Rational factors of D-varieties of dimension 3 with real Anosov flow

Rational factors of -varieties of dimension with real Anosov flow

Rémi Jaoui Rémi Jaoui, Pure Mathematics Department, Waterloo University, Waterloo, Ontario, Canada rjaoui@uwaterloo.ca
July 16, 2019
Abstract.

In this article, we prove that, under some suitable assumptions, the generic type of a real absolutely irreducible -variety of dimension with a mixing Anosov real-analytic flow satisfies exactly one of the two following behaviors: either it is minimal and disintegrated, or it is interalgebraic over with the third power of the generic type of a disintegrated -curve.

For that matter, we prove a conjecture of my thesis on the algebraic dynamical properties of an algebraically presented mixing Anosov flow — namely that, under the same suitable assumptions, a -variety of dimension with a mixing Anosov real-analytic flow does not admit non-trivial rational factors.

Then, the first statement follows from the second — using the Trichotomy Theorem in the theory — and the property of orthogonality to the constants for the aforementionned -varieties.

1. In the end of my thesis, I formulated a conjecture on the algebraic dynamical properties of the geodesic flow of an algebraically presented Riemannian manifold of dimension with negative curvature. The main motivation was that — using techniques from geometric stability theory in a differentially closed fields and the results of [moi] — I proved in the third chapter of my thesis [jaoui], that this conjecture has strong consequences for the nature of the algebraic relations for geodesic motions in negative curvature. The purpose of this note is to prove this conjecture in the more general setting of mixing Anosov flows:

Theorem A.

Let be an absolutely irreducible -variety of dimension over . Assume that the real-analytification of admits a compact (non-empty) connected component contained in the regular locus of .

If the real analytic flow is a mixing Anosov flow, then does not admit any non-trivial rational factor.

Note that since is contained in the regular locus of , it is Zariski-dense in if and only if it is non-empty. Hence — similarly to the criterion of orthogonality to the constants of [moi] — one needs only to describe the dynamics of the vector field on a Zariski-dense connected component of contained in the regular locus of , in order to apply Theorem A.

A first consequence of the compactness and smoothness assumptions on the connected component is that the real-analytic flow of the vector field restricted to is well defined at any time . Then, we require in Theorem A that this real-analytic flow is a (necessarily compact and of dimension ) mixing Anosov flow.

The Anosov flows are uniformly hyperbolic flows defined by Anosov in [Ano]. It is well-known (see for instance [Eber]) that the geodesic flow of a compact Riemannian manifold with negative curvature is an Anosov flow. For an Anosov flow , the various notions of mixing — topologically weakly-mixing, topologically mixing and the mixing properties relatively to an equilibrium measure— collapse into a single one (see for example [Coud]). We will simply say that is a mixing Anosov flow to mean that one of the previous properties is satisfied.

One of the most important classes of mixing Anosov flows is that of geodesic flows of compact Riemannian manifolds with negative curvature (see, for instance, [Dalbo]). In particular, Theorem A can be applied to study the geodesic flow of an algebraically presented compact Riemannian manifold with negative curvature, as stated in my thesis.

Let’s explain in more details the content of the conclusion of Theorem A. A rational factor of is a dominant rational map towards another -variety , that is a -rational map such that . Such a rational factor is called trivial if either is a point or (which happens if and only if is generically finite).

The main heuristic on which Theorem A is based on is the property that mixing Anosov flows should give rise to minimal differential equations. Here — in the same way as simple groups are those who can not be split into simpler (in a naive sense) parts — a differential equation is minimal if its resolution can not be reduced to the resolution of simpler differential equations. Hence, for an absolutely irreducible -variety , a non-trivial rational factor is a very concrete instance of non-minimality.

Conversely, the property for a -variety of admitting no non-trivial rational factors is a special instance concerning the theory of the same notion investigated in [Moosa1] in the more general setting of stable theories. In particular, they noticed that this property — that, in contrast with the stronger notion of minimality, avoid the consideration of base-changes by non constant differential fields — implies a weaker minimality property called semi-minimality.

2. In the next paragraph, we explain the main application of Theorem A that we have in mind — in the formalism of geometric stability theory and then in a more concrete geometric form — for autonomous differential equation of order with a mixing Anosov real-analytic flow. As well as on Theorem A, the next theorem also relies on Hrushovski-Sokolovic classification of minimal types interpretable in the theory (cf. [Sok]) and on the results of [moi] regarding the property of orthogonality to the constants:

Theorem B.

