The algebraic density property for affine toric varieties

The algebraic density property for affine toric varieties

Abstract.

In this paper we generalize the algebraic density property to not necessarily smooth affine varieties relative to some closed subvariety containing the singular locus. This property implies the remarkable approximation results for holomorphic automorphisms of the Andersén-Lempert theory. We show that an affine toric variety satisfies this algebraic density property relative to a closed -invariant subvariety if and only if . For toric surfaces we are able to classify those which posses a strong version of the algebraic density property (relative to the singular locus). The main ingredient in this classification is our proof of an equivariant version of Brunella’s famous classification of complete algebraic vector fields in the affine plane.

2000 Mathematics Subject Classification: 32M05; 32M25; 14M25.
    Key words: Density property, affine toric varieties, locally nilpotent derivations, holomorphic automorphisms, Lie algebras.
    The first and second authors were partially supported by Schweizerischer Nationalfond Grant 200021-140235/1 and the third author was supported by Fondecyt project 11121151

1. Introduction

A remarkable property of the Euclidean space of dimension at least two, that to a great extent compensates for the lack of partition of unity for holomorphic automorphisms, was discovered by Andersén and Lempert in early 1990’s [And90, AL92], see also the work by Forstnerič and Rosay [FR93]. Since then, the theory of Stein manifolds with very large holomorphic automorphism group is called Andersén-Lempert theory.

The property was formalized by Varolin who named it the density property (DP). A Stein manifold has the DP if the Lie algebra generated by complete holomorphic vector fields is dense (in the compact-open topology) in the space of all holomorphic vector fields on . Recall that a vector field is called complete if its flow exits for all complex time and all initial conditions.

The DP allows to construct (global) automorphisms of with prescribed local properties. More precisely, any local phase flow on a Runge domain in can be approximated by (global) automorphisms. This has remarkable applications for geometric questions in complex analysis, we refer the reader to survey articles [Ros99, KK11, Kut14] and the recent book [For11]. For smooth affine algebraic varieties, the algebraic density property (ADP) was also introduced by Varolin. The ADP implies the DP, therefore it is commonly used as a tool to prove the DP.

In this paper we generalize the ADP to not necessarily smooth affine varieties relative to some closed subvariety containing the singular locus as follows: Let be an affine algebraic variety and let be the singular locus. We also let be an algebraic subvariety of containing and let be the ideal of . Let be the -module of vector fields vanishing in , i.e., . Let be the Lie algebra generated by all the complete vector fields in .

Definition 1.

We say that has the strong ADP relative to if . Furthermore, we say that has the ADP relative to if there exists such that . With this definition, the ADP relative to with is just the strong ADP relative to . If we let we simply say that has the strong ADP or the ADP, respectively.

Except for the fact that we consider not necessarily smooth varieties, the strong ADP is a version of Varolin’s Definition 3.1 in [Var01] of DP for the Lie subalgebra of vector fields vanishing on . Whereas for our property is slightly weaker than Varolin’s definition since we generate the Lie subalgebra of vector fields vanishing on of order at least using complete vector fields vanishing on of possibly lower order than . Still this version of the ADP has the same remarkable consequences as in Varolin version of ADP for the group of holomorphic automorphisms of fixing pointwise (see Theorem 3).

In this paper we investigate the ADP for toric varieties. Our first main result is the following theorem (see Theorem 7).

Theorem.

Let be an affine toric variety of dimension at least two and let be a -invariant closed subvariety of containing . Then has the ADP relative to if and only if .

Recall that every smooth affine toric variety is isomorphic . A special case of our theorem where and is the union of up to coordinate hyperplanes has been already proven by Varolin [Var01].

It is well known that every affine toric surface different from or is obtained as a quotient of by the action of a cyclic group. Let be relatively prime positive integers. We denote by the toric surface obtained as the quotient of by the -action , where is a primitive -th root of unity. The following theorem is our second main result (see Corollary 5).

