Commensurating actions of birational groups and groups of pseudo-automorphisms
Pseudo-automorphisms are birational transformations acting as regular automorphisms in codimension . We import ideas from geometric group theory to study groups of birational transformations, and prove that a group of birational transformations that satisfies a fixed point property on cat cubical complexes is birationally conjugate to a group acting by pseudo-automorphisms on some non-empty Zariski-open subset. We apply this argument to classify groups of birational transformations of surfaces with this fixed point property up to birational conjugacy.
2010 Mathematics Subject Classification:Primary 14E07, Secondary 14J50, 20F65
1.1. Birational transformations and pseudo-automorphisms
Let be a quasi-projective variety, over an algebraically closed field . Denote by the group of birational transformations of and by the subgroup of (regular) automorphisms of . For the affine space of dimension , automorphisms are invertible transformations such that both and are defined by polynomial formulas in affine coordinates:
with , . Similarly, birational transformations of are given by rational formulas, i.e. , .
Birational transformations may contract hypersurfaces. Roughly speaking, pseudo-automorphisms are birational transformations that act as automorphisms in codimension . Precisely, a birational transformation is a pseudo-automorphism if there exist Zariski-open subsets and in such that and have codimension and induces an isomorphism from to . The pseudo-automorphisms of form a group, which we denote by . For instance, all birational transformations of Calabi-Yau manifolds are pseudo-automorphisms; and there are examples of such manifolds for which is infinite while is trivial (see [Cantat-Oguiso:2015]). Pseudo-automorphisms are studied in Section 2.
Let be a group of birational transformations of an irreducible projective variety . We say that is pseudo-regularizable if there exists a triple where
is a projective variety and is a birational map;
is a dense Zariski open subset of ;
yields an action of by pseudo-automorphisms on .
More generally if is a homomorphism, we say that it is pseudo-regularizable if is pseudo-regularizable.
One goal of this article is to use rigidity properties of commensurating actions, a purely group-theoretic concept, to show that many group actions are pseudo-regularizable. In particular, we exhibit a class of groups for which all actions by birational transformations on projective varieties are pseudo-regularizable.
1.2. Property (FW)
The class of groups we shall be mainly interested in is characterized by a fixed point property appearing in several related situations, for instance for actions on cat cubical complexes. Here, we adopt the viewpoint of commensurated subsets. Let be a group, and an action of on a set . Let be a subset of . One says that commensurates if the symmetric difference
is finite for every element of . One says that transfixes if there is a subset of such that is finite and is -invariant: for every in .
A group has Property (FW) if, given any action of on any set , all commensurated subsets of are automatically transfixed. For instance, and have Property (FW), but non-trivial free groups do not share this property. Property (FW) is discussed in Section 3.
Let us mention that among various characterizations of Property (FW) (see [Cornulier:Survey-FW]), one is: every combinatorial action of on a cat(0) cube complex fixes some cube. Another, for finitely generated, is that all its infinite Schreier graphs are one-ended.
Let be a projective variety. The group does not really act on , because there are indeterminacy points; it does not act on the set of hypersurfaces either, because some of them may be contracted. As we shall explain, one can introduce the set of all irreducible and reduced hypersurfaces in all birational models (up to a natural identification). Then there is a natural action of the group on this set, given by strict transforms. The rigorous construction of this action follows from a general categorical framework, which is developed in Section 4. Moreover, this action commensurates the subset of hypersurfaces of . This construction leads to the following result.
Let be a projective variety over an algebraically closed field. Let be a subgroup of . If has Property (FW), then is pseudo-regularizable.
There is also a relative version of Property (FW) for pairs of groups , which leads to a similar pseudo-regularization theorem for the subgroup : this is discussed in Section LABEL:par:distorsion, with applications to distorted birational transformations.
Theorem A provides a triple such that conjugates to a group of pseudo-automorphisms on the open subset . There are two extreme cases for the pair depending on the size of the boundary . If this boundary is empty, acts by pseudo-automorphisms on a projective variety . If the boundary is ample, its complement is an affine variety, and then actually acts by regular automorphisms on (see Section 2.4). Thus, in the study of groups of birational transformations, pseudo-automorphisms of projective varieties and regular automorphisms of affine varieties deserve specific attention.
1.4. Classification in dimension
In dimension , pseudo-automorphisms do not differ much from automorphisms; for instance, coincides with if is a smooth projective surface. Thus, for groups with Property (FW), Theorem A can be used to reduce the study of birational transformations to the study of automorphisms of quasi-projective surfaces. Combining results of Danilov and Gizatullin on automorphisms of affine surfaces with a theorem of Farley on groups of piecewise affine transformations of the circle, we will be able to prove the following theorem.
