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 $K$ of characteristic zero is a composite of blowings up and blowings down with nonsingular centers.
0. Introduction
We work over an algebraically closed field $K$ of characteristic 0. We denote the multiplicative group of $K$ by $K^*$.
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 $\phi :X_1 \dashrightarrow X_2$ be a birational map between complete nonsingular algebraic varieties $X_1$ and $X_2$ over an algebraically closed field $K$ of characteristic zero, and let $U\subset X_1$ 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 $U$, namely, there exists a sequence of birational maps between complete nonsingular algebraic varieties
either $\varphi _i:V_{i-1} \dashrightarrow V_{i}$ or $\varphi _i^{-1}:V_{i} \dashrightarrow V_{i-1}$ is a morphism obtained by blowing up a nonsingular irreducible center disjoint from $U$.
Furthermore, there is an index $i_0$ such that for all $i\leq i_0$ the map $V_i\dashrightarrow X_1$ is a projective morphism, and for all $i\geq i_0$ the map $V_i\dashrightarrow X_2$ is a projective morphism. In particular, if $X_1$ and $X_2$ are projective, then all the $V_i$ are projective.
0.2. Strong factorization
If we insist in the assertion above that $\varphi _1^{-1},\mathinner {\ldotp \ldotp \ldotp },\varphi _{i_0}^{-1}$ and $\varphi _{i_0+1},\mathinner {\ldotp \ldotp \ldotp },\varphi _l$ 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
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 $L$ of characteristic 0, and broken arrows $X \dashrightarrow Y$ denote birational $L$-maps, and
(2)
the objects are compact complex manifolds, and broken arrows $X \dashrightarrow Y$ denote bimeromorphic maps.
Given two broken arrows $\phi : X\dashrightarrow Y$ and $\phi ':X'\dashrightarrow Y'$ we define an absolute isomorphism$g:\phi \to \phi '$ as follows:
•
In the case $X$ and $Y$ are algebraic spaces over $L$, and $X'$,$Y'$ are over $L'$, then $g$ consists of an isomorphism $\sigma :\operatorname {Spec}L \to \operatorname {Spec}L'$, together with a pair of biregular $\sigma$-isomorphisms$g_X : X \to X'$ and $g_Y: Y \to Y'$, such that $\phi '\circ g_X = g_Y\circ \phi$.
•
In the analytic case, $g$ simply consists of a pair of biregular isomorphisms $g_X : X \to X'$ and $g_Y: Y \to Y'$, such that $\phi '\circ g_X = g_Y\circ \phi$.
Theorem 0.3.1
Let $\phi :X_1 \dashrightarrow X_2$ be as in case (1) or (2) above. Let $U\subset X_1$ 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 $U$. Namely, to any such $\phi$ we associate a diagram in the corresponding category
either $\varphi _i:V_{i-1} \dashrightarrow V_{i}$ or $\varphi _i^{-1}:V_{i} \dashrightarrow V_{i-1}$ is a morphism obtained by blowing up a nonsingular center disjoint from $U$.
(4)
Functoriality: if $g:\phi \to \phi '$ is an absolute isomorphism, carrying $U$ to $U'$, and $\varphi _i':V_{i-1}' \dashrightarrow V_{i}'$ is the factorization of $\phi '$, then the resulting rational maps $g_i:V_i\dashrightarrow V_i'$ give absolute isomorphisms.
(5)
Moreover, there is an index $i_0$ such that for all $i\leq i_0$ the map $V_i\dashrightarrow X_1$ is a projective morphism, and for all $i\geq i_0$ the map $V_i\dashrightarrow X_2$ is a projective morphism.
(6)
Let $E_i\subset V_i$ be the exceptional divisor of $V_i \to X_1\ ($respectively,$V_i \to X_2)$ in case $i\leq i_0\ ($respectively,$i\geq i_0)$. Then the above centers of blowing up in $V_i$ have normal crossings with $E_i$. If, moreover, $X_1\smallsetminus U\ ($respectively,$X_2\smallsetminus U)$ 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 $G$, of a $G$-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 $p>0$assuming that canonical embedded resolution of singularities holds true for varieties or algebraic spaces of dimension $d+1$ 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 $H^i(X,{\mathcal{O}}_X)$ in characteristic 0; abelian varieties for the birational invariance of $H^1(X, {\mathcal{O}}_X)$ 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$'$)” 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 $X_1$ and $X_2$ 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 $\phi :X_1 \dashrightarrow X_2$, a birational cobordism$B_{\phi }(X_1,X_2)$ is a variety of dimension $\dim (X_1)+1$ with an action of the multiplicative group $K^*$. It is analogous to the usual cobordism $B(M_1,M_2)$ between differentiable manifolds $M_1$ and $M_2$ given by a Morse function $f$ (and in fact in the Kähler case the momentum map of ${\mathbb{C}}^*$ is a Morse function, making the analogy more direct). In the differential setting one can construct an action of the additive real group ${\mathbb{R}}$, where the “time” $t\in {\mathbb{R}}$ acts as a diffeomorphism induced by integrating the vector field $grad(f)$; hence the multiplicative group $({\mathbb{R}}_{>0},\times )= \exp ({\mathbb{R}},+)$ 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 $K^*$-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 $X_1$ or $X_2$. 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 $B_{\phi }(X_1,X_2)$ along torific ideals. This operation induces a toroidal structure in a neighborhood of each connected component $F$ of the fixed point set, on which the action of $K^*$ is a toroidal action (we say that the blowing up torifies the action of $K^*$). Now the birational transformation “passing through $F$” 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 $\geq 2$ 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 $X \to B$, there are modifications $m_X:X' \to X$ and $m_B:B' \to B$, with a lifting $X'\to B'$ which has a toroidal structure. The toroidalization problem (see Reference3, Reference4, Reference43) is that of obtaining such $m_X$ and $m_B$ which are composites of blowings up with nonsingular centers (maybe even with centers supported only over the locus where $X \to B$ 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.
