Mathematical Research Letter 4 (1997), 157-171. †
ON THE INCOMPLETENESS OF THE WEIL-PETERSSON METRIC ALONG
DEGENERATIONS OF CALABI-YAU MANIFOLDS Chin-Lung Wang
Introduction
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, and in the case of abelian varieties and K3 surfaces, the Weil-Petersson metric turns out to coincide with 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 geomet- ric reason. Motivated by the above examples, one may naively think that the completeness of the Weil-Petersson metric still holds true for general Calabi-Yau manifolds (compact K¨ahler manifolds with trivial canonical bundle). However, explicit calculation done by physicists (eg.
Candelas et al. [Ca] for some special nodal degenerations of Calabi-Yau 3-folds) indicated that this may not always be the case.
The notion of completeness depends on the precise definition the
“moduli space”. However, through our analysis, it would become clear that the Weil-Petersson metric is in general incomplete if one sticks
† Some inaccuracies in §3 has been corrected in this reproduced version.
on “moduli” of smooth varieties. In fact, our aim in this paper is to characterize all finite distance degenerations and then to describe the possible picture of the completion. It is found to be closely related to the minimal model program in birational geometry. The results proved here could be regarded as a first attempt toward the completion problem.
There are two parts of this paper. In the first part, §1 – §2, we start with Tian’s description of the Weil-Petersson metric as the Chern form of the Hodge bundle Fn. This fits naturally into the framework of variation of Hodge structures and the Weil-Petersson can be formally defined in this setting. By applying Schmid’s theory of limiting mixed Hodge structures, we obtain in (1.1) our first Hodge-theoretic criterion:
Theorem A. 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.
Here N is the nilpotent monodromy and F∞n is the limiting filtra- tion. In §2, we return to the geometric situation, namely the semi-stable degeneration of polarized Calabi-Yau manifolds. As a simple applica- tion of the Clemens-Schmid exact sequence [C], we get in (2.5) the
Theorem B. The central fiber X has finite Weil-Petersson dis- tance if and only if some irreducible component Xi⊂ X has Hn,06= 0.
This is equivalent to that there is exact one component with hn,0= 1.
This theorem is also claimed in a recent preprint of Hayakawa [H].
However, the proof given there seems to be incomplete and too compli- cate. As a corollary, we deduce in (2.10) the following theorem which we believe to be very close to the final picture of the completion problem:
Theorem C. Let X be a Calabi-Yau varieties which admits a smoothing to Calabi-Yau manifolds. If X has only canonical singu- larities then X has finite Weil-Petersson distance along the base.
In the second part, §3 – §4, we deal with a refined question: up to a base change, is the finite distance degeneration birationally equiv- alent to a smooth family? This is exactly what happens in the case of K3 surfaces [Ku] — this is also one reason that one usually regards the Weil-Petersson metric on the “moduli space” of K3 surfaces to be complete. In general, is there any “essential non-trivial finite distance degeneration”? The simplest examples would be those degenerations with monodromy not of finite order. In §3, we show that it is the case
for nodal degenerations of 3-folds.
In §4, which is perhaps the most technical part of this work, we study the “expected most general cases” of finite distance degenerations, i.e. degenerations with canonical singularities, via various results in 3- fold birational geometry. The main result is:
Theorem D. Let X → ∆ be a projective smoothing of a terminal Gorenstein 3-fold X0 with KX0 nef. Then X → ∆ is not birational to a projective smooth family X0→ ∆ with Xt= Xt0 for t 6= 0.
This applies in particular to terminal degenerations of Calabi-Yau 3-folds (see (4.11)) and hence we conclude that smoothable Calabi-Yau 3-folds with nontrivial terminal singularities all provide essential finite distance points. Since some of these degenerations are known to have C∞ trivial monodromies, Theorem D then gives a negative answer to the so-called “filling-in problem” in dimension 3.
§0 Background Material
Here we briefly recall some basic definitions and well-known prop- erties about Hodge theory and the Weil-Petersson metrics that will be used in this paper. More details could be found in [G, GS, K, S, T1].
0.1. Polarized VHS and the period map. It is well known that from the theory of harmonic forms that the m-th primitive cohomology of a compact K¨ahler manifold (Xn, ω) admits a “polarized real Hodge structure” of weight m. That is, for V := PRm⊗ C ⊂ HCm, one has
(0.2) V =M
p+q=mPp,q, Pp,q = Pq,p.
