H A R V A R D U N I V E R S I T Y
TH E GRADUATE SCHOOL O F ARTS AND SCIENCES
THESIS ACCEPTANCE CERTIFICATE (To be placed in Original Copy)
The undersigned, appointed by the Division
Department o f M athem atics Committee
have examined a thesis entitled
Topology of Birational Manifolds and Applications to Degenerations
presented by Chin-Lung Wang
candidate for the degree of Doctor of Philosophy and hereby certify that it is worthy of acceptance.
Signature . y ..
T ype d name ..Shing—Tui
T ype d name ..Barry. Mazur.
Signature ...
T yped name ...
D ate ...April 9, .1998
PREVIEW
PREVIEW
TOPOLOGY OF BIRATIONAL MANIFOLDS AND APPLICATIONS TO DEGENERATIONS
A thesis presented
by
Chin-Lung Wang
to
The Department of Mathematics
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
in the subject of
Mathematics
Harvard University Cambridge, Massachusetts
March 1998
1998 by Chin-Lung Wangc All rights reserved.
Contents
Acknowledgement . . . iv
Abstract . . . v
Introduction . . . 1
Birational Invariants . . . 6
1. Birational Geometry . . . 6
2. The Weil Conjecture and p-adic Integrals . . . 9
3. The Proof of Theorem A . . . 14
4. Miscellanous Results . . . 16
5. Further Comments . . . 19
Filling in Problem In Dimension Three . . . 22
6. Degenerations with Trivial Monodromy . . . 22
7. Two Key Lemmas . . . 26
8. The Proof of Theorem B . . . 28
9. Weil-Petersson Geometry of Calabi-Yau Moduli . . . 31
10. Speculations . . . 36
References . . . 39
Acknowledgement
I am very grateful to my advisor, Shing-Tung Yau, for his steady encour- agement during my five years of graduate study. His strong enthusiasm and wide scope in mathematics has a great impact on me. The characteristic style and the atmosphere that he has created in his “Student Seminar” gave me many wonderful memories.
I have benefitted from conversation with Bo Cui, Bing Cheng, A. Johan de Jong, Wee Teck Gan, Mark Gross, Brendan Hassett and Kang Zuo. I also received invaluable friendship from Mutao Wang at one decisive point of my graduate study. I would like to take this opportunity to thank all of them.
I am especially indebted to Ching-Li Chai and Wee Teck Gan for sharing their insights on the problems that I have studied and presented here. Several inaccuracies were pointed out and useful comments were provided by them.
I am also grateful to Raoul Bott, H´el`ene Esnault, Benedict H. Gross, Joe Harris, Barry Mazur, Wilfried Schmid, Eckart Viehweg and Jing Yu for their interest in my work. The extremely comfortable environment provided by the Mathematics Department has been determinantal for my study.
Last but not the least, I want to thank my mother Feng Cheng and my wife Hui-Wen Lin for their tremendous love and constant support. Without their tolerance and devotion, I could have never gone this far.
Abstract
The main theme in this thesis is to prove the invariance of Betti numbers of smooth complex projective varieties under certain birational correspondences and to discuss its applications to degeneration problems of smooth minimal models.
There are two parts of it. In the first part, it is shown that if f : X ··→ X0is a birational map between two smooth complex projective varieties such that the canonical bundles are numerically effective along the exceptional loci, then X and X0have the same Betti numbers. In particular, birational smooth minimal models have the same Betti numbers.
The main idea is to use the Weil conjecture. To proceed, we first observe that the whole problem is reduced to the p-adic case, and then use Weil’s formula to identify the number of rational points with certain p-adic integral.
The next key point is to show that the canonical bundles become equivalent after pulled back to a common resolution of the given birational map. Putting this information into the p-adic integrals of both varieties shows that they have the same Jacobian factor in the change of variable formula, hence settles the theorem.
In the second part, it is shown that for a degeneration of three dimensional smooth minimal models acquiring nontrivial terminal singularities, the punc- tured family can not be completed into a smooth projective family. Since there are examples such that the monodromy is trivial in the C∞ sense, this gives a negative answer to the so called “filling in” problem in dimension three.
The proof makes use of various results developed in the Mori theory. The key lemma in the first part and the main result mentioned above also play a very important role here. This degeneration problem is motivated by the study of the Weil-Petersson metric on the moduli spaces of Calabi-Yau manifolds. In fact, we propose the equivalence between incomplete boundary points and de- generations of Calabi-Yau manifolds acquiring at most canonical singularities.
This thesis was written under the supervision of Professor Shing-Tung Yau.
Introduction
I would like to describe personal reflections of my past five years of graduate study, to recall how those problems I was dealing with came to my mind and to explain how they were solved. But overall, I need to first say something about minimal models, which is the main subject that attracted me for most of the time. And perhaps, I hope, that I finally got some feeling of it.
The concept of minimal models goes back to the Italian algebraic geome- ters. It plays a decesive role in the classification theory of algebraic surfaces and is also fundamental in many applications. However, its range of applica- tions are not extended to higher dimensions until S. Mori’s fundamental work on the structure of rational curves appeared in the late 70’s.
During the last two decades, the minimal model theory has been exten- sively developed by S. Mori, M. Reid, Y. Kawamata, E. Viehweg, V. Shokurov, J. Koll´ar and many others. It becomes clear that it forms an important reduc- tion step in the study of higher dimensional algebraic geometry. One of their most significant achievement is that many important conjectures in dimension three were thus solved.
What is a minimal model? It is a birational model with numerically effec- tive canonical divisors and with at most terminal singularities (perhaps with some factoriality assumption). This could make sense only when one glances at Mori’s cone theorem, and its extension by Kawamata, Shokurov and Koll´ar to the singular case. Basically, it says that if the canonical bundle is not nef, then the variety admits further contractions. There are serious problems to continue this process due to the wild singularities that one may encounter after contractions. This was finally resolved by Mori in 1988 by proving the existence of flips, hence settled the existence of minimal models in dimension three.