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 $\phi :X_1 \to X_2$. 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 $B$ is relatively projective over $X_2$, 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 $X_2$.
In section 3 we utilize a factorization of the cobordism $B$ into quasi-elementary pieces $B_{a_i}$, and for each piece construct an ideal sheaf $I$ (Definition 3.1.4) whose blowing up torifies the action of $K^*$ on $B_{a_i}$ (Proposition 3.2.5). In other words, $K^*$ acts toroidally on the variety obtained by blowing up $B_{a_i}$ along $I$.
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 $G$ acts on an algebraic variety $X$. We denote by $X/G$ the space of orbits, and by $X /\!/ G$ 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 $X /\!/ G$ is called a categorical quotient and the space $X/G$ is called a geometric quotient.
A special case where $X /\!/ G$ exists as a scheme is the following: suppose there is an affine $G$-invariant morphism $\pi : X \to Y$. Then we have $X /\!/ G={{\mathcal{S}}}pec_Y \left( (\pi _*{\mathcal{O}}_X)^G \right)$. When this condition holds we say that the action of $G$ on $X$ is relatively affine.
A particular case of this occurs in geometric invariant theory (discussed in section 2.5), where the action of $G$ 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$\widetilde{W}\to W$ we mean a desingularization procedure uniquely associating to $W$ 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 $W\subset U$ is a closed embedding with $U$ nonsingular. Denote by $E_i$ the exceptional divisor at some stage of the blowing up. Then (a)$E_i$ is a normal crossings divisor, and has normal crossings with the center of blowing up, and (b) at the last stage $\widetilde{W}$ has normal crossings with $E_i$.
(2)
“Canonical” means “functorial with respect to smooth morphisms and field extensions”, namely, if $\theta :V\to W$ 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 $\widetilde{\theta }:\widetilde{V}\to \widetilde{W}$.
In particular: (a) if $\theta :W \to W$ is an automorphism (of schemes, not necessarily over $K$), then it can be lifted to an automorphism $\widetilde{W}\to \widetilde{W}$, and (b) the canonical resolution behaves well with respect to étale morphisms: if $V\to W$ is étale, we get an étale morphism of canonical resolutions $\widetilde{V}\to \widetilde{W}$.
An important consequence of these conditions is that all the centers of blowing up lie over the singular locus of $W$.
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 $W \subset U$ is embedded in a nonsingular variety, and $D\subset U$ 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 $D_i+E_i$, where $D_i$ is the inverse image of $D$. 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 $\widetilde{W}$ at the last step.
1.2.4. Elimination of indeterminacies
Now let $\phi :W_1\dashrightarrow W_2$ be a birational map and $U\subset W_1$ an open set on which $\phi$ restricts to a morphism. By elimination of indeterminacies of $\phi$ we mean a morphism $e:W_1'\to W_1$, obtained by a sequence of blowings up with nonsingular centers disjoint from $U$, such that the birational map $\phi \circ e$ is a morphism.
Elimination of indeterminacies can be reduced to principalization of an ideal sheaf: if one is given an ideal sheaf $I$ on $W_1$ with blowing up $W_1'' = Bl_I(W_1)$ such that the birational map $W_1'' \to W_2$ is a morphism, and if $W_1'\to W_1$ is the result of principalization of $I$, then the birational map $W_1' \to W_1''$ is a morphism, therefore the same is true for $W_1' \to W_2$. If the support of the ideal $I$ is disjoint from the open set $U$ where $\phi$ is an morphism, then the centers of blowing up giving $W_1' \to W_1$ are disjoint from $U$.
Proving that such an ideal $I$ exists (say, in the nonprojective case), and in a sufficiently natural manner for proving functoriality (even if $W_i$ are projective), is nontrivial. We make use of Hironaka’s version of Chow’s lemma, as follows.
We may assume that $\phi ^{-1}$ is a morphism; otherwise we replace $W_2$ 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 $I$ on $W_1$ such that the blowing up of $W_1$ along $I$ factors through $W_2$. Hence the canonical principalization of $I$ also factors through $W_2$.
