Torification and factorization of birational maps
Building on work of the fourth author and Morelli’s work, we prove the weak factorization conjecture for birational maps in characteristic zero: a birational map between complete nonsingular varieties over an algebraically closed field of characteristic zero is a composite of blowings up and blowings down with nonsingular centers.
We work over an algebraically closed field of characteristic 0. We denote the multiplicative group of by .
0.1. Statement of the main result
The purpose of this paper is to give a proof for the following weak factorization conjecture of birational maps. We note that another proof of this theorem was given by the fourth author in Reference 82. See section 0.13 for a brief comparison of the two approaches.
0.2. Strong factorization
If we insist in the assertion above that and be morphisms for some we obtain the following strong factorization conjecture. ,
See section 6.1 for further discussion.
0.3. Generalizations of the main theorem
We consider the following categories, in which we denote the morphisms by “broken arrows”:
the objects are complete nonsingular algebraic spaces over an arbitrary field of characteristic 0, and broken arrows denote birational and -maps,
the objects are compact complex manifolds, and broken arrows denote bimeromorphic maps.
Given two broken arrows and we define an absolute isomorphism as follows:
In the case and are algebraic spaces over and , , are over then , consists of an isomorphism together with a pair of biregular , -isomorphisms and such that ,.
In the analytic case, simply consists of a pair of biregular isomorphisms and such that ,.
0.4. Applying the theorem
Suppose one is given a biregular invariant of nonsingular projective varieties and one is interested in the behavior of this invariant under birational transformations. Traditionally, one would (1) study the behavior of the invariant under blowings up with nonsingular centers, (2) form a conjecture according to this study, and finally (3) attempt to prove the conjecture using additional ideas.
Sometimes such additional ideas turn out to be fairly simple (e.g. birational invariance of spaces of differential forms). Sometimes they use known but deep results (e.g. Hodge theory for showing the birational invariance of in characteristic 0; abelian varieties for the birational invariance of in general; or Deligne’s work on the Weil conjectures for the results of Reference 47). Sometimes they lead to the development of beautiful new theories (e.g. Motivic integration for the invariance of Hodge numbers of birational Calabi-Yau varieties, Reference 45, Reference 7, Reference 8, Reference 22, Reference 50; see also Reference 10 where our theorem is applied).
Our theorem implies that, in characteristic 0, step (3) above is no longer necessary: once such a conjecture is compatible with blowings up with nonsingular centers, it holds for any birational map. At the time of the revision of this paper we know of two announced applications for which no alternative methods of proof are known: (a) construction of elliptic genera of singular varieties by L. Borisov and A. Libgober Reference 11, and (b) showing that the algebraic cobordism ring of a field is the Lazard ring, by M. Levine and F. Morel (Reference 48, Théorème 1.1, Reference 49).
When we set out to write this paper, we attempted to give a statement detailed enough and general enough to apply in all applications we had imagined. As soon as the paper was circulated, it became clear that there are applications not covered by Theorem 0.3.1, even though the methods apply. In the preprint Reference 27 of H. Gillet and Ch. Soulé, the authors use the behavior of localized Todd classes under proper birational maps of schemes which are projective over a discrete valuation ring of residue characteristic 0. In their proof they rely on deep (and yet unpublished in complete form) results of J. Franke Reference 25; alternatively, they could have used weak factorization for such maps. While proving this case may be a straightforward exercise using our methods, this would still leave a plethora of other possible applications (more general base schemes, real analytic geometry, analytic geometry, to name a few). -adic
One could imagine a statement of a general “weak factorization – type” result relying on a minimal set of axioms needed to carry out our line of proof of weak factorization. We decided to spare ourselves and the reader such formalism in this paper.
0.5. Early origins of the problem
The history of the factorization problem of birational maps could be traced back to the Italian school of algebraic geometers, who already knew that the operation of blowing up points on surfaces is a fundamental source of richness for surface geometry: the importance of the strong factorization theorem in dimension 2 (see Reference 83) cannot be overestimated in the analysis of the birational geometry of algebraic surfaces. We can only guess that Zariski, possibly even members of the Italian school, contemplated the problem in higher dimension early on, but refrained from stating it before results on resolution of singularities were available. The question of strong factorization was explicitly stated by Hironaka as “Question (F in )”Reference 30, Chapter 0, §6, and the question of weak factorization was raised in Reference 61. The problem remained largely open in higher dimensions despite the efforts and interesting results of many (see e.g. Crauder Reference 15, Kulikov Reference 46, Moishezon Reference 55, Schaps Reference 72, Teicher Reference 76). Many of these were summarized by Pinkham Reference 64, where the weak factorization conjecture is explicitly stated.