Let be a smooth, projective, and irreducible surface, over an algebraically closed field. Let be an infinite subgroup of . If has Property (FW), there is a birational map such that
is the projective plane , a Hirzebruch surface with , or the product of a curve by the projective line . If the characteristic of the field is positive, is the projective plane .
is contained in .
The group has finitely many connected components for all surfaces listed in Assertion (1) of Theorem B. Thus, changing into a finite index subgroup , one gets a subgroup of . Here denotes the connected component of the identity of ; this is an algebraic group, acting algebraically on .
Groups with Kazhdan Property (T) satisfy Property (FW). Thus, Theorem B extends Theorem A of [Cantat:Annals] and the present article offers a new proof of that result.
Theorem B can also be applied to the group , where is a non-square positive integer. Thus, every action of this group on a projective surface by birational transformations is conjugate to an action by regular automorphisms on , the product of a curve by the projective line , or a Hirzebruch surface. Moreover, in this case, Margulis’ superrigidity theorem can be combined with Theorem B to get a more precise result, see §LABEL:scorcorSL2.
In general, for a variety one can ask whether transfixes , or equivalently is pseudo-regularizable. For a surface , this holds precisely when is not birationally equivalent to the product of the projective line with a curve. See §LABEL:surf_birt for more precise results.
This work benefited from interesting discussions with Jérémy Blanc, Vincent Guirardel, Vaughan Jones, Christian Urech, and Junyi Xie.
This preliminary section introduces useful notation for birational transformations and pseudo-automorphisms, and presents a few basic results.
2.1. Birational transformations
Let and be two irreducible and reduced algebraic varieties over an algebraically closed field . Let be a birational map. Choose dense Zariski open subsets and such that induces an isomorphism . Then the graph of is defined as the Zariski closure of in ; it does not depend on the choice of and . The graph is an irreducible variety; both projections and are birational morphisms and .
We shall denote by the indeterminacy set of the birational map .
Theorem 2.1 (Theorem 2.17 in [iitaka1982algebraic]).
Let be a rational map, with a normal variety and a projective variety. Then the indeterminacy set of has codimension .
The transformation of the affine plane is birational, and its indeterminacy locus is the line : this set of co-dimension is mapped “to infinity”. If the affine plane is compactified by the projective plane, the transformation becomes , with two indeterminacy points.
Assume that is normal; in particular, it is smooth in codimension . The jacobian determinant is defined in local coordinates, on the smooth locus of , as the determinant of the differential ; depends on the coordinates, but its zero locus does not. The zeroes of form a hypersurface of the smooth part of ; the zero locus of will be defined as the Zariski closure of this hypersurface in . The exceptional set of is the subset of along which is not a local isomorphism onto its image; by a corollary of Zariski’s main theorem, it coincides with the union of , the zero locus of , and additional parts which are contained in the singular locus of and have therefore codimension . Its complement is the largest open subset on which is a local isomorphism (see [Milne:BookAG, Vakil], for instance).
The total transform of a subset is denoted by . If is not contained in , we denote by its strict transform, defined as the Zariski closure of . We say that a hypersurface is contracted if it is not contained in the indeterminacy set and the codimension of its strict transform is larger than .
A birational map is a pseudo-isomorphism if one can find Zariski open subsets and such that
realizes a regular isomorphism from to and
and have codimension .
Pseudo-isomorphisms from to itself are called pseudo-automorphisms (see § 1.2). The set of pseudo-automorphisms of is a subgroup of .
Start with the standard birational involution which is defined in homogeneous coordinates by Blow-up the vertices of the simplex ; this provides a smooth rational variety together with a birational morphism . Then, is a pseudo-automorphism of , and is an automorphism if .
Let be a birational map between two (irreducible, reduced) normal algebraic varieties. Assume that the codimension of the indeterminacy sets of and is at least . Then, the following properties are equivalent:
The birational maps and do not contract any hypersurface.
The jacobian determinants of and do not vanish on the regular loci of and respectively.
For every smooth point , is a local isomorphism from a neighborhood of to a neighborhood of , and the same holds for .
The birational map is a pseudo-isomorphism from to .
