{∗}. This has important implications for the scalar curvature s_{g} = Tr(r). It can be calculated by the formula

(1.1)  s ω^{n} = 2n ρ ∧ ω^{n−1}.

Since ρ/2π represents the first Chern class c_{1}(X) = c_{1}(T^{1,0}X) = −c_{1}(K), it follows that the average value

(1.2)  s_{0} = \frac{∫_{X} s_{g}dv_{g}}{∫_{X} dv_{g}} = 4nπ \frac{∫_{X} c_{1}(Ω) ∧ ω^{n−1}}{∫_{X} ω^{n}}

of the scalar curvature is a topological invariant, depending only on the Kähler class [ω]. Let E → X be a holomorphic vector bundle with hermitian metric h^{E}. Let

0 → Λ^{0,0}(X, E) ⟶^{\overline{∂}} Λ^{0,1}(X, E) ⟶^{\overline{∂}} ⋯ ⟶^{\overline{∂}} Λ^{0,n}(X, E) → 0

be the Dolbeault complex and let Δ_{0,q}^{E} = \overline{∂} \overline{∂}^{∗}+\overline{∂}^{∗}\overline{∂} be the corresponding Dolbeault-Laplace operator acting in Λ^{0,q}(X, E). Let P_{0,q}^{E} be the projection on harmonic (0, q)-forms and for Re(s) > n/2, let

ζ_{q}(s; E) = \frac{1}{Γ(s)} ∫_{0}^{∞} t^{s−1}Tr(e^{−tΔ_{0,q}^{E}} − P_{0,q}^{E})dt

be the zeta function of Δ_{0,q}^{E}. It admits a meromorphic continuation to ℂ, also denoted by ζ_{q}(s; E), which is regular at s = 0. Then the zeta regularized determinant det Δ_{0,q}^{E} of Δ_{0,q}^{E} is defined by

det Δ_{0,q}^{E} = exp (−\frac{d}{ds}ζ_{q}(s; E)|_{s=0}) .

The Ray-Singer analytic torsion is defined to be

T_{0}(X, E) = (∏_{q=0}^{n}(det Δ_{0,q}^{E})^{(−1)^{q+1}q})^{1/2}

[RS]. This number depends, of course, on the metrics (g, h^{E}) and if it is necessary to indicate this dependence we shall write T_{0}(X, E, g, h^{E}) for the analytic torsion. Let

λ(E) = ⨂_{q=0}^{n}(det H^{q}(X, E))^{(−1)^{q+1}}

be the determinant line associated to the Dolbeault complex. Let H^{0,q}(E) be the space of E-valued harmonic (0, q)-forms. For each q, there is a canonical isomorphism H^{0,q}(E) ⟶^{≅} H^{q}(X, E). Thus, using the L^{2}-metric on H^{0,q}(E), we get a metric ∥ · ∥_{L^{2}} on λ(E). The Quillen metric on λ(E) is then defined by

(1.3)  ∥ · ∥_{Q}=∥ · ∥_{L^{2}} ·T_{0}(X, E).

We shall also consider virtual holomorphic bundles E = ∑_{k=1}^{m} n_{k}E_{k}, where n_{k} ∈ ℤ and E_{k} → X are holomorphic vector bundles. For such E we set

λ(E) = ⨂_{k=1}^{m} λ(E_{k})^{n_{k}}.

Here for n_{k} < 0, λ(E_{k})^{n_{k}} := (λ(E_{k})^{∗})^{−n_{k}}. If each E_{k} is equipped with a hermitian metric, we get a metric ∥ · ∥_{Q} on λ(E) which is the tensor product of the induced Quillen metrics on λ(E_{k})^{n_{k}}. Let v ∈ λ(E), v ≠ 0. Then we define

Q(X, E, g, h) :=∥ v ∥_{Q,g,h} .

If X and E are fixed, we shall write Q(g, h) in place of Q(X, E, g, h).

Let ω be a fixed Kähler form. Put

K_{ω} = {g | g Kähler metric on X with [ω_{g}] = [ω]}.

Let

(1.4)  H_{ω} = \{φ ∈ C^{∞}(X, ℝ) | ω + i∂\overline{∂}φ > 0\}

be the corresponding space of Kähler potentials. If φ ∈ H_{ω}, then

(1.5)  ω(φ) := ω + i∂\overline{∂}φ

is the Kähler form of a Kähler metric g(φ) ∈ K_{ω} and it is well known that the natural map

(1.6)  φ ∈ H_{ω} → g(φ) ∈ K_{ω}

is surjective. If φ ∈ H_{ω} then for each c ∈ ℝ, one has φ+c ∈ H_{ω} and ω(φ) = ω(φ+c). On the other hand, if φ, ψ ∈ H_{ω} are such that ω(φ) = ω(ψ) then ∂\overline{∂}(φ − ψ) = 0 and therefore φ − ψ is constant. Let

H_{ω}^{0} = {φ ∈ H_{ω} | ∫_{X} φω^{n} = 0} .

Then the restriction of the map (1.6) to the subspace H_{ω}^{0} induces an isomorphism

(1.7)  H_{ω}^{0} ≅ K_{ω}.

Furthermore, for any φ ∈ C^{∞}(X) there exists ϵ > 0 such that

ω_{u} = ω + iu∂\overline{∂}φ > 0

for all u ∈ ℝ, |u| < ϵ. The corresponding family g_{u}, |u| < ϵ, of Kähler metrics will be called the variation of g in the φ-direction.

2. The variational formula

Given a smooth family g_{t}, t ∈ ℝ, of Kähler metrics, we will denote by ∗_{t} the Hodge star operator with respect to g_{t} and we set

(2.1)  U_{t} := (g_{t})^{−1} \frac{d}{dt}(g_{t}), α_{t} := ∗_{t}^{−1} \frac{d}{dt}(∗_{t}).

We observe that if g_{t} is the variation of g in the φ-direction, then we have

(2.2)  Tr(U_{t}) = ∑_{α,\overline{β}} g_{t}^{α\overline{β}} \frac{∂^{2}φ}{∂z_{α}∂\overline{z}_{β}} = −\frac{1}{2}Δ_{0,0}^{t}φ.

The variation of Q(X, E, g, h) has been computed by Bismut, Gillet and Soulé [BGS3, Theorem 1.22]. We recall their result. Let g_{t}, t ∈ ℝ, be a smooth family of Kähler metrics on X and h_{t}, t ∈ ℝ, a smooth family of hermitian metrics on E. Let Ω_{t} and L_{t}^{E} be the curvature of the Hermitian holomorphic connection on (T^{(0,1)}X, g_{t}) and (E, h_{t}), respectively. Let Td be the Todd genus and T d_{j} its j-th component. The Todd genus is normalized as in [BGS3]. Then

\frac{∂}{∂t} log Q(X, E, g_{t}, h_{t}) = \frac{1}{2} (\frac{1}{2πi})^{n} ∫_{X} \frac{∂}{∂b}[Td(−Ω_{t} − bU_{t})

× Tr (exp(−L_{t}^{E} − b(h_{t}^{E})^{−1} \frac{∂h_{t}^{E}}{∂t})) ]_{b=0}.

(2.3)

In particular, if we assume that E is trivial, this formula is reduced to

(2.4)  \frac{∂}{∂t} log Q(X, g_{t}) = \frac{1}{2} (\frac{1}{2πi})^{n} ∫_{X} \frac{∂}{∂b} [Td_{n+1}(−Ω_{t} − bU_{t})] |_{b=0}.

Using the definition of Q(X, g) and (1.120) of [BGS3], it follows that

(2.5)  \frac{∂}{∂t} log Q(X, g_{t}) = \frac{∂}{∂t} log T_{0}(X, g_{t}) − \frac{1}{2} ∑_{q=0}^{n}(−1)^{q} Tr(α_{t}P_{0,q}^{t}).

For the application that we have in mind we need a more explicit version of the variational formula. Recall that the Todd genus can be expressed in terms of the Chern classes. Therefore, we have to compute the corresponding derivatives of the Chern classes which are also normalized as in [BGS3].

Lemma 2.1. For all j ≥ 1, we have

(2.6)  \frac{∂}{∂b} (c_{j}(−Ω − bU)) |_{b=0} = ∑_{k=0}^{j−1}(−1)^{j+k} Tr(Ω^{k}U)c_{j−k−1}(Ω).