Let be an absolutely irreducible -variety of dimension over . Assume that the real-analytification of admits a compact (non-empty) connected component contained in the regular locus of .

If the real-analytic flow is a mixing Anosov flow, then exactly one of the two following cases holds:

  • Either the generic type of is minimal and disintegrated.

  • Or there exists a strictly disintegrated type of order over such that the generic type of and are interalgebraic over .

A self-contained exposition of Theorem B (assuming that Theorem A holds) can be found in the third chapter of my thesis (see, for instance, [jaoui, Theorem 5.3.1]). We include a sketch of proof with references to my thesis when needed.

Sketch of proof.

Let be an absolutely irreducible -variety of dimension over satisfying the assumptions of Theorem B. Assume that the real-analytic flow is a mixing Anosov flow and denote by the generic type (in the theory of differentially closed field) of the -variety .

Now, Theorem A ensures that does not admit non-trivial rational factors. As explained in the first paragraph, this property implies that its generic type is semi-minimal (see [jaoui, Proposition 4.1.8]). Hence, there exist a differential fields extension and a minimal type such that is almost internal to the set of -conjugates of the minimal type .

Moreover, since is a mixing flow, Theorem B of [moi] implies that the type is orthogonal to the constants. It follows that is a minimal type, orthogonal to the constants, and non-orthogonal to a type defined over a constant differential field (namely, the type ). Under these assumptions, Hrushvoski-Sokolovic Theorem of [Sok] ensures that the minimal type has to be disintegrated (see [jaoui, Theorem 4.1.6]).

To conclude, one uses standard arguments from (one-based) geometric stability theory to ensure that one can avoid extending parameters to witness almost-internality for in : Since is disintegrated, there exists a minimal type , not orthogonal to such that is interalgebraic (over ) with a power of . Comparing the order of types, we get:

Since is prime, there are only two possibilities:

  • either : the type is interalgebraic with and hence minimal (since we already assumed that it is stationary).

  • or and . In that case, unpublished results of Hrushovski imply that the type has to be -categorical (see also [Moosa2]) and therefore non-orthogonal to a strictly disintegrated type . Since and are interalgebraic over , this concludes the proof. ∎

In [moi2], we proved that the hypotheses of Theorem B are satisfied for unbounded families of real -varieties — namely the geodesic -varieties of real pseudo-Riemannian varieties, whose real-analytification is compact with negative curvature. Consequently, Theorem B provides unbounded families of types satisfying the alternative in the conclusion of Theorem B. When I started working on geodesic flows with negative curvature, I expected a stronger version of Theorem B to hold, namely that the case (ii) never occurs and that the generic type of a -variety satisfying the hypothesis of Theorem B is always minimal and disintegrated. This strengthening of Theorem B would be very interesting and is still open.

3. We now formulate a geometric version of Theorem B, not involving any reference to model-theory. Fix a -variety over some constant differential field . For , denote by the set of closed invariant subvarieties of which project generically on all the factors. We say that the -variety is generically disintegrated if for every , every member can be written as an irreducible component (which projects generically on each factors) of:

where is the projection on the and coordinates and for every . With this terminology in place, Theorem B can be formulated as follows:

Corollary C.

Let be an absolutely irreducible -variety of dimension over . Assume that the real-analytification of admits a compact (non-empty) connected component contained in the regular locus of .

If the real-analytic flow is a mixing Anosov flow, then the -variety is generically disintegrated. Moreover, one of the following two cases holds:

  • Either contains only generically finite-to-finite correspondences and the generic type of is minimal.

  • Or contains at least a closed subvariety of of codimension . Moreover, in that case, is a finite set.

In this very concrete geometric form, Corollary C reduces the classification of the closed invariant subvarieties (projecting generically on all factors) of for to the classification of closed invariant subvarieties of . Hence, stronger forms of Theorem B —such as the minimality property discussed above — are related to the description of the set (and of its “non-generic” counterpart ) of invariant closed subvarieties of projecting generically on both factors. In particular, such a description for mixing Anosov flows would allow a complete description of the algebraic relations for an algebraically presented mixing Anosov flow.

Moreover, the minimal differential equation of order studied by Freitag and Scanlon in [Frei] shows that in the first case, one can not expect to be finite. In the second case, this is a consequence of -categoricity for disintegrated -curves.