Denote by be the inverse of . If the Jacobian determinant of vanishes at some (smooth) point of , then it vanishes along a hypersurface . If (1) is satisfied, the image of is a hypersurface in , and we can find a point such that is not an indeterminacy point of . Since the product of the jacobian determinant of at and of at must be equal to , we get a contradiction. Thus (1) implies (2), and (2) is equivalent to (1). Now, assume that (2) is satisfied. Then does not contract any positive dimensional subset of : is a quasi-finite map from to its image, and so is . Zariski’s main theorem implies that realizes an isomorphism from to (see [Milne:BookAG], Prop. 8.57). Thus, (2) implies (4) and (3). By assumption, and have codimension ; thus, (3) implies (2). Since (4) implies (1), this concludes the proof. ∎
Let be a smooth projective variety with trivial canonical bundle . Let be a non-vanishing section of , and let be a birational transformation of . Then, extends from to and determines a new section of ; this section does not vanish identically because is dominant, hence it does not vanish at all because is trivial. As a consequence, does not vanish, is a pseudo-automorphism of , and . We refer to [Cantat-Oguiso:2015, Fryers:preprint] for families of Calabi-Yau varieties with an infinite group of pseudo-automorphisms.
2.3. Projective varieties
Proposition 2.6 (see [Bedford-Kim:2014]).
Let be a pseudo-isomorphism between two normal projective varieties. Then
the total transform of by is equal to ;
has no isolated indeterminacy point;
if , then is a regular isomorphism.
Let be an indeterminacy point of the pseudo-isomorphism . Then contracts a subset of positive dimension on . Since and are local isomorphisms on the complement of their indeterminacy sets, is contained in . The total transform of a point by is a connected subset of that contains and has dimension . This set is contained in because is a local isomorphism on the complement of ; since , is not an isolated indeterminacy point. This proves Assertions (1) and (2). The third assertion follows from the second one because indeterminacy sets of birational transformations of projective surfaces are finite sets. ∎
Let be a hypersurface of , and let be a pseudo-isomorphism. The divisorial part of the total transform coincides with the strict transform . Indeed, and coincide on the open subset of on which is a local isomorphism, and this open subset has codimension .
Recall that the Néron-Severi group is the free abelian group of codimension cycles modulo cycles which are numerically equivalent to . Its rank is finite and is called the Picard number of .
The action of pseudo-isomorphisms on Néron-Severi groups is functorial: for all pairs of pseudo-isomorphisms and . If is a normal projective variety, the group acts linearly on the Néron-Severi group ; this provides a morphism
The kernel of this morphism is contained in and contains as a finite index subgroup.
As a consequence, if is projective the group is an extension of a discrete linear subgroup of by an algebraic group.
The first statement follows from the equality on divisors. The second follows from the first. To study the kernel of the linear representation , fix an embedding and denote by the polarization given by hyperplane sections in . For every in , is an ample divisor, because its class in coincides with the class of . Now, a theorem of Matsusaka and Mumford implies that is an automorphism of (see [KSC:book] exercise 5.6, and [Matsusaka-Mumford:1964]). To conclude, note that has finite index in the kernel of the action of on (see [Matsusaka:1958, Lieberman:1978]). ∎
2.4. Affine varieties
The group coincides with the group of polynomial automorphisms of the affine space : this is a special case of the following proposition.
Let be an affine variety. If is factorial, the group coincides with the group .
Fix an embedding . Rational functions on are restrictions of rational functions on . Thus, every birational transformation is given by rational formulas where each is a rational function
here, and are relatively prime polynomial functions. Since the local rings are unique factorization domains, we may assume that the hypersurfaces and have no common components. Then, the generic point of is mapped to infinity by . Since is a pseudo-isomorphism, is in fact empty; but if does not vanish on , is a regular map. ∎
3. Groups with Property (FW)
3.1. Commensurated subsets and cardinal definite length functions (see [Cornulier:Survey-FW])
Let be a group, and an action of on a set . Let be a subset of . As in the Introduction, one says that commensurates if the symmetric difference is finite for every element . One says that transfixes if there is a subset of such that is finite and is -invariant: for every in . If is transfixed, then it is commensurated. Actually, is transfixed if and only if the function is bounded on .
A group has Property (FW) if, given any action of on a set , all commensurated subsets of are automatically transfixed. More generally, if is a subgroup of , then has relative Property (FW) if every commensurating action of is transfixing in restriction to . This means that, if acts on a set and commensurates a subset , then transfixes automatically . The case is Property (FW) for .
