Curvature Properties of the Calabi-Yau Moduli

Chin-Lung Wang

Received: September 1, 2003

Communicated by Thomas Peternell

Abstract. A curvature formula for the Weil-Petersson metric on the Calabi-Yau moduli spaces is given. Its relations to the Hodge metrics and the Bryant-Griffiths cubic form are obtained in the threefold case. Asymptotic behavior of the curvature near the boundary of moduli is also discussed via the theory of variations of Hodge structures.

2000 Mathematics Subject Classification: 32G20, 32Q25.

Keywords and Phrases: Weil-Petersson metric, Calabi-Yau manifold, variations of Hodge structures.

Introduction

In a former paper [15], the incompleteness phenomenon of the Weil-Petersson metric on Calabi-Yau moduli spaces was studied. In this note, I shall discuss some curvature properties of it. The first result is a simple explicit formula (Theorem 2.1) for the Riemann curvature tensor. While this problem was treated before in [6] and [10], the approach taken here is more elementary. Two simple proofs of Theorem 2.1 are offered in §2 and both are based on the Hodge-theoretic description of the Weil-Petersson metric [13]. The first one uses a trick to select suitable coordinate system and line bundle section to reduce the computation. The second proof uses Griffiths’ curvature formula for Hodge bundles [2]. Direct consequences of Theorem 2.1 are relations between the Weil-Petersson metric and the Hodge metric for Calabi-Yau threefolds and various positivity results on the curvature tensor (see §3). §4 is devoted to the asymptotic analysis of the curvature near the boundary of moduli spaces. The method is modelled on the first proof and uses Schmid’s theory on the degenerations of Hodge structures [7]. The final section §5 contains some remarks toward the completion and compactification problems of Calabi-Yau moduli spaces.

1. The Weil-Petersson metric

A Calabi-Yau manifold is a compact Kähler manifold with trivial canonical bundle. The local Kuranish family of polarized Calabi-Yau manifolds 𝒳 → S is smooth (unobstructed) by the Bogomolov-Tian-Todorov theorem [13]. One can assign the unique (Ricci-flat) Yau metric g(s) on 𝒳_{s} in the polarization Kähler class [17]. Then, on a fiber 𝒳_{s} =: X, the Kodaira-Spencer theory gives rise to an injective map ρ : T_{s}(S) → H^{1}(X, T_{X}) ≅ H0,1 ; ¯∂(T_{X}) (harmonic representatives). The metric g(s) induces a metric on Λ^{0,1}(T_{X}). For v, w ∈ T_{s}(S), one then defines the Weil-Petersson metric on S by

(1.1)  g_{W P} (v, w) := ∫_{X}⟨ρ(v), ρ(w)⟩_{g(s)}.

Let dim X = n. Using the fact that the global holomorphic n-form Ω(s) is flat with respect to g(s), it can be shown [13] that

(1.2)  g_{W P} (v, w) = −\frac{Q̃(i(v)Ω, \overline{i(w)Ω})}{Q̃(Ω, Ω̄)}.

Here, for convenience, we write Q̃ = \sqrt{−1}^{n}Q(·, ·), where Q is the intersection product. Therefore, Q̃ has alternating signs in the successive primitive cohomology groups P^{p,q} ⊂ H^{p,q}, p + q = n. (1.2) implies that the natural map H^{1}(X, T_{X}) → Hom(H^{n,0}, H^{n−1,1}) via the interior product v ↦ i(v)Ω is an isometry from the tangent space T_{s}(S) to (H^{n,0})^{∗} ⊗ P^{n−1,1}. So the Weil-Petersson metric is precisely the metric induced from the first piece of the Hodge metric on the horizontal tangent bundle over the period domain. A simple calculation in formal Hodge theory shows that

(1.3)  ω_{W P} = RicQ̃(ℋ^{n,0}) = −∂ ¯∂log Q̃(Ω, Ω̄),

where ω_{W P} is the 2-form associated to g_{W P}. In particular, g_{W P} is Kähler and is independent of the choice of Ω. In fact, g_{W P} is also independent of the choice of the polarization. With this background, one can abstract the discussion by considering a polarized variations of Hodge structures ℋ → S of weight n with h^{n,0} = 1 and a smooth base S. In this note, I always assume that it is effectively parametrized in the sense that the infinitesimal period map (also called the second fundamental form [2])

(1.4)  σ : T_{s}(S) → Hom(H^{n,0}, H^{n−1,1}) ⊕ Hom(H^{n−1,1}, H^{n−2,2}) ⊕ ⋯

is bijective in the first piece. Then the Weil-Petersson metric g_{W P} on S is defined by formula (1.2) (or equivalently, (1.3)). One advantage to work with the abstract setting is that, instead of using P^{p,q} in the geometric case, we may write H^{p,q} directly in our presentation.

2. The Riemann curvature tensor formula

Here is the basic formula (compare with [6], [10] and [12]):

Theorem 2.1. For a given effectively parametrized polarized variations of Hodge structures ℋ → S of weight n with h^{n,0} = 1, h^{n−1,1} = d and smooth S, in terms of any holomorphic section Ω of ℋ^{n,0} and the infinitesimal period map σ, the Riemann curvature tensor of the Weil-Petersson metric g_{W P} = g_{ij̄}dt_{i} ⊗dt̄_{j} on S is given by

(2.1)  R_{ij̄kℓ̄} = −(g_{ij̄}g_{kℓ̄} + g_{iℓ̄}g_{kj̄}) + \frac{Q̃(σ_{i}σ_{k}Ω, \overline{σ_{j}σ_{ℓ}Ω})}{Q̃(Ω, Ω̄)}.

