American Mathematical Society

Torification and factorization of birational maps

By Dan Abramovich, Kalle Karu, Kenji Matsuki, Jarosław Włodarczyk

Abstract

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 upper K of characteristic zero is a composite of blowings up and blowings down with nonsingular centers.

0. Introduction

We work over an algebraically closed field upper K of characteristic 0. We denote the multiplicative group of upper K by upper K Superscript asterisk .

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 Reference82. See section 0.13 for a brief comparison of the two approaches.

Theorem 0.1.1 (Weak Factorization).

Let phi colon upper X 1 right dasheD arrow upper X 2 be a birational map between complete nonsingular algebraic varieties upper X 1 and upper X 2 over an algebraically closed field upper K of characteristic zero, and let upper U subset-of upper X 1 be an open set where phi is an isomorphism. Then phi can be factored into a sequence of blowings up and blowings down with nonsingular irreducible centers disjoint from upper U , namely, there exists a sequence of birational maps between complete nonsingular algebraic varieties

upper X 1 equals upper V 0 right dasheD arrow Overscript phi 1 Endscripts upper V 1 right dasheD arrow Overscript phi 2 Endscripts ellipsis right dasheD arrow Overscript phi Subscript i minus 1 Baseline Endscripts upper V Subscript i minus 1 Baseline right dasheD arrow Overscript phi Subscript i Baseline Endscripts upper V Subscript i Baseline right dasheD arrow Overscript phi Subscript i plus 1 Baseline Endscripts ellipsis right dasheD arrow Overscript phi Subscript l minus 1 Baseline Endscripts upper V Subscript l minus 1 Baseline right dasheD arrow Overscript phi Subscript l Baseline Endscripts upper V Subscript l Baseline equals upper X 2

where

(1)

phi equals phi Subscript l Baseline ring phi Subscript l minus 1 Baseline ring ellipsis phi 2 ring phi 1 ,

(2)

phi Subscript i are isomorphisms on upper U , and

(3)

either phi Subscript i Baseline colon upper V Subscript i minus 1 Baseline right dasheD arrow upper V Subscript i Baseline or phi Subscript i Superscript negative 1 Baseline colon upper V Subscript i Baseline right dasheD arrow upper V Subscript i minus 1 Baseline is a morphism obtained by blowing up a nonsingular irreducible center disjoint from upper U .

Furthermore, there is an index i 0 such that for all i less-than-or-equal-to i 0 the map upper V Subscript i Baseline right dasheD arrow upper X 1 is a projective morphism, and for all i greater-than-or-equal-to i 0 the map upper V Subscript i Baseline right dasheD arrow upper X 2 is a projective morphism. In particular, if upper X 1 and upper X 2 are projective, then all the upper V Subscript i are projective.

0.2. Strong factorization

If we insist in the assertion above that phi 1 Superscript negative 1 Baseline comma period period period comma phi Subscript i 0 Superscript negative 1 and phi Subscript i 0 plus 1 Baseline comma period period period comma phi Subscript l Baseline be morphisms for some i 0 , we obtain the following strong factorization conjecture.

Conjecture 0.2.1 (Strong Factorization).

Let the situation be as in Theorem 0.1.1. Then there exists a diagram

StartLayout 1st Row 1st Column Blank 2nd Column Blank 3rd Column upper Y 4th Column Blank 5th Column Blank 2nd Row 1st Column Blank 2nd Column psi 1 down left-arrow 3rd Column Blank 4th Column psi 2 5th Column Blank 3rd Row 1st Column upper X 1 2nd Column Blank 3rd Column right dasheD arrow Overscript phi Endscripts 4th Column Blank 5th Column upper X 2 EndLayout

where the morphisms psi 1 and psi 2 are composites of blowings up of nonsingular centers disjoint from upper U .

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”:

(1)

the objects are complete nonsingular algebraic spaces over an arbitrary field upper L of characteristic 0, and broken arrows upper X right dasheD arrow upper Y denote birational upper L -maps, and

(2)

the objects are compact complex manifolds, and broken arrows upper X right dasheD arrow upper Y denote bimeromorphic maps.

Given two broken arrows phi colon upper X right dasheD arrow upper Y and phi prime colon upper X prime right dasheD arrow upper Y prime we define an absolute isomorphism g colon phi right-arrow phi prime as follows:

In the case upper X and upper Y are algebraic spaces over upper L , and upper X prime , upper Y prime are over upper L prime , then g consists of an isomorphism sigma colon upper S p e c upper L right-arrow upper S p e c upper L prime , together with a pair of biregular sigma -isomorphisms g Subscript upper X Baseline colon upper X right-arrow upper X prime and g Subscript upper Y Baseline colon upper Y right-arrow upper Y prime , such that phi prime ring g Subscript upper X Baseline equals g Subscript upper Y Baseline ring phi .

In the analytic case, g simply consists of a pair of biregular isomorphisms g Subscript upper X Baseline colon upper X right-arrow upper X prime and g Subscript upper Y Baseline colon upper Y right-arrow upper Y prime , such that phi prime ring g Subscript upper X Baseline equals g Subscript upper Y Baseline ring phi .

Theorem 0.3.1

Let phi colon upper X 1 right dasheD arrow upper X 2 be as in case (1) or (2) above. Let upper U subset-of upper X 1 be an open set where phi is an isomorphism. Then phi can be factored, functorially with respect to absolute isomorphisms, into a sequence of blowings up and blowings down with nonsingular centers disjoint from upper U . Namely, to any such phi we associate a diagram in the corresponding category

upper X 1 equals upper V 0 right dasheD arrow Overscript phi 1 Endscripts upper V 1 right dasheD arrow Overscript phi 2 Endscripts ellipsis right dasheD arrow Overscript phi Subscript i minus 1 Baseline Endscripts upper V Subscript i minus 1 Baseline right dasheD arrow Overscript phi Subscript i Baseline Endscripts upper V Subscript i Baseline right dasheD arrow Overscript phi Subscript i plus 1 Baseline Endscripts ellipsis right dasheD arrow Overscript phi Subscript l minus 1 Baseline Endscripts upper V Subscript l minus 1 Baseline right dasheD arrow Overscript phi Subscript l Baseline Endscripts upper V Subscript l Baseline equals upper X 2

where

(1)

phi equals phi Subscript l Baseline ring phi Subscript l minus 1 Baseline ring ellipsis phi 2 ring phi 1 ,

(2)

phi Subscript i are isomorphisms on upper U , and

(3)

either phi Subscript i Baseline colon upper V Subscript i minus 1 Baseline right dasheD arrow upper V Subscript i Baseline or phi Subscript i Superscript negative 1 Baseline colon upper V Subscript i Baseline right dasheD arrow upper V Subscript i minus 1 Baseline is a morphism obtained by blowing up a nonsingular center disjoint from upper U .

(4)

Functoriality: if g colon phi right-arrow phi prime is an absolute isomorphism, carrying upper U to upper U prime , and phi prime Subscript i Baseline colon upper V prime Subscript i minus 1 Baseline right dasheD arrow upper V prime Subscript i is the factorization of phi prime , then the resulting rational maps g Subscript i Baseline colon upper V Subscript i Baseline right dasheD arrow upper V prime Subscript i give absolute isomorphisms.

(5)

Moreover, there is an index i 0 such that for all i less-than-or-equal-to i 0 the map upper V Subscript i Baseline right dasheD arrow upper X 1 is a projective morphism, and for all i greater-than-or-equal-to i 0 the map upper V Subscript i Baseline right dasheD arrow upper X 2 is a projective morphism.

(6)

Let upper E Subscript i Baseline subset-of upper V Subscript i be the exceptional divisor of upper V Subscript i Baseline right-arrow upper X 1 left-parenthesis respectively, upper V Subscript i Baseline right-arrow upper X 2 right-parenthesis in case i less-than-or-equal-to i 0 left-parenthesis respectively, i greater-than-or-equal-to i 0 right-parenthesis . Then the above centers of blowing up in upper V Subscript i have normal crossings with upper E Subscript i . If, moreover, upper X 1 minus upper U left-parenthesis respectively, upper X 2 minus upper U right-parenthesis is a normal crossings divisor, then the centers of blowing up have normal crossings with the inverse images of this divisor.

Remarks

(1)

Note that, in order to achieve functoriality, we cannot require the centers of blowing up to be irreducible.

(2)

Functoriality implies, as immediate corollaries, the existence of factorization over any field of characteristic 0, as well as factorization, equivariant under the action of a group upper G , of a upper G -equivariant birational map. If one assumes the axiom of choice, then a standard argument shows that equivariance implies functoriality. In our proofs we do not use the axiom of choice, with the exceptions of (1) existence of an algebraic closure, and (2) section 5.6, where showing functoriality without the assumption of the axiom of choice would require revising some of the arguments of Reference56. We hope that the interested reader will be able to rework our arguments without the assumption of the axiom of choice if this becomes desirable.

(3)

The same theorem holds true for varieties or algebraic spaces of dimension d over a perfect field of characteristic p greater-than 0 assuming that canonical embedded resolution of singularities holds true for varieties or algebraic spaces of dimension d plus 1 in characteristic p . The proof for varieties goes through word for word as in this paper, while for the algebraic space case one needs to recast some of our steps from the Zariski topology to the étale topology (see Reference38, Reference53).

(4)

While this theorem clearly implies the main theorem as a special case, we prefer to carry out the proof of the main theorem throughout the text, and to indicate the changes one needs to perform for proving Theorem 0.3.1 in section 5.

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 upper H Superscript i Baseline left-parenthesis upper X comma script upper O Subscript upper X Baseline right-parenthesis in characteristic 0; abelian varieties for the birational invariance of upper H Superscript 1 Baseline left-parenthesis upper X comma script upper O Subscript upper X Baseline right-parenthesis in general; or Deligne’s work on the Weil conjectures for the results of Reference47). 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, Reference45, Reference7, Reference8, Reference22, Reference50; see also Reference10 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 Reference11, and (b) showing that the algebraic cobordism ring of a field is the Lazard ring, by M. Levine and F. Morel (Reference48, Théorème 1.1, Reference49).