Proof. We proceed by induction on j. For j = 1 we have

\frac{∂}{∂b} (c_{1}(−Ω − bU)) |_{b=0} = −Tr(U)

proving (2.6) in this case. Now suppose that (2.6) holds for j. We express the Chern classes by the power sums s_{k}. If we formally write

∑_{j=0}^{n} c_{j}(Ω)x^{j} = ∏_{j=1}^{n}(1 + γ_{j}x),

then s_{k} is the k-th elementary symmetric function in γ_{1}, …, γ_{n}. Using Newton’s formula, we obtain

c_{j+1}(−Ω − bU) = \frac{1}{j + 1} ∑_{k=1}^{j+1}(−1)^{k+1}(c_{j+1−k}s_{k})(−Ω − bU).

Since s_{k}(R) = Tr(R^{k}), we get

\frac{∂}{∂b}(s_{k}(−Ω − bU)) |_{b=0}= k(−1)^{k}Tr(Ω^{k−1}U).

Using the induction hypothesis, we deduce

\frac{∂}{∂b}(c_{j+1}(−Ω − bU))|_{b=0}

= \frac{1}{j + 1} ∑_{k=1}^{j+1} \{∑_{l=0}^{j−k}(−1)^{j+l}Tr(Ω^{l}U)c_{j−k−l}(Ω)s_{k}(−Ω)

− kc_{j+1−k}(−Ω)Tr(Ω^{k−1}U)}

= \frac{1}{j + 1} ∑_{l=0}^{j} Tr(Ω^{l}U) \{∑_{k=1}^{j−l}(−1)^{l+j+k}c_{j−k−l}(Ω)s_{k}(Ω)

− (−1)^{j+l}(l + 1)c_{j−l}(Ω)}

= ∑_{l=0}^{j}(−1)^{j+1+l}Tr(Ω^{l}U)c_{j−l}(Ω).

□

Expressing the Todd genus in terms of the Chern classes and using Lemma 2.1, it follows from (2.4) that there exist a_{j_{0}···j_{n+1}} ∈ ℂ, j_{0}, …, j_{n+1} ∈ ℕ, such that for all φ ∈ C^{∞}(X) :

\frac{∂}{∂t} log Q(X, g_{t}) = ∑_{j_{0}+···+j_{n+1}=n} a_{j_{0}···j_{n+1}}

× ∫_{X} Tr(Ω_{t}^{j_{n+1}}U_{t})c_{j_{0}}(Ω_{t}) ⋯ c_{j_{n}}(Ω_{t}).

(2.7)

The coefficients a_{j_{0}···j_{n+1}} ∈ ℂ, j_{0}, …, j_{n+1} ∈ N, can be computed explicitly. For example, if n = 2, then formula (2.7) takes the form

\frac{∂}{∂t} log Q(X, g_{t}) = \frac{1}{48} ∫_{X} Tr(Ω_{t}U_{t})c_{1}(Ω_{t})

− \frac{1}{48} ∫_{X} Tr(U_{t})(c_{2}(Ω_{t}) + c_{1}(Ω_{t})^{2}).

(2.8)

Let φ ∈ C^{∞}(X) and let g_{u}, |u| < ϵ, be the variation of g in the direction of φ. Put

(2.9)  \frac{δ}{δφ} log Q(X, g) = \frac{∂}{∂u} log Q(X, g_{u})|_{u=0}.

If n = 2, it follows from (2.2) and (2.8) that

\frac{δ}{δφ} log Q(X, g) = \frac{1}{48} ∫_{X} Tr(ΩU_{0})c_{1}(Ω)

+ \frac{1}{96} ∫_{X} Δφ(c_{2}(Ω) + c_{1}(Ω)^{2}).

(2.10)

3. Riemann surfaces

In this section we consider the case n = 1. Then X is a compact oriented surface without boundary. First observe that

α(1) = −\frac{Δφ}{2} and α(d\overline{z}) = 0.

Hence, it follows from (2.5) that

\frac{δ}{δφ} log Q(X, g) = \frac{δ}{δφ} log T_{0}(X, g).

Furthermore, \overline{∂} : ∧^{0,0}(X) → ∧^{0,1}(X) is an isomorphism on nonzero eigenspaces. This implies that det Δ_{0,1} = det Δ_{0,0}. Let Δ_{g} = d^{∗}d denote the Laplacian on functions. Then Δ_{g} = 2Δ_{0,0} and therefore we get

(3.1)  2 log T_{0}(X, g) = log det Δ_{g} + (log 2) (1 − \frac{χ(X)}{6}) .

Thus there exists c > 0 such that

T_{0}(X, g) = c(det Δ_{g})^{1/2}.

Next we describe the space K_{ω} for a given Kähler class [ω] ∈ H^{1,1}(X). Since H^{1,1}(X) ≅ ℝ, [ω] is unique up to multiplication by ℝ^{+}. Let g be the metric with Kähler form ω and let Δ = Δ_{g}. Let ψ ∈ C^{∞}(X). Then ω + i∂\overline{∂}ψ > 0 holds if and only if 1 − Δψ > 0. Thus the space H_{ω}, which is defined by (1.4), is given by

(3.2)  H_{ω} ≅ {ψ ∈ C^{∞}(X) | 1 − Δψ > 0}.

Then by using g(ψ) = (1 − Δψ)g, we may regard T_{0}(X, g) as a functional on H_{ω} and by (3.1) we have

2 log T_{0}(X, ψ) = log det Δ_{(1−Δψ)g} + C, ψ ∈ H_{ω}.

Let

Conf(g) = {e^{2φ}g | φ ∈ C^{∞}(X)}

be the conformal equivalence class of metrics on X which contains g. Let

Conf_{0}(g) = {g̃ ∈ Conf(g) | Area(X, g̃) = Area(X, g)}.

Lemma 3.1. Let g be a Kähler metric on X with Kähler form ω. Then

K_{ω} = Conf_{0}(g).

Proof. Let g̃ ∈ K_{ω}. By (3.2) there exists a unique ψ ∈ C^{∞}(X) with ∫ ψ dv_{g} = 0 such that g̃ = (1 − Δψ)g. Since 1 − Δψ > 0, there exists φ ∈ C^{∞}(X) such that e^{2φ} = 1 − Δψ. Moreover

∫_{X} e^{2φ}dv_{g} = ∫_{X} dv_{g} − ∫_{X} Δψdv_{g} = ∫_{X} dv_{g}.

Hence there exists a unique φ ∈ C^{∞}(X) such that g̃ = e^{2φ}g ∈ Conf_{0}(g). On the other hand, let g̃ ∈ Conf_{0}(g). Then g̃ = e^{2φ}g and ∫_{X}(e^{2φ} − 1)dv_{g} = 0. Therefore, there exists a unique ψ ∈ C^{∞}(X) with ∫ ψ dv_{g} = 0 such that e^{2φ} = 1 − Δψ > 0. Hence

g̃ = (1 − Δψ)g ∈ K_{ω}.
□

The functional

(3.3)  g̃ ∈ Conf_{0}(g) ⟼ log det Δg̃ ∈ ℝ

has been studied by Osgood, Philipps and Sarnak [OPS1]. By (3.1) and Lemma 3.1, we see that, up to a constant, (3.3) coincides with g̃ ∈ K_{ω} ⟼ log Q(X, g̃). We shall now derive the main result of [OPS1] by our approach. Let g_{t} ∈ K_{ω}, t ∈ ℝ, be a smooth family of metrics. Then there exists a smooth family ψ_{t} ∈ C^{∞}(X) such that g_{t} = g(ψ_{t}). Using Lemma 2.1, the variational formula (2.4) gives

(3.4)  \frac{∂}{∂t} log Q(X, g_{t}) = −\frac{1}{24} ∫_{X} c_{1}(Ω_{t})Δ_{t}ψ̇_{t} = \frac{1}{12π} ∫_{X} K_{t}Δ_{t}ψ̇_{t} dv_{t},

where K_{t} is the Gauss curvature of g_{t}. We note that the last equality follows from

c_{1}(Ω_{t}) = −\frac{2K_{t}}{π} dv_{t}.

In particular, we get

(3.5)  \frac{δ}{δψ} log Q(X, g) = −\frac{1}{24} ∫_{X} c_{1}(Ω)Δψ = \frac{1}{12π} ∫_{X} KΔψ dv.

