Thi-Lan Le

Monique Thonnat
email: monique.thonnat@sophia.inria.fr

Alain Boucher
email: alain.boucher@auf.org

François Brémond
email: francois.bremond@sophia.inria.fr

A Query Language Combining Object Features and Semantic Events for Surveillance Video Retrieval

In this paper, we propose a novel query language for video indexing and retrieval that (1) enables to make queries both at the image level and at the semantic level (2) enables the users to define their own scenarios based on semantic events and (3) retrieves videos with both exact matching and similarity matching. For a query language, four main issues must be addressed: data modeling, query formulation, query parsing and query matching. In this paper we focus and give contributions on data modeling, query formulation and query matching. We are currently using color histograms and SIFT features at the image level and 10 types of events at the semantic level. We have tested the proposed query language for the retrieval of surveillance videos of a metro station.

In our experiments the database contains more than 200 indexed physical objects and 48 semantic events. The results using different types of queries are promising.

Introduction

Video surveillance is producing huge video databases. While there are many works dedicated to object detection, object tracking and event recognition [START_REF] Hampapur | Smart Video Surveillance: Exploring the concept of multiscale spatiotemporal tracking[END_REF], few works have been done for accessing these databases. For surveillance video indexing and retrieval, apart from object tracking, object classification and event recognition, we have 4 main issues: data modeling, query formulation, query parsing and query matching. The data modeling determines which features are extracted and how they are organized and stored in the database. The query formulation specifies the way in which the user expresses his/her query while the query parsing [START_REF] Vu | Automatic video interpretation: A novel algorithm for temporal scenario recognition[END_REF] specifies the way in which the system analyzes (parses) this query into an internal representation. The aim of query matching is to compare elements stored in the database with the query.

The first works dedicated for the surveillance video indexing and video concentrate on data modeling. In [START_REF] Hampapur | Smart Video Surveillance: Exploring the concept of multiscale spatiotemporal tracking[END_REF], IBM smart surveillance engine successfully detects moving objects, tracks multiple objects, classifies objects and events. However, for retrieving in the database, the queries are done based on only recognized events and metadata. In [START_REF] Durak | Online Surveillance Video Archive System[END_REF], a data model is built for online video surveillance. Various query types are also presented. In [START_REF] Saykol | A database model for querying visual surveillance by integrating semantic and low-level features[END_REF], Saykol et al. presented a framework and a visual surveillance querying language (VSQL) for surveillance video retrieval. These previous works have three main drawbacks. Firstly, they work with an assumption that in the indexing phase objects are perfectly tracked and events are perfectly recognized. It is the reason why video retrieving is based on the exact matching of semantic events and metadata. However, object detection and event recognition are not always successful. The video retrieving must work well under imperfect indexing. In this case, the similarity matching on object features is necessary. Secondly, these approaches limit the search space. Because the videos are only indexed by a set of recognized events, the users' queries are restricted to a limited set of predefined events. Thirdly, it is not flexible and does not take into account various users' interests and users' degrees of knowledge. The users could need more or less information according to their interest and could define differently an 'event' in the form of a query in function of their knowledge in this domain.

Our main contributions are designing a video data model and proposing a novel query language. Our data model is different from the model proposed in [START_REF] Durak | Online Surveillance Video Archive System[END_REF] because it contains object visual features. Our query language overcomes the previous works [START_REF] Hampapur | Smart Video Surveillance: Exploring the concept of multiscale spatiotemporal tracking[END_REF], [START_REF] Durak | Online Surveillance Video Archive System[END_REF], [START_REF] Saykol | A database model for querying visual surveillance by integrating semantic and low-level features[END_REF] because it (1) enables users to make queries both at the image level and at the semantic level (2) allows the users to define their own scenarios based on semantic events and (3) retrieves videos with both exact matching and similarity matching.

The rest of this paper is organized as follows: Section 2 describes the proposed approach including data model, query language and query matching. In section 3, we describe some experimental results and their performance evaluation. We conclude this paper in section 4.