So far, the existence problem is completely open in higher dimensions, but even worse, the minimal model is not unique except in dimension two. It is then important to see what kind of invariants are shared by those birational minimal models. More generally, we would like to know how certain topological invariants change under certain elementary birational transformations. The first main result in this thesis provides the answer for the Betti numbers:
Theorem A. Let f : X · ·→ X0 be a birational map between two smooth complex projective varieties such that the canonical bundles are numerically effective along the exceptional loci, then X and X0have the same Betti numbers.
In particular, birational smooth minimal models have the same Betti numbers.
This type of problem was first studied by Koll´ar in the case of three- folds about ten years ago, by refining Kawamata’s result on three dimensional flops. His method is basically geometrical. In fact he obtained a complete understanding of birational maps between three dimensional minimal models
— they are composed by a sequence of flops and he has a clear local picture of flops. Namely he established, among other things, the invariance of singulari- ties, cohomologies and intersection cohomologies under flops in dimension three (cf. 5.1). In this way, Theorem A only generalizes the “Betti number” state- ment to arbitrary dimensions, and is still under the very restricted smoothness assumption.
However, there are some interesting immedeate consequences of Theorem A. One of them is that the exceptional loci of the given birational map also share the same Betti numbers (Corollary 4.5). This is obtained by applying the Mayer-Vietoris argument to the birational correspondence we may construct via H. Hironaka’s theorem on the resolution of singularities, and then make use of Theorem A. In fact, in all the examples known to the author, the exceptional loci are actually birational to each other componentwise! But we have no proof of this.
The proof of Theorem A is based on some general considerations in bi- rational geometry and Grothendieck-Deligne’s solution to the Weil conjecture [D1, D2]. The bridge to connect these two is the theory of p-adic integrals.
Essentially, all of the algebro-geometric results we need were well developed in the 80’s. And all the arithmetic results we need were done even earlier.
Moreover, instead of the details, we even just need the statements existed in the literatures! It seems that all we need to do is to put them together and to see what happens. However, this is the step that people seemed to ignore. The real intention of this research is an attemption to combine these two theory together. From this point of view, we seem to have a very good start.
I would like to say some words about the development on this problem. In fact, even the idea to use the Weil conjecture via p-adic integrals to compute cohomologies is not new. It has to be dated back to the 70’s to the works of G.
Harder and M.S. Narasimhan [HN], although it was used there in a somewhat different way. p-adic integrals were also studied extensively in the context of Igusa-Weil local zeta functions by J. Denef and F. Loeser since late 80’s [Ig, DF1]. Recently this approach was taken up again by Batyrev, and he first established Theorem A in the special case of projective Calabi-Yau manifolds.
In his case, essentially no minimal model theory needs to be involved. At that time, an even more striking result to me appeared, that was D. Huybrechts’
stronger statement about Hyper-K¨ahler manifolds [Hu] (cf. 5.2).
These results were made famous, at least to me, because it was used by Beauville in explaining Yau-Zaslow’s formula on the number of rational curves on K3 surfaces. Although only hearing this development oversea, from the previous experience in the minimal model theory, notably the abundance con- jecture, I then soon convinced myself that the same result must hold true for general minimal models. I cooked out the first version of Theorem A in Octo- ber 1997 under the further assumption that the canonical bundle is semi-ample (with a help from C.-L. Chai). It is an argument based on p-adic integrals and birational correspondences. At about the same time, Batyrev’s proof in the Calabi-Yau case appeared on the network [Ba] where his “measure theoretic”
argument came to my mind.
By extending these developments further, I then realized that our origi- nal argument based on birational correspondences in fact works equally well without the semi-ampleness assumption. The key point is that the assumption can even be localized to the exceptional loci. This leads to the concept of “K- partial ordering”, which is introduced in §1 and is closely related to interesting geometric situations arising from the minimal model theory. The applicability of the Weil conjecture is largely clarified in terms of this notion (cf. Proposition 2.16 and Theorem 3.1). Moreover, this approach also provides a natural setting in the singular case.
I have tried to develop this, together with the p-adic measure, as far as possible so that it could fit the need of the minimal model theory. In fact, an easy but very interesting fact observed here is that the integral points of a p-adic variety has finite p-adic measure if and only if it has at most terminal singularities (Proposition 2.12). This give me the belief that p-adic integrals fit naturally into the framework of minimal model theory. But due to technical reasons, I have restricted myself to the smooth case when I state and prove Theorem A. (See however 2.17 and 5.3 for more about the singular case.)
I have to point out at least two aspects that Theorem A is still unsatis- factory, the torsion elements are not considered and no natural maps between cohomologies has been even mentioned. Although there is one obvious candi- date for this map, the cohomology correspondence induced from the birational correspondence, it is not clear how to show directly that it induces isomor- phisms. In fact, there is no strong evidence why this should be true.
In the simplest cases, we can show that smooth minimal models mini- mize H2(X, Z) compatible with the Hodge structure among birational smooth projective varieties. And in the singular case, at least we know that minimal models minimize the group of Weil divisors among birational projective vari- eties with at most terminal singularities. The proof is elementary (does not use
the Weil conjecture) and is contained in §4 together with some related results.
In fact, it is simply another application of the notion of K-partial ordering.
Nevertheless, it is worth pointing out that in stating both theorems, what we have in mind is that there should be a “minimal cohomology theory” among birational varieties. Moreover, it should be realized exactly by the minimal models.
* * * * *
Turn to its applications — a basic question in the analysis of boundary point of moduli spaces is the problem about degenerations. There are many powerful tools developed in this area, from the traditional Picard-Lefschetz the- ory to the modern theory entitled with the name “variations of Hodge struc- tures”. However, there are some questions seem to be beyond the scope of Hodge theory. One is the so called “filling in problem”.