4. A major difficulty in the proof of Theorem A is to understand the relative position of (the real-analytification of) the indeterminacy locus of a rational factor and the vector field . In [moi], this issue was solved for rational integrals — that is when the vector field is everywhere zero. Indeed, we proved that the indeterminacy locus of a rational integral of has to be a closed invariant subvariety (see [moi, Lemma 3.3.5]).

For a more general vector , it is probably unreasonable to expect such a strong statement to hold. Indeed, such a statement would be a very strong instance of quantifier elimination for -varieties which is known not to hold in general for -varieties (see for instance [Pillayquantifier]). For a more geometric perspective, this is also reminiscent of the absence of projective models for varieties endowed with vector fields.

In this article, we address this issue by considering (possibly singular) foliations on —which in contrast with vector fields and rational factors — are known to extend to any smooth projective model of . Starting from a rational factor , we consider the relative tangent vector bundle of rank on the biggest open subset where is smooth. Then, using foliation formalism, we extend this (non-singular) foliation to a (possibly singular) foliation on . In contrast with the indeterminacy locus of , the singular locus of the foliation will satisfy nice properties of invariance.

This article is organized as follows: In the first section, we recall various classical facts about Lie-derivative in the setting of algebraic geometry. We try to emphasize the connection with differential algebra by formulating most of the results at the level of -schemes.

The second section is devoted to the notion of (possibly singular) algebraic foliations on a smooth algebraic variety. In fact, apart from the definition itself, we will only be using the extension principle (mentionned above) for algebraic foliations on smooth varieties. We refer to [Bru] in dimension and [Per] in higher dimension for more advanced geometrical results.

In the third section, we define the notion of an invariant foliation on a smooth -variety defined over some constant differential field. Then, we explain how to reduce the study of rational factors of a -variety to the study of its invariant foliations. This fairly general statement, which constitutes the core of our strategy to the proof of Theorem A, might be useful in other contexts than the one of Anosov flows.

In the last section, we prove Theorem A. We start by giving a classification of invariant continuous foliation on a compact Anosov flow of dimension . Apart from the results from the previous section, the main ingredient is then a theorem of Plante in [Plante] that ensures that no invariant continuous foliation appearing in the aforementioned classification can come from a rational factor.

5. The results of this article have been worked out during and slightly after my PhD in Orsay University. They are based on the many interesting ideas and suggestions of my PhD supervisors: Jean Benoît Bost and Martin Hils. I would also like to thank them for their numerous comments on earlier versions of this article. I am also very grateful to Rahim Moosa for many useful discussions and comments while I was writing this article as a post-doctoral fellow in Waterloo University.

1. Lie derivative

In this section, we recall some classical facts on the Lie derivative in the setting of algebraic and analytic geometry.

The Lie derivative of a vector field (or more generally of a tensor field) with respect to another vector field is a standard notion of differential calculus. Surprisingly, it seems that that this notion has not been considered before in the setting of differential algebra à la Buium [Bui].

On the other hand, the effectiveness of the Lie derivative in order to study “non-integrability properties” of a differential equation given by a vector field on a manifold already appears in the work of Morales-Ruiz and Ramis (cf. [Morales] and [MoralesRamis]) by means of the so-called variational equation (see also [Aud, Part 3.1, p.46]). Moreover, the notion of Lie-derivative lies — through Frobenius Integrability Theorem — at the heart of the theory of algebraic foliations of dimension , that we will study in the second section of this article.

1.1. Definition

We fix a field of characteristic . Recall that if is a -algebra then the -module is endowed with a Lie-bracket by the formula:

(1)

The Lie-bracket is compatible with localization. Indeed, if is a -algebra and is a multiplicative system of then the natural isomorphism of -modules:

is also an isomorphism of -Lie algebras.

Definition 1.1.1.

Let be a -scheme. The Lie-bracket defined by the formula (1) on each affine open subset defines a Lie-bracket on the coherent sheaf of derivations of the structural sheaf of .

Similarly, if is a (real or complex) analytic space, then the formula (1) defines a Lie-bracket on the coherent sheaf .

In both cases, under an additional smoothness assumption on (respectively on ), the sheaf is a locally free on and is, indeed, the sheaf of sections of the vector bundle — or in other words, the sheaf of vector fields on .

Lemma 1.1.2.