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 Reference27 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 Reference25; 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, p -adic analytic geometry, to name a few).

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 Reference83) 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 prime )” in Reference30, Chapter 0, §6, and the question of weak factorization was raised in Reference61. The problem remained largely open in higher dimensions despite the efforts and interesting results of many (see e.g. Crauder Reference15, Kulikov Reference46, Moishezon Reference55, Schaps Reference72, Teicher Reference76). Many of these were summarized by Pinkham Reference64, 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 Reference61 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 Reference29, Sally Reference70, Shannon Reference73). Thus Oda’s conjecture presented a substantial challenge and combinatorial difficulty. In dimension 3, Danilov’s proof of Oda’s weak conjecture Reference21 was later supplemented by Ewald Reference24. Oda’s weak conjecture was solved in arbitrary dimension by J. Włodarczyk in Reference80, and another proof was given by R. Morelli in Reference56 (see also Reference57, and Reference4, 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 Reference81.

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 Reference56, R. Morelli also proposed a proof of Oda’s strong conjecture. A gap in this proof, which was not noticed in Reference4, 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 (Reference1, Theorem 3). Christensen Reference13 posed the problem in general and solved it for some special cases in dimension 3. Here the varieties upper X 1 and upper X 2 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 Reference16Reference17. 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 Reference81. As mentioned above, this theory was inspired by the combinatorial notion of polyhedral cobordisms of R. Morelli Reference56, which was used in his proof of weak factorization for toric birational maps.

Given a birational map phi colon upper X 1 right dasheD arrow upper X 2 , a birational cobordism upper B Subscript phi Baseline left-parenthesis upper X 1 comma upper X 2 right-parenthesis is a variety of dimension dimension left-parenthesis upper X 1 right-parenthesis plus 1 with an action of the multiplicative group upper K Superscript asterisk . It is analogous to the usual cobordism upper B left-parenthesis upper M 1 comma upper M 2 right-parenthesis between differentiable manifolds upper M 1 and upper M 2 given by a Morse function f (and in fact in the Kähler case the momentum map of double-struck upper C Superscript asterisk is a Morse function, making the analogy more direct). In the differential setting one can construct an action of the additive real group double-struck upper R , where the “time” t element-of double-struck upper R acts as a diffeomorphism induced by integrating the vector field g r a d left-parenthesis f right-parenthesis ; hence the multiplicative group left-parenthesis double-struck upper R Subscript 0 Baseline comma times right-parenthesis equals exp left-parenthesis double-struck upper R comma plus right-parenthesis acts as well. The critical points of f are precisely the fixed points of the action of the multiplicative group, and the homotopy type of fibers of f changes when we pass through these critical points (see Reference54). Analogously, in the algebraic setting “passing through” the fixed points of the upper K Superscript asterisk -action 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 Reference81: a factorization of phi into certain locally toric birational transformations among varieties with locally toric structures. More precisely, it is shown in Reference81 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 Reference12Reference77Reference78Reference23), 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 upper X 1 or upper X 2 . A variant of our construction using Geometric Invariant Theory, in terms of Thaddeus’s “Master Space”, is given by Hu and Keel in Reference34.

0.9. Locally toric versus toroidal structures

Considering the fact that weak factorization has been proven for toroidal birational maps (Reference80, Reference56, Reference4), 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.

0.10. Torification

In order to bridge over this gap, we follow ideas introduced by Abramovich and de Jong in Reference2, and blow up suitable open subsets, called quasi-elementary cobordisms, of the birational cobordism upper B Subscript phi Baseline left-parenthesis upper X 1 comma upper X 2 right-parenthesis along torific ideals. This operation induces a toroidal structure in a neighborhood of each connected component upper F of the fixed point set, on which the action of upper K Superscript asterisk is a toroidal action (we say that the blowing up torifies the action of upper K Superscript asterisk ). Now the birational transformation “passing through upper F 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 Reference28 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 Reference58, 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 Reference21 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 Reference66Reference67, which appear in an essential way in the minimal model program. The minimal model program is so far proven up to dimension 3 (Reference59, see also Reference39Reference40Reference41Reference44Reference74), and for toric varieties in arbitrary dimension (see Reference68). 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 2 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 Reference71Reference69Reference14, and see Reference52 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 Reference3, Theorem 2.1, it is proven that given a morphism of projective varieties upper X right-arrow upper B , there are modifications m Subscript upper X Baseline colon upper X prime right-arrow upper X and m Subscript upper B Baseline colon upper B prime right-arrow upper B , with a lifting upper X prime right-arrow upper B prime which has a toroidal structure. The toroidalization problem (see Reference3, Reference4, Reference43) is that of obtaining such m Subscript upper X and m Subscript upper B which are composites of blowings up with nonsingular centers (maybe even with centers supported only over the locus where upper X right-arrow upper B is not toroidal).

The proof in Reference3 relies on the work of de Jong Reference36 and methods of Reference2. 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 Reference2. 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 Reference82

Another proof of the weak factorization theorem was given independently by the fourth author in Reference82. 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 Reference82 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 pi -desingularization 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.

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 phi is a projective birational morphism.

Suppose now we have a projective birational morphism phi colon upper X 1 right-arrow upper X 2 . 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 Reference81. Our cobordism upper B is relatively projective over upper X 2 , 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 upper X 2 .

In section 3 we utilize a factorization of the cobordism upper B into quasi-elementary pieces upper B Subscript a Sub Subscript i , and for each piece construct an ideal sheaf upper I (Definition 3.1.4) whose blowing up torifies the action of upper K Superscript asterisk on upper B Subscript a Sub Subscript i (Proposition 3.2.5). In other words, upper K Superscript asterisk acts toroidally on the variety obtained by blowing up upper B Subscript a Sub Subscript i along upper I .

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.

In section 5 we prove Theorem 0.3.1. We then discuss some problems related to strong factorization in section 6.

1. Preliminaries

1.1. Quotients

We use the following definitions for quotients. Suppose a reductive group upper G acts on an algebraic variety upper X . We denote by upper X slash upper G the space of orbits, and by upper X slash slash upper G 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 upper X slash slash upper G is called a categorical quotient and the space upper X slash upper G is called a geometric quotient.

A special case where upper X slash slash upper G exists as a scheme is the following: suppose there is an affine upper G -invariant morphism pi colon upper X right-arrow upper Y . Then we have upper X slash slash upper G equals script upper S p e c Subscript upper Y Baseline left-parenthesis left-parenthesis pi Subscript asterisk Baseline script upper O Subscript upper X Baseline right-parenthesis Superscript upper G Baseline right-parenthesis . When this condition holds we say that the action of upper G on upper X is relatively affine.

A particular case of this occurs in geometric invariant theory (discussed in section 2.5), where the action of upper G 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 Reference9Reference79.

1.2.1. Canonical resolution

Following Hironaka, by a canonical embedded resolution of singularities upper W overTilde right-arrow upper W we mean a desingularization procedure uniquely associating to upper W a composite of blowings up with nonsingular centers, satisfying a number of conditions. In particular:

(1)

“Embedded” means the following: assume the sequence of blowings up is applied when upper W subset-of upper U is a closed embedding with upper U nonsingular. Denote by upper E Subscript i the exceptional divisor at some stage of the blowing up. Then (a) upper E Subscript i is a normal crossings divisor, and has normal crossings with the center of blowing up, and (b) at the last stage upper W overTilde has normal crossings with upper E Subscript i .

(2)

“Canonical” means “functorial with respect to smooth morphisms and field extensions”, namely, if theta colon upper V right-arrow upper W is either a smooth morphism or a field extension, then the formation of the ideals blown up commutes with pulling back by theta ; hence theta can be lifted to a smooth morphism theta overTilde colon upper V overTilde right-arrow upper W overTilde .

In particular: (a) if theta colon upper W right-arrow upper W is an automorphism (of schemes, not necessarily over upper K ), then it can be lifted to an automorphism upper W overTilde right-arrow upper W overTilde , and (b) the canonical resolution behaves well with respect to étale morphisms: if upper V right-arrow upper W is étale, we get an étale morphism of canonical resolutions upper V overTilde right-arrow upper W overTilde .

An important consequence of these conditions is that all the centers of blowing up lie over the singular locus of upper W .

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 upper W subset-of upper U is embedded in a nonsingular variety, and upper D subset-of upper U 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 upper D Subscript i Baseline plus upper E Subscript i , where upper D Subscript i is the inverse image of upper D . This follows since the resolution setup, as in Reference9, allows including such a divisor in “year 0”.

1.2.3. Principalization

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 Reference9, and this implies canonical principalization, as one simply needs to blow up upper W overTilde at the last step.

1.2.4. Elimination of indeterminacies

Now let phi colon upper W 1 right dasheD arrow upper W 2 be a birational map and upper U subset-of upper W 1 an open set on which phi restricts to a morphism. By elimination of indeterminacies of phi we mean a morphism e colon upper W prime 1 right-arrow upper W 1 , obtained by a sequence of blowings up with nonsingular centers disjoint from upper U , such that the birational map phi ring e is a morphism.

Elimination of indeterminacies can be reduced to principalization of an ideal sheaf: if one is given an ideal sheaf upper I on upper W 1 with blowing up upper W double-prime 1 equals upper B l Subscript upper I Baseline left-parenthesis upper W 1 right-parenthesis such that the birational map upper W double-prime 1 right-arrow upper W 2 is a morphism, and if upper W prime 1 right-arrow upper W 1 is the result of principalization of upper I , then the birational map upper W prime 1 right-arrow upper W double-prime 1 is a morphism, therefore the same is true for upper W prime 1 right-arrow upper W 2 . If the support of the ideal upper I is disjoint from the open set upper U where phi is an morphism, then the centers of blowing up giving upper W prime 1 right-arrow upper W 1 are disjoint from upper U .