This is concerned about a degenerating family of smooth projective vari- eties over the disk such that the punctured family is smoothly equivalent to a trivial product. The question is whether this punctured family can be com- pleted into a smooth analytic family. Negative answer to this question is well known in the curve theory, however, it is mainly due to the presence of non- trivial fundamental groups. So it is natural to consider only simply connected varieties. In this setup, V. Kulikov’s classification theorem on semi-stable de- generations of K3 surfaces [Ku] (in the late 70’s) provided the first important class of examples that the filling in problem has a positive answer.
In the 80’s, R. Friedman [F1] and J. Morgan [Mo] had also studies these kind of questions. A negative answer has thus been obtained by them for cer- tain degenerating families of surfaces of general type. From this, they also constructed negative examples for dimensions at least four. But at that mo- ment, people did not know how to answer this question for a given specific family with finite order monodromy, even for the simplest examples – even dimensional nodal degenerations studied in the Picard-Lefschetz theory. The nonfilliability of this was finally proved by C. Voisin in 1990 [Vo].
In his survey paper on Calabi-Yau threefolds [F4], Friedman remarked that for families of quintic hypersurfaces acqriring an A2singularity, the mon- odromy has finite order inside the mapping class group. He also expected that the filling in problem has a negative answer for any finite base change. This question caught my interest for three reasons. One, the fiber dimension is three, which belongs to the unknown zone of the existing list of examples. Two, the singularity is so simple. And more importantly, it is Calabi-Yau, a “natural candidate” for K3 surfaces in three dimensios, and we already know a positive answer for K3’s (sounds like a paradox)!
The second main result of this thesis is to provide a general theorem on terminal degenerations, which in particular answers the filling in problem in negative.
Theorem B. Let X → ∆ be a projective smoothing of a Gorenstein 3-fold X0 with nontrivial terminal singularities and with KX0 nef. Then X → ∆ is not birational to a projective smooth family X0→ ∆ with Xt ∼= Xt0 for t 6= 0.
From this point of view, the above mentioned paradox is simply that there are no terminal singularities in dimension two! Also not a surprise, the proof uses many technical results in the three dimensional minimal model theory.
Friedman’s study on simultaneous resolution of threefold double point [F3] is also fundamental to the proof. And notably, Theorem A is used in an es- sential step. However, we need to make use of its strong form obtained by Koll´ar mentioned above, because from our Theorem A, we don’t know whether the smoothness is preserved between birational Q-factorial minimal models.
Nevertheless, we still expect that further investigation will lead to interesting applications of Theorem A in higher dimensional geometry.
In fact, Theorem B was obtained in 1995, two years before the proof of Theorem A was found. The most exciting thing to me is that in both theorems, the most technical step (to me) is the same! This is what I called the “Key Lemma” in §1. I spent several months in obtaining this lemma when I tried to prove Theorem B. At the end, I found out that a weaker form of it was already in the literature, namely Koll´ar’s paper [Ko]! The remaining step for me is just to generalize it and fortunately this could be done without too much difficulity.
Theorem B is closely related to the study of the Weil-Petersson geometry of Calabi-Yau moduli spaces. This is the original problem that Professor Yau gave me. My original motivation to prove Theorem B is to provide “essential”
metric incomplete boundary point of the moduli space of Calabi-Yau threefolds.
§9 is devoted to this aspect. Needless to say, all of these are somehow related to the study of “Mirror Symmetry” phenomenon.
This article is concluded with certain speculations related to E. Viehweg’s program on the quasi-projectivity of certain moduli spaces and with a question on finite distance degenerations of Calabi-Yau manifolds. The central object in this circle of ideas is an understanding of canonical singularities — as has been introduced to us by M. Reid more than twenty years ago. In fact, it is this concept, together with Mori’s cone theorem and the Kawamata-Viehweg vanishing theorem, that gave the way of the whole development of the minimal model theory started in the early 80’s!
Chapter One — Birational Invariants
§1 Birational Geometry
We begin with some standard definitions. For a complete treatment of minimal model theory, the reader should consult [KMM].
Let X be an n dimensional complex normal Q-Gorenstein variety. That is, the canonical divisor KX is Q-Cartier. Recall that X has (at most) terminal (resp. canonical, resp. log-terminal) singularities if there is a resolution φ : Y → X such that in the canonical bundle relation
(1.1) KY =Q φ∗KX+X
aiEi,
we have that ai > 0 (resp. ai ≥ 0, resp. ai > −1) for all i. Here, the Ei’s vary among the prime components of all the exceptional divisors. Although (1.1) holds only up to Q-linear equivalence, the divisor P aiEi ∈ Zn−1⊗ Q is uniquely determined. Moreover, the condition on ai’s is readily seen to be independent of the chosen resolution. It is also elementary to see that smooth points are all terminal.
Let Z be a proper subvariety of X. A Q-Cartier divisor D is called numer- ically effective (nef) along Z if D.C := degC˜(f∗D) ≥ 0 for all effective curves C ⊂ Z, where f : ˜C → C is the normalization of C. And D is simply called nef if Z = X. A projective variety X is called a minimal model if X is terminal and KX is nef.
Two normal varieties X and X0 are birational if they have isomorphic function fields K(X) ∼= K(X0) (over C). Geometrically, this means that there is a rational map f : X ··→ X0 such that f−1 is also rational. The exceptional loci of f are defined to be the smallest subvarieties Z ⊂ X and Z0⊂ X0 such that f induces an isomorphism X − Z ∼= X0− Z0.
