The classical Weil-Petersson metric on the Teichm¨uller space of compact Riemann surfaces is a K¨ahler metric which is complete only in the case of elliptic curves [Wo]. It has a natural generalization to the deformation spaces of higher dimansional polarized K¨ahler-Einstein manifolds. It is still K¨ahler.
Moreover, in the case of abelian varieties and K3 surfaces, the Weil-Petersson metric turns out to be equal to the Bergman metric of the Hermitian symmetric period domain, hence is in fact “complete” K¨ahler-Einstein [Sc].
The completeness is an important property for differential geometric rea-son. Motivated by the above examples, one may naively think that the com-pleteness of the Weil-Petersson metric still holds true for general Calabi-Yau manifolds. However, explicit calculation done by physicists (eg. Candelas et al.
[CGH] for some special nodal degenerations of Calabi-Yau 3-folds) indicated that this may not always be the case.
Naturally, we need to clarify what do we actually mean that the metric is complete or incomplete. This depends on how we define the “moduli space”, which is already very interesting in the case of K3 surfaces. We will gradually explain what is our understanding of this problem. And it would then become clear that the Weil-Petersson metric is in general incomplete if one sticks on
“moduli” of smooth varieties.
9.1. The Weil-Petersson metric. For a given family of polarized K¨ahler manifolds X → S with K¨ahler metrics g(s) on Xs, one can define a possibly degenerate hermitian metric G on S as follows: at s ∈ S with fiber X = Xs, we consider the Kodaira-Spencer map ρ : TS,s → H1(X, TX) ∼=
When X → S is a polarized K¨ahler-Einstein family and ρ is injective, GW P :=
G is called the Weil-Petersson metric on S.
When X is a Calabi-Yau manifold, we have Yau’s solution to Calabi’s conjecture [Ya] that X has an unique Ricci flat metric in each K¨ahler class and the Bogomolov-Tian-Todorov theorem that the Kuranishi space of X is unobstructed [Ti, To].
Let X → S be a maximal subfamily of the Kuranishi family with a fixed polarization class [ω], then ρ is clearly injective. Let g(s) be the unique Ricci
flat metric in the given polarization. Using the fact that the global holomorphic n-form Ω(s) is flat with respect to g(s), it was shown in [Ti, To] that
(9.3) GW P(v, w) = Q(C(i(v)Ω), i(w)Ω) Q(CΩ, ¯Ω) ,
where H1(X, TX) → Hom(Hn,0, Hn−1,1) ∼= Hn−1,1 via the interior product v 7→ i(v)Ω is the well-known isomorphism. The tangent space TS is mapped to Pn−1,1isomorphically and hence leads to the fact that the n-th flag period map is an local embedding. So the Weil-Petersson metric is induced from the Hodge metric on the n-th piece of the horizontal tangent bundle. For convienence, let’s write eQ =√
where ωW P denotes the fundamental real 2-form of GW P (this formula shows in particular that ωW P is independent of the polarization). The proof is essentially part of Griffiths’ curvature calculation [Gr], hence is purely Hodge theoretic.
So we can extend the definition of GW P to polarized VHS over S with hn,0= 1 by (9.4), although it is only semi-positive. Since it makes sense to talk about geodesics and distances, we will still call it the Weil-Petersson metric.
Clearly, our aim is to characterize all finite distance degenerations and then to describe the possible picture of the completion. We get strong evidence that it is closely related to the minimal model program in birational geome-try. However, the results we can rigorously proved so far are not enough to answer the full question. We do formulate a conjecture in §10 to complete our discussion here.
The result is this section are mostly exercises in Hodge theory. We will recall what we need. Details can be found in [Gr, GS, Cl, Sc].
9.5. Schmid’s theory on limiting MHS. Let D be the period domain for certain polarized Hodge structures and let ˇD be its compact dual. For a polarized VHS φ : ∆× → hT i\D; the map φ lifts to the upper half plane
∆×. The very first part of Schmid’s “nilpotent orbit theorem” says that α(t)
extends holomorphically over t = 0. The special value F∞ := α(0) is called the limiting filtration and is in general outside D. However, the nilpotent operator N uniquely defines a “monodromy weight filtration” on V : 0 ⊂ W0⊂ W1 ⊂ · · · ⊂ W2m−1 ⊂ W2m = V such that N (Wk) ⊂ Wk−2 and induces an isomorphism
(9.7) N`: GWm+`∼= GWm−`,
where GWk := Wk/Wk−1 is the graded piece. These two filtrations F∞p and Wk together define a “polarized mixed Hodge structure” on V in the following sense: the induced Hodge filtration
(9.8) F∞pGWk := F∞p ∩ Wk/F∞p ∩ Wk−1, p = 0, . . . , m
defines a (pure) Hodge structure of weight k on GWk . The operator N acts on them as a morphism of MHS’s of type (−1, −1). That is, N (F∞pGWk ) ⊂ F∞p−1GWk−2. Moreover, for ` ≥ 0, the primitive part Pm+`W := ker N`+1 ⊂ GWm+`
is polarized by Q(·, N`¯·).