Proving that such an ideal upper I exists (say, in the nonprojective case), and in a sufficiently natural manner for proving functoriality (even if upper W Subscript i are projective), is nontrivial. We make use of Hironaka’s version of Chow’s lemma, as follows.

We may assume that phi Superscript negative 1 is a morphism; otherwise we replace upper W 2 by the closure of the graph of phi . Now we use Chow’s lemma, proven by Hironaka in general in Reference31, Corollary 2, p. 504, as a consequence of his flattening procedure: there exists an ideal sheaf upper I on upper W 1 such that the blowing up of upper W 1 along upper I factors through upper W 2 . Hence the canonical principalization of upper I also factors through upper W 2 .

Although it is not explicitly stated by Hironaka, the ideal upper I is the unit ideal in the complement of the open set upper U : the blowing up of upper I consists of a sequence of permissible blowings up (Reference31, Definition 4.4.3, p. 537), each of which is supported in the complement of upper U . Another important fact is that the ideal upper I is invariant, namely, it is functorial under absolute isomorphisms: if phi prime colon upper W prime 1 right dasheD arrow upper W prime 2 is another proper birational map, with corresponding ideal upper I prime , and theta Subscript i Baseline colon upper W Subscript i Baseline right-arrow upper W prime Subscript i are isomorphisms such that phi prime ring theta 1 equals theta 2 ring phi , then theta 1 Superscript asterisk Baseline upper I Superscript prime Baseline equals upper I . 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 upper I , 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 Reference65 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 Reference31, or the algebraic counterpart Reference65).

1.3. Reduction to projective morphisms

We start with a birational map

phi colon upper X 1 right dasheD arrow upper X 2

between complete nonsingular algebraic varieties upper X 1 and upper X 2 defined over upper K and restricting to an isomorphism on an open set upper U .

Lemma 1.3.1 (Hironaka).

There is a commutative diagram

StartLayout 1st Row 1st Column upper X prime 1 2nd Column right-arrow Overscript phi prime Endscripts 3rd Column upper X prime 2 2nd Row 1st Column g 1 down-arrow 2nd Column Blank 3rd Column g 2 3rd Row 1st Column upper X 1 2nd Column right dasheD arrow Overscript phi Endscripts 3rd Column upper X 2 EndLayout

such that g 1 and g 2 are composites of blowings up with nonsingular centers disjoint from upper U , and phi prime is a projective birational morphism.

Proof.

By Hironaka’s theorem on elimination of indeterminacies (see 1.2.4 above), there is a morphism g 2 colon upper X prime 2 right-arrow upper X 2 which is a composite of blowings up with nonsingular centers disjoint from upper U , such that the birational map h colon equals phi Superscript negative 1 Baseline ring g 2 colon upper X prime 2 right-arrow upper X 1 is a morphism:

StartLayout 1st Row 1st Column Blank 2nd Column Blank 3rd Column upper X prime 2 2nd Row 1st Column Blank 2nd Column h down left-arrow 3rd Column g 2 3rd Row 1st Column upper X 1 2nd Column right dasheD arrow Overscript phi Endscripts 3rd Column upper X 2 EndLayout

By the same theorem, there is a morphism g 1 colon upper X prime 1 right-arrow upper X 1 which is a composite of blowings up with nonsingular centers disjoint from upper U , such that phi prime colon equals h Superscript negative 1 Baseline ring g 1 colon upper X prime 1 right-arrow upper X prime 2 is a morphism. Since the composite h ring phi Superscript prime Baseline equals g 1 is projective, it follows that phi prime is projective.

Thus we may replace upper X 1 right dasheD arrow upper X 2 by upper X prime 1 right-arrow upper X prime 2 and assume from now on that phi is a projective morphism.

Note that, by the properties of canonical principalization and Hironaka’s flattening, the formation of phi prime colon upper X prime 1 right-arrow upper X prime 2 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 upper N approximately-equals double-struck upper Z Superscript n be a lattice and sigma subset-of upper N Subscript double-struck upper R a strictly convex rational polyhedral cone. We denote the dual lattice by upper M and the dual cone by sigma Superscript logical-or Baseline subset-of upper M Subscript double-struck upper R . The affine toric variety upper X equals upper X left-parenthesis upper N comma sigma right-parenthesis is defined as

upper X equals upper S p e c upper K left-bracket upper M intersection sigma Superscript logical-or Baseline right-bracket period

For m element-of upper M intersection sigma Superscript logical-or we denote its image in the semigroup algebra upper K left-bracket upper M intersection sigma Superscript logical-or Baseline right-bracket by z Superscript m .

More generally, the toric variety corresponding to a fan normal upper Sigma in upper N Subscript double-struck upper R is denoted by upper X left-parenthesis upper N comma normal upper Sigma right-parenthesis ; see Reference26, Reference62.

If upper X 1 equals upper X left-parenthesis upper N comma normal upper Sigma 1 right-parenthesis and upper X 2 equals upper X left-parenthesis upper N comma normal upper Sigma 2 right-parenthesis are two toric varieties, the embeddings of the torus upper T equals upper S p e c upper K left-bracket upper M right-bracket in both of them define a toric (i.e., upper T -equivariant) birational map upper X 1 right dasheD arrow upper X 2 .

Suppose upper K Superscript asterisk acts effectively on an affine toric variety upper X equals upper X left-parenthesis upper N comma sigma right-parenthesis as a one-parameter subgroup of the torus upper T , corresponding to a primitive lattice point a element-of upper N . If t element-of upper K Superscript asterisk and m element-of upper M , the action on the monomial z Superscript m is given by

t Superscript asterisk Baseline left-parenthesis z Superscript m Baseline right-parenthesis equals t Superscript left-parenthesis a comma m right-parenthesis Baseline dot z Superscript m Baseline comma

where left-parenthesis dot comma dot right-parenthesis is the natural pairing on upper N times upper M . The upper K Superscript asterisk -invariant monomials correspond to the lattice points upper M intersection a Superscript up-tack , hence

upper X slash slash upper K Superscript asterisk Baseline approximately-equals upper S p e c upper K left-bracket upper M intersection sigma Superscript logical-or Baseline intersection a Superscript up-tack Baseline right-bracket period

If a not-an-element-of plus-or-minus sigma , then sigma Superscript logical-or Baseline intersection a Superscript up-tack is a full-dimensional cone in a Superscript up-tack , and it follows that upper X slash slash upper K Superscript asterisk is again an affine toric variety, defined by the lattice pi left-parenthesis upper N right-parenthesis and cone pi left-parenthesis sigma right-parenthesis , where pi colon upper N Subscript double-struck upper R Baseline right-arrow upper N Subscript double-struck upper R Baseline slash double-struck upper R dot a is the projection. This quotient is a geometric quotient precisely when pi colon sigma right-arrow pi left-parenthesis sigma right-parenthesis 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 (Reference42, Reference3) and that of toroidal varieties (see Reference20), which we prefer to call locally toric varieties. A crucial issue in this paper is the distinction between the two notions.

Definition 1.5.1

(1)

A variety upper W is locally toric if for every closed point p element-of upper W there exists an open neighborhood upper V Subscript p Baseline subset-of upper W of p and an étale morphism eta Subscript p Baseline colon upper V Subscript p Baseline right-arrow upper X Subscript p Baseline to a toric variety upper X Subscript p . Such a morphism eta Subscript p is called a toric chart at p .

(2)

An open embedding upper U subset-of upper W is a toroidal embedding if for every closed point p element-of upper W there exists a toric chart eta Subscript p Baseline colon upper V Subscript p Baseline right-arrow upper X Subscript p Baseline at p such that upper U intersection upper V Subscript p Baseline equals eta Subscript p Superscript negative 1 Baseline left-parenthesis upper T right-parenthesis , where upper T subset-of upper X Subscript p is the torus. We call such charts toroidal. Sometimes we omit the open set upper U from the notation and simply say that a variety is toroidal.

(3)

We say that a locally toric (respectively, toroidal) chart on a variety is compatible with a divisor upper D subset-of upper W if eta Subscript p Superscript negative 1 Baseline left-parenthesis upper T right-parenthesis intersection upper D equals normal empty-set , i.e., upper D corresponds to a toric divisor on upper X Subscript p .

A toroidal embedding upper U subset-of upper X can equivalently be specified by the pair left-parenthesis upper X comma upper D Subscript upper X Baseline right-parenthesis , where upper D Subscript upper X is the reduced Weil divisor supported on upper X minus upper U . We will sometimes interchange between upper U subset-of upper X and left-parenthesis upper X comma upper D Subscript upper X Baseline right-parenthesis for denoting a toroidal structure on upper X . A divisor upper D prime is compatible with the toroidal structure left-parenthesis upper X comma upper D Subscript upper X Baseline right-parenthesis if it is supported in upper D Subscript upper X .

For example, the affine line double-struck upper A Superscript 1 is clearly locally toric, double-struck upper A Superscript 1 Baseline minus StartSet 0 EndSet subset-of double-struck upper A Superscript 1 is a toroidal embedding, and double-struck upper A Superscript 1 Baseline subset-of double-struck upper A Superscript 1 is a different toroidal embedding, where a chart at the point 0 can be obtained by translation from the point 1 .

Toroidal embeddings can be naturally made into a category:

Definition 1.5.2

Let upper U Subscript i Baseline subset-of upper W Subscript i left-parenthesis i equals 1 comma 2 right-parenthesis be toroidal embeddings. A proper birational morphism f colon upper W 1 right-arrow upper W 2 is said to be toroidal if, for every closed point q element-of upper W 2 and any p element-of f Superscript negative 1 Baseline q , there is a diagram of fiber squares

StartLayout 1st Row 1st Column upper X Subscript p 2nd Column left-arrow 3rd Column upper V Subscript p 4th Column subset-of 5th Column upper W 1 2nd Row 1st Column phi down-arrow 2nd Column Blank 3rd Column down-arrow 4th Column Blank 5th Column f 3rd Row 1st Column upper X Subscript q 2nd Column left-arrow 3rd Column upper V Subscript q 4th Column subset-of 5th Column upper W 2 EndLayout

where

eta Subscript p Baseline colon upper V Subscript p Baseline right-arrow upper X Subscript p Baseline is a toroidal chart at p ,

eta Subscript q Baseline colon upper V Subscript q Baseline right-arrow upper X Subscript q Baseline is a toroidal chart at q , and

phi colon upper X Subscript p Baseline right-arrow upper X Subscript q Baseline is a toric morphism.