2.1. The first proof. The main trick in the proof is a nice choice of the holomorphic section Ω and special coordinate system on the base S. Since the problem is local, we may assume that S is a disk in ℂ^{d} around t = 0. Specifically, we have

Lemma 2.2. For any k ∈ ℕ, there is a local holomorphic section Ω of ℋ^{n,0} such that in the power series expansion at t = 0

(2.2)  Ω(t) = a_{0} + ∑_{i} a_{i}t_{i} + ⋯ + ∑_{|I|=k} \frac{1}{I!}a_{I}t^{I} + ⋯

we have a_{0} ∈ H^{n,0}, Q̃(a_{0}, ā_{0}) = 1 and Q̃(a_{0}, ā_{I}) = 0 for any multi-index I ≠ 0 and |I| ≤ k. (We always assume that a_{I} = a_{J} if I = J as unordered sets.)

Proof. Only the last statement needs a proof. Let

Ω̃ = ∑_{I} ã_{I}t^{I} = (1 + ∑_{i} λ_{i}t_{i} + ⋯ + ∑_{|I|=k} λ_{I}t^{I})Ω
= a_{0} + ∑_{i}(λ_{i}a_{0} + a_{i})t_{i} + ⋯ + ∑_{|I|=k}(λ_{I}a_{0} + a_{I})t^{I} + ⋯

Set λ_{I} = −Q̃(a_{0}, ā_{I}), then clearly Q̃(ã_{0}, \overline{ã_{I}}) = 0 for I ≠ 0 and |I| ≤ k. □

Lemma 2.3. Pick Ω as in Lemma 2.2. For any k′ ∈ ℕ with 2 ≤ k′ ≤ k, there is a holomorphic coordinate system t such that a_{i} form an orthonormal basis of H^{n−1,1}, i.e. Q̃(a_{i}, ā_{i}) = −δ_{ij}. Moreover, Q̃(a_{i}, ā_{I}) = 0 for all i and I with 2 ≤ |I| ≤ k′.

Proof. The Griffiths transversality says that

a_{i} = \frac{∂}{∂t_{i}} Ω|_{t=0} ∈ H^{n,0} ⊕ H^{n−1,1}.

Lemma 2.2 then implies that a_{i} ∈ H^{n−1,1}. It is also clear that by a linear change of coordinates of t we can make a_{i} to form an orthonormal basis of H^{n−1,1}. For the second statement, consider the following coordinates transformation:

t_{i} = s_{i} + ∑_{1≤j,k≤d} c_{i}^{jk}s_{j}s_{k} + ⋯ + ∑_{|I|=k′} c_{i}^{I}s^{I}, 1 ≤ i ≤ d

with c_{i}^{I} = c_{i}^{J} when I = J as unordered sets. It’s easy to see that the number of coefficients to be determined is the same as the number of equations Q̃(a_{i}, ā_{jk}) = 0, hence the lemma. □

Proof. (of Theorem 2.1) Let Ω and t_{i} be as in the above lemmas. For multiindices I and J, we set q_{I,J} := Q̃(a_{I}, ā_{J}). By Lemma 2.2 and 2.3,

q(t) := Q̃(Ω(t), \overline{Ω(t)})
= 1 − ∑_{i} t_{i}t̄_{i} + ⋯ + ∑_{i,j,k,ℓ} \frac{1}{(ik)!(jℓ)!}q_{ik,jℓ} t_{i}t_{k}t̄_{j}t̄_{ℓ} + O(t^{5}).

To calculate R_{ij̄kℓ̄}, we only need to calculate g_{kℓ̄} up to degree 2 terms:

g_{kℓ̄} = −∂_{k}∂ℓ̄log q = q^{−2}(∂_{k}q∂ℓ̄q − q∂_{k}∂ℓ̄q)
= (1 + 2 ∑_{i} t_{i}t̄_{i} + ⋯) ×
[t_{ℓ}t̄_{k} − (1 − ∑_{i} t_{i}t̄_{i})( − δ_{kℓ} + ∑_{i,j} q_{ik,jℓ}t_{i}t̄_{j}) + ⋯]
= δ_{kℓ} − δ_{kℓ} ∑_{i} t_{i}t̄_{i} + t_{ℓ}t̄_{k} + 2δ_{kℓ} ∑_{i} t_{i}t̄_{i} − ∑_{i,j} q_{ik,jℓ}t_{i}t̄_{j} + ⋯
= δ_{kℓ} + δ_{kℓ} ∑_{i} t_{i}t̄_{i} + t_{ℓ}t̄_{k} − ∑_{i,j} q_{ik,jℓ}t_{i}t̄_{j} + · · · .

Here we have used the fact that degree 3 terms of mixed type (contain t_{k}t̄_{ℓ}) must be 0 by our choice of Ω. As a result, we find that the Weil-Petersson metric g is already in its geodesic normal form, so the full curvature tensor at t = 0 is given by

R_{ij̄kℓ̄} = −\frac{∂^{2}g_{kℓ̄}}{∂t_{i}∂t̄_{j}} = −δ_{ij}δ_{kℓ} − δ_{iℓ}δ_{kj} + q_{ik,jℓ}.