Proposed Approach

Figure 1 shows the general architecture of the proposed approach. This approach is based on an external Video Analysis module and on two internal phases: an indexing phase and a retrieval phase. The external Video Analysis module performs tasks such as mobile object detection, mobile object tracking and event recognition. The results of this module are some Recognized Video Content. These Recognized Video Content can be physical objects, trajectories, events, scenarios, etc. So far, we are only using physical objects and events but the approach can be extended to other types of Recognized Video Content. The indexing phase takes results from the Video Analysis module as input data. The indexing phase has two main tasks: feature extraction and data indexing.

It performs feature extraction to complete the input data by computing missing features and data indexing using a data model. The retrieval phase is divided into five main tasks: query formulation, query parsing, query matching, result ranking and result browsing. In the query formulation task, in order Fig. 1. Global architecture of our approach. This approach is based on an external Video Analysis module and on two internal phases: an indexing phase and a retrieval phase. The indexing phase takes results from the Video Analysis module as input data and performs feature extraction and data indexing using a data model. The retrieval phase takes queries from users (by the query formulation task), analyzes and evaluates them (by query parsing and query matching tasks) using indexed data in the indexing phase. The retrieval results are ranked and returned to the users (by the result ranking and the result browsing tasks). The focus of this paper concerns the parts in blue.

to make the users feel familiar with the query language, we propose a SVSQL (Surveillance Video Structured Query Language) language. The vocabulary and the syntax are described in the next section. In the query, the users can select an image example global or a region in an image from the database (by the image selection task). In this case, the feature extraction task computes some features in the image example which are used by the query matching task. In the query parsing task, queries built with the proposed language are transmitted to a parser. This parser checks the vocabulary, analyzes the syntax and separates the query into several parts. The query matching task searches in the database the elements that satisfy the query. The obtained results are ranked and returned to the users.

Data model

Our data model contains two main components of interest form the user's point of view: physical objects and events.

Physical objects: they are the detected objects in the video database. A physical object can be a static object (e.g. contextual object) or a moving object (e.g. a person, a vehicle). Let P be a physical object, P is defined as follows: P = (Id, [Type], [Name], 2D positions, 3D positions, MBRs (Minimum Bounding Box), Features, Time interval) where Id is the label of the object, the Type and Name attributes are optional. The 2D positions, 3D positions, MBRs are the sets of 2D positions, 3D positions, MBRs of this object during its lifetime indicated by Time interval. The Features currently available in the system are the color histogram and a set of detected keypoints by using SIFT (Scale Invariant Feature Transform) descriptors. These features are computed by the Feature extraction task. We use the SIFT descriptors because the SIFT descriptors are invariant to image scale and rotation and they are shown to provide robust matching across a substantial range of affine distortion change in 3D view point, addition of noise, and change in illumination. Therefore, they help to match efficiently images. However, SIFT descriptors do not focus on color information like the color histogram. The methods for extracting these features can be found in [START_REF] Swain | Color indexing[END_REF] for the color histogram and in [START_REF] Lowe | Distinctive image features from scale-invariant keypoints[END_REF] for the SIFT descriptors.

Events: They are the recognized events in the video database. Let E be an event, E is defined as follows: E = (Id, Name, Confidence value, Involved Physical objects, [Sub events], Time interval). Where Id is the label of the event and Name is the name of the event. The Confidence value specifies the confidence degree of recognized event. The work presented in Section 2 does not take into account this information. In our work, the Confidence value is used to compute the final distance between video frames and the query. The Involved Physical objects specifies which physical objects are involved in this event, while the Sub events are optional, and the Time interval indicates the frames in which the event is recognized.

The advantage of our data model is that it is independent of any application and of any feature extraction, learning and event recognition algorithms. Therefore, we can combine results of different algorithms for feature extraction, learning and event recognition (both descriptive and stochastic approaches) and use it for different application domains.

Proposed language and query syntax

The syntax of a query expressed by our language is the following:

SELECT <Output > FROM < Database > WHERE <Condition >
Where: SELECT, FROM, WHERE are keywords for a query and they are mandatory. A graphic interface can be developed to generate this syntax but it is out of the scope of this paper.