We refer to [Cornulier:Survey-FW] for a detailed study of Property (FW). The next paragraphs present the two main sources of examples for groups with Property (FW) or its relative version, namely Property (T) and distorted subgroups.
Property (FW) should be thought of as a rigidity property. To illustrate this idea, consider a group with Property (PW); by definition, this means that admits a commensurating action on a set , with a commensurating subset such that the function has finite fibers. If is a group with Property (FW), then, every homomorphism has finite image.
3.2. Property (FW) and Property (T)
One can rephrase Property (FW) as follows: has Property (FW) if and only if every isometric action on an “integral Hilbert space” has bounded orbits, where is any discrete set.
A group has Property (FH) if all its isometric actions on Hilbert spaces have fixed points. More generally, a pair of a group and a subgroup has relative Property (FH) if every isometric -action on a Hilbert space has an -fixed point. Thus, the relative Property (FH) implies the relative Property (FW).
By a theorem of Delorme and Guichardet, Property (FH) is equivalent to Kazhdan’s Property (T) for countable groups (see [delaHarpe-Valette:Ast]). Thus, Property (T) implies Property (FW).
Kazhdan’s Property (T) is satisfied by lattices in semisimple Lie groups all of whose simple factors have Property (T), for instance if all simple factors have real rank . For example, satisfies Property (T).
Property (FW) is actually conjectured to hold for all irreducible lattices in semi-simple Lie groups of real rank , such as for . (here, irreducible means that the projection of the lattice modulo every simple factor is dense.) This is known in the case of a semisimple Lie group admitting at least one noncompact simple factor with Kazhdan’s Property (T), for instance in , which admits irreducible lattices (see [Cornulier:MathZ]).
Let be a group. An element of is distorted in if there exists a finite subset of generating a subgroup containing , such that ; here, is the length of with respect to the set . If is finitely generated, this condition holds for some if and only if it holds for every finite generating subset of . For example, every finite order element is distorted.
Let be a field. The distorted elements of are exactly the virtually unipotent elements, that is, those elements whose eigenvalues are all roots of unity; in positive characteristic, these are elements of finite order. By results of Lubotzky, Mozes, and Raghunathan (see [lubotzky1993cyclic, Lubotzky-Mozes-Raghunathan:2001]), the same characterization holds in the group , as soon as ; it also holds in when and is not a perfect square. In contrast, in , every element of infinite order is undistorted.
Lemma 3.3 (see [Cornulier:Survey-FW]).
Let be a group, and a finitely generated abelian subgroup of consisting of distorted elements. Then, the pair has relative Property (FW).
This lemma provides many examples. For instance, if is any finitely generated nilpotent group and is its derived subgroup, then has relative Property (FH); this result is due to Houghton, in a more general formulation encompassing polycyclic groups (see [Cornulier:Survey-FW]). Bounded generation by distorted unipotent elements can also be used to obtain nontrivial examples of groups with Property (FW), including the above examples for , and . The case of is particularly interesting because it does not have Property (T).
3.4. Subgroups of with Property (FW)
If a group acts on a tree by graph automorphisms, then acts on the set of directed edges of ( is non-oriented, so each edge gives rise to a pair of opposite directed edges). Let be the set of directed edges pointing towards a vertex . Then is the set of directed edges lying in the segment between and ; it is finite of cardinality , where is the graph distance. The group commensurates the subset for every , and . As a consequence, if has Property (FW), then it has Property (FA) in the sense that every action of on a tree has bounded orbits. This argument can be combined with Proposition 5.B.1 of [Cornulier:Survey-FW] to obtain the following lemma.
Lemma 3.4 (See [Cornulier:Survey-FW]).
Let be a group with Property (FW), then all finite index subgroups of have Property (FW), and hence have Property (FA). Conversely, if a finite index subgroup of has Property (FW), then so does .
On the other hand, Property (FA) is not stable by taking finite index subgroups.
Let be an algebraically closed field and be a subgroup of .
has a finite orbit on the projective line if and only if it is virtually solvable, if and only if its Zariski closure does not contain .
Assume that all finite index subgroups of have Property (FA) (e.g., has Property FW). If the action of on the projective line preserves a non-empty, finite set, then is finite.
The proof of the first assertion is standard and omitted. The second assertion follows directly from the first one.
In what follows, we denote by the ring of algebraic integers (in some fixed algebraic closure of ).
Theorem 3.6 (Bass [Bass:Pacific]).
Let be an algebraically closed field.
If has positive characteristic, then has no infinite subgroup with Property (FA).
