6.1. Degenerations and 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 (a locally constant K¨ahler class) induced from the K¨ahler form on X .
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. A degeneration X → ∆ is called semi-stable if X0is a reduced divisor with normal crossings in X . By a theorem of Mumford, every degeneration has a semi-stable reduction by a sequence of blow-ups and base-changes.
The diffeomorphism type of the punctured family X× → ∆×depends only on its restriction to a circle. Fix a reference point t 6= 0 in the circle, by using local trivializations along the circle, one obtains a diffeomorphism T : Xt→ Xt
up to isotopies. That is, T is an element in the mapping class group of Xt. We will call T “the monodromy” of the given degeneration.
In the cohomology level, a generator of π1(∆×) ∼= Z induces the so called Picard-Lefschetz transformation – the monodromy T acting 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. The quasi-unipotent statement is also known to be true for any abstract polarized Variation of Hodge Structures [Sc]. In the following, we will usually assume that T is unipotent by allowing a base change implicitly.
6.2. Guiding examples — a preliminary discussion. There ex-ists smoothable Calabi-Yau 3-folds with canonical singularities such that the smoothing comes from a birational contraction of a smooth family over the disk, which induces isomorphisms outside the puncture. These examples are due to Wilson [Wi] 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] they are birational to smooth families possibly after a base change. We will call this knid of degenerations
“trivial” since they do not degenerate at all for certain polarizations.
If the monodromy of a degeneration X → ∆ is not of finite order, the degeneration is clearly “nontrivial” in the above sense. We will however inter-ested in the extremal case, namely degenerations with trivial monodromy. The above examples are of trivial monodromy and are in fact “projectively trivial”
possibly after a base change. By this we simply mean that the punctured family can be filled in smoothly in the projective category.
Is there any degeneration with C∞trivial monodromy but can not be filled in smoothly? As we have already mentioned in the introduction, examples al-ready occurs for curves. However, they are due to the presence of the nontrivial fundamental groups. Simply connected examples were found and studied by Friedman and Morgan in the 80’s. They obtained examples for surfaces of gen-eral type and used them to construct examples for dimensions bigger than or equal to four.
6.3. Picard-Lefschetz theory. We start by recalling the cohomological form of the classical Picard-Lefschetz theorem:
Theorem 6.4. For a nodal degeneration of smooth n-folds, the mon-odromy operator T acting on cohomologies is trivial except possibly in the mid-dle dimensional cohomology. In the midmid-dle dimensional case, we have that
I. (T2− I)2= 0 if n is odd, and that II. T2= I if n is even.
The standard proof is to write down the explicit reflection formula of T in terms of the “vanishing cycles”. However, even to see whether T is of finite order in the cohomology level (in the odd case), one needs to know whether the vanishing cycles represent nontrivial homology classes. Clearly, this is not just a local problem of the singular points. For example, nodal degenerations of odd dimensional quadrics have trivial monodromy on cohomology, since the middle cohomology is trivial! (This was pointed out to the author by J. de Jong.) But this seems to be not the case for general varieties.
In the case that n is even, more is known. Morgan [Mo] proved that the monodromy actually has finite order. That is, after a finite base change, the punctured family is a C∞ product. A nice result proved by Voisin [Vo] says that they are however not filliable by smooth manifolds in the cohomologically K¨ahler category.
6.5. Three dimensional case. Explicit calculations done by Candelas et al. [COGP] shows that there are nodel degenerations of Calabi-Yau 3-folds such that the monodromy is not of finite order. A theoretic proof of this statement 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 threefolds are all simply conected. First of all, a nodal threefold X0always 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 canoni-cal bundle and with h1(Ω) = 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:
(6.6)
X
↓
X0⊂ X ⊃ Xt
It has the following local description: let Vi3 pi be a contrctible neighborhood 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 homotopy type of S2∼ D4× S2.
II. Vi0has the homotopy type of S3× D3. Where the sections σi∼ S3are 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 (6.6) is the Euler number formula:
(6.7) χ(X) − kχ(P1) = χ(X0) − kχ(pt) = χ(Xt) − kχ(S3).
Let W be the “common open set” of X, Xoand Xtaway 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
(6.8) 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 manipulations with (6.7) shows that b3(Xt) = b3(X0) + s. Comparing with the (Mayer-Vietoris) sequence defining the vanishing cycles:
(6.9) 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.
6.10. Filling in problem in dimension three. In [F4], Friedman remarked that a degeneration of quintic hypersurfaces in P4 acquiring an iso-lated A2 singularity (locally of the form: x21+ x22+ x23+ x34 = 0) actually has N = 0 (due to Clemens). Moreover, by Morgan’s result [Mo], the monodromy has finite order in the mapping class group! He asked that whether this punc-tured family can be filled in smoothly in any finite base change. (He expected that the answer in NO.) If not, this will be the first known simply connected example in dimension three.
The main goal of this chapter is to prove a general theorem about the non-filliability of degenerations of three dimansional smooth minimal models acquiring nontrivial terminal singularities. In particular, we obtain in (8.7) a negative answer to Friedman’s queation (as he has expected).