-Output specifies the format of the retrieved results. It can have two values: one is video frames indicating that the retrieved results are video frames in which the <Condition> is satisfied and the other is number of events indicating that the retrieved result is the number of events in the database satisfying the <Condition>. -Database specifies which parts of the video database are used to check the <Condition>. It can be either * for the whole video database or a list of named subparts. This is interesting for surveillance because the video database can be divided into several parts according to time or location. It allows to accelerate the retrieval phase in the case that the users know already which parts of the video database they are interested in. -Condition specifies the conditions that the retrieved results must satisfy.

The users express their requirements by defining this component. The condition may have more than one expression connected together by logic operators (AND, OR, NOT), each expression is started by "(" and ended by ")". This component is the most important component in the language.

There are two types of expression in the condition component: a declaration expression (α d ) which is mandatory and a constraint expression (α r ) which provides additional conditions. The declaration expression indicates the types of variable while the constraint expression specifies constraints the variable must satisfy.

The syntax for a declaration expression is: (v : type) where v is a variable. It is the place where the user specifies if the retrieval is at the image level, the semantic level or both levels. The authorized types are : Physical object and its subtypes (Person, Group, Luggage) and Event. In image and video retrieval applications, users usually want to retrieve indexed data that is similar to an example they have. Therefore, besides Physical objects and Events, we add another type SubImage. SubImage has Features attribute like the Physical objects. In query, (v : SubImage) means that v will be set by users image example.

The syntax for a constraint expression is very rich. The constraint expression can be expressed by using a set of projections, functions, predicates, algebra operators and constants. Currently, the authorized projections are {'s Id, 's Type, 's Name, 's 2D positions, 's 3D positions, 's MBRs, 's Features, 's Time interval } for physical object and {'s Id, 's Name, 's Confidence value, 's Involved Physical objects, 's Sub events, 's Time interval } for event; there are two authorized functions which are histogram distance that returns the distance between color histograms and number matched keypoint that returns the number of matched keypoints between an example image and an indexed object; there are four authorized predicates which are color similarity and keypoints matching that return true if an image example and an indexed object are similar in term of color histogram or keypoints, involved in which verifies whether one indexed object belongs to an event and before which verifies whether an event occurs before another event; the authorized algebra operators are =, <, >, >=, =<, ! =}; the constants can be either numbers or strings.

This language is rich enough to express numerous possible queries. Based on the technique proposed in [START_REF] Vu | Automatic video interpretation: A novel algorithm for temporal scenario recognition[END_REF], we implement a parser to check automatically the syntax of each query. This parser automatically analyzes the syntax of the query. The results of this parsing allow to locate which databases will be used to match query, which variables must be set and which results must be returned.

An example expressed by this language at the semantic level is: Find Close to Gates events occurring in videos of all databases.

SELECT video f rames FROM * WHERE ((e: Events) AND (e's Name = "Close to Gates"))

where e is a variable of Events, e s N ame gets Name of e.

Another example expressed by this language at the image level is: Find indexed persons in the database named Video Database that are similar to a given image.

SELECT video f rames FROM Video Database WHERE ((p: Person) AND (i: SubImage) AND (i color similarity p))

where p is a variable of Person, i is a variable that will be set by an image example, color similarity is predicate that decide whether two images are similar (in color).

Query matching

For each expression α in the condition field, the evaluation of the expression α is performed by matching the indexed database D and the expression α. The results of this process are a set of physical objects and events extracted from D that satisfy α. Let η i = {ς i , µ i , I i } be the ith result instance of the expression α and η be the set of the result instances of α where ς i are the physical objects or the events, µ i is the similarity degree that determines how much ς i satisfy α in a time interval I i . Currently, we have defined the similarity degree for the predicates based on the feature similarity (e.g. nearest neighbors for SIFT descriptors and histogram intersection for color histograms). With the other types the default value of the similarity degree is set to 1.

If the query has more than one expression in the condition, the similarity degrees are computed according to the operator linking these expressions as follows: µ = min(µ j , µ k ), µ = max(µ j , µ k ), µ = (1 -µ k ) for AND, OR, NOT operators where µ j , µ k are the similarity degrees of the expressions α j , α k respectively.