0.6. The toric case
For toric birational maps, the equivariant versions of the weak and strong factorization conjectures were posed in Reference 61 and came to be known as Oda’s weak and strong conjectures. While the toric version can be viewed as a special case of the general factorization conjectures, many of the examples demonstrating the difficulties in higher dimensions are in fact toric (see Hironaka Reference 29, Sally Reference 70, Shannon Reference 73). Thus Oda’s conjecture presented a substantial challenge and combinatorial difficulty. In dimension 3, Danilov’s proof of Oda’s weak conjecture Reference 21 was later supplemented by Ewald Reference 24. Oda’s weak conjecture was solved in arbitrary dimension by J. Włodarczyk in Reference 80, and another proof was given by R. Morelli in Reference 56 (see also Reference 57, and Reference 4, where the result is generalized to the toroidal situation). An important combinatorial notion which Morelli introduced into this study is that of a cobordism between fans. The algebro-geometric realization of Morelli’s combinatorial cobordism is the notion of a birational cobordism introduced in Reference 81.
Our proof of the main theorem relies on toric weak factorization. This remains as one of the most difficult theorems leading to our result.
In Reference 56, R. Morelli also proposed a proof of Oda’s strong conjecture. A gap in this proof, which was not noticed in Reference 4, was recently discovered by K. Karu. As far as we know, Oda’s strong conjecture stands unproven at present even in dimension 3.
0.7. A local version
There is a local version of the factorization conjecture, formulated and proved in dimension 2 by Abhyankar (Reference 1, Theorem 3). Christensen Reference 13 posed the problem in general and solved it for some special cases in dimension 3. Here the varieties and are replaced by appropriate birational local rings dominated by a fixed valuation, and blowings up are replaced by monoidal transforms subordinate to the valuation. The weak form of this local conjecture, as well as the strong version in the threefold case, was recently solved by S. D. Cutkosky in a series of papers Reference 16Reference 17. Cutkosky also shows that the strong version of the conjecture follows from Oda’s strong factorization conjecture for toric morphisms. In a sense, Cutkosky’s result says that the only local obstructions to solving the global strong factorization conjecture lie in the toric case.
0.8. Birational cobordisms
Our method is based upon the theory of birational cobordisms Reference 81. As mentioned above, this theory was inspired by the combinatorial notion of polyhedral cobordisms of R. Morelli Reference 56, which was used in his proof of weak factorization for toric birational maps.
Given a birational map a birational cobordism , is a variety of dimension with an action of the multiplicative group It is analogous to the usual cobordism . between differentiable manifolds and given by a Morse function (and in fact in the Kähler case the momentum map of is a Morse function, making the analogy more direct). In the differential setting one can construct an action of the additive real group where the “time” , acts as a diffeomorphism induced by integrating the vector field hence the multiplicative group ; acts as well. The critical points of are precisely the fixed points of the action of the multiplicative group, and the homotopy type of fibers of changes when we pass through these critical points (see Reference 54). Analogously, in the algebraic setting “passing through” the fixed points of the induces a birational transformation. Looking at the action on the tangent space at each fixed point, we obtain a locally toric description of the transformation. This already gives the main result of -actionReference 81: a factorization of into certain locally toric birational transformations among varieties with locally toric structures. More precisely, it is shown in Reference 81 that the intermediate varieties have abelian quotient singularities, and the locally toric birational transformations can be factored in terms of weighted blowings up. Such birational transformations can also be interpreted using the work of Brion-Procesi, Thaddeus, Dolgachev-Hu and others (see Reference 12Reference 77Reference 78Reference 23), which describes the change of Geometric Invariant Theory quotients associated to a change of linearization. We use such methods in section 2.5 in showing that the intermediate varieties are projective over or A variant of our construction using Geometric Invariant Theory, in terms of Thaddeus’s “Master Space”, is given by Hu and Keel in .Reference 34.
0.9. Locally toric versus toroidal structures
Considering the fact that weak factorization has been proven for toroidal birational maps (Reference 80, Reference 56, Reference 4), one might naïvely think that a locally toric factorization, as indicated in the previous paragraph, would already provide a proof for Theorem 0.1.1.