Rewriting this in its tensor form then gives the formula. □

Remark 2.4. The proof does not require the full condition that ℋ → S is a variation of Hodge structures. The essential part used is the polarization structure on the indefinite metric Q̃ on ℋ. It has a fixed sign on ℋ^{n−1,1} makes possible the definition of the Weil-Petersson metric. For such cases, in terms of the second fundamental form σ, Lemma 2.2 and 2.3 say that under this choice of Ω and t, ordinary differentiations approximate σ up to second order at t = 0. In particular, a_{0} = Ω(0), a_{i} = σ_{i}Ω(0) and a_{ij} = a_{ji} = σ_{i}σ_{j}Ω(0) = σ_{j}σ_{i}Ω(0).

2.2. The second proof. Now we give another proof of Theorem 2.1 via Griffiths’ curvature formula for Hodge bundles.

Proof. Recall the isometry in §1:

(2.3)  T_{s}(S) ≅ (H^{n,0})^{∗} ⊗ H^{n−1,1}

and Griffiths’ curvature formula ([2], Ch.II Prop.4):

(2.4)  ⟨R(e), e′⟩ = ⟨σe, σe′⟩ + ⟨σ^{∗}e, σ^{∗}e′⟩.

Where R is the matrix valued curvature 2-form of ℋ^{p,q}, e and e′ are any two elements of H^{p,q} and ⟨, ⟩ is the Hodge metric.

Let Ω be a holomorphic section of ℋ^{n,0} and consider the basis of H^{n−1,1} given by σ_{i}Ω, then T has a basis e_{i} = Ω^{∗} ⊗ σ_{i}Ω from (2.3). In this basis, the Weil-Petersson metric takes the form

(2.5)  g_{ij̄} = \frac{⟨σ_{i}Ω, σ_{j}Ω⟩}{⟨Ω, Ω⟩}.

Let K, R_{1} and R_{2} be the curvature of T, (ℋ^{n,0})^{∗} and ℋ^{n−1,1} respectively. Using the standard curvature formulae for tensor bundle and dual bundle, we find

K(e_{i}) = (R_{1} ⊗ I_{2} + I_{1} ⊗ R_{2})(Ω^{∗} ⊗ σ_{i}Ω)
= R_{1}(Ω^{∗}) ⊗ σ_{i}Ω + Ω^{∗} ⊗ R_{2}(σ_{i}Ω).

By taking scalar product with e_{j} = Ω^{∗} ⊗ σ_{j}Ω and using the definition of dual metric, we get

⟨K(e_{i}), e_{j}⟩_{W P} = ⟨R_{1}(Ω^{∗}), Ω^{∗}⟩⟨σ_{i}Ω, σ_{j}Ω⟩ + ⟨Ω^{∗}, Ω^{∗}⟩⟨R_{2}(σ_{i}Ω), σ_{j}Ω⟩
= −⟨Ω, Ω⟩^{−2}⟨R(Ω), Ω⟩⟨σ_{i}Ω, σ_{j}Ω⟩ + ⟨Ω, Ω⟩^{−1}⟨R(σ_{i}Ω), σ_{j}Ω⟩.

Now we evaluate this 2-form on e_{k} ∧ē_{ℓ} and apply (2.4), we get (notice the order of ℓ, k and the sign)

(2.6)  −\frac{⟨σ_{k}Ω, σ_{ℓ}Ω⟩}{⟨Ω, Ω⟩} \frac{⟨σ_{i}Ω, σ_{j}Ω⟩}{⟨Ω, Ω⟩} + \frac{⟨σ_{k}σ_{i}Ω, σ_{ℓ}σ_{j}Ω⟩}{⟨Ω, Ω⟩} − \frac{⟨σ_{ℓ}^{∗}σ_{i}Ω, σ_{k}^{∗}σ_{j}Ω⟩}{⟨Ω, Ω⟩}.

Since h^{n,0} = 1, σ_{p}^{∗}σ_{q} acts as a scalar operator on H^{n,0}:

σ_{p}^{∗}σ_{q} = \frac{⟨σ_{p}^{∗}σ_{q}Ω, Ω⟩}{⟨Ω, Ω⟩} = \frac{⟨σ_{q}Ω, σ_{p}Ω⟩}{⟨Ω, Ω⟩}.

Hence the last term in (2.6) becomes

(2.7)  −\frac{⟨σ_{i}Ω, σ_{ℓ}Ω⟩}{⟨Ω, Ω⟩} \frac{\overline{⟨σ_{j}Ω, σ_{k}Ω⟩}}{⟨Ω, Ω⟩} = −\frac{⟨σ_{i}Ω, σ_{ℓ}Ω⟩}{⟨Ω, Ω⟩} \frac{⟨σ_{k}Ω, σ_{j}Ω⟩}{⟨Ω, Ω⟩}.

Using (2.5) and (2.7), then (2.6) gives the formula (2.1). □

3. Some simple consequences of the curvature formula

3.1. Lower bounds of curvature. The immediate consequences of the general curvature formula are various positivity results of different types of curvature. We mention some of them here.

Theorem 3.1. For the Weil-Petersson metric g_{W P}, we have (1) The holomorphic sectional curvature ∑_{i,j,k,ℓ} R_{ij̄kℓ̄}ξ^{i}ξ̄^{j}ξ^{k}ξ̄^{ℓ} ≥ −2|ξ|^{4}.

(2) The Ricci curvature R_{ij̄} ≥ −(d + 1)g_{ij̄}.

(3) The second term in (2.1) is “Nakano semi-positive”.