Implementation and Experimental Results

Video event database

In order to validate our approach, we have used two videos of 10 minutes and 2 hours acquired by two fixed cameras at different positions that record human activities in a metro station. An example of two scenes in two videos is shown in Fig. 2. The scene contains a platform and several gates. The Video Analysis such as automatic object tracking, object classification and event recognition proposed in [START_REF] Vu | Automatic video interpretation: A novel algorithm for temporal scenario recognition[END_REF] have been automatically applied to these videos. As results, we have 221 indexed physical objects (101 for the first video and 120 for the second one) with their labels, 3D positions, 2D positions, MBRs and time intervals. The 120 physical objects in the second video are classified as 29 persons, 27 groups, 16 crowds, 25 luggages et 23 unknowns. One physical object is perfectly tracked and recognized if in all frames in which this object appears, this object is detected and has one sole label. In addition, 10 event types have been defined and recognized for each frame (inside Platform, close to Gates i (i from 1 to 9)) for the second video. These events are defined in the language proposed in [START_REF] Vu | Automatic video interpretation: A novel algorithm for temporal scenario recognition[END_REF] as follows:

Event(close to, PhysicalObjects((p : Person), (eq : Equipment)) Constraints((p distance eq ≤ Close Distance))

where eq is Gate i (i from 1 to 9), Close Distance is a threshold. Event(inside zone, PhysicalObjects((p : Person), (z : Zone)) Constraints((p in z))

where z is a Platform, in is a predicate that checks whether p's center belongs to the polygon z.

Experimental results and performance evaluation

In order to evaluate the retrieval performance, we use the average normalized rank proposed in [START_REF] Müller | The truth about corel -evaluation in image retrieval[END_REF]. We do not use the measures of true positives, true negatives, false positives and false negatives like in classification problem because for the retrieval problem, the retrieval algorithm is good if it return relevant results first.

Rank = 1 N N rel ( N rel i=1 (R i ) - N rel (N rel + 1) 2 ) ( 1
)
where N rel is the number of relevant result for a particular query, N is the size of the tested set, and R i is the rank of the ith relevant result. Rank is zero if all N rel are returned first. The Rank measure lies in the range 0 (good retrieval) to 1 (bad retrieval), with 0.5 corresponding to a random retrieval. Experiment 1: The goal of this experiment is to check whether our proposed language enables users to retrieve effectively persons in the database even though they are not successfully detected and tracked. For this experiment, the query is: Find in the second video the video frames having persons that are similar (in term of keypoint matching) to the person in this example image. This query is expressed as follows:

SELECT video frames FROM Video2 WHERE ((i: SubImage) AND (o: Person) AND (i keypoints matching o))

where keypoints matching is a predefined predicate of our language, i is an example image.

Figure 3.a shows the average normalized rank for this experiment. There are 120 indexed persons in the second video. The retrieval performance is measured over all 120 images using each in turn as a query. Each image query has from 3 to 5 relevant images. The small obtained average normalized ranks (the maximum value being 0.3183) show that the proposed approach retrieve successfully the indexed objects even when they are imperfectly indexed.

Experiment 2: This experiment aims at pointing out the advantage of our language. It allows to retrieve interesting events with a detailed description. For instance, the event close to Gate1 (p) indicates that person p is close to the Gate 1. In the case of many close to Gate1 recognized instances, the user may be interested in only close to Gate1 frames containing a person that is similar to a given example. The user can ask a query as follows:

SELECT video frames FROM Video2 WHERE ((i: SubImage) AND (o: Person) AND (e: Event) AND (e's Name = close to Gate1) AND (o involved in e) AND (i keypoints matching o))

where i is an image example, 's Name is a predefined projection of event's name, involved in is a predefined predicate that determines whether one person is involved in an event and keypoints matching is a predefined predicate described in section 2.

To answer this query, the persons involved in all close to Gate1 events are used for keypoints matching with the given example. The returned results for each query is a list of close to Gate1 events ranked by the number of matched keypoints between the involved persons and the given example. Among 19 close to Gate1 events of the second video, there are several events concerning one sole person.

We have 15 distinct persons concerned to these 19 events. For each person, we have chosen one image example. Totally, we have 15 example images. Each image turns as input for the query. This query is automatically analyzed by the parser presented in the section 2.