Although it is not explicitly stated by Hironaka, the ideal $I$ is the unit ideal in the complement of the open set $U$: the blowing up of $I$ consists of a sequence of permissible blowings up (Reference31, Definition 4.4.3, p. 537), each of which is supported in the complement of $U$. Another important fact is that the ideal $I$ is invariant, namely, it is functorial under absolute isomorphisms: if $\phi ':W_1'\dashrightarrow W_2'$ is another proper birational map, with corresponding ideal $I'$, and $\theta _i:W_i \to W_i'$ are isomorphisms such that $\phi '\circ \theta _1 = \theta _2\circ \phi$, then $\theta _1^*I' = I$. 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 $I$, 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).
such that $g_1$ and $g_2$ are composites of blowings up with nonsingular centers disjoint from $U$, and $\phi '$ is a projective birational morphism.
Proof.
By Hironaka’s theorem on elimination of indeterminacies (see 1.2.4 above), there is a morphism $g_2:X_2' \to X_2$ which is a composite of blowings up with nonsingular centers disjoint from $U$, such that the birational map $h:=\phi ^{-1}\circ g_2 : X_2' \to X_1$ is a morphism:
By the same theorem, there is a morphism $g_1:X_1' \to X_1$ which is a composite of blowings up with nonsingular centers disjoint from $U$, such that $\phi ':=h^{-1}\circ g_1: X_1' \to X_2'$ is a morphism. Since the composite $h\circ \phi ' = g_1$ is projective, it follows that $\phi '$ is projective.
■
Thus we may replace $X_1 \dashrightarrow X_2$ by $X_1' \to X_2'$ 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 $\phi ':X_1' \to X_2'$ 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 $N\cong {\mathbb{Z}}^n$ be a lattice and $\sigma \subset N_{\mathbb{R}}$ a strictly convex rational polyhedral cone. We denote the dual lattice by $M$ and the dual cone by $\sigma ^\vee \subset M_{\mathbb{R}}$. The affine toric variety$X=X(N,\sigma )$ is defined as
For $m\in M\cap \sigma ^\vee$ we denote its image in the semigroup algebra $K[M\cap \sigma ^\vee ]$ by $z^m$.
More generally, the toric variety corresponding to a fan $\Sigma$ in $N_{\mathbb{R}}$ is denoted by $X(N,\Sigma )$; see Reference26, Reference62.
If $X_1=X(N,\Sigma _1)$ and $X_2=X(N,\Sigma _2)$ are two toric varieties, the embeddings of the torus $T=\operatorname {Spec}K[M]$ in both of them define a toric (i.e., $T$-equivariant) birational map $X_1\dashrightarrow X_2$.
Suppose $K^*$ acts effectively on an affine toric variety $X=X(N,\sigma )$ as a one-parameter subgroup of the torus $T$, corresponding to a primitive lattice point $a\in N$. If $t\in K^*$ and $m\in M$, the action on the monomial $z^m$ is given by
If $a\notin \pm \sigma$, then $\sigma ^\vee \cap a^\perp$ is a full-dimensional cone in $a^\perp$, and it follows that $X/\!/K^*$ is again an affine toric variety, defined by the lattice $\pi (N)$ and cone $\pi (\sigma )$, where $\pi : N_{\mathbb{R}}\to N_{\mathbb{R}}/{\mathbb{R}}\cdot a$ is the projection. This quotient is a geometric quotient precisely when $\pi : \sigma \to \pi (\sigma )$ 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 $W$ is locally toric if for every closed point $p\in W$ there exists an open neighborhood $V_p\subset W$ of $p$ and an étale morphism $\eta _p:V_p\to X_p$ to a toric variety $X_p$. Such a morphism $\eta _p$ is called a toric chart at $p$.
(2)
An open embedding $U\subset W$ is a toroidal embedding if for every closed point $p\in W$ there exists a toric chart $\eta _p:V_p\to X_p$ at $p$ such that $U\cap V_p = \eta _p^{-1}(T)$, where $T\subset X_p$ is the torus. We call such charts toroidal. Sometimes we omit the open set $U$ 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 $D\subset W$ if $\eta _p^{-1}(T) \cap D = \emptyset$, i.e., $D$ corresponds to a toric divisor on $X_p$.
A toroidal embedding $U \subset X$ can equivalently be specified by the pair $(X, D_X)$, where $D_X$ is the reduced Weil divisor supported on $X \smallsetminus U$. We will sometimes interchange between $U \subset X$ and $(X, D_X)$ for denoting a toroidal structure on $X$. A divisor $D'$ is compatible with the toroidal structure $(X, D_X)$ if it is supported in $D_X$.
For example, the affine line ${\mathbb{A}}^1$ is clearly locally toric, ${\mathbb{A}}^1\smallsetminus \{0\} \subset {\mathbb{A}}^1$ is a toroidal embedding, and ${\mathbb{A}}^1\subset {\mathbb{A}}^1$ 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 $U_i \subset W_i$$(i=1,2)$ be toroidal embeddings. A proper birational morphism $f: W_1 \to W_2$ is said to be toroidal if, for every closed point $q\in W_2$ and any $p\in f^{-1}q$, there is a diagram of fiber squares
$\eta _{q}:V_{q}\to X_{q}$ is a toroidal chart at $q$, and
•
$\phi : X_p\to X_{q}$ is a toric morphism.
Remarks
(1)
A toroidal embedding as defined above is a toroidal embedding without self-intersection according to the definition in Reference42, and a birational toroidal morphism satisfies the condition of allowability in Reference42.