Proof. This is a pointwise question. For simplicity, let’s use the normal coordinate system given by Lemma 2.2. (1) is obvious since ∑_{i,j,k,ℓ} Q̃(a_{ik}, \overline{a_{jℓ}})ξ^{i}ξ̄^{j}ξ^{k}ξ̄^{ℓ} = Q̃(A, Ā) ≥ 0 for A = ∑_{i,k} a_{ik}ξ^{i}ξ^{k}.

For (2), we need to show that ( ∑_{k,ℓ} g^{kℓ̄}Q̃(a_{ik}, \overline{a_{jl}}))_{i,j} is semi-positive. For any vector ξ = (ξ_{i}), let A_{k} be the vector ∑_{i} a_{ik}ξ_{i}. Then

∑_{i,j,k,ℓ} g^{kℓ̄}Q̃(a_{ik}, ā_{jℓ})ξ_{i}ārξ_{j} = ∑_{k} Q̃(A_{k}, Ā_{k}) ≥ 0.

For (3), it simply means that for any vector u = (u_{pq}) with double indices,

∑_{i,j,k,ℓ} Q̃(a_{ik}, \overline{a_{jℓ}})u_{ik}\overline{u_{jℓ}} = Q̃(A, Ā) ≥ 0

where A := ∑_{p,q} a_{pq}u_{pq}. □

3.2. Relation to the Hodge metric. The period domain has a natural invariant metric induced from the Killing form. The horizontal tangent bundle also has a natural metric induced from the metrics on the Hodge bundles. These two metrics are in fact the same ([2], p.18) and we call it the Hodge metric. The Hodge metric g_{H} on S is defined to be the metric induced from the Hodge metric of the full horizontal tangent bundle. In dimension three, e.g. the moduli spaces of Calabi-Yau threefolds, we can reconstruct the Hodge metric from the Weil-Petersson metric. This result was first deduced by Lu in 1996 through different method, see e.g. [4].

Theorem 3.2. In the case n = 3, we have

g_{H} = (d + 3)g_{W P} + Ric(g_{W P}).

In particular, the Hodge metric g_{H} is Kähler.

Proof. The horizontal tangent bundle is

Hom(H^{3,0}, H^{2,1}) ⊕ Hom(H^{2,1}, H^{1,2}) ⊕ Hom(H^{1,2}, H^{0,3})

The first piece gives the Weil-Petersson metric on S. The third piece is dual to the first one, hence, as one can check easily, gives the same metric. Now

(d + 3)g_{ij̄} + R_{ij̄} = 2g_{ij̄} + ∑_{k,ℓ} g^{kℓ̄} \frac{Q̃(σ_{i}σ_{k}Ω, \overline{σ_{j}σ_{ℓ}Ω})}{Q̃(Ω, Ω̄)}.

The last term gives the Hodge metric of the middle part of the horizontal tangent bundle since σ_{k}Ω form a basis of H^{2,1} and the Hodge metric is defined to be the metric of linear mappings, which are exactly the infinitesimal period maps σ_{i}’s. □

Remark 3.3. Moduli spaces of polarized complex tori (resp. hyperkähler manifolds) correspond to variations of polarized weight one (resp. weight two) Hodge structures. Their universal covering spaces are Hermitian bounded symmetric domains and the invariant (Bergman) metrics are Kähler-Einstein of negative Ricci curvature. In these cases, the weight n polarized VHS are completely determined by the weight one (resp. weight two) polarized VHS. Based on this observation, one can show that g_{W P} and g_{H} both coincide with the Bergman metric up to a positive constant (cf. [9] for the case of g_{W P}). However, as we will see in Theorem 4.4, the negativity of Ricci curvature fails for moduli spaces of general Calabi-Yau manifolds. In fact, the Hodge theory of Calabi-Yau threefolds with h^{1}(𝒪) = 0 may be regarded as the first nontrivial instance of Hodge theory of weight three.

3.3. Relation to the Bryant-Griffiths cubic form. In the case n = 3, Bryant and Griffiths [1] has defined a symmetric cubic form on the parameter space S:

F_{ijk} := \frac{Q̃(σ_{i}σ_{j}σ_{k}Ω, Ω)}{Q̃(Ω, Ω̄)}.

Strominger [12] has obtained a formula for the Riemann curvature tensor through this cubic form F_{ijk} (in physics literature it is called the Yukawa coupling), and it has played important role in the study of Mirror Symmetry. We may derive it from our formula (2.1):

Theorem 3.4. For an effectively parametrized polarized variations of Hodge structures ℋ → S of weight 3, the curvature tensor of g_{W P} is given by

R_{ij̄kℓ̄} = −(g_{ij̄}g_{kℓ̄} + g_{iℓ̄}g_{kj̄}) + ∑_{p,q} g^{pq̄} F_{pik}\overline{F_{qjℓ}}.

Proof. Since Q̃(σ_{i}σ_{j}Ω, Ω) = 0 by the consideration of types, the metric compatibility implies that

0 = ∂_{k}Q̃(σ_{i}σ_{j}Ω, Ω) = Q̃(σ_{k}σ_{i}σ_{j}Ω, Ω) − Q̃(σ_{i}σ_{j}Ω, σ_{k}Ω).