Let be either the field of real or complex numbers and be a -scheme. The (algebraic) Lie-bracket on satisfies the obvious compatibility relation with the analytic one on :

where denotes either the real or the complex analytification and the Lie-bracket on the right-hand side is the (real or complex) analytic one.

Proof.

This can easily be derived from the standard properties of the analytification functor. It is also a direct consequence of the formula (2) of the next paragraph in both analytic and étale coordinates. ∎

Definition 1.1.3.

Let be a -scheme over a constant differential field and a vector field on . The Lie-derivative of , denoted is the -linear morphism defined by:

for every local section .

Lemma 1.1.4.

Let be a -scheme over a constant differential field , let be two local sections defined on the same open set and . We have:

Proof.

These two properties follow immediately from the formula (1). ∎

Before studying, more generally, the -linear operator on a coherent sheaf satisfying the properties of Lemma 1.1.4, we compute, in the next paragraph, the Lie-bracket of two vector fields on a smooth variety in local coordinates (analytic coordinates in the analytic case and étale coordinates in the algebraic one).

1.2. Computation in analytic and étale coordinates

Every analytic manifold can be covered by analytic charts. More precisely, there exists a covering of by open subsets endowed with analytic coordinates (meaning that the map is an analytic isomorphism onto its image).

In order to prove local properties of the Lie-derivative, we will sometimes work locally inside these coordinates. Instead of analytic coordinates, we will use étale coordinates when working with algebraic varieties.

Definition 1.2.1.

Let be a smooth variety over some field of dimension . A system of étale coordinates on is an étale morphism .

In other words, it is a -tuple of regular functions on such that the section of the canonical line bundle does not vanish on .

Remark 1.2.2.

Note that, if is a system of étale coordinates then, by definition, define a trivialization of . It follows that the dual basis of vector fields define a trivialization of the tangent bundle of .

Lemma 1.2.3.

Let be a smooth algebraic variety over some field of characteristic . There exists a covering of by Zariski-open subsets endowed with étale coordinates. ∎

Example 1.2.4.

Let be a smooth algebraic variety over a field of characteristic . Consider an open set and a system of étale coordinates on . Since the vector fields define a trivialization of the tangent bundle of , we may write

for some functions .

Lemma 1.2.5.

With the notation above, the Lie-bracket of the vector fields and is given by:

(2)
Proof.

The lemma follows from Lemma 1.1.4 applied to both and . ∎

Note that when one works with (a subfield of) the field of complex numbers, on may use analytic coordinates instead of étale coordinates and the formula (2) also holds in this analytic setting.

1.3. Lie-derivative and -coherent sheaves

The notion of -coherent sheaves over some -scheme formalizes the notion of a “linear differential equation over ”. This is a straightforward generalization to -schemes of the notion of -module over differential fields, that appears for example in [Pil, Section 3].

Definition 1.3.1.

Let be a -scheme over some constant differential field . A -coherent sheaf over is a pair where is a coherent sheaf over and is a -linear sheaf morphism satisfying the Leibniz-rule with respect to scalar multiplication:

for every local sections and on some open subset of .

If and are both -coherent sheaves over , then a morphism of -coherent sheaves over is a morphism of coherent sheaves over such that

Remark 1.3.2.

The notion of -coherent sheaf is closely related to the more usual notion of a coherent sheaf endowed with a connexion . Recall that if is a scheme over a field , a connexion on a coherent sheaf over is -bilinear morphism:

which satisfies the Leibniz rule with respect to scalar multiplication on and is -linear with respect to scalar multiplication on .

Lemma 1.3.3.

Let be a -scheme over some constant differential field and a coherent sheaf endowed with a connexion on . Then is a -coherent sheaf.

Proof.

By definition, the morphism is -linear and satisfies the Leibniz rule. ∎

In particular, we get the following example:

Example 1.3.4.

Let be a -scheme over some constant differential field and be a free sheaf of rank over . Define the -linear map by the formula:

Then is a -coherent sheaf over .

Lemma 1.3.5.

Let be a -scheme over some constant differential field and let be a coherent sheaf on . If and are both -coherent sheaves then:

Proof.

For a local function and a local section , we have:

It follows that is -linear. ∎

Example 1.3.6.

Let be a smooth -variety over a constant differential field and a locally free -coherent sheaf.