Because of imperfect indexing, one person in the real world may be indexed as different persons within the database. Thanks to similarity matching returned results must consider all inside zone Platform and close to Gate1 events containing indexed persons that are similar (an exact matching that matches the indexed persons by their labels would have returned for this query incomplete results and sometimes empty ones as shown in Figure 4.a). With our technique to answer this query, the system first matches the involved persons in both inside zone Platform and close to Gate1 by keypoint matching. For each person involved in close to Gate1 event, it computes the number of matched keypoints between this person and the persons involved in the inside zone Platform event. A set of persons ordered by their matched keypoints are returned. The inside zone Platform events containing these persons become candidate for retrieval results. Then, these events are used to check whether they satisfy the before constraint with the close to Gate1 events. The before constraint performs based on the starting frames and the ending frames of inside zone Platform and close to Gate1 events.

For each close to Gate1 event, the retrieval result is a list of inside zone Platform events that satisfy the before constraint with close to Gate1 event and that are ranked by the number of matched keypoint between their involved persons and the person involved in close to Gate1 event. The returned result is considered relevant if it contains an inside zone Platform event that satisfies the before constraint and if their involved persons show the same person in the real world.

The average normalized rank for this experiment is given in Fig. 4.b over 19 events of close to Gate1.

This experiment shows the capacity of this language to define new event from the recognized ones with satisfying results (average normalized ranks of all 19 events are smaller than 0.3). As shown in Fig. 3 and Fig. 4, most of the average normalized ranks of the three experiments are small but there are some cases where these measures are quite high (maximum value is respectively 0.3183, 0.3875, 0.262 for the experiment 1, the experiment 2 and the experiment 3) because of the error in object tracking. Therefore, in addition of keypoints and color histogram we intend to use more features or to use a combination of features to solve these problems.

The experiments 1, 2 and 3 show that the proposed approach overcomes the first and the second drawback presented in the introduction. We present another example of query to explain how the proposed approach can do to overcome the third drawback. In the first video, we do not have the results of event recognition. Users may define a new Close to Gates event by stating query as follows:

SELECT video frames FROM Video1 WHERE ((o: Person) AND (z: Gates) AND (o distance z < threshold))

One person is close to gates if the distance between this person and the gates is smaller than a given threshold. The distance is computed based on the 3D position of the persons and the gates. This query takes into account users interest by using a threshold. User can set the value of a threshold as he/she wants. By setting two different values for the threshold, 100 and 150, we have two different results. The first result returns 10 indexed persons with 320 recognized instance of the Close to Gates event. The second one returns 20 indexed persons with 727 recognized instances.

Conclusions

In this paper, we have proposed an approach for video indexing and retrieval for surveillance based on a query language. This new language enables both image and semantic queries and similarity matching. The obtained results for three experiments show that: combining the image level and the semantic level (in experiment 2 and 3) and similarity matching (in experiment 1, 2 and 3) manage imperfect object tracking and imperfect event recognition. New events defined by the user from the recognized ones have been successfully retrieved (in the experiment 3).

Currently, similarity matching has been limited to the color histogram and the keypoints. We plan to study and use more features to enrich the proposed language. In addition, the users may want to make a complex query containing several subqueries. How to combine the results from these subqueries is an issue that we plan to study in the future.

Fig. 2 .

 2 Fig. 2. Two scenes describe human activities in some metro stations(a) in the first video. (b) in the second video.

Fig. 3 .

 3 Fig. 3. (a) The average normalized rank for experiment 1 over 120 queries. (b) The average normalized rank for experiment 2 over 15 queries. The value 0 of average normalized ranks corresponds to good retrieval, value 1 corresponds to bad retrieval and 0.5 corresponds to random retrieval.

Figure 3 . 3 :

 33 b gives the obtained average normalized rank over 15 image examples and 19 events of close to Gate1 in the second video. The ground truth is made by hand for these 15 queries. A returned result is considered relevant if it is a close to Gate1 event whose the involved persons show the same person as in the given image example.Experiment The objective of this experiment is to measure the capacity of this language to define and retrieve new events from the recognized ones. From two recognized events in the database: inside zone Platform and close to Gate1, the user may write a query such as: Find the frames in which one person is going from the Platform to Gate1. This query is expressed as follows:SELECTvideo frames FROM Video2 WHERE ((o 1 : Person) AND (e 1 : Event) AND (e 2 : Event) AND (o 1 involved in e 1 ) AND (o 1 involved in e 2 ) AND (e 1 's Name = inside zone Platform) AND (e 2 's Name = close to Gate1) AND (e 1 before e 2 ))

Fig. 4 .