Remark 3.2. This formula should be compared with the corresponding variational formula (1.12) in [OPS1]. Given ψ ∈ C^{∞}(X), there exists ϵ > 0 such that 1 − uΔψ > 0 for |u| < ϵ. Then by Lemma 3.1, there exists ϕ_{u} ∈ C^{∞}(X) such that

(3.6)  1 − uΔψ = e^{2ϕ_{u}}, |u| < ϵ.

The family g_{u} = e^{2ϕ_{u}}g, |u| < ϵ, of metrics is a deformation of g in Conf_{0}(g). By (3.6), we have

ϕ_{0} = 0, ϕ̇_{0} = −\frac{Δψ}{2} .

Then with respect to the variation defined by g_{u}, in formula (1.12) of [OPS1] we have δφ = ϕ̇_{0}, φ = 0 and δ log A = 0. Hence, up to a constant, (3.5) corresponds to (1.12) of [OPS1].

By (3.5), a metric g̃ is critical for log Q(X, g) if and only if its Gauss curvature K is constant. To investigate the critical point, we use the evolution of a Riemannian metric g on X by Hamilton’s Ricci flow equation with s_{0} as in (1.2)

(3.7)  \frac{∂g_{ij}}{∂t} = (s_{0} − s_{g})g_{ij}.

Note that by the Gauss-Bonnet formula

s_{0} = \frac{4πχ(X)}{A},

where A = Area(X, g). In [Ha], Hamilton proved that if s_{0} ≤ 0 or if s_{g} > 0, then for any initial data, (3.7) has a unique solution g(t) which exists for all time and as t → ∞, converges to a metric of constant curvature. The case s_{0} > 0 was completed by Chow [Ch] who proved that for s_{0} > 0, the scalar curvature s(t) of the solution g(t) of (3.7) becomes positive in finite time. Now if g(0) is hermitian then g(t) is hermitian for all time. Furthermore, the Ricci flow is area preserving [Ha, p.238]. Therefore, the Ricci flow preserves the space K_{ω}. Hence, there exists u(t) ∈ C^{∞}(X) which is a smooth function of t ∈ ℝ^{+} such that, with respect to a local holomorphic parameter z,

g(t) = g(0) + \frac{∂^{2}u(t)}{∂z∂\overline{z}}dz ⊗ d\overline{z}.

Let g(t) = h(t)dz ⊗ d\overline{z}. Then (3.7) implies

(3.8)  Δ_{t}u̇ = −h^{−1} \frac{∂^{2}u̇}{∂z∂\overline{z}} = (s_{g} − s_{0}).

Combined with (3.4) we obtain

\begin{aligned}\frac{∂}{∂t} log Q(g(t)) & = \frac{1}{24π} ∫_{X} s(t)Δ_{t}u̇(t) dv_{t} \\ & = \frac{1}{24π} ∫_{X}(s(t) − s_{0})Δ_{t}u̇(t) dv_{t} \\ & = \frac{1}{24π} ∫_{X}(s(t) − s_{0})^{2} dv_{t} ≥ 0, \end{aligned} ; (3.9)

where s(t) denotes the scalar curvature of the metric g(t). If s_{0} ≤ 0, Hamilton [Ha] proved that there exist ε > 0 and C > 0 such that

(3.10)  |s(t) − s_{0}| ≤ Ce^{−εt}, t ∈ ℝ^{+}.

If s_{0} > 0, he established the same estimate under the additional assumption that s_{g} > 0. By Chow’s results [Ch] it follows that (3.10) holds in the case s_{0} > 0 too. Hence we can integrate (3.9) and we get

log Q(g(∞)) − log Q(g(0)) = \frac{1}{24π} ∫_{0}^{∞} ∫_{X}(s(t) − s_{0})^{2} dv_{t} dt ≥ 0.

Furthermore, if s(0) is not constant then the inequality is strict. Thus we proved

Theorem 3.3. Q has a unique maximum on K_{ω} which is attained at the metric g^{∗} ∈ K_{ω} of constant curvature.

4. Manifolds with constant holomorphic curvature

Let (X, g) be a Kähler manifold and let J denote the complex structure of X. Recall that (X, g) is said to have constant holomorphic curvature H ∈ ℝ, if for any z ∈ X and any ξ ∈ T_{z}X with ⟨ξ, ξ⟩ = 1, the holomorphic curvature

H_{z}(ξ) = ⟨R_{z}(ξ, Jξ)ξ, Jξ⟩

is equal to H. We note that any metric of constant holomorphic curvature is Kähler-Einstein [Go, (6.12)]. Furthermore, compact manifolds with constant holomorphic curvature H can be classified as follows [Be]: If H > 0, then X is isomorphic to ℂP^{n} equipped with the rescaled Fubini-Study metric; if H = 0, then X is isomorphic to a complex torus Γ\ℂ^{n} with the flat metric; if H < 0 then X is isomorphic to a compact quotient Γ\D^{n} of the complex unit ball D^{n} ⊂ ℂ^{n}, equipped with a Bergmann metric, by a discrete group Γ of isometries. Thus manifolds with constant holomorphic curvature may be regarded as higher-dimensional analogues of Riemann surfaces. So, if we think of generalizing the results of Osgood, Philipps and Sarnak [OPS1] to higher dimensions, these are the most obvious candidates for being critical points of the functional log Q(X, g). We wish to apply the variational formula (2.4). For this purpose we need several lemmas.

Lemma 4.1. Suppose that g is a Kähler metric on X with constant holomorphic curvature H, and let Ω denote the curvature form of g. Then

(1) With respect to local holomorphic coordinates (z_{1}, …, z_{n}), Ω is given by

Ω_{k}^{j} = −iHδ_{jk}ω + \frac{H}{2} ∑_{m=1}^{n} g_{k\overline{m}}dz_{j} ∧ d\overline{z}_{m}.

(2) The Chern classes of (X, g) are represented by

c_{j}(Ω) = \binom{n + 1}{j} (−\frac{H}{2π} ω)^{j} .

Proof. 1. The claimed formula for Ω_{k}^{j} is an immediate consequence of the following formula for the Riemann curvature tensor R on a manifold with constant holomorphic curvature H [Go, (6.1.1)]:

R_{j\overline{k}l\overline{m}} = \frac{H}{2} (g_{j\overline{k}}g_{l\overline{m}} + g_{j\overline{m}}g_{l\overline{k}}) .

2. We recall that a representative of the jth Chern class is given in terms of the curvature form Ω by means of the formula

c_{j}(Ω) = (\frac{1}{2πi})^{j} \frac{1}{j!} ∑_{\begin{matrix}k_{1},...,k_{j},l_{1},...,l_{j} \\ ∈\{1,...,n\} \end{matrix}} δ_{k_{1}···k_{j}}^{l_{1}···l_{j}}Ω_{l_{1}}^{k_{1}} ∧ ⋯ ∧ Ω_{l_{j}}^{k_{j}}

[Go, (6.12.1)]. Inserting the expression for Ω_{k}^{j}, given in 1., a straightforward computation leads to

c_{j}(Ω) = (\frac{H}{2πi})^{j} ∑_{p=0}^{j} \binom{n − j + p}{p} (−iω)^{j} = \binom{n + 1}{j} (−\frac{H}{2π} ω)^{j} .

□

Let U = U_{0}, where U_{t} is defined by (2.1).

Lemma 4.2. Let (X, g) be the Kähler manifold with constant holomorphic curvature H. Let φ ∈ C^{∞}(X) and let ω_{u} = ω + iu∂\overline{∂}φ, |u| < ϵ. Then for all j ≥ 1,

Tr(Ω^{j}U) = −\frac{Δφ}{2}(−iHω)^{j} + ∂\overline{∂} (\frac{H}{2} φ(−iHω)^{j−1}) .