(2)
To a toroidal embedding $(U_W\subset W)$ one can associate a polyhedral complex $\Delta _W$, such that proper birational toroidal morphisms to $W$, up to isomorphisms, are in one-to-one correspondence with certain subdivisions of the complex (see Reference42). It follows from this that the composition of two proper birational toroidal morphisms $W_1\to W_2$ and $W_2\to W_3$ is again toroidal: the first morphism corresponds to a subdivision of $\Delta _{W_2}$, the second one to a subdivision of $\Delta _{W_3}$, hence their composition is the unique toroidal morphism corresponding to the subdivision $\Delta _{W_1}$ of $\Delta _{W_3}$.
(3)
Some of the many issues surrounding these definitions we avoided discussing here are addressed in the third author’s lecture notes Reference53.
Let $\psi : W_1 \dashrightarrow W_2$ be a rational map defined on a dense open subset $U$. Denote by $\Gamma _\psi$ the closure of the graph of $\psi _U$ in $W_1 \times W_2$. We say that $\psi$ is proper if the projections $\Gamma _\psi \to W_1$ and $\Gamma _\psi \to W_2$ are both proper.
Definition 1.5.4
Let $U_i \subset W_i$ be toroidal embeddings. A proper birational map $\psi :W_1 \dashrightarrow W_2$ is said to be toroidal if there exists a toroidal embedding $U_Z \subset Z$ and a commutative diagram
where $Z \to W_i$$(i=1,2)$ are proper birational toroidal morphisms. In particular, a proper birational toroidal map induces an isomorphism between the open sets $U_1$ and $U_2$.
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 $W_1 \leftarrow Z_1 \to W_2$ and $W_2 \leftarrow Z_2 \to W_3$ is again toroidal. Indeed, if $Z_1\to W_2$ and $Z_2\to W_2$ correspond to two subdivisions of $\Delta _{W_2}$, then a common refinement of the two subdivisions corresponds to a toroidal embedding $Z$ such that $Z\to Z_1$ and $Z\to Z_2$ are toroidal morphisms. For example, the coarsest refinement corresponds to taking for $Z$ the normalization of the closure of the graph of the birational map $Z_1\dashrightarrow Z_2$. The composite maps $Z \to W_i$ 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 $Y$. 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 $\psi :W_1 \dashrightarrow W_2$ together with a diagram of birational maps$$\begin{equation*} \begin{array}{ccccc} W_1 & &\stackrel {\psi }{\dashrightarrow } & & W_2\\& \searrow & & \swarrow & \\& & Y & & \\\end{array} \end{equation*}$$
between locally toric varieties $W_1$ and $W_2$ satisfying the following condition:
For every closed point $q\in Y$ there exist a toric chart $\eta _q:V_q\to X_q$ at $q$, and a diagram of fibered squares$$\begin{equation*} \begin{array}{rcccl} W_1 & \rightarrow & Y & \leftarrow & W_2 \\\cup & & \cup & & \cup \\V_1 & \rightarrow & V_q & \leftarrow & V_2 \\\downarrow & & \downarrow & & \downarrow \\X_1 & \rightarrow & X_q & \leftarrow & X_2 \end{array} \end{equation*}$$
such that
(a)
$V_i\to X_i$ are toric charts for $W_i$,$i=1,2$, and
(b)
$X_i \to X_q$ are toric morphisms
(2)
Analogously, let $U_i \subset W_i$ be toroidal embeddings. A tightly toroidal birational transformation between them is a tightly locally toric birational transformation $\psi :W_1 \dashrightarrow W_2$ 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 $W_1\leftarrow Z \to W_2$ can be deduced from this, as follows. By toroidal resolution of singularities we may assume $Z$ is nonsingular. We apply toroidal weak factorization to the morphisms $Z\to W_i$, to get a sequence of toroidal birational maps
consisting of toroidal blowings up and down with nonsingular centers.
We state this result for later reference:
Theorem 1.6.1
Let $U_1\subset W_1$ and $U_2\subset W_2$ be nonsingular toroidal embeddings. Let $\psi : W_1\dashrightarrow W_2$ 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.
Let $V$ and $X$ be varieties with relatively affine $K^*$-actions, and let $\eta : V\to X$ be a $K^*$-equivariant étale morphism. Then $\eta$ is said to be strongly étale if
(i)
the quotient map $V/\!/K^*\to X/\!/K^*$ is étale, and
Let $W$ be a locally toric variety with a $K^*$-action, such that $W /\!/ K^*$ exists. We say that the action is locally toric if for any closed point $p\in W$ we have a toric chart $\eta _p:V_p\to X_p$ at $p$ and a one-parameter subgroup $K^*\subset T_p$ of the torus in $X_p$, satisfying
•
$V_p= \pi ^{-1} \pi V_p$, where $\pi :W\to W/\!/K^*$ is the projection;
•
$\eta _p$ is $K^*$-equivariant and strongly étale.
(2)
If $U\subset W$ is a toroidal embedding, we say that $K^*$ acts toroidally on $W$ if the charts above can be chosen toroidal.