Let us write \overline{σ_{j}σ_{ℓ}Ω} = ∑_{p} a^{p} σ_{p}Ω, then

∑_{p} a^{p} g_{pq̄} = − ∑_{p} a^{p} \frac{Q̃(σ_{p}Ω, \overline{σ_{q}Ω})}{Q̃(Ω, Ω̄)} = −\frac{Q̃(\overline{σ_{j}σ_{ℓ}Ω}, \overline{σ_{q}Ω})}{Q̃(Ω, Ω̄)} = \overline{F_{qjℓ}}.

So a^{p} = ∑_{q} g^{pq̄}\overline{F_{qjℓ}} and the second term in (2.1) becomes

\frac{Q̃(σ_{i}σ_{k}Ω, \overline{σ_{j}σ_{ℓ}Ω})}{Q̃(Ω, Ω̄)} = ∑_{p,q} g^{pq̄} \overline{F_{qjℓ}} Q̃(σ_{i}σ_{k}Ω, σ_{p}Ω) = ∑_{p,q} g^{pq̄} F_{pik} \overline{F_{qjℓ}}.

□

Remark 3.5. In the geometric case, namely moduli of Calabi-Yau threefolds, the cubic form is usually written as

F_{ijk} = e^{−K} ∫_{X} ∂_{i}∂_{j}∂_{k} Ω ∧ Ω,

where K = log Q̃ and Ω is a relative holomorphic three-form over S.

4. Asymptotic behavior of the curvature along degenerations

To study the asymptotic behavior of the curvature, we may localize the problem and study degenerations of polarized Hodge structures. By taking a holomorphic curve transversal to the degenerating loci, or equivalently we study the limiting behavior of the holomorphic sectional curvature, we may consider the following situation (consult [2], [7] for more details): a period mapping

ϕ : Δ^{×} → ⟨T ⟩\D → ⟨T ⟩\ℙ(V)

which corresponds to the degeneration. Here V = H^{n} is a reference vector space with a quadratic form Q as in §1, and with T ∈ Aut(V, Q) the Picard-Lefschetz monodromy. Assume that T is unipotent and let N = log T. There is an uniquely defined weight filtration W : 0 = W_{−1} ⊂ W_{0} ⊂ ⋯ ⊂ W_{2n} = V such that

NW_{i} ⊂ W_{i−2} and N^{k} : Gr_{n+k}^{W} ≅ Gr_{n−k}^{W}

where Gr_{i}^{W} := W_{i}/W_{i−1}. This W, together with the limiting Hodge filtration F_{∞} := lim_{t→0} e^{−zN}F_{t} (z = log t/2π\sqrt{−1} is the coordinates on the upper half plane, t ∈ Δ^{×}) constitute Schmid’s polarized limiting mixed Hodge structures. This means that Gr_{i}^{W} admits a polarized Hodge structure ⨁_{p+q=i} H_{∞}^{p,q} of weight i induced from F_{∞} and Q such that for k ≥ 0, the primitive part P_{n+k}^{W} := KerN^{k+1} ⊂ Gr_{n+k}^{W} is polarized by Q(·, N^{k}¯·). Notice that N is a morphism of type (−1, −1) in the sense that N(H_{∞}^{p,q}) ⊂ H_{∞}^{p−1,q−1}. This allows one to view the mixed Hodge structure in terms of a Hodge diamond and view N as the operator analogous to “contraction by the Kähler form”. By Schmid’s nilpotent orbit theorem ([7], cf. [15], §0-§1), we can pick the (multivalued) holomorphic section Ω of ℱ^{n} over Δ^{×} by

Ω(t) = Ω(z) := e^{zN}a(t) = e^{log tÑ}a(t) ∈ F_{t}^{n},

where a(t) = ∑ a_{i}t^{i} is holomorphic over Δ with value in V and Ñ := N/2π\sqrt{−1}. Also 0 ≠ a(0) = a_{0} ∈ F_{∞}^{n}. (Notice that while Ω(z) is singlevalued, Ω(t) is well-defined only locally or with its value mod T.) Now we may summarize the computations done in [15], §1 in the following form:

Theorem 4.1. The induced Weil-Petersson metric g_{W P} on Δ^{×} is incomplete at t = 0 if and only if F_{∞}^{n} ⊂ Ker N. In the complete case, i.e. Na_{0} ≠ 0, let k := max{ i | N^{i}a_{0} ≠ 0 }. Then Q̃(Ω, Ω̄) blows up to +∞ with order c| log |t|^{2}|^{k} and the metric g_{W P} blows up to +∞ with order

\frac{kdt ⊗ dt̄}{|t|^{2}| log |t|^{2}|^{2}} ,

i.e. it is asymptotic to the Poincaré metric, where c = (k!)^{−1}|Q̃(Ñ^{k}a_{0}, ā_{0})| > 0. In the incomplete case, i.e. Na_{0} = 0, the holomorphic section Ω(t) extends continuously over t = 0.

Idea of proof. We have the following well-known calculation: for any k ∈ ℝ,

(4.1)  −∂ ¯∂log | log |t|^{2}|^{k} = \frac{kdt ∧ dt̄}{|t|^{2}| log |t|^{2}|^{2}} ,

which is also true asymptotically if t is a holomorphic section of a Hermitian line bundle as the defining section of certain divisor, in a general smooth base S of arbitrary dimension. The main point is to prove that lower order terms are still of lower order whenever we take derivatives. This is done in [15] when S = Δ^{×}. This is also the main point of the remaining discussion in this section. □

To achieve the goal, we define operators S_{k} = k + Ñ for any k ∈ ℤ. Then all S_{k} commute with each other and S_{k} is invertible if k ≠ 0. By Theorem 4.1, we only need to study the incomplete case, i.e. Na_{0} = 0. As in Lemma 2.2., we may assume that Q̃(a_{0}, ā_{0}) = 1 and Q̃(a_{0}, ā_{i}) = 0 for all i ≥ 1.