Proof. Choose local holomorphic coordinates (z_{1}, …, z_{n}). Then we claim that for all j ≥ 1:

\begin{aligned}(Ω^{j}U)_{q}^{p} = & ∑_{k=1}^{n} \{(−iHω)^{j}g^{p\overline{k}}φ_{q\overline{k}} \\ & + (−iHω)^{j−1} \frac{H}{2} φ_{q\overline{k}}dz_{p} ∧ d\overline{z}_{k}\} , \end{aligned} ; (4.1)

where φ_{q\overline{k}} = ∂^{2}/∂z_{q}∂\overline{z}_{k}φ. To prove (4.1), we proceed by induction on j. If j = 1, then using Lemma 4.1, we get

\begin{aligned}(ΩU)_{q}^{p} & = ∑_{l=1}^{n} Ω_{l}^{p}U_{q}^{l} = ∑_{l,k} (−iHδ_{pl}ω + \frac{H}{2} ∑_{m=1}^{n} g_{l\overline{m}}dz_{p} ∧ dz\overline{m}) g^{l\overline{k}}φ_{q\overline{k}} \\ & = ∑_{k=1}^{n} (−iHωg^{p\overline{k}}φ_{q\overline{k}} + \frac{H}{2} φ_{q\overline{k}}dz_{p} ∧ \overline{z}_{k}) . \end{aligned}

Suppose that (4.1) holds for j. Then using Lemma 4.1 combined with the induction hypothesis, we obtain

(Ω^{j+1}U)_{q}^{p} = ∑_{l=1}^{n} Ω_{l}^{p}(Ω^{j}U)_{q}^{l} = ∑_{kl} (−iHδ_{pl}ω + \frac{H}{2} ∑_{m=1}^{n} g_{l\overline{m}}dz_{p} ∧ d\overline{z}_{m})

∧ ((−iHω)^{j}g^{l\overline{k}}φ_{q\overline{k}} + (−iHω)^{j−1} \frac{H}{2} φ_{q\overline{k}}dz_{l} ∧ d\overline{z}_{k})

= ∑_{k=1}^{n} \{(−iHω)^{j+1}g^{p\overline{k}}φ_{q\overline{k}} + (−iHω)^{j} \frac{H}{2} φ_{q\overline{k}}dz_{p} ∧ d\overline{z}_{k}\} .

This proves (4.1). Thus in local holomorphic coordinates, we get

\begin{aligned}Tr(Ω^{j}U) & = (−iHω)^{j} ∑_{p,k=1}^{n} g^{p\overline{k}} \frac{∂^{2}φ}{∂z_{p}∂\overline{z}_{k}} + \frac{H}{2} (−iHω)^{j−1} ∑_{p,k=1}^{n} \frac{∂^{2}φ}{∂z_{p}∂\overline{z}_{k}} dz_{p} ∧ d\overline{z}_{k} \\ & = −\frac{Δφ}{2}(−iHω)^{j} + \frac{H}{2} (−iHω)^{j−1}∂\overline{∂}φ. \end{aligned}

Since ω is closed, this implies the lemma.

□

Theorem 4.3. Let g_{KE} ∈ K_{ω} be a metric of constant holomorphic curvature. Then g_{KE} is a critical point of Q: K_{ω} → ℝ.

Proof. By (1.6) it suffices to prove that

\frac{δ}{δφ} log Q(X, g_{KE}) = 0

for all φ ∈ C^{∞}(X). Let φ ∈ C^{∞}(X) and let g_{u}, |u| < ε, be the corresponding variation of g_{KE} in the φ-direction. We use the variational formula (2.7) and apply Lemma 4.1 and Lemma 4.2 to the integrant. Then it follows that there exist A, B ∈ ℂ such that

\begin{aligned}\frac{δ}{δφ} log Q(X, g_{KE}) & = ∫_{X} \{AΔφω^{n} + B∂\overline{∂}(φω^{n−1})\} \\ & = A ∫_{X} Δφω^{n} + B ∫_{X} ∂\overline{∂}(φω^{n−1}). \end{aligned}

Hence, we get

\frac{δ}{δφ} log Q(X, g_{KE}) = 0.

This concludes the proof. □

In the case of a complex torus, Theorem 0.1 can be restated in terms of the analytic torsion. Namely, we have

Corollary 4.4. Let X = Γ\ℂ^{n} be a complex torus and let g_{KE} be the flat metric on X. Then

\frac{δ}{δφ}T_{0}(X, g_{KE}) = 0

for all φ ∈ C^{∞}(X).

Proof. Let φ ∈ C^{∞}(X) and let g_{u}, |u| < ϵ, be the variation of g_{KE} in the φdirection. Let α = α_{0} denote the operator (2.1) at u = 0 where ∗_{u} is the Hodge star operator with respect to g_{u}. Let P_{0,q}, 0 ≤ q ≤ n, denote the harmonic projections with respect to g_{KE}. By Theorem 4.3 and (2.5) it is sufficient to prove that

∑_{q=0}^{n}(−1)^{q}T r(αP_{0,q}) = 0

for all φ ∈ C^{∞}(X). Let (z_{1}, …, z_{n}) denote the standard coordinates on X induced from ℂ^{n}. Recall that an orthonormal basis for the space H^{0,q}(X) of harmonic (0, q)-forms is given by

\{(vol(X))^{−1/2}d\overline{z}^{B} | B ⊂ \{1, …, n\}, |B| = q\} .

Using the definition of the Hodge star operator and the assumption that (g_{KE})_{i\overline{j}} = δ_{ij}, it follows that

\overline{T r(αP_{0,q})} = c ∑_{|B|=q} ∫ \frac{∂}{∂u} \{det ((g_{u}^{a\overline{b}})_{a,b∈B}) det ((g_{u})_{i\overline{j}})\} |_{u=0} dv_{0}

= c ∑_{|B|=q} ∫_{X} \{∑_{b∈B} ġ^{b\overline{b}} − \frac{Δφ}{2}\} dv_{0}

= c_{1} ∫_{X} Δφ dv_{0} = 0.

□

Remark 4.5. It remains an open problem whether or not a Kähler metric g_{KE} with constant holomorphic curvature is the unique critical point of log Q(X, g̃) for g̃ ∈ K_{ω}. We also do not know if g_{KE} is a maximum or minimum of log Q(X, g̃).

5. K3 surfaces

Another class of examples are K3 surfaces. Recall that a K3 surface is a compact, connected complex analytic surface X that is regular, meaning h^{1}(X, O) = 0, and its canonical bundle K = Λ^{2,0}(T^{∗}X) is trivial. By Siu [Si], every K3 surface admits a Kähler metric. Furthermore, by Yau [Y], every Kähler class [ω] of X contains a unique Kähler-Einstein metric g_{KE}. This metric should be a natural candidate for a critical point of Q and we shall now investigate under what conditions this is the case.

Proposition 5.1. Let X be a K3 surface and let [ω] be a Kähler class of X. Then the unique Kähler-Einstein metric g_{KE} ∈ K_{ω} is a critical point of Q : K_{ω} → ℝ if and only if Δc_{2}(Ω_{KE}) = 0, where Ω_{KE} is the curvature form of g_{KE}.

Proof. Since c_{1}(X) = 0, the Kähler-Einstein metric g_{KE} is Ricci flat. Therefore we have c_{1}(Ω_{KE}) = 0. Then it follows from the variational formula (2.10) that

\frac{δ}{δφ} log Q(g_{KE}) = \frac{1}{96} ∫_{X} Δφc_{2}(Ω_{KE}) = \frac{1}{96} ∫_{X} φΔc_{2}(Ω_{KE})

for all φ ∈ C^{∞}(X). Hence g_{KE} is a critical point of Q if and only if Δc_{2}(Ω_{KE}) = 0. □

The condition Δc_{2}(Ω_{KE}) = 0 can be reformulated in terms of the curvature tensor.

Lemma 5.2. Let g be a Ricci flat metric on X. Let R be the curvature tensor of g. Then we have

c_{2}(Ω) = \frac{1}{4π^{2}} |R|^{2}ω^{2},

where |R| is the pointwise norm of R.

Proof. Let W denote the Weyl curvature tensor of g. Since g is Ricci flat, it follows from [Be, (1.116)] that R = W. Furthermore, let L : Λ^{p,q}(T^{∗}X) → Λ^{p+1,q+1}(T^{∗}X) be the operator L(η) = ω ∧ η and let Λ be its adjoint. Then it follows from equation (2.80) in [Be] that

Λ^{2}(c_{2}(Ω)) = \frac{1}{4π^{2}} |W |^{2}.

This implies the lemma. □

Combining Proposition 5.1 and Lemma 5.2, we obtain

Corollary 5.3. The Kähler-Einstein metric g_{KE} ∈ K_{ω} is a critical point of Q if and only if |R(x)| is constant.

The condition that |R(x)| is constant is very stringent and we do not know if there exists any K3 surface which admits a Kähler-Einstein metric satisfying this condition. However, by a result of S. Kobayashi [Ko], one knows that on certain K3 surfaces there exist Kähler-Einstein metrics with curvature concentrated near some divisor. These are examples of Kähler-Einstein metrics on K3 surfaces such that |R| is not constant.