When φ comes from geometric situations, namely the period map of a degeneration X → ∆, by adding together with the non-primitive part, the total cohomology Hm(Xt, C) still admits non-polarized MHS.
We now give the basic criterion for finite Weil-Petersson distance in the case of one parameter degenerations of polarized Hodge structures φ : ∆× → hT i\D with hn,0= 1:
Theorem 9.9. The center of a degeneration of polarized Hodge structures of weight n with Fn ∼= C has finite Weil-Petersson distance if and only if N F∞n = 0.
Proof. Let Φ : H → D be the lifting. To start the computation, all we need is a good choice of a holomorphic section Ω of Hn,0. Let pn: D → P(V ) be the projection to the Fn part. we have Φn(z) = (ezNα(t))n = ezNαn(t).
Here ∗n := pn(∗) ∈ P(V ) means the n-th flag. Near t = 0, we can consider a vector (local homogeneous coordinates) representation a of αn in V . Then a(t) = a0+ a1t + · · · is holomorphic in t. We have orrespondingly
(9.10) A(z) = a0+ a1e2π
√−1z+ a2e4π
√−1z+ · · · .
The crucial point here is that the function e2π
√−1z = e2π
√−1xe−2πy has the property that all the partial derivatives in x and y decay to 0 exponentially
as y → ∞, with rate of decay independent of x. For ease of notation, let h be the function class satisfying the above property and h the corresponding function class with values in V .
Now let Ω(z) = ezNA(z). This is the desired section because vector rep-resentations correspond to sections of the tautological line budle of Pn which pull back to Hn,0by Φ. So the K¨ahler form ωW P of the induced Weil-Petersson metric GW P on H is given by where p(y) is a polynomial in y with
(9.13) d = deg p(y) = max{ ` | N`α06= 0 }.
This a consequence of the polarization condition for the mixed Hodge structure (9.5) and the fact that a0∈ Gn+d. So
Return to the geometric situation, namely the semi-stable degeneration of polarized Calabi-Yau manifolds. As a simple application of the Clemens-Schmid exact sequence [Cl], we have
Theorem 9.15. The central fiber X has finite Weil-Petersson distance if and only if some irreducible component Xi ⊂ X has Hn,0 6= 0. This is equivalent to that there is exact one component with hn,0= 1.
Proof. By the results of Schmid in (9.5), F∞ and N defines a MHS on Hn(Xt) for a reference fiber Xt with t 6= 0. It follows from (9.7) that (ker N ) ∩ F∞n ≡ GWn F∞n. So N F∞n = 0 if and only if F∞n = GWn F∞n.
Recall that the “geometric genus formula” [Cl] says that
(9.16) hn,0(Xt) ≥X
ihn,0(Xi),
and the RHS corresponds to all the invariant cycles in F∞n, that is, (ker N )∩F∞n. Since the LHS of (9.16) corresponds to F∞n, the eqality holds if and only if F∞n = (ker N ) ∩ F∞n = GWn F∞n, that is, if and only if N F∞n = 0.
In our case, Theorem 9.9 says that finite distance is equivalent to N F∞n = 0. Since hn,0(Xt) = 1, this is equivalent to that there exist some (and so at most one) component with hn,06= 0 (and so in fact it must be 1). The proof is now complete. Q.E.D.
As a corollary, we deduce the following theorem which we believe to be very close to the final answer of the completion problem:
Theorem 9.17. Let X be a Calabi-Yau varieties which admits a smooth-ing to Calabi-Yau manifolds. If X has only canonical ssmooth-ingularities then X has finite Weil-Petersson distance along the base.
Proof. For any resolution f : eX → X, we have as in the above that Hn,0( eX, C) = Γ( eX, K
Xe) = Γ( eX,P eiEi) (notice that ei’s are integers). Since Ei’s are exceptional, it follows easily that Hn,0( eX, C) 6= 0 precisely when X has at most canonical singularities.
Now let X → ∆ be a smoothing of X. Take a semi-stable reduction of it, then there is a component in the central fiber of the semi-stable reduction which corresponds to the proper transform of X. Then it has hn,0 = 1. Now apply Theorem 9.15 and notice that finite distance in a special smoothing implies finite distance in the whole smoothing component. Q.E.D.
Example 9.18. According to [Re], hypersurface singularities of monomial typeP
ixdi = 0 is canonical if and only ifP
i1/di> 1. In the three dimensional case, the finiteness of the Weil-Petersson distance with singularities of this type were known to Candelas et al. [CGH] via direct calculations. Theorem 9.17 seems to indicate that canonical singularities may also play significant role in certain physics problems.