Lemma 4.2. If a_{k} is the first nonzero term other than a_{0}, then S_{k}a_{k} ∈ F_{∞}^{n−1}. If moreover Na_{k} = 0 then a_{k} ∈ H_{∞}^{n−1,1}, and for the next nonzero term a_{k+ℓ} we have S_{ℓ}S_{k+ℓ}a_{k+ℓ} ∈ F_{∞}^{n−2}.

Proof. By the Griffiths transversality, we have

(4.2)  Ω′(t) = e^{zN}[\frac{1}{t}Ña + a′] ∈ F_{t}^{n−1}.

Since Na_{0} = 0, this implies Ña_{k}t^{k−1} + ka_{k}t^{k−1} + ⋯ ∈ e^{−zN}F_{t}^{n−1}. Take out the factor t^{k−1} and let t → 0, we get

S_{k}a_{k} ∈ F_{∞}^{n−1}.

In fact, we have from (4.2), e^{zN}(S_{k}a_{k} + S_{k+1}a_{k+1}t + ⋯) ∈ F_{t}^{n−1}. Taking derivative and by transversality again, we get

e^{zN}[\frac{1}{t}ÑS_{k}a_{k} + S_{1}S_{k+1}a_{k+1} + S_{2}S_{k+2}a_{k+2}t + ⋯] ∈ F_{t}^{n−2}.

So, if Na_{k} = 0, then for the next nonzero term a_{k+ℓ}, we have by the same way

S_{ℓ}S_{k+ℓ}a_{k+ℓ} ∈ F_{∞}^{n−2}.

We also know the following equivalence: Na_{k} = 0 iff NS_{k}a_{k} = 0 iff S_{k}a_{k} ∈ H_{∞}^{n−1,1} (because S_{k}a_{k} ∈ F_{∞}^{n−1}). Since N is a morphism of type (−1, −1) and the only nontrivial part of F_{∞}^{n} is in H_{∞}^{n,0}, this is equivalent to a_{k} ∈ H_{∞}^{n−1,1}. (This follows easily from the Hodge diamond.) □

Define q_{ij} = Q̃(e^{zN}a_{i}, \overline{e^{zN}a_{j}}) ≡ Q̃(e^{log |t|^{2}Ñ}a_{i}, ā_{j}), which are functions of log |t|. The following basic lemma is the key for all the computations, which explains “lower order terms are stable under differentiations”.

Lemma 4.3. Let Q^{S_{k}}(a, b̄) := Q(S_{k}a, b̄) ≡ Q(a, \overline{S_{k}b}), then for all k, ℓ ∈ ℤ,

t\frac{∂}{∂t}(q_{ij}t^{k}t̄^{ℓ}) = q_{ij}^{S_{k}}t^{k}t̄^{ℓ} and t̄\frac{∂}{∂t̄}(q_{ij}t^{k}t̄^{ℓ}) = q_{ij}^{S_{ℓ}}t^{k}t̄^{ℓ}.

Proof. Straightforward. □

Now we state the main result of this section:

Theorem 4.4. For any degeneration of polarized Hodge structures of weight n with h^{n,0} = 1, the induced Weil-Petersson metric g_{W P} has finite volume. For one parameter degenerations with finite Weil-Petersson distance, if Na_{1} ≠ 0 then g_{W P} blows up to +∞ with order c| log |t|^{2}|^{k} and the curvature form K_{W P} = Kdt ∧ dt̄ blows up to +∞ with order

\frac{kdt ∧ dt̄}{|t|^{2}| log |t|^{2}|^{2}} ,

where k = max\{ i | N^{i}S_{1}a_{1} ≠ 0 \} ≤ n − 1, c = (k!)^{−1}|Q̃(Ñ^{k}S_{1}a_{1}, \overline{S_{1}a_{1}})| > 0.

Remark 4.5. The statement about finite volume is a standard fact in Hodge theory. By the nilpotent orbit theorem, the Hodge metrics on the Hodge bundles degenerate at most logarithmically. So g_{H}, and hence g_{W P}, has finite volume. In fact, in view of (4.1), the method of the following proof implies that g_{H} is asymptotic to the Poincaré metric of the punctured disk along transversal directions toward boundary divisors of the base space.

Proof. Pick a(t) and Ω(t) as before, (i.e. Na_{0} = 0, q_{00} = 1, q_{0i} = 0 for i ≥ 1 and all other q_{ij} are functions of log |t|). We have

q(t) = Q̃(Ω(t), Ω̄(t))
= 1 + q_{11}tt̄+ (q_{12}tt̄^{2} + q_{21}t^{2}t̄) + (q_{22}t^{2}t̄^{2} + q_{31}t^{3}t̄+ q_{13}tt̄^{3}) + ⋯

Applying Lemma 4.3, we can compute

\frac{∂q}{∂t} \frac{∂q}{∂t̄} − q \frac{∂^{2}q}{∂t∂t̄} = q_{11}^{S_{1}} t̄q_{11}^{S_{1}}t − (1 + q_{11}tt̄)(q_{11}^{S_{1}S_{1}} + q_{12}^{S_{1}S_{2}} t̄+ q_{21}^{S_{2}S_{1}}t
+ q_{22}^{S_{2}S_{2}}tt̄+ q_{31}^{S_{3}S_{1}}t^{2} + q_{13}^{S_{1}S_{3}} t̄^{2}) + ⋯