Theorem.

has the strong ADP if and only if divides and .

Furthermore, for every affine toric surface our methods allow to determine the values of from Definition 1 for which . The main ingredient in the proof of this theorem is an equivariant version of Brunella’s famous classification of complete algebraic vector fields in the affine plane (see [Bru04]) or, equivalently, classification of complete algebraic vector fields on affine toric surfaces (see Theorem 10). This result might be of independent interest.

2. Vector fields and the algebraic density property

In this section we prove a general method for establishing the ADP that we later will use to show the ADP for toric varieties.

Definition 1.

Let be an affine algebraic variety and be a subvariety containing .

  1. Let be the subgroup of automorphism of stabilizing . We say that is homogeneous with respect to if acts transitively on .

  2. We also let . A finite subset of the tangent space is called a generating set if the image of under the action of the isotropy group of in generate the whole tangent space .

The following is our main tool to establish the ADP for toric varieties. It is a generalization of [KK08, Theorem 1].

Theorem 2.

Let be an algebraic variety homogeneous with respect to some subvariety . Let also be a finitely generated submodule of the -module of vector fields vanishing on . Assume that . If the fiber of over some contains a generating set, then has the ADP relative to .

Proof.

Let be a finite set of vector fields in such that is a generating set. Let now be a finite collection of automorphisms fixing such that span the tangent space at . Since change of coordinates does not change completeness of a vector field, for , the finitely generated module is again contained in . By replacing with , we can assume that span the tangent space at .

We let . We also let be the decomposition of in irreducible components and we pick . Since is homogeneous with respect to , we can choose sending to . We also put . Let now

By construction and so we can proceed by induction on dimension to obtain a finite collection of automorphisms such that the collection span the tangent space at every point .

We let . With the same argument as before, is a finitely generated -submodule of contained in . By construction, we have that the fiber of at every is trivial. Hence, the support of is contained in .

We define

By construction . This yields . Furthermore, by [Har77, Ch. II Ex 5.6] we have that . Recall that is the ideal of and let so that . Let now be a finite set of generators of . Since , we have that there exists such that for all . Letting we obtain

Hence the theorem follows. ∎

3. The algebraic density property for affine toric varieties

We first recall the basic facts from toric geometry that will be needed in this section. They can be found in any text about toric geometry such as [Ful93, Oda88, CLS11].

Let and be mutually dual lattices of rank with duality pairing , where . We also let and . Letting be the algebraic torus . A toric variety is a normal variety endowed with an effective action of having an open orbit. Since the -action is effective, the open orbit is equal to .

It is well known that affine toric varieties can be described by means of strongly convex polyhedral cones (pointed cones) in the vector space . Indeed, let be a pointed cone in , then is an affine toric variety and every affine toric variety arises this way. Here is the semigroup algebra . In the following, we denote by .

There is a one to one correspondence between the faces of the cone and the orbits of the -action on (usually called the Orbit-Cone correspondence). The dimension of an orbit is given by and its closure is given by where runs over all faces of containing . The ideal of an orbit closure is given by

where is the orthogonal of . Furthermore, the ideal of is

where denotes the relative interior.

As usual, we identify a ray with its primitive vector. The set of all the rays of is denoted by . A cone is called smooth if is part of a basis of the lattice . Let be any face. The orbit is contained in if and only if is smooth.

Let now and . The linear map

is a homogeneous derivation of the algebra and so it is a homogeneous vector field on . By the exponential map, the tangent space of at the identity is isomorphic to and the evaluation of the vector field at the smooth point is .

Let be a pointed cone. The following proposition gives a description of all the homogeneous vector fields on . The first statement of the following result can be found in [Dem70]. For the convenience of the reader we provide a short argument.

Proposition 1.

The homogeneous vector field on extends to a homogeneous vector field in if and only if

  1. , or

  2. There exists such that

    1. ,

    2. , and

    3. for all .

Furthermore, is locally nilpotent if and only if it is of type II, and is semisimple if and only if it is of type I and .