Equivalently, this can be expressed in terms of the Hodge filtration F : V = F0⊃ F1 ⊃ · · · ⊃ Fm with Fp=L
i≥pPi,m−i. Moreover, for m ≤ n, the Hodge-Riemann bilinear form
(0.3) Q(u, v) := (−1)m(m−1)2 Z
X
u ∧ v ∧ ωn−m
polarizes V in the sense that F satisfies the Hodge-Riemann bilinear relations:
(0.4) I. Q(Fp, Fm+1−p) = 0 and II. Q(Cv, ¯v) > 0,
where C, the Weil operator, acts on Pp,q by multiplying√
−1p−q. Varying the above data (V, F, Q), a family of polarized K¨ahler man- ifolds X → S, ie. [ω(s)] is locally constant, gives rise to a “polarized VHS over S”: (V, F, ∇, Q) with ∇ the flat connection. This data sat- isfies the Griffiths transversality relation
(0.5) ∇Fp ⊂ Fp−1⊗ Ω1S.
It follows that ¯∂Fp ⊂ Fp and hence that Fp’s form holomorphic sub- bundles of the flat bundle V, which is polarized by the flat bilinear form Q. This can also be described in terms of the “period map”:
The period domain D is the classifying space of all Hodge filtra- tions F of V that are polarized by Q. We have D = G/K where G = Aut(PR, Q) and K the stabilizer of a point. It comes naturally with the tautological homogeneous vector bundle Fp ⊂ V := V × D.
The compact dual ˇD is the set of all the F ’s which satisfy only the first Hodge-Riemann relation. It contains D as an open subvariety.
The family X → S gives rise to the period map φ : S → Γ\D with Γ the representation of π1(S) in G. It is clear that φ∗Fp gives the holo- morphic vector bundle mentioned above and (0.5) translates into the horizontality of φ:
(0.6) dφ : TS →M
p+q=mHom(Pp,q, Pp−1,q+1) =: TDh, where TDh is called the horizontal tangent bundle.
We can formalize the above situation and define the polarized VHS as a locally liftable horizontal holomorphic map φ : S → Γ\D with Γ a representation of π1(S) in G ∩ Aut(HZ), where HZ is an integral lattice such that PR ⊂ HR. In the case Fn ∼= C, we will also consider the
“n-th flag period map” φn : S → Γ\P(V ) which in fact contains almost all the information that we will need in this paper.
0.7. Semi-stable degenerations and the monodromies. We are interested in the case of a degeneration X → ∆ of polarized K¨ahler n-folds. By this we mean that X is a K¨ahler (n+1)-fold and X → ∆ is a proper flat holomorphic map with the general fiber Xt, t 6= 0, a smooth K¨ahler n-fold. Notice that the resulting family over the punctured disk has a polarization induced from the K¨ahler form on X .
X → ∆ is called semi-stable if X0 is a reduced divisor with normal crossings in X . By a theorem of Mumford [K], every degeneration has
a semi-stable reduction by a sequence of blow-ups and base-changes. In general, X → ∆ is called a degeneration of certain type if X0 has only singularities of that type. And by “X → ∆ is a smoothing of X0”, we will mean that X → ∆ is a proper flat family with smooth Xt for t 6= 0 but without assuming the complex space X to be smooth.
Now a generator of π1(∆×) ∼= Z induces the so called Picard- Lefschetz transformation – the monodromy T on HZm, which is known to be quasi-unipotent. Under the semi-stable asssumption, T will be unipotent and we will consider the associated nilpotent operator N :=
log T acting on HQm (and therefore on V ⊂ HCm since the polarization class is invariant under T ). The quasi-unipotent statement is also known to be true for any polarized VHS [S]. In this paper, we will usually assume that T is unipotent by allowing a base change implicitly.
0.8. Schmid’s theory on limiting MHS. For a polarized VHS φ : ∆× → hT i\D; the map φ lifts to the upper half plane Φ : H → D with the coordinates t ∈ ∆× and z ∈ H related by t = e2π√−1z. Set (0.9) A(z) = e−zNΦ(z) : H → ˇD,
(instead of D). Since A(z + 1) = A(z), A descends to a function α(t) on
∆×. 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
(0.10) 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
(0.11) 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, by adding together with the non-primitive part, the total cohomology Hm(Xt, C) still admits non-polarized MHS.
0.12. The Weil-Petersson metric. For a given family of po- larized 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) ∼= H0,1∂¯ (TX) into harmonic forms with respect to g(s); so for v, w ∈ Ts(S), we may define
(0.13) G(v, w) :=
Z
X
hρ(v), ρ(w)ig(s).