6. Algebraic manifolds

In this section we introduce a twisted version of our functional Q which has a simpler variational formula. For this purpose we now assume that X is a projective algebraic manifold. Note that by the Kodaira embedding theorem [GH] any compact complex manifold with definite Chern class is a projective algebraic manifold.

A polarization of X is the choice of an ample line bundle L → X up to isomorphism. The pair (X, L) is called a polarized algebraic manifold. Since L is ample, a certain power L^{⊗k}, k ∈ ℕ, defines an embedding i_{L} : X ↪ ℂP^{n} [GH]. Let g be the pullback of the Fubini-Study metric on ℂP^{n} and let [ω] be the Kähler class determined by g. Then [ω] is a rational multiple of c_{1}(L). So we can normalize g such that [ω] = c_{1}(L). This implies that [ω] ∈ H^{2}(X, ℤ) ∩ H^{1,1}(X, ℝ). We fix a hermitian metric h on L such that the curvature Θ_{h} satisfies Θ_{h} = −2πiω. Such metrics h exist [GH, pp.163,191] and, up to multiplication by a positive real number, h is uniquely determined by ω. Given φ ∈ C^{∞}(X, ℝ), put

h(φ) = e^{−2πφ}h.

Then h(φ) is a hermitian metric on L whose curvature is given by

Θ_{h(φ)} = Θ_{h} + 2π∂\overline{∂}φ = −2πi(ω + i∂\overline{∂}φ).

Thus, if φ ∈ H_{ω}, we have

(6.1)  Θ_{h(φ)} = −2πiω(φ).

In this section we consider the analytic torsion with coefficients in a certain virtual vector bundle E associated to L. For its definition we introduce the following numbers

κ_{1} = ∫_{X} c_{1}(Ω) ∧ ω^{n−1} and κ_{2} = ∫_{X} ω^{n}.

Since [ω] ∈ H^{2}(X, ℤ), it follows that κ_{1} and κ_{2} are integers. Then by (1.2), we have

s_{0} = 4nπ \frac{κ_{1}}{κ_{2}} .

Following Donaldson [D] and Tian [Ti1], we define the virtual vector bundle E over X to be

E = ⨁^{4κ_{2}(n+1)}(L − L^{−1})^{⊗n} ⊕ ⨁^{−κ_{1}n}(L − L^{−1})^{⊗(n+1)}.

If h̃ is any hermitian metric on L, we denote by h̃^{E} the induced hermitian metric on E. Using this notation we introduce the following functional on H_{ω}:

(6.2)  Q_{L}(X, φ) = Q(X, E, g(φ), h(φ))^{(−1)^{n}}, φ ∈ H_{ω}.

Since X is fixed, we set Q_{L}(φ) = Q_{L}(X, φ). To begin with we compute the variation of Q_{L}(φ).

Theorem 6.1. Let φ_{u}, u ∈ [a, b], be a smooth family of functions in H_{ω}. Let ω_{u} = ω + i∂\overline{∂}φ_{u} and let s(u) be the scalar curvature of the metric g_{u} = g(φ_{u}). Then

\frac{∂}{∂u} log Q_{L}(φ_{u}) = c_{n} ∫_{X} φ̇_{u}(s(u) − s_{0})ω_{u}^{n},

where

c_{n} = κ_{2}(n + 1)2^{n−1}.

Proof. To compute the variation, we apply formula (2.3). Let F_{u}^{E} denote the curvature form of the metric h_{u}^{E}, induced on E by h(φ_{u}), and let V_{u} = (h_{u}^{E})^{−1} \frac{d}{du}(h_{u}^{E}) . Then by (2.3) we have

\frac{∂}{∂u} log Q_{L}(φ_{u}) = \frac{(−1)^{n}}{2} (\frac{1}{2πi})^{n} ∫_{X} [\frac{∂}{∂b} Td(−Ω_{u} − bU_{u})|_{b=0} ch(E, h_{u}^{E})

+ Td(−Ω_{u})\frac{∂}{∂b} ch(−F_{u}^{E} − bV_{u})|_{b=0}] .

(6.3)

We start with the computation of the first term. Using that Θ_{u} = Θ_{h_{u}} = −2πiω_{u}, it follows that

\begin{aligned}ch(E, h_{u}^{E}) & = 4κ_{2}(n + 1)(e^{2πiω_{u}} − e^{−2πiω_{u}})^{n} − κ_{1}n(e^{2πiω_{u}} − e^{−2πiω_{u}})^{n+1} \\ & = 4κ_{2}(n + 1)(4πiω_{u})^{n}. \end{aligned}

Hence, by Lemma 2.1 we obtain

\begin{aligned}∫_{X}[\frac{∂}{∂b} Td( & − Ω_{u} − bU_{u})|_{b=0} ch(E, h_{u}^{E})]^{(n)} \\ & = 2κ_{2}(n + 1)(4πi)^{n} ∫_{X} \frac{∂}{∂b}c_{1}(−Ω_{u} − bU_{u})|_{b=0}ω_{u}^{n} \\ & = κ_{2}(n + 1)(4πi)^{n} ∫_{X}(Δ_{u}φ_{u})ω_{u}^{n} = 0. \end{aligned} ; (6.4)

As for the second term, we have

\frac{∂}{∂b} ch( − F_{u}^{E} − bV_{u})|_{b=0}
= (n + 1) {4nκ_{2} ch(L_{u}^{−1} − L_{u})^{n−1} − nκ_{1} ch(L_{u}^{−1} − L_{u})^{n}}

· \frac{∂}{∂b} \{ch(2πiω_{u} + b2πφ̇_{u}) − ch(−2πiω_{u} − b2πφ̇_{u})\} |_{b=0}

= 2π(n + 1) {4nκ_{2}(e^{−2πiω_{u}} − e^{2πiω_{u}})^{n−1}

−nκ_{1}(e^{−2πiω_{u}} − e^{2πiω_{u}})^{n}} φ̇_{u}(e^{2πiω_{u}} + e^{−2πiω_{u}})

= 4π(n + 1)κ_{2}(−4πi)^{n} \{−\frac{n}{πi}ω_{u}^{n−1} − n\frac{κ_{1}}{κ_{2}} ω_{u}^{n}\} φ̇_{u}.

By (1.1), we have c_{1}(Ω_{u}) = ρ/2π. Hence, using (1.2), we get

[Td(−Ω_{u})\frac{∂}{∂b} ch(−F_{u}^{E} − bV_{u})|_{b=0}]^{(n)}

= 4π(n + 1)κ_{2}(−4πi)^{n} (nc_{1}(Ω_{u}) ∧ ω_{u}^{n−1} − n\frac{κ_{1}}{κ_{2}} ω_{u}^{n}) φ̇_{u}

= 2^{n}(n + 1)κ_{2}(2πi)^{n}(−1)^{n}φ̇_{u}(s(u) − s_{0})ω_{u}^{n}.

Together with (6.3) and (6.4) this leads to

\frac{∂}{∂u} log Q_{L}(φ_{u}) = c_{n} ∫_{X} φ̇_{u}(s(u) − s_{0})ω_{u}^{n}.

□

Let φ_{1}, φ_{2} ∈ H_{ω}. Let {φ_{u} | a ≤ u ≤ b} be a piecewise smooth path in H_{ω} such that φ_{a} = φ_{1} and φ_{b} = φ_{2}. Then it follows from Theorem 6.1 that

(6.5)  log (\frac{Q_{L}(φ_{1})}{Q_{L}(φ_{2})}) = −c_{n} ∫_{a}^{b} \{∫_{X} φ̇_{u}(s(u) − s_{0})ω_{u}^{n}\} dt.