Proof.

The vector field extends to if and only if . Since is spanned by for all , it is enough to show that . In combinatorial terms, this corresponds to the condition:

(1)

Assume first that is not proportional to any . Then for every there exists such that and . Hence, (1) implies that and so is of type I.

Assume now that there exists such that . With the same argument as above we can show that for all . Let now such that . Then (1) implies that . This yields . If then is of type II. If then and is of type I.

To prove the second assertion, we let be a homogeneous vector field. A straightforward computation shows that

(2)

Assume first that is of type I and that . If then (2) yields

and so is not locally finite since is not finite dimensional. If then let be such that . In this case (2) implies

and again is not locally finite with a similar argument.

Assume now that is of type I and that . The vector field is the infinitesimal generator of the algebraic -action on given by the -grading on induced by the degree function . Hence, the vector field is semisimple.

Finally, assume that is of type II. For every we let . Now, is locally nilpotent since by (2). ∎

In the following corollary, we give an explicit description of the homogeneous complete vector fields on an affine toric variety.

Corollary 2.

The vector field is complete if and only if it is of type II, or it is of type I and .

Proof.

The vector fields of type II are locally nilpotent, hence complete. In the following, we assume that is of type I. First, assume that . Then and since belongs to the kernel of , we have that is complete.

Assume now that . Let be the ideal of , i.e.,

Since , we have that . Hence, is invariant by and so is also invariant by . In the following, we show that is not complete when restricted to . Since , is complete if and only if is complete, we will assume that is a primitive vector in and .

Without loss of generality, we choose mutually dual bases of and such that and , with and . We will also denote the standard coordinates of the torus , where is the base of . In this coordinates, the vector field restricted to is given by

which is not complete on since . Indeed the vector fields on are not complete for . ∎

Remark that in Corollary 2 complete vector fields of type I are extensions of complete vector fields on the big torus while complete vector fields of type II are locally nilpotent, hence not complete in . In the next lemma, we give a criterion for a homogeneous vector field to vanish in an orbit closure.

Lemma 3.

Let be a non-zero homogeneous vector field on and let be a face. Then vanishes at the orbit closure if and only if

  1. or for some .

  2. for some .

Proof.

The vector field does not vanish at the orbit closure if and only if . In combinatorial terms this happens if and only if

(3)

Case of type I. In this case, we have so for all if and only if and for all . This is the case if and only if and . Such and exists if and only if , i.e., if and only if . Hence, we conclude that does not vanish at the orbit closure if and only if and for all .

Case of type II. In this case we have that there exists such that , , and for all .

Assume first that . An argument similar to case I yields that does not vanish at the orbit closure if and only if and for all . Since , we have that and so the vector field does not vanish at the orbit closure if and only if for all .

Assume now that . If there exists such that , then for all and so vanishes at the orbit by (3). Assume for all and let be such that and for all . We have so and for all . By (3), we conclude that does not vanish at the orbit closure . ∎

Remark 4.

The degree of a homogeneous locally nilpotent vector fields (of type II) is called a root of . The set of all roots of is denoted by . For a root , the ray is called the distinguished ray of and the -action generated by the homogeneous locally nilpotent vector field is denoted by .

In order to show the ADP for toric varieties, we need to show that is homogeneous with respect to some closed subvariety . In [AZK12], the authors prove that is homogeneous with respect to . In fact, they show that the group of special automorphisms acts infinite-transitively with respect to . In the following, we will show how their methods can be applied to show that is homogeneous with respect to any -invariant closed subvariety .

Proposition 5.

Let be a pointed cone and let be any -invariant closed subvariety of containing . Then is homogeneous relative to .

Proof.

Using the -action and the Orbit-Cone correspondence, to prove the theorem it is enough to find, for every orbit in different from the open orbit, an automorphism that

  1. sends a point in into an orbit of higher dimension, and

  2. leaves stable every orbit not containing in its closure.