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 (1) Yau’s solution to Calabi’s conjecture [Y] that X has an unique Ricci flat metric in each K¨ahler class and (2) the Bogomolov-Tian-Todorov theorem that the Kuranishi space of X is unobstructed [T1, 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 [T1, To] that
(0.14) 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−1,1 isomorphically 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 =√
−1nQ (= Q(C·,¯·) on Hn,0= Pn,0). Tian observed that eQ is a K¨ahler potential of GW P, that is,
(0.15) ωW P =
√−1
2 RiceQ(Hn,0) = −
√−1
2 ∂ ¯∂ log 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 [G], hence is purely Hodge theoretic. So we can extend the definition of GW P to polarized VHS over S with hn,0= 1 by (0.15), 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.
§1 Hodge Theoretic Criterion for Finite Distance Points
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 1.1. 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. We will keep the notation from §0. 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 correspondingly (1.2) 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 representations correspond to sections of the tautological line budle of Pn which pull back to Hn,0 by Φ. So the K¨ahler form ωW P of the induced Weil-Petersson metric GW P on H is given by
(1.3) ωW P = −
√−1
2 ∂ ¯∂ log eQ(ezNA(z), ezN¯ A(z)).
Since we are in one complex variable, write GW P = G|dz|2, then G = −(1/4)4 log eQ. We have Q(T u, T v) = Q(u, v), it follows easily that Q(N u, v) = −Q(u, N v) and Q(ezNu, v) = Q(u, e−zNv). Since A = a0+ h, we have
(1.4)
Q(ee zNA, ezN¯ A) = e¯ Q(ezNa0, ezN¯ a¯0) + h
= eQ(e2√−1yNa0, ¯a0) + h = p(y) + h, where p(y) is a polynomial in y with
(1.5) d = deg p(y) = max{ ` | N`α06= 0 }.
This a consequence of the polarization condition for the mixed Hodge structure (0.8) and the fact that a0∈ Gn+d. So
(1.6)
4G = (p0+ h)2− (p + h)(p00+ h)
(p + h)2 = (p02− pp00) + h p2+ h
∼ p02− pp00
p2 + h ∼ d2− d(d − 1)
y2 + h = d y2 + h.
Here we have used the fact that p−2h ∈ h. Obviously, if N F∞n = 0 then d = 0 and G = h, so R∞
t
√G |dz| < ∞ for some curve (e.g. x = c).
When N F∞n 6= 0 we have d ≥ 1 and for y large enough we can make h < 1/y3uniformly in x, then clearly R∞
t
√G |dz| ∼ 2 log y |∞t = ∞ for any path with y → ∞. Q.E.D.
Remark 1.7. From the proof of (1.1), we know that in the case of infinite distance, the Weil-Petersson metric is exponentially asymptotic to a scaling of the Poincar´e metric. In particular, the (holomorphic sectional) curvature approaches to a negative constant when t → 0.
This is exactly the situation what we are familiar with for the moduli space of elliptic curves.
§2 Geometric Criterion for Finite Distance Points
For a semi-stable degeneration, there is a well-known procedure to relate the limiting MHS and Deligne’s canonical MHS on the singular cohomology of the central fiber, namely the Clemens-Schmid exact se- quence, which generalizes the classical Picard-Lefschetz theory (cf. §3).
Let’s briefly recall the constructions (see [C, GS] for more details). Let X = ∪iXi be a simple normal crossing variety, for I = {i0, · · · , ip}, XI := Xi0 ∩ · · · ∩ Xip. Also let X[p] be the disjoint union of all XI with |I| = p + 1. There is a spectral sequence which computes H∗(X) = H∗(X, C) with Ep,q0 = Ωq(X[p]), d0 := d (the exterior dif- ferentiation of forms) and d1 := δ: the restriction operator of forms defined by
(2.1) (δφ)(Xi0···ip+1) :=Xp+1
j=0(−1)jφ(Xi0··· ˆij···ip+1)¯
¯Xi0···ip+1.
Clearly Ep,q1 = Hq(X[p]) and the Ep,q2 term is computed from (2.2) Hq(X[p−1])→ Hδ q(X[p])→ Hδ q(X[p+1]),
where the δ’s respect Hodge structures. Moreover, it degenerates at E2. The weight filtration for the resulting MHS on Hm(X) is W` :=
L
s≤`Em−s,s2 , and the Hodge filtration is the usual one for each factor induced from E1. Notice that in contrast to the limiting MHS, the canonical MHS has terms G`Hm(X) only for 0 ≤ ` ≤ m.