Remarks

(1)

A toroidal embedding as defined above is a toroidal embedding without self-intersection according to the definition in Reference42, and a birational toroidal morphism satisfies the condition of allowability in Reference42.

(2)

To a toroidal embedding left-parenthesis upper U Subscript upper W Baseline subset-of upper W right-parenthesis one can associate a polyhedral complex normal upper Delta Subscript upper W , such that proper birational toroidal morphisms to upper W , up to isomorphisms, are in one-to-one correspondence with certain subdivisions of the complex (see Reference42). It follows from this that the composition of two proper birational toroidal morphisms upper W 1 right-arrow upper W 2 and upper W 2 right-arrow upper W 3 is again toroidal: the first morphism corresponds to a subdivision of normal upper Delta Subscript upper W 2 , the second one to a subdivision of normal upper Delta Subscript upper W 3 , hence their composition is the unique toroidal morphism corresponding to the subdivision normal upper Delta Subscript upper W 1 of normal upper Delta Subscript upper W 3 .

(3)

Some of the many issues surrounding these definitions we avoided discussing here are addressed in the third author’s lecture notes Reference53.

We now turn to birational maps:

Definition 1.5.3 (Reference30, Reference35).

Let psi colon upper W 1 right dasheD arrow upper W 2 be a rational map defined on a dense open subset upper U . Denote by normal upper Gamma Subscript psi the closure of the graph of psi Subscript upper U in upper W 1 times upper W 2 . We say that psi is proper if the projections normal upper Gamma Subscript psi Baseline right-arrow upper W 1 and normal upper Gamma Subscript psi Baseline right-arrow upper W 2 are both proper.

Definition 1.5.4

Let upper U Subscript i Baseline subset-of upper W Subscript i be toroidal embeddings. A proper birational map psi colon upper W 1 right dasheD arrow upper W 2 is said to be toroidal if there exists a toroidal embedding upper U Subscript upper Z Baseline subset-of upper Z and a commutative diagram

StartLayout 1st Row 1st Column Blank 2nd Column Blank 3rd Column upper Z 4th Column Blank 5th Column Blank 2nd Row 1st Column Blank 2nd Column down left-arrow 3rd Column Blank 4th Column down right-arrow 5th Column Blank 3rd Row 1st Column upper W 1 2nd Column Blank 3rd Column right dasheD arrow Overscript psi Endscripts 4th Column Blank 5th Column upper W 2 EndLayout

where upper Z right-arrow upper W Subscript i left-parenthesis i equals 1 comma 2 right-parenthesis are proper birational toroidal morphisms. In particular, a proper birational toroidal map induces an isomorphism between the open sets upper U 1 and upper U 2 .

Remarks

(1)

It follows from the correspondence between proper birational toroidal morphisms and subdivisions of polyhedral complexes that the composition of toroidal birational maps given by upper W 1 left-arrow upper Z 1 right-arrow upper W 2 and upper W 2 left-arrow upper Z 2 right-arrow upper W 3 is again toroidal. Indeed, if upper Z 1 right-arrow upper W 2 and upper Z 2 right-arrow upper W 2 correspond to two subdivisions of normal upper Delta Subscript upper W 2 , then a common refinement of the two subdivisions corresponds to a toroidal embedding upper Z such that upper Z right-arrow upper Z 1 and upper Z right-arrow upper Z 2 are toroidal morphisms. For example, the coarsest refinement corresponds to taking for upper Z the normalization of the closure of the graph of the birational map upper Z 1 right dasheD arrow upper Z 2 . The composite maps upper Z right-arrow upper W Subscript i are all toroidal birational morphisms.

(2)

It can be shown that a morphism between toroidal embeddings which is a toroidal birational map in the sense of Definition 1.5.4 is a toroidal morphism in the sense of Definition 1.5.2. In other words, Definitions 1.5.2 and 1.5.4 are compatible.

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 Reference82 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 upper Y . 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.

Definition 1.5.5

(1)

A tightly locally toric birational transformation is a proper birational map psi colon upper W 1 right dasheD arrow upper W 2 together with a diagram of birational maps StartLayout 1st Row 1st Column upper W 1 2nd Column Blank 3rd Column right dasheD arrow Overscript psi Endscripts 4th Column Blank 5th Column upper W 2 2nd Row 1st Column Blank 2nd Column down right-arrow 3rd Column Blank 4th Column down left-arrow 5th Column Blank 3rd Row 1st Column Blank 2nd Column Blank 3rd Column upper Y 4th Column Blank 5th Column Blank EndLayout

between locally toric varieties upper W 1 and upper W 2 satisfying the following condition:

For every closed point q element-of upper Y there exist a toric chart eta Subscript q Baseline colon upper V Subscript q Baseline right-arrow upper X Subscript q Baseline at q , and a diagram of fibered squares StartLayout 1st Row 1st Column upper W 1 2nd Column right-arrow 3rd Column upper Y 4th Column left-arrow 5th Column upper W 2 2nd Row 1st Column union 2nd Column Blank 3rd Column union 4th Column Blank 5th Column union 3rd Row 1st Column upper V 1 2nd Column right-arrow 3rd Column upper V Subscript q 4th Column left-arrow 5th Column upper V 2 4th Row 1st Column down-arrow 2nd Column Blank 3rd Column down-arrow 4th Column Blank 5th Column down-arrow 5th Row 1st Column upper X 1 2nd Column right-arrow 3rd Column upper X Subscript q 4th Column left-arrow 5th Column upper X 2 EndLayout

such that

(a)

upper V Subscript i Baseline right-arrow upper X Subscript i are toric charts for upper W Subscript i , i equals 1 comma 2 , and

(b)

upper X Subscript i Baseline right-arrow upper X Subscript q are toric morphisms

(2)

Analogously, let upper U Subscript i Baseline subset-of upper W Subscript i be toroidal embeddings. A tightly toroidal birational transformation between them is a tightly locally toric birational transformation psi colon upper W 1 right dasheD arrow upper W 2 where the toric charts above can be chosen to be toroidal.

Remark

While tightly locally toric birational transformations are essential in our arguments, tightly toroidal transformations are not: the argument used before to show that a composition of toroidal birational maps is toroidal shows that a tightly toroidal birational transformation gives a toroidal birational map. This is the only property of such transformations we will use.

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 Reference4 for toroidal morphisms, using the correspondence between birational toroidal morphisms and subdivisions of polyhedral complexes. The general case of a toroidal birational map upper W 1 left-arrow upper Z right-arrow upper W 2 can be deduced from this, as follows. By toroidal resolution of singularities we may assume upper Z is nonsingular. We apply toroidal weak factorization to the morphisms upper Z right-arrow upper W Subscript i , to get a sequence of toroidal birational maps

upper W 1 equals upper V 1 right dasheD arrow upper V 2 right dasheD arrow ellipsis right dasheD arrow upper V Subscript l minus 1 Baseline right dasheD arrow upper V Subscript l Baseline equals upper Z right dasheD arrow upper V Subscript l plus 1 Baseline right dasheD arrow ellipsis right dasheD arrow upper V Subscript k minus 1 Baseline right dasheD arrow upper V Subscript k Baseline equals upper W 2

consisting of toroidal blowings up and down with nonsingular centers.

We state this result for later reference:

Theorem 1.6.1

Let upper U 1 subset-of upper W 1 and upper U 2 subset-of upper W 2 be nonsingular toroidal embeddings. Let psi colon upper W 1 right dasheD arrow upper W 2 be a proper toroidal birational map. Then phi can be factored into a sequence of toroidal birational maps consisting of toroidal blowings up and down of nonsingular centers in nonsingular toroidal embeddings.

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.

1.7. Locally toric and toroidal actions

Definition 1.7.1 (see Reference60, p. 198).

Let upper V and upper X be varieties with relatively affine upper K Superscript asterisk -actions, and let eta colon upper V right-arrow upper X be a upper K Superscript asterisk -equivariant étale morphism. Then eta is said to be strongly étale if

(i)

the quotient map upper V slash slash upper K Superscript asterisk Baseline right-arrow upper X slash slash upper K Superscript asterisk is étale, and

(ii)

the natural map upper V right-arrow upper X times Underscript upper X slash slash upper K Superscript asterisk Baseline Endscripts upper V slash slash upper K Superscript asterisk

is an isomorphism.

Definition 1.7.2

(1)

Let upper W be a locally toric variety with a upper K Superscript asterisk -action, such that upper W slash slash upper K Superscript asterisk exists. We say that the action is locally toric if for any closed point p element-of upper W we have a toric chart eta Subscript p Baseline colon upper V Subscript p Baseline right-arrow upper X Subscript p Baseline at p and a one-parameter subgroup upper K Superscript asterisk Baseline subset-of upper T Subscript p of the torus in upper X Subscript p , satisfying

upper V Subscript p Baseline equals pi Superscript negative 1 Baseline pi upper V Subscript p , where pi colon upper W right-arrow upper W slash slash upper K Superscript asterisk is the projection;

eta Subscript p is upper K Superscript asterisk -equivariant and strongly étale.

(2)

If upper U subset-of upper W is a toroidal embedding, we say that upper K Superscript asterisk acts toroidally on upper W if the charts above can be chosen toroidal.