However, in the locally toric structure obtained from a cobordism, the embedded tori chosen may vary from point to point, while a toroidal structure (see Definition 1.5.1) requires the embedded tori to be induced from one fixed open set. Thus there is still a gap between the notion of locally toric birational transformations and that of toroidal birational maps. Developing a method for bridging over this gap is the main contribution of this paper.
In order to bridge over this gap, we follow ideas introduced by Abramovich and de Jong in Reference 2, and blow up suitable open subsets, called quasi-elementary cobordisms, of the birational cobordism along torific ideals. This operation induces a toroidal structure in a neighborhood of each connected component of the fixed point set, on which the action of is a toroidal action (we say that the blowing up torifies the action of Now the birational transformation “passing through ). is toroidal. We use canonical resolution of singularities to desingularize the resulting varieties, bringing ourselves to a situation where we can apply the factorization theorem for toroidal birational maps. This completes the proof of Theorem ”0.1.1.
0.11. Relation with the minimal model program
It is worthwhile to note the relation of the factorization problem to the development of Mori’s program. Hironaka Reference 28 used the cone of effective curves to study the properties of birational morphisms. This direction was further developed and given a decisive impact by Mori Reference 58, who introduced the notion of extremal rays and systematically used it in an attempt to construct minimal models in higher dimension, called the minimal model program. Danilov Reference 21 introduced the notion of canonical and terminal singularities in conjunction with the toric factorization problem. This was developed by Reid into a general theory of these singularities Reference 66Reference 67, which appear in an essential way in the minimal model program. The minimal model program is so far proven up to dimension 3 (Reference 59, see also Reference 39Reference 40Reference 41Reference 44Reference 74), and for toric varieties in arbitrary dimension (see Reference 68). In the steps of the minimal model program one is only allowed to contract a divisor into a variety with terminal singularities, or to perform a flip, modifying some codimension loci. This allows a factorization of a given birational morphism into such “elementary operations”. An algorithm to factor birational maps among uniruled varieties, known as Sarkisov’s program, has been developed and carried out in dimension 3 (see Reference 71Reference 69Reference 14, and see Reference 52 for the toric case in arbitrary dimension). Still, we do not know of a way to solve the classical factorization problem using such a factorization.
0.12. Relation with the toroidalization problem
In Reference 3, Theorem 2.1, it is proven that given a morphism of projective varieties there are modifications , and with a lifting , which has a toroidal structure. The toroidalization problem (see Reference 3, Reference 4, Reference 43) is that of obtaining such and which are composites of blowings up with nonsingular centers (maybe even with centers supported only over the locus where is not toroidal).
The proof in Reference 3 relies on the work of de Jong Reference 36 and methods of Reference 2. The authors of the present paper have tried to use these methods to approach the factorization conjectures, so far without success; one notion we do use in this paper is the torific ideal of Reference 2. It would be interesting if one could turn this approach on its head and prove a result on toroidalization using factorization.
More on this in section 6.2.
0.13. Relation with the proof in Reference 82
Another proof of the weak factorization theorem was given independently by the fourth author in Reference 82. The main difference between the two approaches is the following: in the current paper we are using objects such as torific ideals defined locally on each quasi-elementary piece of a cobordism. The blowing up of a torific ideal gives the quasi-elementary cobordism a toroidal structure. These toroidal modifications are then pieced together using canonical resolution of singularities. In contrast, in Reference 82 one works globally: a new combinatorial theory of stratified toroidal varieties and appropriate morphisms between them is developed, which allows one to apply Morelli’s algorithm directly to the entire birational cobordism. This stratified toroidal variety structure on the cobordism is somewhere in between our notions of locally toric and toroidal structures. -desingularization
0.14. Outline of the paper
In section 1 we discuss locally toric and toroidal structures. We also use elimination of indeterminacies of a rational map to reduce the proof of Theorem 0.1.1 to the case where is a projective birational morphism.
Suppose now we have a projective birational morphism In section .2 we apply the theory of birational cobordisms to obtain a slightly refined version of factorization into locally toric birational maps, first proven in Reference 81. Our cobordism is relatively projective over and using a geometric invariant theory analysis, inspired by Thaddeus’s work, we show that the intermediate varieties can be chosen to be projective over ,.