Let be the rays of . In [AZK12, Lemma 2.3] and its proof, the authors show that for every smooth orbit there exists a root such that

(4)

Furthermore, they show that a generic automorphism in the -action corresponding to the root satisfies .

Let be any orbit that does not contain in its closure. In combinatorial terms, this means that is a face of that is not contained in . We claim that leaves point-wise invariant and so satisfies which proves the proposition.

In terms of the vector field , our claim is equivalent to vanishes at . Since is not contained in there exists a ray of that is not a ray of . By (4) we have . Now the claim follows from Lemma 3. ∎

For our next theorem we need the following lemma that follows by direct computation.

Lemma 6.

Let and be two homogeneous vector fields. Then , where and .

Theorem 7.

Let be a affine toric variety of dimension at least two and let be a -invariant closed subvariety of containing . Then has the ADP relative to if and only if .

Proof.

Let be the toric variety given by the pointed cone and let . There is at least one codimension one -orbit not contained in . Assume it is for some ray . Let be a root with as distinguished ray. By (4), we can assume that for all . By Lemma 3, the locally nilpotent vector field vanishes at and so .

Letting be such that , we let

The set is contained in since vanishes in . In fact, is a submodule of since for every and every , we have . Furthermore, the fiber over the identity is given by

(5)

and so contains a generating set. We claim that . Hence has the ADP relative to by Theorem 2 and Proposition 5.

By Corollary 2, the vector field is complete if . Hence, to prove our claim it is enough to show that for every , there exists such that and .

Indeed, let and choose be such that and which implies that belongs to . This is possible since lies in and is a root of . By Lemma 6 we have

A routine computation shows that

proving the claim.

Assume now that . The converse of the theorem follows from the fact that for all affine toric varieties and all there is a vector field , where . Indeed, Andersén [And00] proved that any complete algebraic vector field on does preserve the Haar form

Thus if we find in whose restriction to does not preserve we are done.

After a change of coordinates one can assume that . Then is a regular vector field on contained in for big enough which does not preserve . ∎

Remark 8.

Lárusson proved in [Lár11, For13] that all smooth toric varieties are Oka-Forstnerič manifolds, however it is still unknown if they are elliptic, see [For11, Kut14] for definitions. The proof of Theorem 7 can be adapted to prove the following: every smooth quasi-affine toric variety is elliptic (and thus an Oka-Forstnerič manifold). Indeed, the torus is well known to be elliptic. Let be a smooth quasi-affine toric variety different from . Let also be an affine toric variety such that is an equivariant open embedding and let . Now, Proposition 5 and (5) imply that is elliptic [For11, Example 5.5.13 (B)].

4. Classification of complete vector fields on affine toric surfaces

In this section we classify all complete algebraic vector fields on a given affine toric surface . The classification works essentially the same as the classification of complete vector fields on done by Brunella [Bru04].

From now on we will use the fact that each affine toric surface different from or can be seen as the quotient of by the action of a cyclic group. Let be the order of the group and let be a co-prime number and consider the action of given by where is a primitive -th root of unity. We obtain the projection onto our toric surface which is a ramified covering of ramified only over the unique singular point. Certainly each vector field on pulls back to an invariant vector field of by using the fiber-wise isomorphism on the tangent space. A complete vector field on will pull back to an invariant complete vector field on .

Definition 1.

Let be a regular function on . The function is called -preserved if the fibers of are sent to fibers of by the -action. It is called -homogeneous of degree if for all . Let denote the space of -homogeneous polynomials of degree then we obtain a decomposition of the ring of regular functions on into -homogeneous parts . In particular is the ring of invariant functions .

It is clear from the definition that is spanned by all monomials with . Clearly invariant vector fields are of the form with and . Moreover we have the following easy lemma:

Lemma 2.

Let be a regular function then the following are equivalent:

  1. is -homogeneous,

  2. is -preserved with ,

  3. is -invariant.

Proof.