Suppose that has characteristic zero and that is a countable subgroup with Property (FA), and is not virtually abelian. Then acts irreducibly on , and is conjugate to a subgroup of . If moreover for some subfield containing , then we can choose the conjugating matrix to belong to .
On the proof.
The original statement [Bass:Pacific, Theorem 6.5] yields this statement, except the last fact, and assumes that is contained in with a finitely generated field. The latter condition is actually automatic: indeed, being a countable group with Property (FA), is finitely generated [Serre:AASL2, §6, Th. 15], and one can choose to be the field generated by entries of a finite generating subset.
For the last assertion, we have for some such that ; we claim that this implies that . First, since is absolutely irreducible, this implies that . The conclusion follows from Lemma 3.7 below, which can be of independent interest. ∎
Let be fields. Then the normalizer is reduced to .
Since for the three elementary matrices , we deduce by a plain computation that for all , , , such that . In particular, for all indices and such that and are nonzero, the quotient belongs to . It follows that . ∎
Let be an algebraically closed field. Let be a projective curve over , and let be the field of rational functions on the curve . Let be an infinite subgroup of . If has Property (FA), then
the field has characteristic ;
there is an element of that conjugates to a subgroup of .∎
4. A categoral limit construction
The purpose of this section is to describe a general categorical construction, which can be used to construct various actions of groups of birational transformations, such as Manin’s construction of the Picard-Manin space (see [Manin:cubic-forms, Cantat:Annals]), as well as the commensurating action which is the main construction of this paper. A closely related construction is performed by V. Jones in [jones2016no] to construct representations of Thompson’s groups, although it does not directly apply here.
4.1. Categories of projective varieties
Here, in a category , arrows between any two objects and are assumed to form a set . Recall that a category is small if its objects form a set, and is essentially small if it is equivalent to a small category, or equivalently if there is a subset of the collection of objects meeting every isomorphism class. A subcategory of a category is full if all arrows of between objects of are also arrows of .
Our main example will be the following. Fix an algebraically closed field . Let be the category whose objects are irreducible (reduced) projective -varieties and whose arrows are birational morphisms. Let be the category with the same objects, but whose arrows are birational maps. Similarly, one can consider the category of irreducible (reduced) normal projective varieties, with arrows given by birational morphisms, and the category with the same objects but whose arrows are all birational maps. By construction, is a full subcategory of , which is a subcategory of .
4.2. Relative thinness and well-cofiltered categories
Given a category and an object , let us define the category whose objects are pairs with and , and whose arrows are given by arrows such that . A category is thin if there is at most one arrow between any two objects. Let us say that a category is relatively thin if the category is thin for all .
A category in which every arrow is invertible is relatively thin, and so are all its subcategories. This applies to the categories of Example 4.1: the category of birational maps between irreducible projective varieties, and to its subcategory , and similarly to and its subcategory .
Recall that a category is cofiltered if it satisfies the following two properties (a) and (b):
for any pair of objects , , there exists an object with arrows ;
for any pair of objects , and arrows , , there exists an object and an arrow such that .
Note that (b) is automatically satisfied when the category is thin. We say that a category is well-cofiltered if it is relatively thin and for every object , the category is cofiltered (note that we do not require to be cofiltered).
Coming again to the categories of Example 4.1, the category is essentially small and well-cofiltered. It is relatively thin, as mentioned in Example 4.2. To show that is cofiltered, consider two birational morphisms and , and denote by the composition . The graph is a projective subvariety of . One can compose the projection of onto with (resp. onto with ) to get a birational morphism ; this birational morphism is an object in that dominates and , as in property (a).
The full subcategory of enjoys the same properties. When has characteristic zero, the resolution of indeterminacies implies that its full subcategory of non-singular varieties (and birational morphisms) is also well-cofiltered.
4.3. Filtering inductive limits
We shall say that a category admits filtering inductive limits if for every small, thin and cofiltered category and every contravariant functor , the colimit of exists (and then it also exists when “small” is replaced with “essentially small”). For example, the category of sets and the category of groups admit filtering inductive limits (see [Vakil], § 1.4, for colimits).
Let us consider an essentially small category , a category admitting filtering inductive limits, and a contravariant functor ; we denote the functor by on objects and on arrows. Assume that is well-cofiltered. Then, for every object , we can restrict the functor to and take the colimit of this restriction . Roughly speaking, is the inductive limit in of all for . So, for every arrow in , there is an arrow in , in ; and for every arrow in , the following diagram commutes