In section 3 we utilize a factorization of the cobordism into quasi-elementary pieces and for each piece construct an ideal sheaf , (Definition 3.1.4) whose blowing up torifies the action of on (Proposition 3.2.5). In other words, acts toroidally on the variety obtained by blowing up along .
In section 4 we prove the weak factorization theorem by putting together the toroidal birational maps obtained from the torification of the quasi-elementary cobordisms (Proposition 4.2.1), and applying toroidal weak factorization. The main tool in this step is canonical resolution of singularities.
We use the following definitions for quotients. Suppose a reductive group acts on an algebraic variety We denote by . the space of orbits, and by the space of equivalence classes of orbits, where the equivalence relation is generated by the condition that two orbits are equivalent if their closures intersect; such a space is endowed with a scheme structure which satisfies the usual universal property, if such a structure exists. In such a case, the space is called a categorical quotient and the space is called a geometric quotient.
A special case where exists as a scheme is the following: suppose there is an affine morphism -invariant Then we have . When this condition holds we say that the action of . on is relatively affine.
A particular case of this occurs in geometric invariant theory (discussed in section 2.5), where the action of on the open set of points which are semistable with respect to a fixed linearization is relatively affine.
1.2. Canonical resolution of singularities and canonical principalization
In the following (especially Lemma 1.3.1, section 4.2, section 5), we will use canonical versions of Hironaka’s theorems on resolution of singularities and principalization of an ideal, proved in Reference 9Reference 79.
1.2.1. Canonical resolution
Following Hironaka, by a canonical embedded resolution of singularities we mean a desingularization procedure uniquely associating to a composite of blowings up with nonsingular centers, satisfying a number of conditions. In particular:
“Embedded” means the following: assume the sequence of blowings up is applied when is a closed embedding with nonsingular. Denote by the exceptional divisor at some stage of the blowing up. Then (a) is a normal crossings divisor, and has normal crossings with the center of blowing up, and (b) at the last stage has normal crossings with .
“Canonical” means “functorial with respect to smooth morphisms and field extensions”, namely, if is either a smooth morphism or a field extension, then the formation of the ideals blown up commutes with pulling back by hence ; can be lifted to a smooth morphism .
In particular: (a) if is an automorphism (of schemes, not necessarily over then it can be lifted to an automorphism ), and (b) the canonical resolution behaves well with respect to étale morphisms: if , is étale, we get an étale morphism of canonical resolutions .
An important consequence of these conditions is that all the centers of blowing up lie over the singular locus of .
We note that the resolution processes in the work of Bierstone and Milman and of Villamayor commute with arbitrary formally smooth morphisms (in particular smooth morphisms, field extensions, and formal completions), though the treatment in any of the published works does not seem to state that explicitly.
1.2.2. Compatibility with a normal crossings divisor
If is embedded in a nonsingular variety, and is a normal crossings divisor, then a variant of the resolution procedure allows one to choose the centers of blowing up to have normal crossings with where , is the inverse image of This follows since the resolution setup, as in .Reference 9, allows including such a divisor in “year 0”.
By canonical principalization of an ideal sheaf in a nonsingular variety we mean “the canonical embedded resolution of singularities of the subscheme defined by the ideal sheaf making it a divisor with normal crossings”; i.e., a composite of blowings up with nonsingular centers such that the total transform of the ideal is a divisor with simple normal crossings. Canonical embedded resolution of singularities of an arbitrary subscheme, not necessarily reduced or irreducible, is discussed in section 11 of Reference 9, and this implies canonical principalization, as one simply needs to blow up at the last step.
1.2.4. Elimination of indeterminacies
Now let be a birational map and an open set on which restricts to a morphism. By elimination of indeterminacies of we mean a morphism obtained by a sequence of blowings up with nonsingular centers disjoint from , such that the birational map , is a morphism.
Elimination of indeterminacies can be reduced to principalization of an ideal sheaf: if one is given an ideal sheaf on with blowing up such that the birational map is a morphism, and if is the result of principalization of then the birational map , is a morphism, therefore the same is true for If the support of the ideal . is disjoint from the open set where is an morphism, then the centers of blowing up giving are disjoint from .
Proving that such an ideal exists (say, in the nonprojective case), and in a sufficiently natural manner for proving functoriality (even if are projective), is nontrivial. We make use of Hironaka’s version of Chow’s lemma, as follows.