 4 Fig. 4. (a)Three indexed persons with labels 10, 8, 7 describe the same person in the real world. These indexed persons belong to close to Gate1 and inside zone Platform events that satisfy before constraint. The exact matching based on persons' label gives only one result of person label 10 while our similarity matching gives three persons label 10, 8, 7 with respectively 1, 2, 5 of rank (b) The average normalized rank for experiment 3 over 19 close to Gate1 events. The value 0 of average normalized ranks corresponds to good retrieval, value 1 corresponds to bad retrieval and 0.5 corresponds to random retrieval.


Frank Loray

AND Jorge Vitório Pereira

Frédéric Touzet
email: frederic.touzet@univ-rennes1.fr

Deformation of Rational Curves Along Foliations

Keywords: 1991 Mathematics Subject Classification, 37F75, Key words and phrases, Foliation, Transverse Structure, We acknowledge financial support from ANR-16-CE40-0008 project Foliage

published or not. The documents may come   L'archive ouverte pluridisciplinaire

INTRODUCTION

1.1. Motivation. Singular holomorphic codimension one foliations on projective spaces of dimension at least three have been widely studied in recent years. Much of the recent activity on the subject was spurred by the classification of irreducible components of the space of foliations of degree two by Cerveau and Lins Neto [START_REF] Cerveau | Irreducible components of the space of holomorphic foliations of degree two in CP(n), n ≥ 3[END_REF]. Despite of the growing literature on the subject, not much of it is devoted to the next simplest case: irreducible components of the space of foliations of degree three on P 3 . A notable exception is [START_REF] Cerveau | A structural theorem for codimension-one foliations on P n , n ≥ 3, with an application to degree-three foliations[END_REF] where it is proven that they are either transversely affine foliations, or are rational pullbacks of foliations on surfaces. In this paper we refine [START_REF] Cerveau | A structural theorem for codimension-one foliations on P n , n ≥ 3, with an application to degree-three foliations[END_REF] by means of the following result.

Theorem A. If F is a codimension one singular holomorphic foliation on P 3 of degree three then

(1) F is defined by a closed rational 1-form without codimension one zeros; or [START_REF] Casale | Suites de Godbillon-Vey et intégrales premières[END_REF] there exists a degree one foliation by algebraic curves tangent to F ; or (3) F is a linear pull-back of a degree 3 foliation on P 2 ; or (4) F admits a rational first integral.

The foliations defined by closed rational 1-forms without codimension one zeros lie at the so called logarithmic components of the space of foliations, see [START_REF] Loray | Foliations with trivial canonical bundle on Fano 3-folds[END_REF]Section 2.5] or [4, Lemma 8]. It is also known the existence of irreducible components, for the space of codimension one foliations of every degree on projective spaces of any dimension n ≥ 3, with general element equal to a linear pull-back of a foliation on P 2 . Theorem A reduces the problem of determining/classifying the irreducible components of the space of degree three foliations on P 3 to the problem of determining/classifying irreducible components with general element fitting into the descriptions given by items [START_REF] Casale | Suites de Godbillon-Vey et intégrales premières[END_REF] and (4) of Theorem A.

Theorem A is proved through the study of deformations of morphisms from P 1 to P 3 along leaves of the codimension one foliation.

1.2. Tangential foliations. As it was observed in [START_REF] Loray | Singular foliations with trivial canonical class[END_REF], foliations on a given uniruled manifold X naturally induce foliations on the space of morphisms Mor(P 1 , X), the so called tangential foliations. A leaf L of a tangential foliation correspond to a maximal analytic family of morphisms which do not move the points outside the starting leaves. In other words, if L is a leaf of a tangential foliation and L × P 1 → X is the restriction of the evaluation morphism then the pull-back of the foliation on X to L × P 1 is the foliation given by the fibers of the projection L × P 1 → P 1 , or the foliation with just one leaf. A more precise definition of tangential foliation can be found in Section 4. In this paper, we refine our previous result on the subject [START_REF] Loray | Singular foliations with trivial canonical class[END_REF]Theorem 6.5].