The definition above is equivalent to the existence of the following diagram of fiber squares:
where the horizontal maps provide toric (resp. toroidal) charts in $W$ and $W/\!/K^*$. 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 $X = \operatorname {Spec}\ K[x, x^{-1},y]$ with the action $t(x,y) = (t^2x, t^{-1}y)$. The quotient is $X / K^*=\operatorname {Spec}\ K[xy^2].$ There is an equivariant étale cover $V = \operatorname {Spec}\ K[u, u^{-1},y]$ with the action $t(u,y) = (tu, t^{-1}y)$, where the map is defined by $x = u^2$. The quotient is $V / K^*=\operatorname {Spec}\ K[uy],$ which is a branched cover of $X/K^*$, since $xy^2 = (uy)^2$.
The following lemma shows that locally toric $K^*$-actions are ubiquitous. We note that it can be proven with fewer assumptions; see Reference81, Reference53.
Lemma 1.7.3
Let $W$ be a nonsingular variety with a relatively affine $K^*$-action, that is, the scheme $W/\!/K^*$ exists and the morphism $W\to W/\!/K^*$ is an affine morphism. Then the action of $K^*$ on $W$ is locally toric.
Proof.
Taking an affine open set in $W/\!/K^*$, we may assume that $W$ is affine. We embed $W$ 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 $\overline {W}$ is a nonsingular projective variety with a $K^*$-action, and $W\subset \overline {W}$ is an affine invariant open subset.
Let $p\in W$ be a closed point. Since $\overline {W}$ is complete, the orbit of $p$ has a limit point $q=\lim _{t\to 0} t(p)$ in $\overline {W}$. Now $q$ is fixed by $K^*$, hence $K^*$ acts on the cotangent space $m_q/m_q^2$ at $q$. Since $K^*$ is reductive, we can lift a set of eigenvectors of this action to semi-invariant local parameters $x_1,\mathinner {\ldotp \ldotp \ldotp }, x_n$ at $q$. These local parameters define a $K^*$-equivariant étale morphism $\eta _q: V_q\to X_q$ from an affine $K^*$ invariant open neighborhood $V_q$ of $q$ to the tangent space $X_q = \operatorname {Spec}({\operatorname {Sym}}\ m_q/m_q^2)$ at $q$. The latter has a structure of a toric variety, where the torus is the complement of the zero set of $\prod x_i$.
Separating the parameters $x_i$ into $K^*$-invariants and noninvariants, we get a factorization $X_q = X_q^0 \times X_q^1$, where the action of $K^*$ on $X_q^1$ is trivial and the action on $X_q^0$ has $0$ as its unique fixed point. Thus we get a product decomposition $X_q/\!/K^* = X_q^0/\!/K^* \times X_q^1$.
By Luna’s Fundamental Lemma (Reference51, Lemme 3), there exist affine $K^*$-invariant neighborhoods $V_q'$ of $q$ and $X_q'$ of $0$, such that the restriction $\eta _q': V_q' \to X_q'$ is strongly étale. Consider first the case $q\in W$, in which case we may replace $p$ by $q$. Denote $Z=X_q^{K^*}\cap X_q'$. Then $Z \subset X_q^{K^*}\simeq X_q^1$ is affine open, and, using the direct product decomposition above, $X_q^0\times Z \subset X_q$ is affine open. Denote $X_q'' = X_q'\cap X_q^0\times Z$. This is affine open in $X_q$, and it is easy to see that $X_q''/\!/K^* \to X_q/\!/K^*$ is an open embedding: an orbit in $X_q''$ is closed if and only if it is closed in $X_q$. Writing $V_q'' = {\eta _q'}^{-1}X_q''$, it follows that $V_q'' \to X_q$ is a strongly étale toric chart.
In the case $q\notin W$, replace $V_q$ by $V_q''$ . Now $\eta _q$ is injective on any orbit, and therefore it is injective on the orbit of $p$. Let $X_p\subset X_q$ be the affine open toric subvariety in which the torus orbit of $\eta _q(p)$ is closed, and let $V_p = \eta _q^{-1}X_p\cap W$. Now consider the restriction $\eta : V_p\to X_p$, where the $K^*$-orbits of $p$ and $\eta (p)$ are closed. By Luna’s Fundamental Lemma there exist affine open $K^*$-invariant neighborhoods $V_p'\subset V_p$ and $X_p'\subset X_p$ of $\eta _p(p)$ such that the restriction $\eta : V_p'\to X_p'$ is a strongly étale morphism. Since $X_p/K^*$ is a geometric quotient, we have an open embedding $X_p'/K^*\subset X_p/K^*$ and we have a strongly étale toric chart $V_p \to X_p$.
It remains to show that the charts can be chosen saturated with respect to the projection $\pi : W\to W/\!/K^*$. If the orbit of $p$ has a limit point $q=\lim _{t\to 0} t\cdot p$ or $q=\lim _{t\to \infty } t\cdot p$ in $W$, 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 $\pi (W\smallsetminus V_p)$ is closed and does not contain $\pi (p)$, so we can choose an affine neighborhood $Y$ in its complement, and replace $V_p$ by $\pi ^{-1} Y$.