(Since we will assume that Na_{1} ≠ 0, q_{11} will be the only term needed. However we have calculated more terms in order for later use.) So the metric ds^{2} = g|dt|^{2} is given by

g = −∂_{t}∂t̄log q = q^{−2}(∂_{t}q∂t̄q − q∂_{t}∂t̄q)
= −q_{11}^{S_{1}S_{1}} − (q_{12}^{S_{1}S_{2}}t + q_{21}^{S_{2}S_{1}} t̄)

+ (q_{11}q_{11}^{S_{1}S_{1}} + |q_{11}^{S_{1}}|^{2} − q_{22}^{S_{2}S_{2}})tt̄− q_{31}^{S_{3}S_{1}}t^{2} − q_{13}^{S_{1}S_{3}} t̄^{2} + ⋯ (4.3)

As t → 0, the first term determines the behavior of g. If Na_{1} ≠ 0 (so a_{1} ≠ 0), let ξ = S_{1}a_{1} ∈ F_{∞}^{n−1} (by Lemma 4.2) and let k = max{ i | N^{i}ξ ≠ 0 }. (k ≤ n−1 simply because Gr_{2n}^{W} = 0.) Then the highest order term of −q_{11}^{S_{1}S_{1}}, with respect to log |t|^{2}, is given by (for this purpose we can ignore all operators S_{i} with i ≠ 0)

(4.4)  −\frac{1}{k!}( log |t|^{2})^{k}Q̃(Ñ^{k}ξ, ξ̄).

This term has nontrivial coefficient and is in fact positive. This follows from the fact that Q̃ polarizes the limiting mixed Hodge structures. Hence g blows up with the expected order, with c = (k!)^{−1}|Q̃(Ñ^{k}S_{1}a_{1}, \overline{S_{1}a_{1}})|. For the curvature form Kdt ∧ dt̄ = −∂ ¯∂log g, the most singular term of K is given by

( − q_{11}^{S_{1}S_{1}})^{−2}[( − q_{11}^{S_{1}S_{1}S_{0}}t^{−1})(q_{11}^{S_{1}S_{1}S_{0}} t̄^{−1}) − ( − q_{11}^{S_{1}S_{1}})( − q_{11}^{S_{1}S_{1}S_{0}S_{0}}t^{−1}t̄^{−1})]

= ( − q_{11}^{S_{1}S_{1}})^{−2}[|q_{11}^{S_{1}S_{1}S_{0}}|^{2} − q_{11}^{S_{1}S_{1}}q_{11}^{S_{1}S_{1}S_{0}S_{0}}]|t|^{−2}.

We need to show that this term is nontrivial. As before, we may ignore all S_{i} with i ≠ 0. So the highest order terms of (|q_{11}^{S_{1}S_{1}S_{0}}|^{2} − q_{11}^{S_{1}S_{1}}q_{11}^{S_{1}S_{1}S_{0}S_{0}}) are

\frac{1}{(k − 1)!}( log |t|^{2})^{2(k−1)}|Q̃(Ñ^{k}ξ, ξ̄)|^{2} − \frac{1}{k!(k − 2)!}( log |t|^{2})^{k+(k−2)}|Q̃(Ñ^{k}ξ, ξ̄)|^{2}.

It is clear that the coefficient \frac{1}{(k−1)!} − \frac{1}{k!(k−2)!} = \frac{1}{k!(k−1)!} > 0. Taking into account the order of (−q_{11}^{S_{1}S_{1}})^{−2} given by (4.4) shows that K blows up to +∞ with the expected order k|t|^{−2}| log |t|^{2}|^{−2}. □

Question 4.6. In the original smooth case in §2, a_{1} corresponds to the Kodaira-Spencer class of the variation, so we know that a_{1} ≠ 0. For the degenerate case, what is the geometric meaning of a_{1}? Is it some kind of Kodaira-Spencer class for singular varieties?

Remark 4.7. Assume that a_{1} ≠ 0. If Na_{1} = 0, then by Lemma 4.2, a_{1} ∈ H_{∞}^{n−1,1} and

−q_{11}^{S_{1}S_{1}} = −Q̃(e^{log |t|^{2}Ñ}S_{1}a_{1}, \overline{S_{1}a_{1}}) ≡ −Q̃(a_{1}, ā_{1}) > 0.

From (4.3), g_{W P} has a non-degenerate continuous extension over t = 0. In the case n = 3, if the moduli is one-dimensional (h^{2,1} = 1), then a_{1} ≠ 0 and Na_{1} = 0 imply that N ≡ 0. That is, the variation of Hodge structures does not degenerate at all. More generally, if we have a positive answer to Question 4.6, then we also have a similar statement for multi-dimensional moduli. Namely, N_{i}a_{0} = 0, N_{i}a_{1,j} = 0 for all i, j implies that N_{i} ≡ 0 for all i, where N_{i}’s are the local monodromies, a_{1,j}’s are coefficients of the linear terms in a(t).

We conclude this section by a partial result:

Proposition 4.8. Assume that a_{1} ≠ 0, Na_{1} = 0 (so g_{W P} is continuous over t = 0). If Na_{2} = 0 then the curvature tensor has a continuous extension over t = 0. The converse is true for n = 3. If the curvature tensor does not extend continuously over t = 0, it has a logarithmic blowing-up.