The Clemens-Schmid exact sequence for a semi-stable degeneration
(2.3)
smooth Xt ⊂ X ⊃ X0 (= X = ∪iXi)
↓ ↓ ↓
(0 6=) t ∈ ∆ 3 0
is an exact sequence of MHS’s:
(2.4)
· · · → H2n+2−m(X0)→ Hj m(X0)→ Hi m N→ Hm k→ H2n−m(X0) → · · · Notice that the inclusion X0 ⊂ X is a homotopy equivalence, j is induced by inclusion and duality and i is induced by the inclusion Xt ⊂ X ∼ X0. Also Hm denotes the cohomology for the general fiber Xt and N is the nilpotent monodromy oprator. Moreover, this exact sequence is compatible with the MHS’s with types of morphisms (n + 1, n + 1), (0, 0), (−1, −1) and (−n, −n) respectively, where type (p, q) means F∗G∗ → F∗+pG∗+2q. Here MHS for homology is defined by duality: G−`Hq := G`(Hq)∗ and F−pG−`Hq := Ann(Fp+1G`Hq).
When the degeneration of Hodge structures in (1.1) comes from a semi-stable degeneration of Calabi-Yau manifolds X → ∆, we have:
Theorem 2.5. The central fiber X = X0has finite Weil-Petersson distance if and only if some irreducible component Xi ⊂ X has Hn,06=
0. This is equivalent to that there is exact one component with hn,0= 1.
Proof. By the results of Schmid in (0.8), F∞ and N defines a MHS on Hn(Xt) for a reference fiber Xt with t 6= 0. It follows from (0.10) 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” [C] says that
(2.6) 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 (2.6) has the same dimension as 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 1.1 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,0 6= 0 (and so in fact it must be 1). The proof is now complete.
For the reader’s convienence, we sketch the well-known argument for the geometric genus formula. Apply the Clemens-Schmid exact se- quences to FnGnHn, we get
(2.7) → F−1G−n−2Hn+2(X0) → FnGnHn(X0) → F∞nGWn Hn N→ 0 We know by definition that Gn+2Hn+2(X0) = E0,n+22 = ker δ with δ : Hn+2(X[0]) → Hn+2(X[1]) and that F2Hn+2(X0) = Hn+2(X0).
From this we conclude that F2(ker δ) ≡ ker δ. So
(2.8) F−1G−n−2Hn+2(X0) = Ann(F2Gn+2Hn+2(X0)) = 0, and (2.7) implies that FnGnHn(X0) ∼= F∞nGWn Hn.
Now E0,n2 is computed from 0 → E0,n1 → E1,n1 ≡ Hn(X[1]), and the Fn part of the right term is zero. So FnGnHn(X0) = FnE0,n2 = FnE0,n1 = FnHn(X[0]) = Hn,0(X[0]).
The resulting isomorphism F∞nGWn Hn ∼= Hn,0(X[0]) clearly gives what we want. Q.E.D.
Remark 2.9. Both (1.1) and (2.5) are stated in the one parameter case, but the Weil-Petersson metric distance should be evaluated in
the corresponding smoothing component of the central fiber, which is in general of many dimensions. However, finite distance in a special direction implies finite distance in the whole component, so (2.5) indeed provides a suifficient condition for the existence of finite distance points.
The converse is not obvious in case the base dimension is bigger than one. We plan to discuss this issue in a forthcoming work.
Now we apply (2.5) to smoothable singular Calabi-Yau varieties.
A Calabi-Yau variety is by definition a normal projective variety with trivial canonical (cartier) divisor. Recall that a normal variety X is has canonical (resp. terminal) singularities if KX is Q-Cartier and there is a (equivalently for any) resolution f : eX → X such that K eX =Q f∗KX +P
eiEi with ei ≥ 0 (resp. ei > 0), where Ei’s are the exceptional divisors. Canonical singularities in dimension two are exactly RDP’s (also called Du Val, A-D-E, Kleinian singularities). Ter- minal singularities must be of codimension three. In dimension three, they are completely classified by Reid and Mori [R].
Canonical singularities play an important role in birational geome- try. In the case of Calabi-Yau 3-folds, birational primitive contractions [W] will create at most canonical singularities. It has been conjectured that the moduli spaces of Calabi-Yau 3-folds (with h1(O) = 0) of dif- ferent topological types can be “connected” by performing primitive contractions and smoothings. This statement is also known to be in- terestng from the point of view of physics. Our next result implies that this can happen only within finite Weil-Petersson distance.