Let $\phi :X_1 \dashrightarrow X_2$ be a birational map between two algebraic varieties $X_1$ and $X_2$ over $K$, isomorphic on an open set $U$. A normal algebraic variety $B$ is called a birational cobordism for $\phi$ and denoted by $B_{\phi }(X_1,X_2)$ if it satisfies the following conditions:
(1)
The multiplicative group $K^*$ acts effectively on $B = B_{\phi }(X_1,X_2)$.
(2)
The sets$$\begin{equation*} \begin{array}{rcrl} B_- &:=& \{x \in B:&\lim _{t \to 0\ }t(x) \mbox{ does not exist in }B\} \\\mbox{and } B_+ &:=& \{x \in B:&\lim _{t \to \infty }t(x) \mbox{ does not exist in }B\} \end{array} \end{equation*}$$
are nonempty Zariski open subsets of $B$.
(3)
There are isomorphisms$$\begin{equation*} B_-/K^* \stackrel {\sim }{\to } X_1 \quad \mbox{ and } \quad B_+/K^* \stackrel {\sim }{\to } X_2. \end{equation*}$$
(4)
Considering the rational map $\psi :B_-\dashrightarrow B_+$ induced by the inclusions $(B_-\cap B_+) \subset B_-$ and $(B_-\cap B_+) \subset B_+$, the following diagram commutes:$$\begin{equation*} \begin{array}{ccc} B_- & \stackrel {\psi }{\dashrightarrow }& B_+ \\\downarrow & & \downarrow \\X_1 &\stackrel {\phi }{\dashrightarrow } & X_2\end{array} \end{equation*}$$
We say that $B$respects the open set $U$ if $U$ is contained in the image of $(B_-\cap B_+)/K^*$.
Let $B=B_\phi (X_1,X_2)$ be a birational cobordism. We define a relation $\prec$ among connected components of $B^{K^*}$ as follows: let $F_1, F_2\subset B^{K^*}$ be two connected components, and set $F_1 \prec F_2$ if there is a point $x\notin B^{K^*}$ such that $\lim _{t\to 0 } t(x) \in F_1$ and $\lim _{t\to \infty }t(x)\in F_2$.
Definition 2.1.4
A birational cobordism $B=B_\phi (X_1,X_2)$ is said to be quasi-elementary if any two connected components $F_1, F_2\subset B^{K^*}$ are incomparable with respect to $\prec$.
Note that this condition prohibits, in particular, the existence of a “loop”, namely a connected component $F$ and a point $y\notin F$ such that both $\lim _{t\to 0 } t(x) \in F$ and $\lim _{t\to \infty }t(x)\in F$.
We assume $K^*$ acts effectively, namely $\gcd (\alpha _1,\mathinner {\ldotp \ldotp \ldotp },\alpha _n) = 1$. We regard ${\mathbb{A}}^n$ as a toric variety defined by a lattice $N\cong {\mathbb{Z}}^n$ and a nonsingular cone $\sigma \in N_{\mathbb{R}}$ generated by the standard basis
The dual cone $\sigma ^\vee$ is generated by the dual basis $v_1^*,\mathinner {\ldotp \ldotp \ldotp },v_n^*$, and we identify $z^{v_i^*}=z_i$. The $K^*$-action then corresponds to a one-parameter subgroup
$$\begin{equation*} a = (\alpha _1, \mathinner {\ldotp \ldotp \ldotp }, \alpha _n) \in N. \end{equation*}$$
We assume that $a\notin \pm \sigma$. We have the obvious description of the sets $B_+$ and $B_-$:
$$\begin{equation*} \begin{array}{rcl} B_- &=& \{(z_1, \dotsc , z_n); z_i \neq 0 \mbox{ for some }i \mbox{ with }\alpha _i = (v_i^*,a) < 0\}, \\B_+ &=& \{(z_1, \dotsc , z_n); z_i \neq 0 \mbox{ for some }i \mbox{ with }\alpha _i = (v_i^*,a) > 0\}. \end{array} \end{equation*}$$
We define the upper boundary and lower boundary fans of $\sigma$ to be
$$\begin{equation*} \begin{array}{rcl} \partial _-\sigma &=& \{x \in \sigma ;x + \epsilon \cdot a \not \in \sigma \mbox{ for all }\epsilon > 0\}, \\\partial _+\sigma &=& \{x \in \sigma ;x + \epsilon \cdot (- a) \not \in \sigma \mbox{ for all }\epsilon > 0\}. \end{array} \end{equation*}$$
Then we obtain the description of $B_+$ and $B_-$ as the toric varieties corresponding to the fans $\partial _+\sigma$ and $\partial _-\sigma$ in $N_{\mathbb{R}}$.
Let $\pi : N_{\mathbb{R}}\to N_{\mathbb{R}}/{\mathbb{R}}\cdot a$ be the projection. Then $B/\!/K^*$ is again an affine toric variety defined by the lattice $\pi (N)$ and cone$\pi (\sigma )$. Similarly, one can check that the geometric quotients $B_-/K^*$ and $B_+/K^*$ are toric varieties defined by fans$\pi (\partial _+\sigma )$ and $\pi (\partial _-\sigma )$. Since both $\pi (\partial _+\sigma )$ and $\pi (\partial _-\sigma )$ are subdivisions of $\pi (\sigma )$, we get a diagram of birational toric maps
It is easy to see (see, e.g., Reference81) that the varieties $B_\pm /K^*$ 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 $\Sigma$ is a subdivision of a convex polyhedral cone in $N_{\mathbb{R}}$ with lower boundary $\partial _-\Sigma$ and upper boundary $\partial _+\Sigma$ relative to an element $a \in N \smallsetminus \pm \Sigma$, then the toric variety corresponding to $\Sigma$, with the $K^*$-action given by the one-parameter subgroup $a \in N$, is a birational cobordism between the two toric varieties corresponding to $\pi (\partial _-\Sigma )$ and $\pi (\partial _+\Sigma )$ as fans in $N_{\mathbb{R}}/{\mathbb{R}}\cdot a$.