The definition above is equivalent to the existence of the following diagram of fiber squares:

StartLayout 1st Row 1st Column upper X Subscript p 2nd Column left-arrow 3rd Column upper V Subscript p 4th Column subset-of 5th Column upper W 2nd Row 1st Column down-arrow 2nd Column Blank 3rd Column down-arrow 4th Column Blank 5th Column f 3rd Row 1st Column upper X slash slash upper K Superscript asterisk 2nd Column left-arrow 3rd Column upper V Subscript p Baseline slash slash upper K Superscript asterisk 4th Column subset-of 5th Column upper W slash slash upper K Superscript asterisk EndLayout

where the horizontal maps provide toric (resp. toroidal) charts in upper W and upper W slash slash upper K Superscript asterisk . It follows that the quotient of a locally toric variety by a locally toric action is again locally toric; the same holds in the toroidal case.

Remark

If we do not insist on the charts being strongly étale, then the morphism of quotients may fail to be étale. Consider, for instance, the space upper X equals upper S p e c upper K left-bracket x comma x Superscript negative 1 Baseline comma y right-bracket with the action t left-parenthesis x comma y right-parenthesis equals left-parenthesis t squared x comma t Superscript negative 1 Baseline y right-parenthesis . The quotient is upper X slash upper K Superscript asterisk Baseline equals upper S p e c upper K left-bracket x y squared right-bracket period There is an equivariant étale cover upper V equals upper S p e c upper K left-bracket u comma u Superscript negative 1 Baseline comma y right-bracket with the action t left-parenthesis u comma y right-parenthesis equals left-parenthesis t u comma t Superscript negative 1 Baseline y right-parenthesis , where the map is defined by x equals u squared . The quotient is upper V slash upper K Superscript asterisk Baseline equals upper S p e c upper K left-bracket u y right-bracket comma which is a branched cover of upper X slash upper K Superscript asterisk , since x y squared equals left-parenthesis u y right-parenthesis squared .

The following lemma shows that locally toric upper K Superscript asterisk -actions are ubiquitous. We note that it can be proven with fewer assumptions; see Reference81, Reference53.

Lemma 1.7.3

Let upper W be a nonsingular variety with a relatively affine upper K Superscript asterisk -action, that is, the scheme upper W slash slash upper K Superscript asterisk exists and the morphism upper W right-arrow upper W slash slash upper K Superscript asterisk is an affine morphism. Then the action of upper K Superscript asterisk on upper W is locally toric.

Proof.

Taking an affine open set in upper W slash slash upper K Superscript asterisk , we may assume that upper W is affine. We embed upper W equivariantly into a projective space and take its completion (see, e.g., Reference75). After applying equivariant resolution of singularities to this completion (see section 1.2) we may also assume that upper W overbar is a nonsingular projective variety with a upper K Superscript asterisk -action, and upper W subset-of upper W overbar is an affine invariant open subset.

Let p element-of upper W be a closed point. Since upper W overbar is complete, the orbit of p has a limit point q equals limit Underscript t right-arrow 0 Endscripts t left-parenthesis p right-parenthesis in upper W overbar . Now q is fixed by upper K Superscript asterisk , hence upper K Superscript asterisk acts on the cotangent space m Subscript q Baseline slash m Subscript q Superscript 2 at q . Since upper K Superscript asterisk is reductive, we can lift a set of eigenvectors of this action to semi-invariant local parameters x 1 comma period period period comma x Subscript n Baseline at q . These local parameters define a upper K Superscript asterisk -equivariant étale morphism eta Subscript q Baseline colon upper V Subscript q Baseline right-arrow upper X Subscript q Baseline from an affine upper K Superscript asterisk invariant open neighborhood upper V Subscript q of q to the tangent space upper X Subscript q Baseline equals upper S p e c left-parenthesis upper S y m m Subscript q Baseline slash m Subscript q Superscript 2 Baseline right-parenthesis at q . The latter has a structure of a toric variety, where the torus is the complement of the zero set of product x Subscript i .

Separating the parameters x Subscript i into upper K Superscript asterisk -invariants and noninvariants, we get a factorization upper X Subscript q Baseline equals upper X Subscript q Superscript 0 Baseline times upper X Subscript q Superscript 1 , where the action of upper K Superscript asterisk on upper X Subscript q Superscript 1 is trivial and the action on upper X Subscript q Superscript 0 has 0 as its unique fixed point. Thus we get a product decomposition upper X Subscript q Baseline slash slash upper K Superscript asterisk Baseline equals upper X Subscript q Superscript 0 Baseline slash slash upper K Superscript asterisk Baseline times upper X Subscript q Superscript 1 .

By Luna’s Fundamental Lemma (Reference51, Lemme 3), there exist affine upper K Superscript asterisk -invariant neighborhoods upper V prime Subscript q of q and upper X prime Subscript q of 0 , such that the restriction eta prime Subscript q Baseline colon upper V prime Subscript q Baseline right-arrow upper X prime Subscript q is strongly étale. Consider first the case q element-of upper W , in which case we may replace p by q . Denote upper Z equals upper X Subscript q Superscript upper K Super Superscript asterisk Baseline intersection upper X prime Subscript q . Then upper Z subset-of upper X Subscript q Superscript upper K Super Superscript asterisk Baseline asymptotically-equals upper X Subscript q Superscript 1 is affine open, and, using the direct product decomposition above, upper X Subscript q Superscript 0 Baseline times upper Z subset-of upper X Subscript q is affine open. Denote upper X double-prime Subscript q Baseline equals upper X prime Subscript q Baseline intersection upper X Subscript q Superscript 0 Baseline times upper Z . This is affine open in upper X Subscript q , and it is easy to see that upper X double-prime Subscript q Baseline slash slash upper K Superscript asterisk Baseline right-arrow upper X Subscript q Baseline slash slash upper K Superscript asterisk is an open embedding: an orbit in upper X double-prime Subscript q is closed if and only if it is closed in upper X Subscript q . Writing upper V double-prime Subscript q Baseline equals eta prime Subscript q Baseline Superscript negative 1 Baseline upper X double-prime Subscript q , it follows that upper V double-prime Subscript q Baseline right-arrow upper X Subscript q is a strongly étale toric chart.

In the case q not-an-element-of upper W , replace upper V Subscript q by upper V double-prime Subscript q . Now eta Subscript q is injective on any orbit, and therefore it is injective on the orbit of p . Let upper X Subscript p Baseline subset-of upper X Subscript q be the affine open toric subvariety in which the torus orbit of eta Subscript q Baseline left-parenthesis p right-parenthesis is closed, and let upper V Subscript p Baseline equals eta Subscript q Superscript negative 1 Baseline upper X Subscript p Baseline intersection upper W . Now consider the restriction eta colon upper V Subscript p Baseline right-arrow upper X Subscript p Baseline , where the upper K Superscript asterisk -orbits of p and eta left-parenthesis p right-parenthesis are closed. By Luna’s Fundamental Lemma there exist affine open upper K Superscript asterisk -invariant neighborhoods upper V prime Subscript p Baseline subset-of upper V Subscript p and upper X prime Subscript p Baseline subset-of upper X Subscript p of eta Subscript p Baseline left-parenthesis p right-parenthesis such that the restriction eta colon upper V prime Subscript p Baseline right-arrow upper X prime Subscript p is a strongly étale morphism. Since upper X Subscript p Baseline slash upper K Superscript asterisk is a geometric quotient, we have an open embedding upper X prime Subscript p Baseline slash upper K Superscript asterisk subset-of upper X Subscript p slash upper K Superscript asterisk and we have a strongly étale toric chart upper V Subscript p Baseline right-arrow upper X Subscript p .

It remains to show that the charts can be chosen saturated with respect to the projection pi colon upper W right-arrow upper W slash slash upper K Superscript asterisk . If the orbit of p has a limit point q equals limit Underscript t right-arrow 0 Endscripts t dot p or q equals limit Underscript t right-arrow normal infinity Endscripts t dot p in upper W , which is necessarily unique as pi is affine, then an equivariant toric chart at q also covers p . So we may replace p by q and assume that the orbit of p is closed. Now pi left-parenthesis upper W minus upper V Subscript p Baseline right-parenthesis is closed and does not contain pi left-parenthesis p right-parenthesis , so we can choose an affine neighborhood upper Y in its complement, and replace upper V Subscript p by pi Superscript negative 1 Baseline upper Y .

2. Birational cobordisms

2.1. Definitions

Definition 2.1.1 (Reference81).

Let phi colon upper X 1 right dasheD arrow upper X 2 be a birational map between two algebraic varieties upper X 1 and upper X 2 over upper K , isomorphic on an open set upper U . A normal algebraic variety upper B is called a birational cobordism for phi and denoted by upper B Subscript phi Baseline left-parenthesis upper X 1 comma upper X 2 right-parenthesis if it satisfies the following conditions:

(1)

The multiplicative group upper K Superscript asterisk acts effectively on upper B equals upper B Subscript phi Baseline left-parenthesis upper X 1 comma upper X 2 right-parenthesis .

(2)

The sets StartLayout 1st Row 1st Column upper B Subscript minus 2nd Column colon equals 3rd Column left-brace x element-of upper B colon 4th Column limit Underscript t right-arrow 0 Endscripts t left-parenthesis x right-parenthesis does not exist in upper B right-brace 2nd Row 1st Column and upper B Subscript plus Baseline 2nd Column colon equals 3rd Column left-brace x element-of upper B colon 4th Column limit Underscript t right-arrow normal infinity Endscripts t left-parenthesis x right-parenthesis does not exist in upper B right-brace EndLayout

are nonempty Zariski open subsets of upper B .

(3)

There are isomorphismsupper B Subscript minus Baseline slash upper K Superscript asterisk Baseline ModifyingAbove right-arrow With tilde upper X 1 and upper B Subscript plus Baseline slash upper K Superscript asterisk Baseline ModifyingAbove right-arrow With tilde upper X 2 period

(4)

Considering the rational map psi colon upper B Subscript minus Baseline right dasheD arrow upper B Subscript plus Baseline induced by the inclusions left-parenthesis upper B Subscript minus Baseline intersection upper B Subscript plus Baseline right-parenthesis subset-of upper B Subscript minus and left-parenthesis upper B Subscript minus Baseline intersection upper B Subscript plus Baseline right-parenthesis subset-of upper B Subscript plus , the following diagram commutes: StartLayout 1st Row 1st Column upper B Subscript minus 2nd Column right dasheD arrow Overscript psi Endscripts 3rd Column upper B Subscript plus 2nd Row 1st Column down-arrow 2nd Column Blank 3rd Column down-arrow 3rd Row 1st Column upper X 1 2nd Column right dasheD arrow Overscript phi Endscripts 3rd Column upper X 2 EndLayout

We say that upper B respects the open set upper U if upper U is contained in the image of left-parenthesis upper B Subscript minus Baseline intersection upper B Subscript plus Baseline right-parenthesis slash upper K Superscript asterisk .