Theorem B. Let X be a uniruled projective manifold and let F be a codimension one foliation on X. Fix an irreducible component M of the space of morphisms from P 1 to X containing a free morphism and let F tang be the tangential foliation of F defined on this irreducible component M ⊂ Mor(P 1 , X). Let δ be the codimension of a general leaf L of F tang inside its Zariski closure L, i.e. δ = dim L -dim L. Then δ ≤ 3. Furthermore, if L is not algebraic (i.e. δ > 0), then the following assertions hold true.

(1) δ = 3 if and only if F is transversely projective but not transversely affine; and

(2) δ = 2 if and only if F is transversely affine but not virtually transversely Euclidean; and (3) δ = 1 if and only if F is virtually transversely Euclidean.

We refer the reader to Section 2 for the definition of transversely projective, transversely affine, and virtually transversely Euclidean foliations.

Whereas [START_REF] Loray | Singular foliations with trivial canonical class[END_REF]Theorem 6.5] provides necessary conditions for a foliation to be transversely projective/transversely affine/virtually transversely Euclidean; Theorem B above establishes an equivalence between the codimension δ and the dimension of the corresponding transverse Lie algebra: sl 2 for transversely projective, aff for transversely affine, and C for virtually transversely Euclidean. Even more importantly, Theorem B improves on the previous result by dropping the hypothesis on the Zariski denseness of the general leaf of the tangential foliation. This is achieved by means of Theorem 3.1, a result which might has independent interest. 1.3. Bootstrapping. On simply connected uniruled manifolds carrying rational curves in general position with respect to the foliation in question, one can push further the analysis carried out to prove Theorem B in order to achieve the more precise result below. Its proof consists in successive applications of Theorem B, where in each step one gains more constraints on the nature of transverse structure.

Theorem C. Let X be a simply connected uniruled projective manifold and let F be a codimension one foliation on X. Fix an irreducible component M of the space of morphisms Mor(P 1 , X) containing a free morphism and let F tang be the tangential foliation of F defined on M . If the general leaf of F tang is not algebraic and the general morphism f : P 1 → X in M intersects non-trivially and transversely all the algebraic hypersurfaces invariant by F then F is defined by a closed rational 1-form without divisorial components in its zero set.

1.4. Structure of the paper. In Section 2 we recall the basic definitions concerning transverse structure of codimension one foliations and present the key properties which will be used throughout. Section 3 is devoted to the proof of extension of transverse structures from fibers of a fibration to the whole ambient manifold. Section 4 studies the tangential foliations and contain the proofs of Theorems B and C. Finally, Theorem A is proved in Section 5.

1.5. Acknowledgements. This work was partially supported by the Brazil-France Agreement in Mathematics. F. Loray and F. Touzet acknowledge the support of ANR-16-CE40-0008 project Foliage and MATH-AmSud project n o 88881.117598. J. V. Pereira acknowledges the support of CNPq, Faperj, and Freiburg Institute for Advanced Studies (FRIAS). The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union's Seventh Programme (FP7/2007-2013) under REA grant agreement n o [609305].

TRANSVERSE STRUCTURES FOR CODIMENSION ONE FOLIATIONS

2.1. Transversely affine and virtually transversely Euclidean foliations. Let F be a codimension one foliation on a projective manifold X. Since any line bundle on X admits rational sections, we may choose a rational 1-form ω 0 defining F . The foliation F is transversely affine if there exists a rational 1-form ω 1 such that dω 0 = ω 0 ∧ ω 1 and dω 1 = 0 .

We say that the pair (ω 0 , ω 1 ) defines a transverse affine structure for F . Although this definition is made in terms of a particular 1-form

ω 0 defining F , if ω ′ 0 = hω 0 is another rational 1-form defining F then ω ′ 1 = ω 1 -dh h is a closed rational 1-form which satisfies dω ′ 0 = ω ′ 0 ∧ ω ′ 1 .
It is therefore natural to say that two pairs of rational 1-forms (ω 0 , ω 1 ) and (ω ′ 0 , ω ′ 1 ) define the same transverse affine structure for F if there exists a non-zero rational function h such that (ω ′ 0 , ω ′ 1 ) = (hω 0 , ω 1 -dh h ). A transverse affine structure for F determines a local system of first integrals given by the branches of the multi-valued function

F = exp ω 1 ω 0 .
Different branches of F at a common domain of definition differ by left composition with an element of the affine group Aff(C) ≃ C ⋊ C * .