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 $X_1 \dashrightarrow X_2$. Here we deal with a very special case.
Theorem 2.3.1
Let $\phi :X_1 \to X_2$ be a projective birational morphism between complete nonsingular algebraic varieties, which is an isomorphism on an open set $U$. Then there is a complete nonsingular algebraic variety $\overline {B}$ with an effective $K^*$-action, satisfying the following properties:
(1)
There exist closed embeddings $\iota _1:X_1 \hookrightarrow \overline {B}^{K^*}$ and $\iota _2:X_2 \hookrightarrow \overline {B}^{K^*}$ with disjoint images.
(2)
The open subvariety $B = \overline {B} \smallsetminus \left(\iota _1(X_1)\cup \iota _2(X_2)\right)$ is a birational cobordism between $X_1$ and $X_2$ respecting the open set $U$.
(3)
There is a coherent sheaf $E$ on $X_2$, with a $K^*$-action, and a closed $K^*$-equivariant embedding $\overline {B}\subset {\mathbb{P}}(E):= {{\mathcal{P}}}roj_{X_2}\ {\operatorname {Sym}}E$.
Proof.
Let $J\subset {\mathcal{O}}_{X_2}$ be an ideal sheaf such that $\phi :X_1\to X_2$ is the blowing up morphism of $X_2$ along $J$ and $J_U={\mathcal{O}}_U$. Let $I_0$ be the ideal of the point $0\in {\mathbb{P}}^1$. Consider $W_0 = X_2 \times {\mathbb{P}}^1$ and let $p:W_0 \to X_2$ and $q:W_0 \to {\mathbb{P}}^1$ be the projections. Let $I = (p^{-1} J +q^{-1} I_0) {\mathcal{O}}_{W_0}$. Let $W$ be the blowing up of $W_0$ along $I$. (Paolo Aluffi has pointed out that this $W$ is used when constructing the deformation to the normal cone of $J$.)
We claim that $X_1$ and $X_2$ lie in the nonsingular locus of $W$. For $X_2\cong X_2\times \{\infty \}\subset X_2\times {\mathbb{A}}^1\subset W$ this is clear. Since $X_1$ is nonsingular, embedded in $W$ as the strict transform of $X_2\times \{0\}\subset X_2\times {\mathbb{P}}^1$, to prove that $X_1$ lies in the nonsingular locus it suffices to prove that $X_1$ is a Cartier divisor in $W$. We look at local coordinates. Let $A=\Gamma (V,{\mathcal{O}}_V)$ for some affine open subset $V\subset X_2$, and let $y_1,\mathinner {\ldotp \ldotp \ldotp },y_m$ be a set of generators of $J$ on $V$. Then on the affine open subset $V\times {\mathbb{A}}^1\subset X_2\times {\mathbb{P}}^1$ with coordinate ring $A[x]$, the ideal $I$ is generated by $y_1,\mathinner {\ldotp \ldotp \ldotp },y_m,x$. The charts of the blowing up containing the strict transform of $\{x=0\}$ are of the form
where $K^*$ acts on the second factor. The strict transform of $\{x=0\}$ is defined by $\frac {x}{y_i}$, hence it is Cartier.
Let $\overline {B} \to W$ be a canonical resolution of singularities. Then conditions (1) and (2) are clearly satisfied. For condition (3), note that $\overline {B} \to X_2\times {\mathbb{P}}^1$, being a composition of blowings up of invariant ideals, admits an equivariant ample line bundle. Twisting by the pullback of ${\mathcal{O}}_{{\mathbb{P}}^1}(n)$ we obtain an equivariant line bundle which is ample for $\overline {B} \to X_2$. Replacing this by a sufficiently high power and pushing forward we get $E$.
■
We refer the reader to Reference81 for more details.
We call a variety $\overline {B}$ as in the theorem a compactified, relatively projective cobordism.
2.4. Collapsibility and projectivity
Let $B=B_\phi (X_1,X_2)$ be a birational cobordism. We seek a criterion for collapsibility of $B$.
Let ${\mathcal{C}}$ be the set of connected components of $B_\phi (X_1,X_2)^{K^*}$, and let $\chi :{\mathcal{C}}\to {\mathbb{Z}}$ be a function. We say that $\chi$ is strictly increasing if $F \prec F' \Rightarrow \chi (F) < \chi (F')$. The following lemma is obvious:
Lemma 2.4.1
Assume there exists a strictly increasing function $\chi$. Then $\prec$ is a strict pre-order, and $B$ is collapsible. Conversely, suppose $B$ 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 $a_1<a_2<\cdots <a_m\in {\mathbb{Z}}$ be the values of $\chi$.