Among the class of birational Q-Gorenstein varieties, We have the notion of K-partial ordering (where the “K” is for canonical divisors):
Definition 1.2. For two Q-Gorenstein varieties X and X0, we say that X ≤K X0(resp. X <K X0) if there is a birational correspondence (φ, φ0) : X ← Y → X0 with Y smooth, such that φ∗KX ≤Q φ0∗KX0 (resp. “<Q”). Moreover,
“X ≤K X0” plus “X ≥K X0” implies that “X =K X0”, ie. φ∗KX =Q φ0∗KX0. In this case, we say that X and X0 are K-equivalent.
The well-definedness of this notion follows from the canonical bundle re-
lations
(1.3) KY =Q φ∗KX+ E =Qφ0∗KX0+ E0,
since we know that X ≤K X0 if and only if E ≥ E0. In the terminal case, this means that φ has more exceptional divisors than φ0 (so heuristically, X is
“smaller” than X0).
Here is the typical geometric situation that we can compare their K-partial order:
Key Lemma 1.4. Let f : X · · → X0 be a birational map between two varieties with canonical singularities. Suppose that the exceptional locus Z ⊂ X is proper and that KX is nef along Z, then X ≤K X0. Moreover, if X0 is terminal, then Z has codimension at least two.
Proof. Let φ : Y → X and φ0 : Y → X0 be a good common resolution of singularities of f so that the union of the exceptional set of φ and φ0 is a normal crossing divisor of Y . This can be done by considering ¯Γf ⊂ X × X0, the closure of the graph of f , blowing up the exceptional set of ¯Γf → X and Γ¯f → X0 and then taking Y to be a Hironaka (embedded) resolution [Hi].
Consider the canonical bundle relations:
(1.5) KY =Qφ∗KX+ E ≡ φ∗KX+ F + G
=Qφ0∗KX0+ E0≡ φ0∗KX0+ F0+ G0.
Here F and F0denote the sum of divisors (with coefficients ≥ 0) which are both φ and φ0 exceptional. G (resp. G0) denotes the part which is φ exceptional but not φ0 exceptional (resp. φ0 but not φ exceptional). Notice that φ(G0) ⊂ Z.
To proceed, we write
(1.6) φ0∗KX0 =Qφ∗KX+ G + (F − F0− G0).
It is enough to prove that F − F0− G0 ≥ 0, because this implies that F − F0≥ 0 and G0= 0, and so E ≥ E0.
By taking a generic hyperplane section H of Y n − 2 times, the problem is reduced to a problem on surfaces. Namely
(1.7) Hn−2.φ0∗KX0 =Q Hn−2.φ∗KX + ζ + (ξ − ξ0− ζ0),
where ξ = Hn−2.F and ζ = Hn−2.G etc. If ξ −ξ0−ζ0is not effective, write it as Hn−2.(A − B) = a − b with A and B effective. Then by taking the intersection of (1.7) with b, we get
(1.8) B.Hn−2.φ0∗KX0 =Q B.Hn−2.φ∗KX + b.ζ + b.a − b2.
The left hand side is always zero since B is φ0exceptional. Moreover, if B ⊂ F0 then B.Hn−2.φ∗KX = 0 too. If B ⊂ G0 then the curve φ(B.Hn−2) ⊂ φ(G0) ⊂ Z is inside the exceptional locus. So the first three terms in the right hand side are non-negative since KX is nef along Z and a, b and ζ are different components. However, since b is a nontrivial combination of φ0 exceptional curves in Hn−2, we have from the Hodge index theorem for surfaces that b2< 0, a contradiction! Hence F − F0− G0≥ 0.
For the second statement, from the construction of Y , we know that all components of the exceptional sets, denoted by Exc φ and Exc φ0 respectively, are divisors. If X0 is assumed to be terminal, then all φ0 exceptional divisors occur as components of E0. So G0 = 0 implies that Exc φ0⊂ Exc φ. With this understood, from
(1.9) X − φ(Exc φ) ∼= Y − Exc φ ∼= X0− φ0(Exc φ) ⊂ X0− φ0(Exc φ0), we conclude that Z ⊂ φ(Exc φ) is of codimension at least two. Q.E.D.
Corollary 1.10. Let f : X ··→ X0be a birational map between two varieties with at most canonical singularities such that KX (resp. KX0) is nef along the exceptional locus Z ⊂ X (resp. Z0⊂ X0), then X =K X0. Moreover, f extends to an isomorphism in codimension one if X and X0 are terminal. This applies, in particular, if both X and X0 are minimal models.
Variant 1.11. Instead of assuming that the exceptional locus in X is proper, one can generalize Key Lemma 1.4 to the relative case, namely f is a S-birational map and that X → S and X0 → S are proper S-schemes. The proof is identical to the one given above by changing notation.
Remark 1.12. This type of argument is familiar in the minimal model theory. Notably, in analyzing the log-flip diagram (eg. [KMM; 5-1-11]) or more specially, the flops. Key Lemma 1.4 implies that if X0 is a flip of X, then X ≥K X0 (in fact, more is true: X >K X0). Corollary 1.10 implies that flop induces K-equivalence. Since flip/flop will not be used in any essential way in this paper, we will refer the interested reader to [KMM] for the definitions.
The proof given above is inspired by Koll´ar’s treatment of flops in [K1].
§2 The Weil Conjecture and p-adic Integrals
To prove Theorem A, we will show that X and X0 have the same number of rational points over certain finite fields when a suitable good reduction is taken. That is, we prove that they have the same “zeta function”. The theorem will then follow from the statement of the Weil conjecture.
2.1. The reduction procedure. This is standard in algebraic geometry and in number theory: as long as we perform only a finite number of “algebraic constructions” in the complex case, e.g. consider morphisms, since all the objects involved can by defined by a finite number of polynomials, we can take S ⊂ C a finitely generated subring over Z so that everything is defined over S.
S has the property that the residue field S/m of any maximal ideal m ⊂ S is finite.