Consider an open set for which is free. Using Example 1.3.4 and Lemma 1.3.5, there are functions for such that:

(3)

where is the -matrix with coefficients .

Conversely, for every matrix with coefficients , defines a -coherent sheaf on .

Example 1.3.7.

Let be a -scheme over some constant differential field . By Lemma 1.1.4, the pair is a -coherent sheaf over .

Assume now that is a smooth variety. The derivation is associated to a vector field on . Consider an open subset and a system of étale coordinates on . Combining the formulas (3) and (2), we get:

Since the coefficients of the matrix do not depend linearly on the vector field , this kind of -coherent sheaves never come by Lemma 1.3.3 from a connexion on .

1.4. Categories of -coherent sheaves over -schemes

Lemma 1.4.1.

Let be a -scheme over some constant differential field . The category of -coherent sheaves over is an Abelian category.

Proof.

If is a morphism of -coherent sheaves, it is easy to check that the kernel and the image of are respective -subcoherent sheaves of and respectively.

Moreover, if is a -coherent sheave of , then there exists a unique -coherent sheaf structure which makes the canonical projection into a morphism of -coherent sheaves.

Using the forgetful functor to the category of coherent sheaves on , which is an Abelian category, it is easy to check that the axioms of an Abelian category are satisfied. ∎

Definition 1.4.2.

Let be a morphism of -schemes over some constant differential field and a -coherent sheaf over .

The pull-back of by , denoted , is the coherent sheaf over endowed with the derivation:

Example 1.4.3.

Let be a morphism of -schemes over some constant differential field and a locally free -coherent sheaf over .

Consider an open subset such that the restriction of to is free. Using formula (3), we can write:

where is an matrix with coefficients .

Set . Note that the restriction of to is also free.

Lemma 1.4.4.

With the notations above, the -coherent structure on is given by:

where is the matrix with coefficients

1.5. Main Proposition

Proposition 1.5.1.

Let be a morphism of smooth -varieties over some constant differential field . The derivative of defines a morphism of -coherent sheaves over :

Proof.

It is sufficient to work locally in the Zariski topology. Consider étale coordinates and be étale coordinates on such that .

We want to show that . Using Example 1.3.7 and Lemma 1.4.4, we can write:

Using this notations, the previous equality translates into:

which is an identity between two matrices of size with coefficients in .

Now, since is a morphism of -varieties, we have , which — after denoting , the coordinate function of — translates in these coordinates by:

For and , the chain rule for derivation as well as the Leibniz rule imply that:

Moreover, since , we have that:

This concludes the proof of the proposition. ∎

We now gather two corollaries of Proposition 1.5.1 which deal respectively with two different geometric situations.

Corollary 1.5.2.

Let be a dominant morphism of smooth -varieties over a constant differential field . The coherent subsheaf is invariant under the Lie-derivative of .

Corollary 1.5.3.

Let be a smooth -variety and a closed smooth invariant submanifold. We have an exact sequence of sheaves over :

where denotes the normal bundle of in . The Lie-derivative induces a well-defined -coherent sheaf structure on .

1.6. Cauchy formula on an analytic manifold

Construction 1.6.1.

Let be a differential ring of characteristic . Consider the differential ring of formal power series over endowed with the derivation sending to and the morphism of differential rings given by the expansion in power series:

We denote by the morphism of evaluation at and we denote by , the unique extension of to satisfying . Note that with this derivation, the morphism of rings becomes a morphism of differential rings:

Lemma 1.6.2.

With the notation above for every , may be described as the unique solution of the differential equation:

Remark 1.6.3.

Consider an analytic manifold, an analytic vector field on and . The vector field induces a derivation on . In [moi, Lemme 3.1.21], we proved — using Cauchy integral formula the bound the norm of — that in that case, the morphism may be factored as:

As noted in [moi], one easily checks that if denotes the local analytic flow of the vector field at , this implies that:

This property was then used to translate — for a closed submanifold of — invariance properties with respect to the vector field in terms of invariance properties with respect to the local analytic flow (see [moi, Proposition 3.1.20]).

Construction 1.6.4.

Let be a differential ring of characteristic and be a -module over . One may define a morphism of -modules by the formula:

When is given by the Lie-derivative of a vector field on an analytic manifold , we will use once again Cauchy integral formula to bound the sequence :

Let be a smooth analytic manifold and let be a vector field on . For every point , the Lie-derivative of vector field defines a derivation . We denote by the ring of local analytic functions on . Pull-back along the local flow of defines a morphism of -modules:

We extend the derivation on to a derivation on still denoted by setting .