(1) implies (2) since if is constant on a curve then also is constant and follows directly from the homogeneity. The fiber contains the -fixed point thus (3) follows from (2). If the zero fibers of and coincide then we have that for some . By we see that is a -th root of unity and thus (3) implies (1). ∎

The following lemma is the crucial step in the classification of invariant complete algebraic vector fields and hence of complete algebraic vector fields on the toric variety . Recall that a rational first integral of a vector field is a rational function such that its fibers are tangential to the vector field.

Lemma 3.

Let be a -invariant complete algebraic vector field on then preserves either a -homogeneous fibration with general fibers or or has a reduced rational first integral .

Proof.

By [Bru04] there is fibration with or fibers which is preserved by the flow of . We may assume that . If is -homogeneous then we are done. If is not -homogeneous then we construct a rational first integral. The map acts by multiplication with some on the set of fibers of parametrized by so we have (indeed is a fixed point of ). Since is invariant the same holds true for and hence the rational map is a rational first integral for . By Stein factorization has a reduced first integral. Recall that every rational function can be decomposed into such that has connected regular fibers, or equivalently is reduced. This factorization is called Stein factorization. ∎

The next step will be the classification of -homogeneous fibrations with general fibers or and rational first integrals for invariant vector fields. The classification will be done up to equivariant automorphisms of which will lead to a classification of the vector fields on up to automorphism of since equivariant automorphisms clearly project down to automorphims of the quotient. Equivariant automorphisms of are given by invertible maps with and .

First we establish an equivariant version of the Abhyankar-Moh Theorem. We provide a proof using the classical verion of the theorem. See [AZ13] for a different proof.

Lemma 4.

Let be a line which is invariant by the group action. Then there is an equivariant automorphism of mapping to or . Moreover a cross of two invariant lines can be mapped to .

Proof.

By the classical Abhyankar-Moh Theorem we know that is given by a polynomial which is a component of an automorphism of . In order to find the other component of the automorphism we have to find an invariant section of the trivial line bundle given by . We start with an arbitrary trivialization and get an invariant section taking the average over images of the zero section by the group action. Each image is another section because the action sends fibers of to fibers of since the zero fiber is invariant. We denote the polynomial giving this invariant section by . The map given by is an automorphism of since it is the composition of the trivialization we started with and the map where is the invariant section. Because the zero sets of and are invariant they are -homogeneous by Lemma 2 and since they are the two components of an automorphism their homogeneity degrees coincide with and so either or is an equivariant automorphism and the claim follows. The second statement is trivial since there we already have an invariant section by assumption. ∎

We get the following corollary as an immediate consequence, see also [FKZ08].

Corollary 5.

Let be a -homogeneous fibration with fibers and then or up to equivariant automorphism of .

For the classification of -homogeous fibration with fibers we first state the non-equivariant version used in [Bru04], see also [Suz77].

Lemma 6.

Let be a fibration with fibers then has one special fiber (say ) and it is isomorphic to or and is up to automorphism of of the form or for coprime , and .

The equivariant version of this lemma is given by the two following lemmas.

Lemma 7.

Let be a -homogeneous fibration with fibers and then there are coprime and an invariant polynomial with and such that up to equivariant automorphism with or with .

Proof.

By Lemma 6 we know that there exists a not necessary equivariant automorphism such that is as in Lemma 6. Clearly, the curve is invariant by the group action since it is the only fiber component isomophic to . By Lemma 4 we may assume the or . In the first case this implies that, up to equivariant automorphism, and for some and and hence is of the form with . Since is -homogeneous we have and hence the map was equivariant after all and has the desired standard form up to equivariant automorphism. The equality follows from the fact that . The case leads similarly to the second possibility. ∎

Lemma 8.

Let be a -homogeneous fibration with fibers and then there are coprime such that up to equivariant automorphism. If is divisible by 4 (say ) and then can also be of the form .

Proof.

By Lemma 6 there is an automorphism such that . Clearly the 0-fiber