Definition 2.1.2 (Reference81).

Let upper B equals upper B Subscript phi Baseline left-parenthesis upper X 1 comma upper X 2 right-parenthesis be a birational cobordism, and let upper F subset-of upper B Superscript upper K Super Superscript asterisk be a subset of the fixed-point set. We define

StartLayout 1st Row 1st Column upper F Superscript plus 2nd Column equals 3rd Column StartSet x element-of upper B vertical-bar limit Underscript t right-arrow 0 Endscripts t left-parenthesis x right-parenthesis element-of upper F EndSet comma 2nd Row 1st Column upper F Superscript minus 2nd Column equals 3rd Column StartSet x element-of upper B vertical-bar limit Underscript t right-arrow normal infinity Endscripts t left-parenthesis x right-parenthesis element-of upper F EndSet comma 3rd Row 1st Column upper F Superscript plus-or-minus 2nd Column equals 3rd Column upper F Superscript plus Baseline union upper F Superscript minus Baseline period EndLayout

Definition 2.1.3 (Reference81).

Let upper B equals upper B Subscript phi Baseline left-parenthesis upper X 1 comma upper X 2 right-parenthesis be a birational cobordism. We define a relation precedes among connected components of upper B Superscript upper K Super Superscript asterisk as follows: let upper F 1 comma upper F 2 subset-of upper B Superscript upper K Super Superscript asterisk be two connected components, and set upper F 1 precedes upper F 2 if there is a point x not-an-element-of upper B Superscript upper K Super Superscript asterisk such that limit Underscript t right-arrow 0 Endscripts t left-parenthesis x right-parenthesis element-of upper F 1 and limit Underscript t right-arrow normal infinity Endscripts t left-parenthesis x right-parenthesis element-of upper F 2 .

Definition 2.1.4

A birational cobordism upper B equals upper B Subscript phi Baseline left-parenthesis upper X 1 comma upper X 2 right-parenthesis is said to be quasi-elementary if any two connected components upper F 1 comma upper F 2 subset-of upper B Superscript upper K Super Superscript asterisk are incomparable with respect to precedes .

Note that this condition prohibits, in particular, the existence of a “loop”, namely a connected component upper F and a point y not-an-element-of upper F such that both limit Underscript t right-arrow 0 Endscripts t left-parenthesis x right-parenthesis element-of upper F and limit Underscript t right-arrow normal infinity Endscripts t left-parenthesis x right-parenthesis element-of upper F .

Definition 2.1.5 (Reference81).

A quasi-elementary cobordism upper B is said to be elementary if the fixed point set upper B Superscript upper K Super Superscript asterisk is connected.

Definition 2.1.6 (cf. Reference56, Reference81).

We say that a birational cobordism

upper B equals upper B Subscript phi Baseline left-parenthesis upper X 1 comma upper X 2 right-parenthesis

is collapsible if the relation precedes is a strict pre-order, namely, there is no cyclic chain of fixed point components

upper F 1 precedes upper F 2 precedes period period period precedes upper F Subscript m Baseline precedes upper F 1 period

2.2. The main example

We now recall a fundamental example of an elementary birational cobordism in the toric setting, discussed in Reference81:

Example 2.2.1

Let upper B equals double-struck upper A Superscript n Baseline equals upper S p e c upper K left-bracket z 1 comma ellipsis comma z Subscript n Baseline right-bracket and let t element-of upper K Superscript asterisk act by

t left-parenthesis z 1 comma period period period comma z Subscript i Baseline comma period period period comma z Subscript n Baseline right-parenthesis equals left-parenthesis t Superscript alpha 1 Baseline z 1 comma period period period comma t Superscript alpha Super Subscript i Superscript Baseline z Subscript i Baseline comma period period period comma t Superscript alpha Super Subscript n Superscript Baseline z Subscript n Baseline right-parenthesis period

We assume upper K Superscript asterisk acts effectively, namely gcd left-parenthesis alpha 1 comma period period period comma alpha Subscript n Baseline right-parenthesis equals 1 . We regard double-struck upper A Superscript n as a toric variety defined by a lattice upper N approximately-equals double-struck upper Z Superscript n and a nonsingular cone sigma element-of upper N Subscript double-struck upper R generated by the standard basis

sigma equals mathematical left-angle v 1 comma period period period comma v Subscript n Baseline mathematical right-angle period

The dual cone sigma Superscript logical-or is generated by the dual basis v 1 Superscript asterisk Baseline comma period period period comma v Subscript n Superscript asterisk , and we identify z Superscript v Super Subscript i Super Superscript asterisk Baseline equals z Subscript i . The upper K Superscript asterisk -action then corresponds to a one-parameter subgroup

a equals left-parenthesis alpha 1 comma period period period comma alpha Subscript n Baseline right-parenthesis element-of upper N period

We assume that a not-an-element-of plus-or-minus sigma . We have the obvious description of the sets upper B Subscript plus and upper B Subscript minus :

StartLayout 1st Row 1st Column upper B Subscript minus 2nd Column equals 3rd Column StartSet left-parenthesis z 1 comma ellipsis comma z Subscript n Baseline right-parenthesis semicolon z Subscript i Baseline not-equals 0 for some i with alpha Subscript i Baseline equals left-parenthesis v Subscript i Superscript asterisk Baseline comma a right-parenthesis less-than 0 EndSet comma 2nd Row 1st Column upper B Subscript plus 2nd Column equals 3rd Column StartSet left-parenthesis z 1 comma ellipsis comma z Subscript n Baseline right-parenthesis semicolon z Subscript i Baseline not-equals 0 for some i with alpha Subscript i Baseline equals left-parenthesis v Subscript i Superscript asterisk Baseline comma a right-parenthesis greater-than 0 EndSet period EndLayout

We define the upper boundary and lower boundary fans of sigma to be

StartLayout 1st Row 1st Column partial-differential Subscript minus Baseline sigma 2nd Column equals 3rd Column StartSet x element-of sigma semicolon x plus epsilon dot a not-an-element-of sigma for all epsilon greater-than 0 EndSet comma 2nd Row 1st Column partial-differential Subscript plus Baseline sigma 2nd Column equals 3rd Column StartSet x element-of sigma semicolon x plus epsilon dot left-parenthesis negative a right-parenthesis not-an-element-of sigma for all epsilon greater-than 0 EndSet period EndLayout

Then we obtain the description of upper B Subscript plus and upper B Subscript minus as the toric varieties corresponding to the fans partial-differential Subscript plus Baseline sigma and partial-differential Subscript minus Baseline sigma in upper N Subscript double-struck upper R .

Let pi colon upper N Subscript double-struck upper R Baseline right-arrow upper N Subscript double-struck upper R Baseline slash double-struck upper R dot a be the projection. Then upper B slash slash upper K Superscript asterisk is again an affine toric variety defined by the lattice pi left-parenthesis upper N right-parenthesis and cone pi left-parenthesis sigma right-parenthesis . Similarly, one can check that the geometric quotients upper B Subscript minus Baseline slash upper K Superscript asterisk and upper B Subscript plus Baseline slash upper K Superscript asterisk are toric varieties defined by fans pi left-parenthesis partial-differential Subscript plus Baseline sigma right-parenthesis and pi left-parenthesis partial-differential Subscript minus Baseline sigma right-parenthesis . Since both pi left-parenthesis partial-differential Subscript plus Baseline sigma right-parenthesis and pi left-parenthesis partial-differential Subscript minus Baseline sigma right-parenthesis are subdivisions of pi left-parenthesis sigma right-parenthesis , we get a diagram of birational toric maps

StartLayout 1st Row 1st Column upper B Subscript minus Baseline slash upper K Superscript asterisk 2nd Column Blank 3rd Column right dasheD arrow Overscript phi Endscripts 4th Column Blank 5th Column upper B Subscript plus Baseline slash upper K Superscript asterisk 2nd Row 1st Column Blank 2nd Column down right-arrow 3rd Column Blank 4th Column down left-arrow 5th Column Blank 3rd Row 1st Column Blank 2nd Column Blank 3rd Column upper B slash slash upper K Superscript asterisk 4th Column Blank 5th Column Blank EndLayout

It is easy to see (see, e.g., Reference81) that the varieties upper B Subscript plus-or-minus Baseline slash upper K Superscript asterisk have only abelian quotient singularities. Moreover, the map phi can be factored as a weighted blowing up followed by a weighted blowing down.

More generally, one can prove that if normal upper Sigma is a subdivision of a convex polyhedral cone in upper N Subscript double-struck upper R with lower boundary partial-differential Subscript minus Baseline normal upper Sigma and upper boundary partial-differential Subscript plus Baseline normal upper Sigma relative to an element a element-of upper N minus plus-or-minus normal upper Sigma , then the toric variety corresponding to normal upper Sigma , with the upper K Superscript asterisk -action given by the one-parameter subgroup a element-of upper N , is a birational cobordism between the two toric varieties corresponding to pi left-parenthesis partial-differential Subscript minus Baseline normal upper Sigma right-parenthesis and pi left-parenthesis partial-differential Subscript plus Baseline normal upper Sigma right-parenthesis as fans in upper N Subscript double-struck upper R Baseline slash double-struck upper R dot a .

For the details, we refer the reader to Reference56, Reference81 and Reference4.

2.3. Construction of a cobordism

It was shown in Reference81 that birational cobordisms exist for a large class of birational maps upper X 1 right dasheD arrow upper X 2 . Here we deal with a very special case.