If we start with “smooth objects”, general reduction theory then says that for an infinite number of “good primes” (in fact, Zariski dense in Spec (S)), we may get good reductions so that everything is defined smoothly over the finite residue field Fq with q = pr for some prime number p. We may also assume that this reduction has a lifting such that everything is defined smoothly over R, the maximal compact subring of a p-adic local field K, i.e. a finite extension field of Qp, with residue field Fq.
More precisely, let F be the quotient field of S. Based on the fact (and others) that Qp has infinite transcendence degree, the “embedding theorem”
(see for example [Ca; p.82]) says that for an infinite number of p’s, there is an embedding of fields i : F → Qp such that i(S) ⊂ Zp. Moreover, i may be chosen so that a prescribed finite subset of S, say the coefficients of those defining polynomials, is mapped into the set of p-adic units. This embedding then gives the desired lifting.
Let P be the unique maximal ideal of R (so R/P ∼= Fq). We denote by X, ¯¯ U , . . . those objects constructed from X, U . . . via reductions mod P . That is, objects lie over the point Spec R/P → Spec R — they are defined over Fq. We also denote the reduction map by π : X(R) → ¯X(Fq) etc.
2.2. The Weil conjecture. Let ¯X be a variety defined over a finite field Fq. After fixing an algebraic closure, the Weil zeta function of ¯X is defined by
(2.3) Z( ¯X, t) := exp X
k≥1| ¯X(Fqk)|tk k
! .
In 1949, Weil conjectured several nice properties of this zeta function for smooth projective varieties and expalined how some of these would follow once a
suitable cohomology theory exists [W1]. This lead Grothendieck to his creation of ´etale cohomology theory.
More precisely, Grothendieck proved a “Lefschetz fixed point formula” in a very general context (eg. constructible sheaves over seperated schems of finite type . . .) [D2], which in particular implies that the zeta function is a rational function:
(2.4) Z( ¯X, t) = P1(t) · · · P2n−1(t) P0(t)P2(t) · · · P2n(t),
where Pj(t) is a polynomial with integer coefficients such that Pj(0) = 1 and deg Pj(t) = hj, the j-th Betti number of compactly supported `-adic ´etale cohomologies (for a prime ` 6= p). Moreover, when ¯X comes from a good reduction of a smooth complex projective variety X in the sense described in (2.1), hj coincides with the j-th Betti number of the singular cohomologies of X(C).
Deligne [D1] completed the proof of the Weil conjecture by proving the important “Riemann Hypothesis” that all roots of Pj(t) have absolute value q−j/2. In particular, the complete information about the Fqk-rational points determines the hj’s and all the roots.
2.5. Counting points via p-adic integrals. How do we count ¯X(Fq)?
If ¯X comes from the good reduction of a smooth R-scheme, we will see that such a counting can be achieved by using p-adic integrals (cf. Theorem 2.8). We will first recall some elementary aspects of the p-adic integral over K-analytic manifolds and over R-schemes.
Consider the Haar measure on the locally compact field K, normalized so that the compact open “disk” R has volume 1:
(2.6)
Z
R
|dz| = 1.
We may extend this to the multivariable case and define the p-adic integral of any regular n form Ψ = ψ(z1, · · · , zn)dz1∧ · · · ∧ dzn by
(2.7)
Z
Rn
|Ψ| :=
Z
Rn
|ψ(z)||dz1∧ · · · ∧ dzn|.
Here |a| := q−νp(NK/Qp(a)) is the usual p-adic norm.
We may define an integral slightly more general than (2.7): suppose that Ψ is a r-pluricanonical form such that in local analytic coordinates we have
Ψ = ψ(z1, · · · , zn)(dz1∧ · · · ∧ dzn)⊗r. We define the integration of a “r-th root of |Ψ|” by
(2.70)
Z
Rn
|Ψ|1/r :=
Z
Rn
|ψ(z)|1/r|dz1∧ · · · ∧ dzn|.
This is independent of the choice of coordinates, as can be checked easily by the same method as in [W2; p.14]. So we can extend the definition to (not necessarily complete) K-analytic manifolds with Ψ a (possibly meromorphic) pluricanonical form. Certainly then the integral defined may not be finite.
The key property we need is the following (slightly more general form of a) formula of Weil [W2; 2.2.5]. We briefly sketch its proof.
Theorem 2.8. Let U be a smooth R-scheme and Ω a nowhere zero r- pluricanonical form on U , then
Z
U (R)
|Ω|1/r = | ¯U (Fq)|
qn .
Proof. The proof given by Weil in [W2] goes through without difficulties
— one first observes that the reduction map π: U (R) → ¯U (Fq) induces an isomorphism between π−1(¯t) and P Rn for any ¯t ∈ ¯U (Fq) (Hensel’s lemma) such that there is a function ψ with |ψ(z)| = 1 and
(2.9) Ω = ψ(z) · (dz1∧ · · · ∧ dzn)⊗r in the K-analytic chart P Rn. This implies that R
π−1(¯t)|Ω|1/r = 1/qn for any
¯t ∈ ¯U (Fq). Summing over ¯t then gives the result. Q.E.D.
The right hand side of (2.8) shows that the integral is independent of the choice of the form Ω. One may also see this by observing that any two such forms differ by a nowhere vanishing function on U (over R) which takes values in the units on all R-points. This allows one to define a canonical p-adic measure on the R-points of smooth R-schemes by “gluing” the local integrals.
We will define it in the singular case with the hope that it may be useful for later development.
2.10. Canonical measure on Q-Gorenstein R-schemes. We will only consider those R-schemes, eg. X, that come from complex Q-Gorenstein varieties as in (2.1). Let r ∈ N such that rKX is Cartier (locally free). We may assume that we have a R-resolution of singularities φ: Y → X, which is
a projective R-morphism, so that the reduced part of the exceptional set is a simple normal crossing R-variety. We will define a measure on X(R) such that the measurable sets are exactly the compact open subsets in the K-analytic topology.