The right hand side of (6.5) is equal to c_{n} times the functional M(φ_{1}, φ_{2}) introduced by Mabuchi [Ma1, (2.2.2)]. It is defined for any compact Kähler manifold X. For a projective algebraic manifold X, (6.5) implies that M(φ_{1}, φ_{2}) is independent of the path {φ_{u} | a ≤ u ≤ b} in H_{ω} connecting φ_{1} and φ_{2}. This was proved by Mabuchi [Ma1, Theorem 2.4] in general, using different methods. Let φ ∈ H_{ω} and c ∈ ℝ. Set φ_{u} = φ + uc, u ∈ [0, 1]. Then by (6.5) and (1.2) we get

\begin{aligned}log (\frac{Q_{L}(φ + c)}{Q_{L}(φ)}) & = c_{n}c ∫_{0}^{1} \{∫_{X}(s(u) − s_{0})ω_{u}^{n}\} du \\ & = \frac{c_{n}c}{n!} ∫_{0}^{1} \{∫_{X} s(u)dv_{g} − s_{0}Vol(X)\} du = 0. \end{aligned}

Hence for all φ ∈ H_{ω} and c ∈ ℝ we have

Q_{L}(φ + c) = Q_{L}(φ).

Thus by (1.6), Q_{L} : H_{ω} → ℝ factors through K_{ω}. The induced functional on K_{ω} will be denoted by the same letter:

g̃ ∈ K_{ω} ⟼ Q_{L}(g̃) ∈ ℝ.

By (6.5), the map μ: K_{ω_{0}} → ℝ defined by

(6.6)  μ(g̃) = c_{n}^{−1} log \frac{Q_{L}(g)}{Q_{L}(g̃)}

coincides with the K-energy map defined by Mabuchi [Ma1, Section3].

Remark 6.2. Tian [Ti1, p.258] was the first to observe the connection between the Quillen metric and the K-energy. He used a virtual bundle of the form

⨁^{a} ((K_{X}^{−1} − K_{X}) ⊗ (L − L^{−1})^{n}) ⊕ ⨁^{b}(L − L^{−1})^{n+1}.

Up to a constant, it gives the same variational formula.

Let g_{u} ∈ K_{ω}, |u| < ϵ, be a smooth path. Then by (1.6) there exists a smooth path φ_{u} ∈ H_{ω} such that g_{u} = g(φ_{u}). It follows from Theorem 6.1 that

\frac{∂}{∂u} log Q_{L}(g_{u})|_{u=0} = c_{n} ∫_{X} φ̇_{0}(s_{g_{0}} − s_{0})ω_{0}^{n}.

In particular, if g_{u}, |u| < ϵ, is the variation of g in the direction of some φ ∈ C^{∞}(X), then this formula implies

\frac{δ}{δφ} log Q_{L}(g) = c_{n} ∫_{X} φ(s_{g} − s_{0})ω^{n}.

This proves the following theorem.

Theorem 6.3. A Kähler metric g_{0} ∈ K_{ω} is a critical point of Q_{L}: K_{ω} → ℝ if and only if the scalar curvature s_{g_{0}} of g_{0} is constant. In this case s_{g_{0}} = s_{0}.

This holds for the K-energy in general [Ma1].

In [Ca], Calabi introduced the notion of extremal Kähler metrics where a Kähler metric g̃ ∈ K_{ω} is called extremal if g̃ is a critical point of the functional

S(g̃) = ∫_{X} s2 ; g̃ dvg̃, g̃ ∈ K_{ω}.

Any metric g̃ of constant scalar curvature is extremal, but as shown by Calabi [Ca], the converse is not true, i.e., there are extremal Kähler metrics with nonconstant scalar curvature. However, if X is non-uniruled, then the extremal Kähler metrics are precisely the metrics with constant scalar curvature.

Now we collect some further information about the functional Q_{L}. By Theorem 6.3, the critical points of Q_{L} are the metrics of constant scalar curvature. If K_{ω} contains a Kähler-Einstein metric, then the subset K_{E} ⊂ K_{ω} of all Kähler-Einstein metrics coincides with the set of critical points of Q_{L}. In particular, if c_{1}(X) ≤ 0, then by [A], [Y], Q_{L} has critical points. On the other hand, the Futaki invariant [Fu], [Ca2] obstructs the existence of constant scalar curvature metrics in certain Kähler classes. Therefore, critical points do not always exist.

The K-energy map μ: K_{ω} → ℝ has been studied by Mabuchi and Bando [Ma1], [Ma2], [B], [BM], mainly in connection with the Futaki obstruction to the existence of Kähler-Einstein metrics on compact complex manifolds with c_{1}(X) > 0. Mabuchi [Ma2] defined a natural Riemannian structure on K_{ω}, i.e., K_{ω} can be equipped canonically with the structure of an infinite-dimensional Riemannian manifold. The tangent space Tω̃K_{ω} of K_{ω} at ω̃ ∈ K_{ω} can be identified with

{η ∈ C^{∞}(X, ℝ) | ∫_{X} ηω̃^{n} = 0}

and the Riemannian structure is given by

⟨η_{1}, η_{2}⟩ = \frac{1}{Vol(X)} ∫_{X} η_{1}η_{2} \frac{ω̃^{n}}{n!} , η_{1}, η_{2} ∈ Tω̃K_{ω}.

One of the main results of [Ma2, Theorem 5.3] states that μ: K_{ω} → ℝ is a convex function, meaning that Hessμ is positive semidefinite. Therefore, by (6.6) the same holds for log Q_{L}. This resembles the situation in the Riemann surface case.

If K_{ω} contains a Kähler-Einstein metric, then by Theorem 0.1, Q_{L} is bounded from above. In general, we do not know anything about boundedness of Q_{L}.

Finally, we note that by Theorem 6.1 the gradient flow of log Q_{L} is given by

\frac{∂φ}{∂t}(t) = s(t) − s_{0},

where s(t) is the scalar curvature of the metric g(t) = g(φ(t)). Using that

s = 2 ∑_{kl} g^{k\overline{l}}r_{k\overline{l}} and r_{k\overline{l}} = −\frac{∂^{2}}{∂z_{k}∂\overline{z}_{l}} log det(g_{α\overline{β}}),

we get the following fourth order nonlinear parabolic equation

(6.7)  \frac{∂φ}{∂t} = −2 ∑_{k,l} g(t)^{k\overline{l}} \frac{∂^{2}}{∂z_{k}∂\overline{z_{l}}} (log det (g_{α\overline{β}} + \frac{∂^{2}φ}{∂z_{α}∂\overline{z_{β}}} )) − s_{0}.

Differentiating this equation with respect to t, we get

\frac{∂}{∂t} (\frac{∂φ}{∂t}) = −2Δ_{t}^{2} (\frac{∂φ}{∂t}) + ∑_{k,l} r(t)^{k\overline{l}} \frac{∂^{2}}{∂z_{k}∂\overline{z}_{l}} (\frac{∂φ}{∂t}) ,

where r(t) is the Ricci tensor of the metric g(t). So, from standard theory we know that the solution of the initial value problem for (6.7) exists for short time. The crucial question is to prove existence for all time. If φ(t) is any solution of (6.7), then by Theorem 6.1 we get

\frac{∂}{∂t} log Q_{L}(g(φ(t)) = c_{n} ∫_{X}(s(t) − s_{0})^{2}ω_{t}^{n} ≥ 0.

Thus log Q_{L} is increasing along the gradient flow.

7. Kähler-Einstein metrics

By Theorem 6.3, the critical points of Q_{L} are exactly the metrics of constant scalar curvature. In general, it seems to be difficult to determine the nature of the critical points. Much more can be said if [ω] contains a Kähler-Einstein metric ω_{KE}. Recall that ω is said to be Kähler-Einstein if the Ricci form ρ_{ω} is proportional to the Kähler form

(7.1)  ρ_{ω} = λω

for some λ ∈ ℝ. Since 2πc_{1}(X) = [ρ_{ω}], the first Chern class has to satisfy either one of the following conditions

c_{1}(X) = 0 or c_{1}(X) > 0 or c_{1}(X) < 0.

If c_{1}(X) = 0, it follows from Yau [Y] that each Kähler class [ω] contains a unique Kähler-Einstein metric ω_{KE}. If c_{1}(X) < 0, i.e., if the canonical line bundle K_{X} is ample, then it was shown by T. Aubin [A] and S.-T. Yau [Y], that up to multiplication by a positive scalar, there exists a unique Kähler-Einstein metric g_{KE} on X which can be normalized such that 2πc_{1}(X) = −[ω_{KE}]. In the case c_{1}(X) > 0 there are obstructions for the existence of Kähler-Einstein metrics [Fu], [Ti2] and the existence problem has not been completely settled yet. See [Bo] for a review of the results. First we make the following elementary observation.