The following is an immediate extension of Proposition 1 of Reference81.
Proposition 2.4.3
(1)
$B_{a_i}$ is a quasi-elementary cobordism.
(2)
For $i=1,\mathinner {\ldotp \ldotp \ldotp }, m-1$ we have $(B_{a_i})_+ = (B_{a_{i+1}})_-$.
The following is an analogue of Lemma 1 of Reference81 in the case of the cobordisms we have constructed.
Proposition 2.4.4
Let $E$ be a coherent sheaf on $X_2$ with a $K^*$-action, and let $\overline {B} \subset {\mathbb{P}}(E)$ be a compactified, relatively projective cobordism embedded $K^*$-equivariantly. Then there exists a strictly increasing function $\chi$ for the cobordism $B = \overline {B} \smallsetminus (X_1 \cup X_2)$. In particular, the cobordism is collapsible.
Proof.
Since $K^*$ acts trivially on $X_2$, and since $K^*$ is reductive, there exists a direct sum decomposition
$$\begin{equation*} E = \mathop {\bigoplus }\limits _{b\in {\mathbb{Z}}} E_b \end{equation*}$$
where $E_b$ is the subsheaf on which the action of $K^*$ is given by the character $t\mapsto t^b$. Denote by $b_0,\mathinner {\ldotp \ldotp \ldotp },b_k$ the characters which figure in this representation. Note that there are disjoint embeddings ${\mathbb{P}}(E_{b_j}) \subset {\mathbb{P}}(E)$.
■
Let $p\in B$ be a fixed point lying in the fiber ${\mathbb{P}}(E_q)$ over $q\in X_2$. We choose a basis
Then there is a $j_p$ such that $p_{b_j,\nu }=0$ whenever $j\neq j_p$. In particular, $p\in {\mathbb{P}}(E_{b_{j_p}})\subset {\mathbb{P}}(E)$.
If $F\subset B^{K^*}$ is a connected component of the fixed point set, then it follows from the lemma that $F \subset {\mathbb{P}}(E_{b_j})$ for some $j$. We define
To check that $\chi$ is strictly increasing, consider a point $p\in B$ such that $\lim _{t\to 0} t(p) \in F_1$ and $\lim _{t\to \infty } t (p) \in F_2$ for some fixed point components $F_1$ and $F_2$. Let the coordinates of $p$ in the fiber over $q\in X_2$ be $(p_{b_0,1},\mathinner {\ldotp \ldotp \ldotp }, p_{b_0,d_0},\mathinner {\ldotp \ldotp \ldotp }, p_{b_k,1},\mathinner {\ldotp \ldotp \ldotp }, p_{b_k,d_k})$. Now
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 $E$ and its decomposition according to the character. Let $\{b_j\}$ be the characters of the action of $K^*$ on $E$ and $\{a_i\}$ the subset of those $b_j$ that are in the image of $\chi$. If we use the Veronese embedding $\overline {B} \subset {\mathbb{P}}({\operatorname {Sym}}^2(E))$ and replace $E$ by ${\operatorname {Sym}}^2(E)$, we may assume that $a_i$ are even, in particular $a_{i+1} > a_i+1$ (this is a technical condition which comes in handy in what follows).
Denote by $\rho _0(t)$ the action of $t\in K^*$ on $E$. For any $r\in {\mathbb{Z}}$ consider the “twisted” action $\rho _r(t)= t^{-r}\cdot \rho _0(t)$. Note that the induced action on ${\mathbb{P}}(E)$ does not depend on the “twist” $r$. Considering the decomposition $E = \mathop {\bigoplus } E_{b_j}$, we see that $\rho _r(t)$ acts on $E_{b_j}$ by multiplication by $t^{b_j-r}$.
We can apply geometric invariant theory in its relative form (see, e.g., Reference63, Reference33) to the action $\rho _r(t)$ of $K^*$. Recall that a point $p\in {\mathbb{P}}(E)$ is said to be semistable with respect to $\rho _r$, written $p\in ({\mathbb{P}}(E),\rho _r)^{ss}$, if there is a positive integer $n$ and a $\rho _r$-invariant local section $s\in ({\operatorname {Sym}}^n(E))^{\rho _r}$, such that $s(p)\neq 0$. The main result of geometric invariant theory implies that
moreover, the quotient map $({\mathbb{P}}(E),\rho _r)^{ss} \to ({\mathbb{P}}(E),\rho _r)^{ss} /\!/ K^*$ is affine. We can define $(\overline {B},\rho _r)^{ss}$ analogously, and we automatically have $(\overline {B},\rho _r)^{ss} = \overline {B} \cap ({\mathbb{P}}(E),\rho _r)^{ss}$.
The numerical criterion of semistability (see Reference60) immediately implies the following:
Let $B$ be a collapsible nonsingular birational cobordism. Then we can write $B$ as a union of quasi-elementary cobordisms $B=\bigcup _i B_{a_i}$, with $(B_{a_i})_+ = (B_{a_{i+1}})_-$. By Lemma 1.7.3 each $B_{a_i}$ has a locally toric structure such that the action of