Theorem 2.3.1

Let phi colon upper X 1 right-arrow upper X 2 be a projective birational morphism between complete nonsingular algebraic varieties, which is an isomorphism on an open set upper U . Then there is a complete nonsingular algebraic variety upper B overbar with an effective upper K Superscript asterisk -action, satisfying the following properties:

(1)

There exist closed embeddings iota 1 colon upper X 1 right-arrow with hook upper B overbar Superscript upper K Super Superscript asterisk and iota 2 colon upper X 2 right-arrow with hook upper B overbar Superscript upper K Super Superscript asterisk with disjoint images.

(2)

The open subvariety upper B equals upper B overbar minus left-parenthesis iota 1 left-parenthesis upper X 1 right-parenthesis union iota 2 left-parenthesis upper X 2 right-parenthesis right-parenthesis is a birational cobordism between upper X 1 and upper X 2 respecting the open set upper U .

(3)

There is a coherent sheaf upper E on upper X 2 , with a upper K Superscript asterisk -action, and a closed upper K Superscript asterisk -equivariant embedding upper B overbar subset-of double-struck upper P left-parenthesis upper E right-parenthesis colon equals script upper P r o j Subscript upper X 2 Baseline upper S y m upper E .

Proof.

Let upper J subset-of script upper O Subscript upper X 2 be an ideal sheaf such that phi colon upper X 1 right-arrow upper X 2 is the blowing up morphism of upper X 2 along upper J and upper J Subscript upper U Baseline equals script upper O Subscript upper U . Let upper I 0 be the ideal of the point 0 element-of double-struck upper P Superscript 1 . Consider upper W 0 equals upper X 2 times double-struck upper P Superscript 1 and let p colon upper W 0 right-arrow upper X 2 and q colon upper W 0 right-arrow double-struck upper P Superscript 1 be the projections. Let upper I equals left-parenthesis p Superscript negative 1 Baseline upper J plus q Superscript negative 1 Baseline upper I 0 right-parenthesis script upper O Subscript upper W 0 . Let upper W be the blowing up of upper W 0 along upper I . (Paolo Aluffi has pointed out that this upper W is used when constructing the deformation to the normal cone of upper J .)

We claim that upper X 1 and upper X 2 lie in the nonsingular locus of upper W . For upper X 2 approximately-equals upper X 2 times StartSet normal infinity EndSet subset-of upper X 2 times double-struck upper A Superscript 1 Baseline subset-of upper W this is clear. Since upper X 1 is nonsingular, embedded in upper W as the strict transform of upper X 2 times StartSet 0 EndSet subset-of upper X 2 times double-struck upper P Superscript 1 , to prove that upper X 1 lies in the nonsingular locus it suffices to prove that upper X 1 is a Cartier divisor in upper W . We look at local coordinates. Let upper A equals normal upper Gamma left-parenthesis upper V comma script upper O Subscript upper V Baseline right-parenthesis for some affine open subset upper V subset-of upper X 2 , and let y 1 comma period period period comma y Subscript m Baseline be a set of generators of upper J on upper V . Then on the affine open subset upper V times double-struck upper A Superscript 1 subset-of upper X 2 times double-struck upper P Superscript 1 with coordinate ring upper A left-bracket x right-bracket , the ideal upper I is generated by y 1 comma period period period comma y Subscript m Baseline comma x . The charts of the blowing up containing the strict transform of StartSet x equals 0 EndSet are of the form

upper S p e c upper A left-bracket StartFraction y 1 Over y Subscript i Baseline EndFraction comma period period period comma StartFraction y Subscript m Baseline Over y Subscript i Baseline EndFraction comma StartFraction x Over y Subscript i Baseline EndFraction right-bracket equals upper S p e c upper A left-bracket StartFraction y 1 Over y Subscript i Baseline EndFraction comma period period period comma StartFraction y Subscript m Baseline Over y Subscript i Baseline EndFraction right-bracket times upper S p e c upper K left-bracket StartFraction x Over y Subscript i Baseline EndFraction right-bracket comma

where upper K Superscript asterisk acts on the second factor. The strict transform of StartSet x equals 0 EndSet is defined by StartFraction x Over y Subscript i Baseline EndFraction , hence it is Cartier.

Let upper B overbar right-arrow upper W be a canonical resolution of singularities. Then conditions (1) and (2) are clearly satisfied. For condition (3), note that upper B overbar right-arrow upper X 2 times double-struck upper P Superscript 1 , being a composition of blowings up of invariant ideals, admits an equivariant ample line bundle. Twisting by the pullback of script upper O Subscript double-struck upper P Sub Superscript 1 Baseline left-parenthesis n right-parenthesis we obtain an equivariant line bundle which is ample for upper B overbar right-arrow upper X 2 . Replacing this by a sufficiently high power and pushing forward we get upper E .

We refer the reader to Reference81 for more details.

We call a variety upper B overbar as in the theorem a compactified, relatively projective cobordism.

2.4. Collapsibility and projectivity

Let upper B equals upper B Subscript phi Baseline left-parenthesis upper X 1 comma upper X 2 right-parenthesis be a birational cobordism. We seek a criterion for collapsibility of upper B .

Let script upper C be the set of connected components of upper B Subscript phi Baseline left-parenthesis upper X 1 comma upper X 2 right-parenthesis Superscript upper K Super Superscript asterisk , and let chi colon script upper C right-arrow double-struck upper Z be a function. We say that chi is strictly increasing if upper F precedes upper F prime right double arrow chi left-parenthesis upper F right-parenthesis less-than chi left-parenthesis upper F prime right-parenthesis . The following lemma is obvious:

Lemma 2.4.1

Assume there exists a strictly increasing function chi . Then precedes is a strict pre-order, and upper B is collapsible. Conversely, suppose upper B is collapsible. Then there exists a strictly increasing function chi .

Remark

It is evident that every strictly increasing function can be replaced by one which induces a strict total order. However, it will be convenient for us to consider arbitrary strictly increasing functions.

Let chi be a strictly increasing function, and let a 1 less-than a 2 less-than ellipsis less-than a Subscript m Baseline element-of double-struck upper Z be the values of chi .

Definition 2.4.2

We denote

(1)

upper F Subscript a Sub Subscript i Baseline equals union StartSet upper F vertical-bar chi left-parenthesis upper F right-parenthesis equals a Subscript i Baseline EndSet .

(2)

upper F Subscript a Sub Subscript i Superscript plus Baseline equals union StartSet upper F Superscript plus Baseline vertical-bar chi left-parenthesis upper F right-parenthesis equals a Subscript i Baseline EndSet .

(3)

upper F Subscript a Sub Subscript i Superscript minus Baseline equals union StartSet upper F Superscript minus Baseline vertical-bar chi left-parenthesis upper F right-parenthesis equals a Subscript i Baseline EndSet .

(4)

upper F Subscript a Sub Subscript i Superscript plus-or-minus Baseline equals union StartSet upper F Superscript plus-or-minus Baseline vertical-bar chi left-parenthesis upper F right-parenthesis equals a Subscript i Baseline EndSet .

(5)

upper B Subscript a Sub Subscript i Baseline equals upper B minus left-parenthesis union StartSet upper F Superscript minus Baseline vertical-bar chi left-parenthesis upper F right-parenthesis less-than a Subscript i Baseline EndSet union union StartSet upper F Superscript plus Baseline vertical-bar chi left-parenthesis upper F right-parenthesis greater-than a Subscript i Baseline EndSet right-parenthesis .

The following is an immediate extension of Proposition 1 of Reference81.

Proposition 2.4.3

(1)

upper B Subscript a Sub Subscript i is a quasi-elementary cobordism.

(2)

For i equals 1 comma period period period comma m minus 1 we have left-parenthesis upper B Subscript a Sub Subscript i Subscript Baseline right-parenthesis Subscript plus Baseline equals left-parenthesis upper B Subscript a Sub Subscript i plus 1 Subscript Baseline right-parenthesis Subscript minus .

The following is an analogue of Lemma 1 of Reference81 in the case of the cobordisms we have constructed.

Proposition 2.4.4

Let upper E be a coherent sheaf on upper X 2 with a upper K Superscript asterisk -action, and let upper B overbar subset-of double-struck upper P left-parenthesis upper E right-parenthesis be a compactified, relatively projective cobordism embedded upper K Superscript asterisk -equivariantly. Then there exists a strictly increasing function chi for the cobordism upper B equals upper B overbar minus left-parenthesis upper X 1 union upper X 2 right-parenthesis . In particular, the cobordism is collapsible.

Proof.

Since upper K Superscript asterisk acts trivially on upper X 2 , and since upper K Superscript asterisk is reductive, there exists a direct sum decomposition

upper E equals circled-plus Underscript b element-of double-struck upper Z Endscripts upper E Subscript b

where upper E Subscript b is the subsheaf on which the action of upper K Superscript asterisk is given by the character t right-arrow from bar t Superscript b . Denote by b 0 comma period period period comma b Subscript k Baseline the characters which figure in this representation. Note that there are disjoint embeddings double-struck upper P left-parenthesis upper E Subscript b Sub Subscript j Subscript Baseline right-parenthesis subset-of double-struck upper P left-parenthesis upper E right-parenthesis .

