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.