Proof. Following Remark 4.7 and Lemma 2.3, we may assume that −q_{11} = 1, q_{12} = 0 and q_{1j}’s are constants for j ≥ 2. So among the degree two terms of g, only the tt̄ term contributes to the curvature. Namely from (4.3),

g = 1 + (2 − q_{22}^{S_{2}S_{2}})tt̄+ · · · .

Again we are in the “normal coordinates”, so

K = −2 + q_{22}^{S_{2}S_{2}S_{1}S_{1}} + · · · .

It is clear that Na_{2} = 0 implies that q_{22}^{S_{2}S_{2}S_{1}S_{1}} is a constant (= 4Q̃(a_{2}, ā_{2})). We will show that the converse is true if n = 3. We may assume that a_{2} ≠ 0. By Lemma 4.2 we have a_{1} ∈ H_{∞}^{2,1} and S_{1}S_{2}a_{2} ∈ F_{∞}^{1}, so we may write (from the Hodge diamond) S_{1}S_{2}a_{2} = λa_{1} + α + β with α ∈ H_{∞}^{2,2} and β ∈ H_{∞}^{1,1}. Now the constancy of q_{22}^{S_{2}S_{2}S_{1}S_{1}} implies that

Q̃(N(λa_{1} + α + β), \overline{λa_{1} + α + β}) = 0

as it is the highest order term (N^{2} ≡ 0). So the fact Na_{1} = 0 and Nβ = 0 implies that Q̃(Nα, ᾱ) = 0. This in turn forces α = 0 by the polarization condition. So S_{1}S_{2}a_{2} = λa_{1} + β, and S_{1}S_{2}Na_{2} = λNa_{1} + Nβ = 0. That is, Na_{2} = 0. The remaining statement about the logarithmic blowing-up is clear. □

5. Concluding Remarks

Based on Yau’s solution to the Calabi Conjecture [17], the existence of the coarse moduli spaces of polarized Calabi-Yau manifolds in the category of separated analytic spaces was proved by Schumacher [8] in the 80’s. Combined with the Bogomolov-Tian-Todorov theorem [13], these moduli spaces are smooth Kähler orbifolds equipped with the Weil-Petersson metrics. In the algebraic category, the coarse moduli spaces can also be constructed in the category of Moishezon spaces. Moreover, for polarized (projective) Calabi-Yau manifolds, the quasi-projectivity of such moduli spaces has been proved by Viehweg [14] in late 80’s. From the analytic viewpoint, there is also a theory originated from Siu and Yau [11] dealing with the projective compactification problem for complete Kähler manifolds with finite volume. Results of Mok, Zhong [5] and Yeung [18] say that a sufficient condition is the negativity of Ricci curvature and the boundedness of sectional curvature. In general, the Weil-Petersson metric does not satisfy these conditions. This leads to some puzzles since the ample line bundle constructed by Viehweg ([14], Corollary 7.22) seems to indicate that ω_{W P} will play important role in the compactification problem. Since the curvature tends to be negative in the infinite distance boundaries, the puzzle occurs only at the finite distance part. This leads to two different aspects: The first one is the geometrical metric completion problem. In [15], it is proposed that degenerations of Calabi-Yau manifolds with finite Weil-Petersson distance should correspond to degenerations with at most canonical singularities in a suitable birational model. Now this is known to follow from the minimal model conjecture in higher dimensions [16]. With this admitted, one may then go ahead to analyze the structure of these completed spaces. Are they quasi-affine varieties? Another aspect is the usage of Hodge metric. Näively, since the Ricci curvature of the Weil-Petersson metric has a lower bound −(d + 1)g_{W P} and blows up to ∞ at some finite distance boundary points, for any k > 0 one may pull these points out to infinity by considering the following new Kähler metric

g̃_{ij̄} = (d + 1 + k)g_{ij̄} + R_{ij̄} = kg_{ij̄} + ∑_{k,ℓ} g^{kℓ̄} \frac{Q̃(σ_{i}σ_{k}Ω, \overline{σ_{j}σ_{ℓ}Ω})}{Q̃(Ω, Ω̄)}.

When the blowing-up is faster than logarithmic growth (e.g. Na_{1} ≠ 0), g̃ is then complete (at these boundaries). Otherwise one may need to repeat this process. This inductive structure is implicit in Theorem 4.1 and 4.4 and is explicitly expressed in Theorem 3.2 in the case n = 3, k = 2. It suggests that the resulting metric will be quasi-isometric to the Hodge metric. If we start with the Hodge metric directly, the coarse moduli spaces being Moishezon allows us to assume that the boundary has a local model as normal crossing divisors. Then a similar asymptotic analysis as in §4 implies that the metric behaves like the Poincaré metric in the transversal direction toward the codimension one boundaries (i.e. points with N ≢ 0 where N is the local monodromy). In particular, it admits bounded sectional curvature. The main problem here is the higher codimensional boundaries. In this direction, we should mention that the negativity of the Ricci curvature for Hodge metrics has recently been proved by Lu [3]. We expect that the Hodge metric would eventually provide projective compactifications of the moduli spaces through the recipe of [5], [18].

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Chin-Lung Wang Mathematics Division National Center for Theoretic Sciences National Tsing-Hua University Hsinchu, Taiwan dragon@math.cts.nthu.edu.tw