Lemma 1.6.5 (Cauchy formula for the Lie derivative).

With the notation above for every vector fields in a neighborhood of , we have:

Moreover, we have:

Proof.

The relation between the Lie-derivative of a vector field and the pull-back by the local flow of the vector field , given in the first part is well-known and holds more generally on smooth manifolds (see any textbook of differential geometry).

For the second part, one needs to check that the right hand-side converges normally to an analytic function in a neighborhood of . By formal derivation of power series, it follows easily that the left-hand side satisfies the differential equation given by the first part and therefore must be equal to the local flow . Similar formulas already appear in the work of Cauchy.

Given a vector field on , we need a uniform bound for in a neighborhood of , in order to prove normal convergence. Since is smooth and that we work locally on , we may assume that and .

Fix are the radii of two complex polydisks of and a natural number. By applying Cauchy integral formula to the holomorphic coordinates of , there exists a constant such that for all , all radii and all vector fields , we have:

where denotes the supremum norm on the polydisk with radius . It then follows from successive applications of the previous inequality to the radii that:

The normal convergence of the left-hand side follows from this inequality. ∎

2. Foliations on a smooth algebraic variety

In this section, we recall the standard definition of an algebraic foliation — in the setting of algebraic varieties over a field of characteristic — that we will use in this article.

Let be a smooth irreducible algebraic variety over . Intuitively, a foliation on is the data of a subspace of the tangent space at for every point of that depends algebraically on the point and such that the sheaf of sections of is stable under Lie-bracket.

The dimension of the subspace at the generic point of is called the rank of the foliation. A singularity of the foliation is simply a point of where the dimension of the fibre is less than the rank of the foliation.

More restrictively, we will require the foliations to satisfy an additional assumption of saturation. On the one hand, the involutivity property — that is the stability under Lie-bracket — ensures the local analytic integrability of the foliation, outside of the singular locus. On the other hand, the saturation hypothesis ensures that the singular locus of is “small”, namely of codimension at least in .

The main motivation for these additional requirement is the extension result (Proposition 2.3.1) for any algebraic foliation on a quasi-projective smooth variety to a foliation on any projective closure.

2.1. Algebraic foliations and their singular locus

Let be an irreducible and smooth variety over a field of characteristic . The coherent sheaf is a locally free sheaf of rank .

Definition 2.1.1.

A foliation on is a sub-sheaf of the tangent bundle which satisfies:

  • The sub-sheaf is involutive, that is, stable under the Lie-bracket.

  • The sub-sheaf is saturated, that is, the quotient does not have torsion.

The coherent sheaf is the sheaf of vector fields tangent to the foliation . The rank of the foliation is the (generic) rank of the coherent sheaf . Alternatively, we say that is a -foliation to mean that is a foliation with rank .

Remark 2.1.2.

We first comment on the two main assumptions in Definition 2.1.1, namely involutivity and saturation.

  • In this article, we are mainly interested in smooth algebraic varieties defined over the field of real or complex numbers. In that case, the involutivity assumption is crucial to ensure local analytic integrability of the algebraic foliation under study (see section 2.5). However, for a coherent sub-sheaf of of rank , the condition of involutivity is automatically satisfied.

  • The property of saturation for (inside a locally free sheaf) implies that the coherent sheaf is reflexive (namely, isomorphic to its bidual). This property, which is extensively studied in [Har2], is weaker than being locally free but stronger than being torsion-free.

    In particular, a -foliation is always defined by an invertible sheaf whereas for a -foliation on a smooth algebraic variety of dimension is always defined by a coherent sheaf which is locally free outside a finite set of points (see [Har2]) of . Since the main result of this article deals with -variety of dimension , these two examples are the most important ones for the proof of Theorem A.

Lemma 2.1.3.

Let be a smooth algebraic variety over and let be a coherent sub-sheaf of the tangent sheaf of . The following properties are equivalent:

  • does not have torsion.

  • There exists an open set such that and

    is an exact sequence of locally free-sheaves on .

Proof.

Indeed, fix of codimension . Since is a smooth algebraic variety, the local ring is a principal local ring. It follows that the exact sequence of torsion-free -modules:

is in fact an exact sequence of free -module which therefore have to split. Since this is true for every point of codimension , the lemma follows. ∎

Definition 2.1.4.