Let Ui’s be a Zariski open cover of X such that rKX|Ui is actually free.
Then for a compact open subset S ⊂ Ui(R) ⊂ X(R), we define its measure by
(2.11) mX(S) ≡
Z
S
|Ωi|1/r :=
Z
φ−1(S)
|φ∗Ωi|1/r,
where Ωi is an arbitrary generator of rKX|Ui. Notice that the properness of φ implies that φ−1(S) ⊂ Y (R). This allows us to operate the integral entirely on R-points.
For general compact open S ⊂ X(R), we may break S into disjoint pieces Sj so that Sj is contained in some Ui(R) (in fact, Sj may be chosen to lie entirely in a fiber of the reduction map π), and then define mX(S) = P
imX(Si).
Notice that mX(S) is again independent of the choice of Ui, Ωi and Y .
The following proposition explains the possible connection between the canonical measure and the minimal model theory:
Proposition 2.12. For a Q-Gorenstein R-variety X, X(R) has finite measure if and only if X has at most log-terminal singularities.
Proof. Consider the canonical bundle relation for φ: Y → X
(2.13) rKY = φ∗rKX+X
ieiEi
with rKX being Cartier and ei∈ Z. To determine the finiteness of mX(X(R)), we only need to consider those R-points on the exceptional fibers. Locally, div φ∗Ω = P
ieiEi for a generator Ω of rKX. So the integral is a product of one dimensional integrals of the form
(2.14) Ii:=
Z
R
|zeidz⊗r|1/r = Z
R
|z|ei/r|dz|.
If this is finite, then (2.15) Ii=
Z
P R
|z|ei/r|dz| + (q − 1)1
q = q−(ei/r+1)Ii+q − 1 q .
Since Ii> 0, this makes sense only if qei/r+1> 1. That is, ei/r > −1, which is exactly the definition of log-terminal singularities. Q.E.D.
Since the measure is defined Zariski-locally via p-adic integrals, for smooth X, we have from Weil’s formula (2.8) that:
Corollary 2.16. Let X be an n-dimensional smooth R-variety with finite residue field Fq, then
mX(X(R)) = | ¯X(Fq)|
qn .
Remark 2.17. If X is singular, mX((X(R)) is a weighted counting of the rational points. By definition, if φ: Y → X is a crepant R-morphism, ie.
KY =Q φ∗KX, then mX((X(R)) = mY((Y (R)). In particular, mX((X(R)) counts the rational points of ¯Y if Y is smooth! This applies to many inter- esting “pure canonical” singularities and to terminal singularities having small resolutions. However, further investigation on the precise “geometric meaning”
of this weighted counting is still needed for the general case (cf. 5.3).
§3 The Proof of Theorem A
We will in fact prove a result which connects the notion of K-partial or- dering and the canonical measure. This will largely clarify the role played by the Weil conjecture.
Theorem 3.1. Let X and X0 be two birational log-terminal R-varieties.
Then mX(X(R)) ≤ mX0(X0(R)) if X ≤K X0. In particular, K-equivalence implies measure equivalence.
Proof. Consider as before, a birational correspondence (φ, φ0) : X ← Y → X0 over R with Y a smooth R-variety. Let r ∈ N be such that both rKX
and rKX0 are Cartier. Then X ≤K X0 if and only if in the canonical bundle relations rKY = φ∗rKX+ E = φ0∗rKX0+ E0, we have E ≥ E0.
From the properness of φ and φ0, we have that φ−1(X(R)) = Y (R) = φ0−1(X0(R)). So from the definition of the measure (2.11), it suffices to show that for any compact open subset T ⊂ Y (R) with π(T ) a single point ¯y ∈ Y (F¯ q), we have
(3.2)
Z
T
|φ∗Ω|1/r ≤ Z
T
|φ0∗Ω0|1/r.
Here Ω is an arbitrary local generator of rKX on a Zariski open set U where rKX is actually free and such that ¯φ(¯y) ∈ ¯U (and with similar conditions for Ω0).
Clearly, (3.2) can fail to be an equality only if ¯y ∈ ¯E ∪ ¯E0. However, in this case E ≥ E0 says that the order of φ∗Ω is no less than that of φ∗Ω. (3.2) then follows from the definition of the p-adic integral (2.70) (see also (2.15)).
Q.E.D.
If X and X0 are smooth, combining this with (2.16) gives
Corollary 3.3. Let X and X0 be two birational smooth R-schemes. Then
| ¯X(Fq)| ≤ | ¯X0(Fq)| if X ≤K X0.
With this done, by working on cyclotomic extensions of K, the same proof shows that | ¯X(Fqk)| ≤ | ¯X0(Fqk)| for all k ∈ N. In particular, Z( ¯X, t) ≤ Z( ¯X0, t) for all t > 0. The same is true for all the derivatives, but it is not clear how to make use of these. The simplest application is given by:
Corollary 3.4. Let X and X0 be two birational complex smooth varieties.
They have the same Euler number for the compactly supported cohomologies if X =K X0.
Proof. Apply the reduction procedure (2.1) to reduce this to the p-adic case. The statement then follows from Grothendieck’s Lefschetz fixed point formula (2.4) and the above comparison of zeta functions. Q.E.D.
So far we have not used Deligne’s theorem on the “Riemann Hypothesis”.
To use it, we need to impose the projective assumption.
Theorem 3.5. Let X and X0 be two birational smooth projective R- schemes. If X =K X0 then mX(X(R)) = mX0(X0(R)). This is equivalent to Z( ¯X, t) = Z( ¯X0, t). In particular, they have the same “Betti numbers” by the Weil conjecture.
Now we may come back to our original geometric situation:
Theorem A. Let f : X · ·→ X0 be a birational map between two smooth complex projective varieties such that the canonical bundles are numerically effective along the exceptional loci, then X and X0have the same Betti numbers.