Theorem 2.10. Let X be a Calabi-Yau varieties which admits a smoothing to Calabi-Yau manifolds. If X has only canonical singulari- ties 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 eX) = 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 2.1 and notice that finite distance in a special smoothing implies finite distance in the whole smoothing component. Q.E.D.
Example 2.11. According to [R], hypersurface singularities of monomial type P
ixdi = 0 is canonical if and only if P
i1/di > 1.
In the three dimensional case, the finiteness of the Weil-Petersson dis- tance with singularities of this type were known to Candelas et al. [Ca]
via direct calculations. Theorem 2.9 seems to indicate that canonical singularities may also play significant role in certain physics problems.
Question 2.12. Is the converse of (2.10) true? More precisely, if a degeneration of Calabi-Yau manifolds has finite Weil-Petersson dis- tance, is that true this degeneration is birational to another degenera- tion such that the central fiber is an irreducible Calabi-Yau variety with only canonical singularities? This would be an important step toward the completion program.
Remark 2.13. The problem of whether a singular Calabi-Yau variety X with canonical singularities has a flat deformation into non- singular Calabi-Yau’s Xt has already been studied extensively in di- mension 3. The first step was taken by Friedman [F1] in the case of ODP’s (see also Tian [T2] and [F2]). Recent preprints of Namikawa- Steenbrink and M. Gross have provided quite satisfactory results in this direction. These developments are closely related to Z. Ran’s extension of the Bogomolov-Tian-Todorov theorem to the singular case.
Remark 2.14. In fact, all the statements in §1 and §2 are true in the following more general setting. Given a smooth polarized fam- ily of varieties X → S parametrized by a smooth base S and with h0(Xs, KXs) ≥ 2, we may consider the semi-definite metric ω on S given by the Chern form of det Fn, that is, det f∗KX /S. Using this metric, the main results (1.1), (2.5) and (2.10) generalize immediately. In fact, the same proofs work except notationally more complicated. However, even in the K¨ahler-Einstein case with KX ample, this metric is not the Weil-Petersson metric defined in (0.12). There is a complicated relation between the two in terms of certain “Quillen metrics”.
§3 Incompleteness I: Nontrivial Monodromy
In §3 and §4, we work in the projective category. For Calabi-Yau varieties, it will be assumed that h1(O) = 0. In particular, we have excluded the case of abelian varieties.
There exists smoothable Calabi-Yau 3-folds with canonical singu-
larities such that the smoothing comes from a birational contraction of a smooth family over the disk, which induces isomorphisms out- side the puncture. These examples are due to Wilson [W] in his deep study of the jumping phenomenon of K¨ahler cones. More precisely, his proposition 4.4 says that the “type III primitive contraction” with the exceptional divisor a quasi-ruled surface over an elliptic curve provides such an example.
In the surface case, these correspond to smoothings of K3 surfaces with RDP’s. By Kulikov’s classification theorem [Ku] and the fact that
“N F∞2 = 0 implies N = 0”, they are birational to smooth families (up to a base change).
For our purpose, the above examples should not be considered as incomplete points by the following reason: one can include these points by hand – just replace the degeneration by the smooth family by allowing the polarization line bundle to be only big and nef. In the case of K3 surfaces, an equivalent way is to add these points by allowing Ricci-flat orbifold metrics. In fact, this process leads to the (metric) completion of the K3 moduli! However, The situation changes if dim X ≥ 3. We will see that there are “nontrivial” examples. By this we mean a degeneration such that the complement of the central fiber can not be completed into a smooth family.
If the monodromy T is not of finite order (N 6= 0) then the degen- eration is clearly “nontrivial” in the above sense. In this connection, we mention the following classical result of Picard, Lefschetz and Poincar´e:
Theorem 3.1. For a nodal degeneration of smooth n-folds, the monodromy T is trivial except possibly in the middle dimensional coho- mology. In the middle dimensional case, we have that
I. N2= 0 if n is odd, and that II. T2= I (so N = 0) if n is even.
The standard proof of (3.1) is to write down the explicit formula of T in terms of the “vanishing cycles”. However, in order to see whether N 6= 0 in the odd case one needs to know whether the vanishing cycle represent nontrivial homology classes, and this is clearly not just a local problem near the singular points. (The fact that the vanishing cycle can be homologically trivial was kindly pointed out to me by J. de Jong.)
As an exercise, We will show that how Theorem 3.1 follows easily from the topological version of the Clemens-Schmid exact sequence and then make some remarks on it.
First of all, a semi-stable reduction can be obtained by first doing a degree 2 base change and then blowing up the ODP’s of the total space.
So X[0] is the union of n-quadrics and the proper transform X0 of the original central fiber, X[1]is the union of (n − 1)-quadrics and X[2]= ∅.