Let be a foliation on . The singular locus of , denoted is the set of points such that is not locally-free in a neighborhood of , i.e.

Definition 2.1.5.

Let be a foliation on . The foliation on is called non-singular if .

Proposition 2.1.6.

Let be a foliation on . The singular locus of is a closed subset of of codimension .

Proof.

The fact that the singular locus of a foliation is closed follows from the general properties of morphisms of coherent sheaves. This also follows from the computations of Example 2.1.7. Moreover, the previous lemma shows that its codimension is greater than . ∎

Example 2.1.7.

Let be a foliation on and an open subset endowed with a system of étale coordinates . Recall that the tangent sheaf may be identified with the free sheaf with basis .

Since the coherent sub-sheaf is finitely generated, it can be described as the coherent sheaf on spanned by some vector fields written as:

We can now explicit the system of equations describing the singular locus of restricted to in terms of these datas:

Hence, if is a foliation of rank , the system of equations describing the singular locus of is given by the vanishing of the minors of the matrix .

2.2. Analytification of an algebraic foliation

Assume that is the field of real or complex number. We have the following counterpart for Definition 2.1.1 in the analytic setting:

Definition 2.2.1.

Let is a real or complex analytic manifold. An analytic foliation on is a coherent subsheaf of the tangent sheaf of which is both involutive and saturated.

Similarly to the algebraic case, for an analytic foliation on an analytic manifold , one can define its rank and its singular locus . The singular locus of is a closed analytic subspace of of codimension .

Lemma 2.2.2.

Let be a complex (resp. a real) smooth algebraic variety and a foliation on . Through the complex-analytication (resp. real-analytification) functor , defines an analytic foliation on .

Moreover, the rank of is the rank of and the singular locus of is the analytification of the singular locus of .

Remark 2.2.3.

When we will work in the analytic setting, we will mainly work with non-singular foliation. The only exception to that rule is the proof of Proposition 3.2.2 where we work with the local analytic flow of a vector field to prove invariance properties for its singular locus.

Once this has been established, we will simply throw away the singular locus of the foliation at the level of scheme before applying the analytification functor.

2.3. Saturation of an algebraic foliation on a open subset

Proposition 2.3.1 (Saturation).

Let be a smooth algebraic variety over and let be a dense open subset. Any algebraic foliation on extends uniquely to an algebraic foliation on .

Proof.

Let be an algebraic foliation on . There exists a coherent subsheaf of such that (namely, the sheaf of vector fields such that , which, as a simple verification shows, is quasi-coherent, hence coherent).

Let be the saturation of in (see [Har2]). By definition, is a saturated subsheaf of , whose restriction to is . Therefore, it suffices to check that is involutive.

Let be a local section of . The Lie-bracket with defines a morphism of -modules:

Since the algebraic foliation is involutive, this morphism is zero on . Consequently, the image of this morphism is a coherent sheaf whose support is a proper closed subvariety of , hence a torsion sheaf. Since the coherent sheaf has no torsion, this morphism is zero.

The uniqueness is a direct consequence of the saturation hypothesis, since two saturated subsheaves of which have the same generic fibre are equal (see [Har2]). ∎

Proposition 2.3.1 is useful to construct algebraic foliations on smooth algebraic variety : it shows that one only needs to construct them on a dense open subset.

Construction 2.3.2.

Let be a rational dominant morphism of smooth irreducible varieties over a field . Denote by the dimension of and the dimension of . Let be the biggest open set where is defined and smooth. The restriction of the differential of to has constant rank .

Lemma 2.3.3.

With the notation above, defines a non-singular algebraic foliation on of rank .

Proof.

Since is a sub-vector bundle of , it suffices to prove that is involutive. Let be a local section of on a open set . Then, is a morphism of smooth -varieties. By Corollary 1.5.2, is stable under . Since this is true for every local section, is involutive. ∎

Definition 2.3.4.

Let be a rational dominant morphism of smooth irreducible varieties over a field . Denote by the dimension of and the dimension of .

By Proposition 2.3.1, the non-singular foliation uniquely extends to a (possibly singular) -foliation on denoted and called the foliation tangent to .

2.4. -Foliation tangent to a vector field

Definition 2.4.1.

Let be a smooth and irreducible algebraic variety and be a non-zero rational vector field on . Let