Lemma 7.1. Let (X, g) be a compact Kähler manifold with definite or vanishing first Chern class. Suppose that there is a Kähler-Einstein metric g_{KE} on X with [ω_{KE}] = [ω_{g}]. Then g has constant scalar curvature iff g is Kähler-Einstein.

Proof. It follows from (1.1) that a Kähler-Einstein metric has constant scalar curvature. Now assume that g has constant scalar curvature s_{g}. Then s_{g} = s_{0}. Let g_{KE} be a Kähler-Einstein metric with [ω_{KE}] = [ω]. By (1.1), we have

ρ_{KE} = \frac{s_{0}}{2n}ω_{KE}.

Since [ρ_{KE}] = [ρ], we get \frac{s_{0}}{2n}[ω] = [ρ]. Hence there exists φ ∈ C^{∞}(X, ℝ) such that

ρ = \frac{s_{0}}{2n}ω + i∂\overline{∂}φ.

Taking the trace of both sides of this equation yields

\frac{s_{0}}{2} − \frac{1}{2}Δφ = tr(r) = \frac{s_{g}}{2} = \frac{s_{0}}{2} .

This implies that φ is constant. Therefore, we get ρ = \frac{s_{0}}{2n}ω. □

Corollary 7.2. Let K_{KE} ⊂ K_{ω} denote the set of all Kähler-Einstein metrics contained in K_{ω}. Suppose that K_{KE} ≠ ∅. Then K_{KE} is precisely the set of critical points of Q_{L}.

Proof of Theorem 0.1. First assume that c_{1}(X) ≤ 0. Since by our assumption, the set of Kähler-Einstein metrics which are contained in K_{ω} is nonempty, it follows from [A] and [Y] that K_{ω} contains a unique Kähler-Einstein metric g_{KE}, and if c_{1}(X) < 0, then \frac{1}{2π} [ω_{KE}] represents −c_{1}(X). Now it follows from Corollary 7.2 that g_{KE} is the unique critical point of Q_{L} : K_{ω} → ℝ. To prove that g_{KE} is the absolute maximum of Q_{L}, we employ the complex analogue of the Ricci flow. We normalize g_{KE} such that

ρ_{KE} = \frac{s_{0}}{2n}ω_{KE}.

Let g ∈ K_{ω}. By H.-D. Cao [C], the evolution equation

(7.2)  \frac{∂g̃_{i\overline{j}}}{∂t} = −r̃_{i\overline{j}} + \frac{s_{0}}{2n}g̃_{i\overline{j}}, g̃_{i\overline{j}}(0) = g_{i\overline{j}}, i, j = 1, …, n,

has a unique solution g̃(t) which exists for all t ≥ 0 and satisfies

(7.3)  lim_{t→∞} g̃(t) = g_{KE}

in the C^{∞}-topology. Moreover, there exists u ∈ C^{∞}(ℝ^{+} × X, ℝ) with

(7.4)  g̃_{i\overline{j}}(t, z) = g_{i\overline{j}}(z) + \frac{∂^{2}u(t, z)}{∂z_{i}∂\overline{z}_{j}}.

Let Δ_{t} be the Laplace operator with respect to g̃(t). By (7.2) and (7.4) we obtain

\begin{aligned}\frac{1}{2}Δ_{t}(u̇(t)) & = − ∑_{i,j} g^{i\overline{j}}(t)\frac{∂^{2}u̇(t)}{∂z_{j}∂\overline{z}_{j}} = − ∑_{i,j} g^{i\overline{j}}(t)\frac{∂g̃_{i\overline{j}}}{∂t} (t) \\ & = ∑_{i,j} g^{i\overline{j}}(t) (r̃_{i\overline{j}}(t) − \frac{s_{0}}{2n}g̃_{i\overline{j}}(t)) = \frac{s̃(t)}{2} − \frac{s_{0}}{2} . \end{aligned}

Thus u(t) satisfies

(7.5)  Δ_{t}(u̇(t)) = s̃(t) − s_{0}.

Together with Theorem 6.1 we get

\begin{aligned}\frac{∂}{∂t} log Q(g̃_{t}) & = c_{n} ∫_{X} u̇(t)(s̃(t) − s_{0})ω_{t}^{n} \\ & = c_{n} ∫_{X} u̇(t)Δ_{t}(u̇(t))ω_{t}^{n} \\ & = \frac{c_{n}}{n!} ∫_{X} |∇u̇(t)|_{t}^{2} dv_{t}. \end{aligned} ; (7.6)

Hence, by the definition of c_{n}, we obtain

(7.7)  \frac{∂}{∂t} log Q(g̃_{t}) ≥ 0.

Suppose that s̃(0) is not constant. Then by (7.5) there exists ε > 0 such that for all t ≤ ε, u̇(t, z) is not constant as a function of z. By (7.6) it follows that

(7.8)  \frac{∂}{∂t} log Q(g̃_{t}) > 0 for all t ≤ ε.

By [C, Proposition 2.2], u̇(t) converges to a constant in the C^{∞}-topology as t → ∞. Also g̃_{i\overline{j}}(t) converges to (g_{KE})_{i\overline{j}} in the C^{∞}-topology as t → ∞ [C, Main Theorem]. This implies that there exists C_{1} > 0 such that

(7.9)  sup_{z∈X} |Δ_{t}(u̇(t, z))| ≤ C

for all t ≥ 0. Put

a(t) = \frac{1}{Vol(X)} ∫_{X} u̇(t)dv_{t}, t ≥ 0.

By (2.11) and (2.18) of [C] there exist C_{2}, C_{3} > 0 such that

(7.10)  ∫_{X} |u̇_{t} − a(t)|dv_{t} ≤ C_{2}e^{−C_{3}t}.

Using (7.9) and (7.10), we obtain

(7.11)  |∫_{X} u̇_{t}Δ(u̇_{t})dv_{t}| = |∫_{X}(u̇_{t} − a(t))Δ_{t}(u̇_{t})dv_{t}| ≤ C_{4}e^{−C_{3}t}.

Since for t → ∞, g̃_{i\overline{j}}(t) converges to (g_{KE})_{i\overline{j}} in the C^{∞}-topology, it follows that

(7.12)  log Q_{L}(g_{KE}) = lim_{t→∞} log Q_{L}(g̃_{t}).

Combining (7.6), (7.11) and (7.12), it follows that for all t > 0, we have

log Q(g_{KE}) − log Q(g̃_{t}) = c_{n} ∫_{t}^{∞} \frac{∂}{∂v} log Q(g̃_{v})dv.

Hence, if g ≠ g_{KE}, (7.7) together with (7.8) imply that

log Q_{L}(g_{KE}) > log Q_{L}(g).

This completes the proof of the first part of the theorem. Another proof of this result can be given by using (3.8) of [Ti1]. Now assume that c_{1}(X) > 0. Let K_{E} ⊂ K_{0} be the subset of Kähler-Einstein metrics. By our assumption K_{E} ≠ ∅. Then it follows from [B, Theorem 1] that the K-energy map μ: K_{ω} → ℝ takes its absolute minimum on K_{E}. Hence, by (6.6), Q_{L} attains its maximum on K_{E}.

8. Moduli spaces and Quillen metric

So far, we considered the Quillen norm Q_{L} of a fixed vector v ∈ λ(E) as a function on K_{ω}. In this section we investigate the behaviour of the Quillen metric with respect to variations of the complex structure. In [FS], Fujiki and Schumacher defined the moduli space of extremal Kähler metrics on a fixed C^{∞} manifold.

First we consider the local problem. Let (π : 𝔛 → S, ω̃) be a metrically polarized family of compact Kähler manifolds [FS, Definition 3.2] over a reduced complex space S and let p: F → 𝔛 be a holomorphic hermitian vector bundle with metric h^{F}. Let λ(F) be the holomorphic line bundle on S associated to (det Rπ_{∗}F)^{−1}, where F is the locally free sheaf corresponding to F. Given s ∈ S, let X_{s} = π^{−1}(s) and F_{s} = F |X_{s}. We recall that there exists a canonical isomorphism

(8.1)  λ(F)_{s} ≅ λ(F_{s}) = ⨂_{q≥0} (det H^{q}(X_{s}, F_{s}))^{(−1)^{q+1}}

[BGS3]. For each s ∈ S, ω̃ defines a Kähler metric g_{s} on X_{s} and we get a family g̃ = {g_{s}}_{s∈S} of Kähler metrics. Let h_{s}^{F} be the hermitian metric on F_{s} induced by h^{F}. Let h^{λ(F_{s})} be the Quillen metric on λ(F_{s}) associated to (g_{s}, h_{s}^{F}). Using the isomorphism (8.1), {h^{λ(F_{s})}}_{s∈S} defines a metric h^{λ(F)} on λ(F). It is proved in [BGS3] that the metric h^{λ(F)} is smooth. This metric is called the Quillen metric on λ(F) associated to (g̃, h^{F}).