For m < n, we claim that for ` ≤ m − 1 , G`Hm = 0. Suppose it has been proved up to ` − 1, then
(3.2) G`−2n−2H2n+2−m(X0) → G`Hm(X0) → GW` Hm N→ 0.
If m − ` ≥ 2, since X[2] = ∅ we have that G`Hm(X0) = Em−`,`2 = 0. If m − 1 = `, since m < n we have E1,m−12 = coker δ with δ : Hm−1(X[0]) → Hm−1(X[1]), which is surjective by explicit cohomolo- gies of quadrics (or use hyperplane section theorem). This proves that G`Hm(X0) = 0 and so GW` Hm = 0 up to ` = m − 1. This means that the MHS is pure and so N = 0.
For Hn, the same argument shows that GW` Hn = 0 up to ` = n − 2, so N2 = 0. Since ` < n, G`−2n−2H2n+2−n(X0) is the dual of G2(n+1)−`H2(n+1)−n(X0), which is zero. So Gn−1Hn(X0) ∼= GWn−1Hn is given by coker δ with
(3.3) δ : Hn−1(X[0]) → Hn−1(X[1]).
Now the middle cohomology of an (n − 1)-quadric is zero if n is even, so N = 0 in this case. Q.E.D.
For the case n is odd, N 6= 0 if and only if δ is not surjective.
The middle cohomology has rank 2 for an even dimensional quadric.
The image of these n-quadrics under δ consist of suitable powers of the hyperplane class, which is also in the image of Hn−1(X0) if n ≥ 5 (because for any of these (n − 1)-quadrics E, E |E generates H2(E), which is only one dimensional). Therefore δ is surjective if and only if the induced map
(3.4) δ0 : Hn−1(X0) → Hn−1(X[1])
is surjective. That is, the surjectivity of (3.3) (or (3.4) for n ≥ 5) is equivalent to the triviality of monodromy N . It is immedeate from this that if the monodromy is trivial then the number of ODP’s of the central fiber X has an upper bound given by bn−1(X). However, it is not clear how to get anything more without specifying the varieties under consideration.
In the three dimensional case, there are some explicit computations done by Candelas et al. showing that certain nodal degenerations have indeed monodromies not of finite order. Hence one obtains nontrivial finite distance examples. A theoretic proof of this statement for any nodal degenerations turns out to be delicate (even for Calabi-Yau 3- folds). We will give a sketch of it by showing the existence of nontrivial vanishing cycles, following a suggestion by Mark Gross.
Let us assume that our 3-folds are all simply connected. First of all, a nodal threefold X0 always admits (not necessarily projective) small resolutions X → X0 with smooth rational curves X ⊃ Ci → pi ∈ X0 contracted to ODP’s. In the case of Calabi-Yau threefolds (Gorenstein threefolds with trivial canonical bundle and with h1(O) = 0), the existence of global smoothing X → ∆ of X0 forces that there are nontrivial relations of [Ci] ∈ H2(X) by Friedman’s result [F3, F4].
That is, the canonical map e : L
iZ[Ci] → H2(X, Z) has nontrivial kernel dimension s > 0. Consider the resulting surgery diagram:
(3.5)
X
↓
X0⊂ X ⊃ Xt
It has the following local description: let Vi3 pi be a contrctible neigh- borhood of an ODP, Vi0 ⊂ Xt be the smoothing of Vi and Ui ⊂ X be the inverse image of Vi. Then
I. Uiis a deformation retract neighborhood Ciand so has the homo- topy type of S2∼ D4× S2.
II. Vi0has the homotopy type of S3× D3. Where the sections σi∼ S3 are the so called vanishing cycles.
III. The surgery from X to Xt is induced from ∂(D4× S2) = S3× S2=
∂(S3× D3).
Let us assume that there are k ODP’s.
An immedeate consequence of (3.5) is the Euler number formula:
(3.6) χ(X) − kχ(P1) = χ(X0) − kχ(pt) = χ(Xt) − kχ(S3).
Let W be the “common open set” of X, Xo and Xt away from all points pi’s such that W and Vi’s cover Xt etc. A portion of the Mayer-Vietoris sequence of the covering {W, Vi0} of Xt gives
(3.7) 0 → H3(W ) → H3(Xt) →M
iZ[Ci] → H2(X) → H2(Xt) → 0.
Hence that b2(X) = b2(Xt) + (k − s).