In particular, birational smooth minimal models have the same Betti numbers.
Proof. By Corollary 1.10, X and X0 are K-equivalent. So Theorem A simply follows from the reduction procedure (2.1) and Theorem 3.5. Q.E.D.
Remark 3.6. In the preliminary version of this paper (dated October 1997), Theorem A was stated with the assumption that the canonical bundle is semi-ample, that is, rKX is generated by global sections for some r ∈ N. The proof proceeds by cutting out the pluri-canonical divisors and applying p-adic integrals to the birational correspondence, where the notion of K-equivalence is essential for this step to work.
By using Weil’s formula (2.8), the proof is then concluded by induction on dimensions. In this approach, the usage of integration of a r-th root of the absolute value of a pluricanonical form was suggested to the author by C.-L.
Chai in order to deal with the case that r > 1. Happily enough, as the author realized later, the semi-ample assumption can be removed once we observed that the problem can even be localized to the exceptional loci.
Remark 3.7. The equivalence of zeta functions is a stronger statement than the equivalence of Betti numbers. Moreover, we have in fact established the equivalence of zeta functions for a dense set of primes. From the theory of motives, this suggests that we may in fact have the equivalence of Hodge structures. Further investigation in this should be interesting and important.
Question 3.8. Is Theorem A true for K¨ahler manifolds?
§4 Miscellaneous Results
Now we come back to the complex number field and begin with an ele- mentary observation:
Lemma 4.1. If the exceptional loci of a birational map f : X · · → X0 between two smooth projective varieties have codimension at least two then for i ≤ 2 we have πi(X) ∼= πi(X0) and Hi(X, Z) ∼= Hi(X0, Z) which is compatible with the rational Hodge structures.
Proof. The real codimension four statement plus the transversality ar- gument shows that πi(X) ∼= πi(X0), Hi(X, Z) ∼= Hi(X0, Z) and Hi(X, Z) ∼= Hi(X0, Z) canonically for i ≤ 2. Moreover, by Hartog’s extension we know that the Hodge groups H0(Ωi) are all birational invariants among smooth va- rieties. The orthogonality of Hodge filtrations then shows that Hi(X, Q) and Hi(X0, Q) share the same rational Hodge structures for i ≤ 2. Q.E.D.
A slightly deeper result is given by
Proposition 4.2. If the exceptional loci Z ⊂ X and Z0 ⊂ X0 of a bira- tional map f between two smooth varieties have codimension at least two, then hi(X) − hi(Z) = hi(X0) − hi(Z0).
Proof. Construct a birational correpondence X ← Y → X as in §1 and denote the exceptional divisor of φ: Y → X (resp. φ0: Y → X0) by E (resp.
E0). Since Hironaka’s resolution process only blows up smooth centers inside the singular set of the graph of f , the isomorphism X − Z ∼= X0− Z0 implies that φ(E ∪ E0) ⊂ Z and φ0(E ∪ E0) ⊂ Z0, hence that Ered = Ered0 , Z = φ(E) and Z0= φ0(E0).
Consider an open cover {V, W } of X by letting V := X − Z and W ⊃ Z be a deformation retract neighborhood. Let ˜V := φ−1(V ) and ˜W := φ−1(W ) ⊃ E be the corresponding open cover of Y . Then we have the following commutative diagram of integral cohomologies
(4.3)
Hi−1( ˜V ∩ ˜W ) → Hi(Y ) → Hi( ˜V ) ⊕ Hi(E) → Hi( ˜V ∩ ˜W )
↑ ↑ ↑ ↑
Hi−1(V ∩ W ) → Hi(X) → Hi(V ) ⊕ Hi(Z) → Hi(V ∩ W ) It is a general fact that φ∗: Hi(X) → Hi(Y ) is injective (by the projection formula, that φ is proper of degree one implies that φ! ◦ φ∗(a) = a for all a ∈ Hi(X)). Since ˜V ∼= V and ˜V ∩ ˜W ∼= V ∩ W , simple diagram chasing shows that Hi(Z) → Hi(E) is also injective. We may then break (4.3) into short
exact sequences
(4.4) 0 → φ∗Hi(X) → Hi(Y ) → Hi(E)/φ∗Hi(Z) → 0.
Similarly, we have for φ0: Y → X0:
(4.40) 0 → φ0∗Hi(X0) → Hi(Y ) → Hi(E0)/φ0∗Hi(Z0) → 0.
Since Ered= Ered0 , the proposition follows immedeately. Q.E.D.
Combining this with Theorem A gives
Corollary 4.5. Let f : X ··→ X0 be a birational map between two smooth complex projective varieties such that the canonical bundles are numerically effective along the exceptional loci, then the exceptional loci also have the same Betti numbers. In particular, this applies to birational smooth minimal models.
Remark 4.6. The proof of Theorem A in fact also shows that ¯Z and ¯Z0 have the same number of Fq-rational points. This is simply because | ¯X(Fq)| =
| ¯X0(Fq)| and ¯X − ¯Z ∼= ¯X0− ¯Z0. In particular, if Z and Z0 are smooth then they have the same Betti numbers. Although this argument apparently only works for smooth Z and Z0, which is very restricted, it is more than just a special case of (4.5) — since it carries certain nontrivial arithmetic information.
We are now in a position to show that minimal models are really minimal in the sense of cohomologies:
Theorem 4.7. Smooth minimal models minimize H2(X, Z) compatible with the Hodge structure among birational smooth projective varieties. In the singular case, the minimal models minimize the group of Weil divisors among birational projective varieties with at most terminal singularities.