Let p element-of upper B be a fixed point lying in the fiber double-struck upper P left-parenthesis upper E Subscript q Baseline right-parenthesis over q element-of upper X 2 . We choose a basis

left-parenthesis x Subscript b 0 comma 1 Baseline comma period period period comma x Subscript b 0 comma d 0 Baseline comma period period period comma x Subscript b Sub Subscript k Subscript comma 1 Baseline comma period period period comma x Subscript b Sub Subscript k Subscript comma d Sub Subscript k Subscript Baseline right-parenthesis

of upper E Subscript q where x Subscript b Sub Subscript j Subscript comma nu Baseline element-of upper E Subscript b Sub Subscript j and use the following lemma:

Lemma 2.4.5

Suppose p element-of double-struck upper P left-parenthesis upper E Subscript q Baseline right-parenthesis Superscript upper K Super Superscript asterisk is a fixed point with homogeneous coordinates

left-parenthesis p Subscript b 0 comma 1 Baseline comma period period period comma p Subscript b 0 comma d 0 Baseline comma period period period comma p Subscript b Sub Subscript k Subscript comma 1 Baseline comma period period period comma p Subscript b Sub Subscript k Subscript comma d Sub Subscript k Subscript Baseline right-parenthesis period

Then there is a j Subscript p such that p Subscript b Sub Subscript j Subscript comma nu Baseline equals 0 whenever j not-equals j Subscript p . In particular, p element-of double-struck upper P left-parenthesis upper E Subscript b Sub Subscript j Sub Sub Subscript p Sub Subscript Subscript Baseline right-parenthesis subset-of double-struck upper P left-parenthesis upper E right-parenthesis .

If upper F subset-of upper B Superscript upper K Super Superscript asterisk is a connected component of the fixed point set, then it follows from the lemma that upper F subset-of double-struck upper P left-parenthesis upper E Subscript b Sub Subscript j Subscript Baseline right-parenthesis for some j . We define

chi left-parenthesis upper F right-parenthesis equals b Subscript j Baseline period

To check that chi is strictly increasing, consider a point p element-of upper B such that limit Underscript t right-arrow 0 Endscripts t left-parenthesis p right-parenthesis element-of upper F 1 and limit Underscript t right-arrow normal infinity Endscripts t left-parenthesis p right-parenthesis element-of upper F 2 for some fixed point components upper F 1 and upper F 2 . Let the coordinates of p in the fiber over q element-of upper X 2 be left-parenthesis p Subscript b 0 comma 1 Baseline comma period period period comma p Subscript b 0 comma d 0 Baseline comma period period period comma p Subscript b Sub Subscript k Subscript comma 1 Baseline comma period period period comma p Subscript b Sub Subscript k Subscript comma d Sub Subscript k Subscript Baseline right-parenthesis . Now

StartLayout 1st Row 1st Column limit Underscript t right-arrow 0 Endscripts t left-parenthesis p right-parenthesis 2nd Column element-of double-struck upper P left-parenthesis upper E Subscript b Sub Subscript m i n Subscript Baseline right-parenthesis comma 2nd Row 1st Column limit Underscript t right-arrow normal infinity Endscripts t left-parenthesis p right-parenthesis 2nd Column element-of double-struck upper P left-parenthesis upper E Subscript b Sub Subscript m a x Subscript Baseline right-parenthesis comma EndLayout

where

StartLayout 1st Row b Subscript m i n Baseline equals min left-brace b Subscript j Baseline colon p Subscript b Sub Subscript j Subscript comma nu Baseline not-equals 0 for some nu right-brace comma 2nd Row b Subscript m a x Baseline equals max left-brace b Subscript j Baseline colon p Subscript b Sub Subscript j Subscript comma nu Baseline not-equals 0 for some nu right-brace period EndLayout

Thus, if p is not fixed by upper K Superscript asterisk , then

chi left-parenthesis upper F 1 right-parenthesis equals b Subscript m i n Baseline less-than b Subscript m a x Baseline equals chi left-parenthesis upper F 2 right-parenthesis period

2.5. Geometric invariant theory and projectivity

In this section we use geometric invariant theory and ideas (originating in symplectic geometry) developed by M. Thaddeus and others (see, e.g., Reference78), in order to obtain a result about relative projectivity of quotients.

We continue with the notation of the last section. Consider the sheaf upper E and its decomposition according to the character. Let StartSet b Subscript j Baseline EndSet be the characters of the action of upper K Superscript asterisk on upper E and StartSet a Subscript i Baseline EndSet the subset of those b Subscript j that are in the image of chi . If we use the Veronese embedding upper B overbar subset-of double-struck upper P left-parenthesis upper S y m squared left-parenthesis upper E right-parenthesis right-parenthesis and replace upper E by upper S y m squared left-parenthesis upper E right-parenthesis , we may assume that a Subscript i are even, in particular a Subscript i plus 1 Baseline greater-than a Subscript i Baseline plus 1 (this is a technical condition which comes in handy in what follows).

Denote by rho 0 left-parenthesis t right-parenthesis the action of t element-of upper K Superscript asterisk on upper E . For any r element-of double-struck upper Z consider the “twisted” action rho Subscript r Baseline left-parenthesis t right-parenthesis equals t Superscript negative r Baseline dot rho 0 left-parenthesis t right-parenthesis . Note that the induced action on double-struck upper P left-parenthesis upper E right-parenthesis does not depend on the “twist” r . Considering the decomposition upper E equals circled-plus upper E Subscript b Sub Subscript j , we see that rho Subscript r Baseline left-parenthesis t right-parenthesis acts on upper E Subscript b Sub Subscript j by multiplication by t Superscript b Super Subscript j Superscript minus r .

We can apply geometric invariant theory in its relative form (see, e.g., Reference63, Reference33) to the action rho Subscript r Baseline left-parenthesis t right-parenthesis of upper K Superscript asterisk . Recall that a point p element-of double-struck upper P left-parenthesis upper E right-parenthesis is said to be semistable with respect to rho Subscript r , written p element-of left-parenthesis double-struck upper P left-parenthesis upper E right-parenthesis comma rho Subscript r Baseline right-parenthesis Superscript s s , if there is a positive integer n and a rho Subscript r -invariant local section s element-of left-parenthesis upper S y m Superscript n Baseline left-parenthesis upper E right-parenthesis right-parenthesis Superscript rho Super Subscript r , such that s left-parenthesis p right-parenthesis not-equals 0 . The main result of geometric invariant theory implies that

script upper P times r times o times j Underscript upper X 2 Endscripts circled-plus Underscript n greater-than-or-equal-to 0 Overscript normal infinity Endscripts left-parenthesis upper S y m Superscript n Baseline left-parenthesis upper E right-parenthesis right-parenthesis Superscript rho Super Subscript r Superscript Baseline equals left-parenthesis double-struck upper P left-parenthesis upper E right-parenthesis comma rho Subscript r Baseline right-parenthesis Superscript s s Baseline slash slash upper K Superscript asterisk Baseline semicolon

moreover, the quotient map left-parenthesis double-struck upper P left-parenthesis upper E right-parenthesis comma rho Subscript r Baseline right-parenthesis Superscript s s Baseline right-arrow left-parenthesis double-struck upper P left-parenthesis upper E right-parenthesis comma rho Subscript r Baseline right-parenthesis Superscript s s Baseline slash slash upper K Superscript asterisk is affine. We can define left-parenthesis upper B overbar comma rho Subscript r Baseline right-parenthesis Superscript s s analogously, and we automatically have left-parenthesis upper B overbar comma rho Subscript r Baseline right-parenthesis Superscript s s Baseline equals upper B overbar intersection left-parenthesis double-struck upper P left-parenthesis upper E right-parenthesis comma rho Subscript r Baseline right-parenthesis Superscript s s .

The numerical criterion of semistability (see Reference60) immediately implies the following:

Lemma 2.5.1

For 0 less-than-or-equal-to i less-than-or-equal-to m we have

(1)

left-parenthesis upper B overbar comma rho Subscript a Sub Subscript i Subscript Baseline right-parenthesis Superscript s s Baseline equals upper B Subscript a Sub Subscript i .

(2)

left-parenthesis upper B overbar comma rho Subscript a Sub Subscript i Subscript plus 1 Baseline right-parenthesis Superscript s s Baseline equals left-parenthesis upper B Subscript a Sub Subscript i Subscript Baseline right-parenthesis Subscript plus .

(3)

left-parenthesis upper B overbar comma rho Subscript a Sub Subscript i Subscript minus 1 Baseline right-parenthesis Superscript s s Baseline equals left-parenthesis upper B Subscript a Sub Subscript i Subscript Baseline right-parenthesis Subscript minus .

In other words, the triangle of birational maps

StartLayout 1st Row 1st Column left-parenthesis upper B Subscript a Sub Subscript i Subscript Baseline right-parenthesis Subscript minus Baseline slash upper K Superscript asterisk 2nd Column Blank 3rd Column right dasheD arrow Overscript phi Subscript i Endscripts 4th Column Blank 5th Column left-parenthesis upper B Subscript a Sub Subscript i Subscript Baseline right-parenthesis Subscript plus Baseline slash upper K Superscript asterisk 2nd Row 1st Column Blank 2nd Column down right-arrow 3rd Column Blank 4th Column down left-arrow 5th Column Blank 3rd Row 1st Column Blank 2nd Column Blank 3rd Column upper B Subscript a Sub Subscript i Baseline slash slash upper K Superscript asterisk 4th Column Blank 5th Column Blank EndLayout

is induced by a change of linearization of the action of upper K Superscript asterisk .

In particular we obtain:

Proposition 2.5.2

The morphisms left-parenthesis upper B Subscript a Sub Subscript i Subscript Baseline right-parenthesis Subscript plus Baseline slash upper K Superscript asterisk Baseline right-arrow upper X 2 , left-parenthesis upper B Subscript a Sub Subscript i Subscript Baseline right-parenthesis Subscript minus Baseline slash upper K Superscript asterisk Baseline right-arrow upper X 2 and upper B Subscript a Sub Subscript i Baseline slash slash upper K Superscript asterisk Baseline right-arrow upper X 2 are projective.

2.6. The main result of Reference81

Let upper B be a collapsible nonsingular birational cobordism. Then we can write upper B as a union of quasi-elementary cobordisms upper B equals union Underscript i Endscripts upper B Subscript a Sub Subscript i , with left-parenthesis upper B Subscript a Sub Subscript i Subscript Baseline right-parenthesis Subscript plus Baseline equals left-parenthesis upper B Subscript a Sub Subscript i plus 1 Subscript Baseline right-parenthesis Subscript minus . By Lemma 1.7.3 each upper B Subscript a Sub Subscript i has a locally toric structure such that the action of