Take into account of b2(X0) = b2(Xt) and b4(X0) = b4(X) (which also follows from suitable Mayer-Vietoris sequences), simple manipula- tions with (6.7) shows that b3(Xt) = b3(X0) + s. Comparing with the (Mayer-Vietoris) sequence defining the vanishing cycles:
(3.8) M
iZ[σi] → H3(Xt) → H3(X0) → 0,
we conclude that s > 0 is the dimension of the sapce of vanishing cycles.
Q.E.D.
Remark (3.9) We do not know whether every Calabi-Yau three- fold admits nontrivial finite distance degenerations, not to say nodal degenerations. It is also possible for a nontrivial degeneration to have N = 0! In [F2], Friedman remarked that there exists families of quintic hypersurfaces in P4 aquiring an A2singularity and have N = 0 (due to Clemens) and in fact are even C∞ trivial outside the puncture after a finite base change. He then asked whether this family can be filled in smoothly up to a base change. Since an A2singularity is terminal, the degeneration has finite Weil-Petersson distance from the smooth fibers.
Thus we need better method to detect nontrivial finite distance points.
This is the main issue of the following section.
§4 Incompleteness II: Birational Geometry
Now we go to the most technical part of this paper. In this section, by using several results of Reid, Kawamata and Koll´ar in the theory of 3-fold birational geometry along with Friedman’s result on the simul- taneous resolution of 3-fold double points, a negative answer to the
“filling problem” as stated at the end of §3 is given for any projective smoothing of a terminal Gorenstein 3-fold with numerical effective (nef) canonical bundle. As a consequence, any smoothable terminal Calabi- Yau 3-fold provides nontrivial incomplete points of the Weil-Petersson metric. Even if the monodromy is completely (eg. C∞) trivial! Similar statement is true for the general setting in remark (2.14). Here is the main theorem:
Theorem 4.1. Let X → ∆ be a projective smoothing of a terminal Gorenstein 3-fold X0 with KX0 nef. Then X → ∆ is not birational to a projective smooth family X0→ ∆ with Xt= Xt0 for t 6= 0.
We start with the following important fact (true in any dimension):
Theorem 4.2. Let X → ∆ and X0→ ∆ be two projective families with smooth general fiber Xt= Xt0 for t 6= 0. Assume that
I. X and X0 have at most terminal singularities and II. KX (resp. KX0) is nef on the central fiber,
then the bimeromorphic map which identifies all fibers outside t = 0 extends to a map which is an isomorphism in codimension one. In particular, X0 and X00 are birational to each other.
Proof. This is essentially the same as in [K1, lemma 4.3], except that they deal with the case where X and X0 are both compact projec- tive (and so ∆ is not involved). The same proof applies to our relative situation basically because our families are assumed to be projective.
So we will just give a sketch of the proof:
Let φ be the given bimeromorphic map and Z be a desingulariza- tion of the closure of the graph of φ with projection maps p : Z → X and p0: Z → X0 over ∆. Clearly we have that Zt∼= Xt ∼= Xt0 for t 6= 0.
If p and p0have the same exceptional divisors then the p-exceptional set and p0-exceptional set differ only in codimension two or higher, let E be the union of both set. Then we have the following isomorphisms
(4.3) X − p(E) ∼= Z − E ∼= X0− p0(E), which is the extension of φ we want.
To see p and p0 have the same exceptional divisors, consider the relation between canonical divisors:
(4.4) KZ = p∗KX + E1+ F = p0∗KX0+ E2+ G,
where Ei (resp. F , resp. G) denotes the part which are p and p0 (resp.
p but not p0, resp. p0 but not p) exceptional. We can then write (4.5) p∗KX = p0∗KX0+ G + (E2− E1− F ).
Because of the existence of relative hyperplane sections over ∆, the key reduction lemma in [K3, (5.2.5.3)] can be adapted for our purpose – it says that we only need to prove the above statement for the surface case; the nef condition is used here. It implies that E2− E1− F ≥ 0, hence F = 0 and E2≥ E1. Reversing the role of p and p0 gives G = 0 and E1 ≥ E2, so we have in fact E1= E2. Since both X and X0 have
terminal singularities, all exceptional divisors must appear in Ei. So the theorem is proved. Q.E.D.
Proof of (4.1). Assume such a smooth family X0 → ∆ exists. We will check the conditions needed in (4.2). II is clearly satisfied since KX|X0 = KX0, which is nef. To see I, first notice that by a simple fact in commutative algebra, the total space of a small smoothing of Gorenstein singularities is again Gorenstein. We then need the following nontrivial theorem. (Although the statement is not explicitly appeared in [K3], the proof is actually contained in [K3, (17.4), (17.6)], and so will not be given here.)