We may assume that is a morphism; otherwise we replace by the closure of the graph of Now we use Chow’s lemma, proven by Hironaka in general in .Reference 31, Corollary 2, p. 504, as a consequence of his flattening procedure: there exists an ideal sheaf on such that the blowing up of along factors through Hence the canonical principalization of . also factors through .
Although it is not explicitly stated by Hironaka, the ideal is the unit ideal in the complement of the open set the blowing up of : consists of a sequence of permissible blowings up (Reference 31, Definition 4.4.3, p. 537), each of which is supported in the complement of Another important fact is that the ideal . is invariant, namely, it is functorial under absolute isomorphisms: if is another proper birational map, with corresponding ideal and , are isomorphisms such that then , This follows simply because at no point in Hironaka’s flattening procedure is there a need for any choice. .
It must be pointed out that Hironaka’s flattening procedure, and therefore the choice of the ideal does not commute with smooth morphisms in general — in fact Hironaka gives an example where it does not commute with localization. ,
The same results hold for analytic and algebraic spaces. While Hironaka states his result only in the analytic setting, the arguments hold in the algebraic setting as well. See Reference 65 for an earlier treatment of the case of varieties.
We emphasize again that Chow’s lemma in the analytic setting, and its delicate properties in both the algebraic and analytic settings, rely on Hironaka’s difficult flattening theorem (see Reference 31, or the algebraic counterpart Reference 65).
1.3. Reduction to projective morphisms
We start with a birational map
between complete nonsingular algebraic varieties and defined over and restricting to an isomorphism on an open set .
Thus we may replace by and assume from now on that is a projective morphism.
Note that, by the properties of canonical principalization and Hironaka’s flattening, the formation of is functorial under absolute isomorphisms, and the blowings up have normal crossings with the appropriate divisors. This will be used in the proof of Theorem 0.3.1 (see section 5).
1.4. Toric varieties
Let be a lattice and a strictly convex rational polyhedral cone. We denote the dual lattice by and the dual cone by The affine toric variety . is defined as
For we denote its image in the semigroup algebra by .
If and are two toric varieties, the embeddings of the torus in both of them define a toric (i.e., birational map -equivariant).
Suppose acts effectively on an affine toric variety as a one-parameter subgroup of the torus corresponding to a primitive lattice point , If . and the action on the monomial , is given by
where is the natural pairing on The . monomials correspond to the lattice points -invariant hence ,
If then , is a full-dimensional cone in and it follows that , is again an affine toric variety, defined by the lattice and cone where , is the projection. This quotient is a geometric quotient precisely when is a bijection.
1.5. Locally toric and toroidal structures
There is some confusion in the literature between the notion of toroidal embeddings and toroidal morphisms (Reference 42, Reference 3) and that of toroidal varieties (see Reference 20), which we prefer to call locally toric varieties. A crucial issue in this paper is the distinction between the two notions.
For example, the affine line is clearly locally toric, is a toroidal embedding, and is a different toroidal embedding, where a chart at the point can be obtained by translation from the point .
Toroidal embeddings can be naturally made into a category:
We now turn to birational maps:
For locally toric varieties, there are no satisfactory analogues of the definitions of toroidal morphisms and birational maps. One can define a “locally toric morphism” to be one which is toric on suitable toric charts, but this notion is neither stable under composition nor amenable to combinatorial manipulations. An extensive and quite delicate theory involving stratifications of locally toric varieties is developed in Reference 82 in order to resolve this issue. Here we use a different remedy. We define a restrictive class of birational transformations between locally toric and toroidal varieties, in which all charts are “uniform” over a common base These are still not stable under composition, but their local combinatorial nature suffices for our goals. These are the only transformations we will need in the considerations of the current paper. .
1.6. Weak factorization for toroidal birational maps
The weak factorization theorem for proper birational toric maps can be extended to the case of proper birational toroidal maps. This is proved in Reference 4 for toroidal morphisms, using the correspondence between birational toroidal morphisms and subdivisions of polyhedral complexes. The general case of a toroidal birational map can be deduced from this, as follows. By toroidal resolution of singularities we may assume is nonsingular. We apply toroidal weak factorization to the morphisms to get a sequence of toroidal birational maps ,
consisting of toroidal blowings up and down with nonsingular centers.
We state this result for later reference:
This does not immediately imply that one can choose a factorization satisfying a projectivity statement as in the main theorem, or in a functorial manner. We will show these facts in sections 2.7 and 5, respectively. It should be mentioned that if toric strong factorization is true, then the toroidal case follows.