Proof. Let f : X · ·→ X0 be a birational map between two n dimensional smooth projective varieties where only X is assumed to be minimal. In the notation of §1, Key Lemma 1.4 says that E ≥ E0. So we obtain canonical morphisms Hi(E) → Hi(E0) induced from E0 ⊂ E. Since Z := φ(E) and Z0:= φ0(E0) are of codimension at least two, H2n−2(Z) = 0 = H2n−2(Z0). By comparing (4.4) and (4.40) via the surjective map H2n−2(E) → H2n−2(E0), we obtain a canonical embedding:
(4.8) φ∗H2n−2(X, Z) ⊂ φ0∗H2n−2(X0, Z).
which respects the Hodge structures. This induces an injective map (4.9) φ0!◦ φ∗: H2n−2(X, Z) → H2n−2(X0, Z),
which by the projection formula is easily seen to be independent of the choices of Y , hence canonical. Poincar´e duality then concludes the first statement of 4.7.
For the second statement, we may simply copy the above proof by replacing (4.4) with the similar formula for the Weil divisors. Q.E.D.
One can also interpret this result in terms of the Picard group if the ter- minal varieties considered are assumed to be factorial or Q-factorial.
§5 Further Comments
We conclude this chapter with two historical remarks and three technical remarks:
5.1. Birational geometry. A version of Key Lemma 1.4, or rather the Corollary 1.10, was used before by Koll´ar in his study of three dimensional flops. In fact, he proved that three dimensional birational Q-factorial minimal models all share the same singularities, singular cohomologies and intersection cohomologies with pure Hodge structures (via deep results due to Saito). See [K1] for the details.
More recently, the author used a relative version of (1.10), namely vari- ant 1.11, to study degenerations of minimal projective threefolds [W; §4] and obtained a negative answer to the so called “filling-in problem” in dimension three. This result is now included in chapter two with some refinement of the original proof.
5.2. Previous results. After Koll´ar’s result on threefolds, the problem on the equivalence of Betti numbers seemed to be ignored for a while until recently when Batyrev treated the case of projective Calabi-Yau manifolds [Ba].
In the special case of projective hyper-K¨ahler manifolds, Theorem A has also been proved recently by Huybrechts [Hu] using quite different methods. In fact, he proved more — these manifolds are all inseparable points in the moduli space (hence are diffeomorphic and share the same Hodge structures)!
This problem on general minimal models, to the best of the author’s knowl- edge, has not been studied until the present work. In our case, the homotopy types will generally be different. In fact, it is well known that for a single elementary transform of threefolds, although the singular cohomologies are canonically identified, the cup product must change. However, inspired by Koll´ar’s result and Remark 3.7, we still expect that the (non-polarized) Hodge structures will turn out to be the same.
5.3. Singular case. In order to generalize Theorem A to the singular case, our approach works equally well in the log-terminal case, with the only problem being that we need a good interpretation like Weil’s formula (2.8) for the precise meaning of the weighted counting, which is the key to relate p-adic integrals to the Weil conjecture.
Since a suitable version of the Weil conjecture for singular varieties has already been proved by Deligne in [BBD] in terms of the intersection coho-
mologies introduced by Goresky and MacPherson [GM], this problem is thus reduced to the calculation of local Lefschetz numbers.
More precisely, one needs to evaluate the p-adic integrals over a singu- lar point and to reconstruct the “constructible complexes of sheaves” which it may correspond to. If luckily enough, it is the intersection cohomology com- plexes, then we may get our conclusion again via Deligne’s theorem. A detailed discussion on this will be continued in a subsequent paper.
5.4. Minimal cohomology. For Theorem 4.7, it is likely that a similar argument works for proving that terminal minimal models also minimize the second intersection cohomology groups and that they all share the same pure Hodge structures. The important injectivity of φ∗: IHi(X) → IHi(Y ) needed to conclude (4.4) is now a consequence of the so called “decomposition theorem”
of projective morphisms. ([BBD] again!)
An interesting question arises: is the Picard number (or the second Betti number) of a non-minimal model always strictly bigger than the one attained by the minimal models?
Mazur raised the following question: can one extract the expected “min- imal cohomology piece” directly from any smooth model without refering to the minimal models?
5.5. Recent development. We first notice that the proof of Theorem A can be formally seperated into three parts:
1. Geometric situations lead to the conclusion of K-equivalence. This is done Theorem 1.4, or Corollary 1.10. In particular, this applies to birational minimal models.
2. A reasonable integration/measure theory attached to a variety. Here we deal with p-adic integrals, or equivalently, the number of rational points in the case of smooth varieties. Theorem 3.1 shows that K-equivalence implies measure equivalent. In the notation used there, E and E0 are exactly the Jacobian factor occuring in the changing of variables formula from X and X0 to Y respectively.
3. Topological/geometrical interpretation of the integral. In our case, this corresponds to Grothendieck-Deligne’s solution to the Weil conjecture.
We can then formulate a meta theorem via the above steps by considering more general integrals.
Recently, based on an idea of Kontsevich, Denef and Loeser [DL2] has constructed a motivic integration on the space of arcs of an algebraic variety, which generalizes the p-adic integral. Using this new integration theory in step 2 and Deligne’s theorem on the existence of functorial mixed Hodge structures
on compactly supported cohomologies of algebraic varieties in step 3, Theorem A can be strengthened to the statement that X and X0 also have the same Hodge numbers. Moreover, the usage of motivic integration allows much better understanding of the exceptional loci. However, like the case of p-adic integrals, the topological meaning of the full measure in the singular case is still not well understood.
After the present work was completed, their preprint [DL2] and then the preprint version of this chapter became avaliable in the network. Afterwards, the above implication was also observed and pointed out to the author by Loeser. Since their construction of motivic integration is quite delicate, we will not try to say anything about it here. The interested reader is referred to [DL2]
for the details of this wonderful theory.
Chapter Two — Filling in Problem in Dimension Three
§6 Degenerations with Trivial Monodromy
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 middle 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).