Theorem 4.6. The total space of a small smoothing of Gorenstein canonical singularities has at most Gorenstein terminal singularities.
Since both conditions in (4.2) are satisfied, we know that X0 is birational to X00. We will show that this is impossible.
If X0 is Q-factorial then X0 and X00 are birationally equivalent minimal models. Recall that a minimal model is a normal variety which is Q-factorial, terminal and has nef canonical class. By Koll´ar’s theorem on flops [K1], they are related by a sequence of flops. But a flop does not change the singularities in the terminal case, so we get a contradiction.
If X0is not Q-factorial, a theorem of Reid-Kawamata (see e.g. [K3, (6.7.4)]) says that we still have a projective small morphism X → X0
from a (Q-factorial) minimal model X to X0. X is birational to X0and so is birational to X00. As before, this implies that X is smooth and it is related to X00 by a sequence of flops. By Koll´ar’s result again [K1], X and X00 have the same integral homologies and hence have the same homologies as the general fiber Xt in X .
Consider the following “small contraction-smoothing” diagram:
(4.7)
X
↓
X0⊂ X ⊃ Xt
If X0has only ODP singularities, (4.7) is nothing but a “surgery di- agram” appeared in the Picard-Lefschetz theory. There is a well known explicit formula which relates the homologies of X and Xt and shows in particular that they can not be the same. We will state this for- mula in a form suitable for our purpose. The proof is basically a simple
Mayer-Vietoris argument which is entirely the same as that sketched in [F2].
Lemma 4.8. Given a diagram as above in the C∞ category such that near each singular point of X0it is a “small contraction-smoothing”
diagram of a germ of ODP. Let Ci be the rational curves contracted to those ODP’s and let e : L
iZ[Ci] → H2(X, Z) be the map which associates to each Ci its class in X, then H2(Xt) = coker e.
So, H2(Xt) ∼= H2(X) means the image of e is zero, which is impos- sible because X is projective. This is the desired contradiction in the case when X0 has only ODP’s as singular points.
In the general case, since the singularities are of index one, by Reid’s classification they are exactly isolated cDV singular points, that is, one parameter deformation of surface RDP’s. By Friedman’s result [F1], if p ∈ V is a germ of an isolated cDV point and C ⊂ U is the cor- responding germ of the exceptional set (which is a curve) contracted to p, then the versal deformation spaces Def(p, V ) and Def(C, U ) are both smooth and there is an inclusion map of complex spaces Def(C, U ) → Def(p, V ). Moreover, one can deform the complex structure of a small neighborhood of C so that in this new complex structure, C decom- poses into several P1’s and the contraction map deforms to a nontrivial contraction of these P1’s down to ODP’s, while keeping a neighborhood of these ODP’s to remain in the versal deformations of the germ p ∈ V . We can preform this analytic process for all C’s and p’s simultane- ously in each corresponding small neighborhoods and then patch them together smoothly. As a result, we obtain a deformed diagram which satisfies the conditions stated in lemma 4.8:
(4.9)
Xe
↓
Xe0⊂ eX ⊃ eXt
By our construction, eX is diffeomorphic to X and eXt is diffeomorphic to Xt for t 6= 0. The later is true because Def (p, V ) is smooth and the constructiuon is local. Now we have again,
(4.10) H2( eXt) ∼= H2(Xt) ∼= H2(X) ∼= H2( eX).
This implies that the image of e is zero. Since the original excep- tional curve has nontrivial homology class, at least one deformed ra-
tional curve has nontrivial homology class. This leads to the desired contradiction again and we are done. Q.E.D.
In the case of Calabi-Yau 3-folds with at most canonical singulari- ties, h1(O) = 0 implies h2(O) = 0. Hence any smoothing X → ∆ must be projective by the semi-continuity of h2(O) = 0, and in fact Xt must still be Calabi-Yau. So we conclude the following:
Theorem 4.11. Let X → ∆ be a smoothing of a terminal Calabi- Yau 3-fold. Then X → ∆ is not birational to a smooth family.
Acknowledgement
The author would like to thank Professor S.T. Yau for his steady encouragement. Also special thanks to Professor W. Schmid for his interest in this work and to B. Hassett, J. de Jong and X. Chen for many helpful discussions. B. Cheng and M. Wang carefully read the preliminary version of this paper and suggested many improvements in the presentation. I really appreciate their help.
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Chin-Lung Wang
Harvard University, Department of Mathematics One Oxford Street 325, Cambridge, MA 02138 Email: [email protected]