If S is a complex manifold, then the curvature of the holomorphic connection on λ(F) was computed by [BGS3, Theorem 1.27]. Namely the first Chern form is given by

(8.2)  c_{1}(λ(F), h^{λ(F)}) = − [∫_{𝔛/S} Td(𝔛/S, g̃) ch(F, h^{F})]^{(2)} ,

where Td(𝔛/S, g̃) is the Todd class of the relative tangent bundle. This formula was extended by Fujiki and Schumacher [FS, Theorem 10.1] to the case where S is a reduced complex space.

Now let (π : 𝔛 → S, ω̃, L) be a metrically polarized family of compact Hodge manifolds [FS, Definition 3.8]. Thus (π : 𝔛 → S, L) is a polarized family of Hodge manifolds and (π : 𝔛 → S, ω̃) is a metrically polarized family of Kähler manifolds such that for each s ∈ S, ω_{s} represents c_{1}(L_{s}) on X_{s}. Furthermore, there exists a hermitian metric h on L such that the restriction of c_{1}(L, h) to each fibre X_{s} represents ω_{s} [FS, Proposition 3.10]. Any such metric is called admissible.

Let

κ_{1} = ∫_{X_{s}} c_{1}(X_{s}) ∧ ω_{s}^{n−1} and κ_{2} = ∫_{X_{s}} ω_{s}^{n}.

These are integers and therefore they are independent of s ∈ S. Using L, we introduce the family version of the virtual holomorphic bundle used in Section 6. Let n be the fibre dimension and set

E = ⨁^{4(n+1)κ_{2}}(L − L^{−1})^{n} ⊕ ⨁^{−nκ_{1}}(L − L^{−1})^{n+1}.

Let h be an admissible hermitian metric on L for (π, ω̃). Let h^{E} be the induced hermitian metric on E. Then the Chern character of (E, h^{E}) is given by

\begin{aligned}ch(E, h^{E}) = 4(n + 1)κ_{2}\{2^{n} & c_{1}(L, h)^{n} + \frac{n2^{n−1}}{3} c_{1}(L, h)^{n+2} + ⋯ \} \\ & − nκ_{1} \{2^{n+1}c_{1}(L, h)^{n+1} + ⋯ \} . \end{aligned}

Hence by (8.2) we get the following expression for the first Chern form of (λ(E), h^{λ(E)}):

\begin{matrix}c_{1}(λ(E), h^{λ(E)}) = −κ_{2}(n + 1) & 2^{n+1}(∫_{𝔛/S} c_{1}(L, h)^{n}c_{1}(𝔛/S, g̃) \\ − \frac{n}{n + 1} \frac{κ_{1}}{κ_{2}} ∫_{𝔛/S} c_{1}(L, h)^{n+1}). \end{matrix} ; (8.3)

Assume that the family (π : 𝔛 → S, ω̃, L) is effective which means that the Kodaira-Spencer map associated to (π, L) is injective [FS, Definition 4.1]. Then by [FS, Definition 7.2], there exists a generalized Weil-Petersson metric ĥ_{W P} = {ĥ_{s}}_{s∈S} on S. Let ω̂_{W P} denote the corresponding Weil-Petersson form. Using Theorem 7.8 of [FS] together with (8.3), one gets the following version of Theorem 10.3 of [FS].

Theorem 8.1. Let (π : 𝔛 → S, ω̃, L) be an effective family of extremal compact Hodge manifolds of constant scalar curvature with S being connected, and let h be an admissible hermitian metric on L for (π, ω̃). Then the first Chern form of the determinant line bundle (λ(E), h^{λ(E)}) is given by

c_{1}(λ(E), h^{λ(E)}) = a_{n}ω̂_{W P} ,

where a_{n} = −2^{n+1}κ_{2}(n + 1)!.

If we recall the isomorphism (8.1) and the construction of the Quillen metric on λ(E), we get an expression of the curvature in terms of the analytic torsion. For each s ∈ S, the fibre X_{s} is equipped with an extremal Hodge metric g_{s} and L_{s} is equipped with a hermitian metric h_{s} such that Θ_{h_{s}} = −2πiω_{s}. Let h^{E_{s}} be the associated hermitian metric in the virtual bundle

E_{s} = ⨁^{4(n+1)κ_{2}}(L_{s} − L_{s}^{−1})^{n} ⊕ ⨁^{−nκ_{1}}(L_{s} − L_{s}^{−1})^{n+1}

and put T_{0}(X_{s}, E_{s}) = T_{0}(X_{s}, E_{s}, g_{s}, h^{E_{s}}). Furthermore, let ∥ · ∥_{λ(E_{s})} denote the metric induced by (g_{s}, h^{E_{s}}) on the determinant line

λ(E_{s}) = ⨂_{q≥0} (det H^{q}(X_{s}, E_{s}))^{(−1)^{q+1}} .

Now let ϕ : S′ → λ(E) be a local holomorphic section of λ(E) over some open subset S′ ⊂ S. Using the definition of the Quillen metric in λ(E), it follows from Theorem 8.1 that

(8.4)  ∂_{s}\overline{∂}_{s} log (∥ ϕ(s) ∥_{λ(E_{s})}^{2} T_{0}(X_{s}, E_{s})) = a_{n}ω̂_{W P} .

In other words, log (∥ ϕ(s) ∥_{λ(E_{s})}^{2} T_{0}(X_{s}, E_{s})) is a potential for the generalized Weil-Petersson metric on S.

This construction can be globalized [FS, §11]. Fix a compact connected C^{∞} manifold X and an integral class α ∈ H^{2}(X, ℝ). Let 𝔐_{H,e} be the moduli space of extremal Hodge manifolds with underlying C^{∞} structure (X, α). Then 𝔐_{H,e} is a complex V -manifold and the generalized Weil-Petersson metric is a V -form ω_{W P} on 𝔐_{H,e}. As shown in [FS], the local determinant line bundles can be patched together to a holomorphic line bundle F on 𝔐_{H,e} and the Quillen metric on the local bundles patches together to a V -metric h^{F} on F. The global version of Theorem 8.1 implies that the Chern V -form of (F, h^{F}) satisfies

(8.5)  c_{1}(F, h^{F}) = aω_{W P}

for some integer a ≠ 0. In particular, this implies that the cohomology class [ω_{W P}] is an integral cohomology class. Hence, any compact analytic subspace of 𝔐_{H,e} is a projective algebraic manifold.

We note that this is well known for Riemann surfaces. Let X be a compact Riemann surface of genus g ≥ 2. Then there exists a discrete cocompact subgroup Γ ⊂ SL(2, ℝ) such that X = Γ\H, where H is the upper half-plane. Let g be the Riemannian metric on X which is induced by the Poincaré metric on H and let Δ = \overline{∂}^{∗}\overline{∂} be the corresponding Laplace operator. Fix a marking of X and let τ be the corresponding period matrix of X. We regard both Δ and τ locally as functions on the moduli space 𝔐_{g} of compact Riemann surfaces of genus g. Then one has

∂_{z}\overline{∂}_{z} log (\frac{det Δ_{z}}{det τ_{z}} ) = \frac{i}{6π} ω_{W P} ,

where ω_{W P} is the usual Weil-Petersson metric on 𝔐_{g}. This is essentially (8.5). Furthermore, Wolpert [Wo] has shown that ω_{W P} extends to a closed form with singularities on the compactification \overline{𝔐}_{g} of the moduli space and in the sense of currents, \frac{1}{π^{2}} ω_{W P} is the Chern form of a continuous metric h_{W P} on a certain line bundle λ_{W P}. This gives a projective embedding of \overline{𝔐}_{g}.

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Universität Bonn, Mathematisches Institut, Beringstrasse 1, D – 53115 Bonn, Germany

    E-mail address: mueller@@math.uni-bonn.de

Universität Bonn, Physikalisches Institut, Nußallee 12, D – 53115 Bonn, Germany

    E-mail address: wendland@@avz109.physik.uni-bonn.de