If ω 1 is logarithmic with all its periods integral multiples of 2π √ -1 then (exp ω 1 )ω 0 is a closed rational 1-form defining F . In this case we will say that the transverse affine structure defined by (ω 0 , ω 1 ) is transversely Euclidean.

Transverse affine structures for which ω 1 is logarithmic with all its periods commensurable to 2π √ -1 are called virtually transversely Euclidean structures. In this case, the 1-form (exp ω 1 )ω 0 is not necessarily a rational 1-form, but after passing to a finite ramified covering of X it becomes one.

Transversely projective foliations.

Similarly, a foliation F on X is called transversely projective if for any rational 1-form ω 0 defining F there exists rational 1-forms ω 1 and ω 2 such that

dω 0 = ω 0 ∧ ω 1 dω 1 = ω 0 ∧ ω 2 dω 2 = ω 1 ∧ ω 2 .
The triple of rational 1-forms (ω 0 , ω 1 , ω 2 ) defines a projective structure for F . Two triples (ω 0 , ω 1 , ω 2 ) and (ω ′ 0 , ω ′ 1 , ω ′ 2 ) define the same projective structure for F if there exists rational functions f, g ∈ C(X) such that

ω ′ 0 = f ω 0 ω ′ 1 = ω 1 - df f + gω 0 ω ′ 2 = f -1 ω 2 + gω 1 + g 2 ω 0 -dg .
As in the case of transversely affine foliations, the transversely projective foliations admit a canonical collection of local holomorphic first integrals defined on the complement of the polar set of ω 0 , ω 1 , ω 2 , see for instance [START_REF] Casale | Suites de Godbillon-Vey et intégrales premières[END_REF].

For a thorough discussion about transversely affine and transversely projective foliations of codimension one on projective manifolds including a description of their global structure we invite the reader to consult [START_REF] Cousin | Transversely affine foliations on projective manifolds[END_REF] and [START_REF] Loray | Representations of quasi-projective groups, flat connections and transversely projective foliations[END_REF] respectively. Here we will review two important features of transversely homogeneous foliations/structures of codimension one in a formulation slightly more general than what is currently available in the literature. These two features play an essential role in this paper.

2.3.

Behaviour under dominant rational maps. Transversely homogenous structures for codimension one foliations behave rather well with respect to dominant rational maps as the result below shows.

Lemma 2.1. Let G be a codimension one foliation on a projective manifold Y , F a codimension one foliation on a projective manifold X, and F : Y X a dominant rational map such that G = F * F . If G is transversely projective, transversely affine, virtually transversely Euclidean then F is, respectively, transversely projective, transversely affine, virtually transversely Euclidean.

Proof. This is essentially the content of [8, Lemma 6.2], a geometric translation of [2, Lemma 2.1 and Lemma 3.1]. The only part of the statement not explicit proved there is that when G is virtually transversely Euclidean then the same holds true for F . For the sake of completeness let us present a proof of this implication.

Notice that we can assume from the beginning that F does not admit a rational first integral, as otherwise there would be nothing to prove. After cutting Y with hyperplane sections we can assume that Y and X have the same dimension. We can further assume that G is transversely Euclidean and that the generically finite rational map F : Y X is Galois. Let ω be a rational 1-form defining F . Let η be a closed rational 1-form defining G. If ϕ is one of the deck transformation of F then ϕ * η = λη for some root of the unity λ. Since η and F * ω define the same foliation, there exists a rational function h such that

F * ω = hη. Of course ϕ * h = λ -1 h and consequently ϕ * dh h = dh h . The closedness of η implies dF * ω = dh h ∧ F * ω.
The invariance of dh h under the deck transformations allows us to find a logaritmic 1-form β on X with rational residues such that dh h = F * β. Hence dω = β ∧ ω and F is virtually transversely Euclidean.

The converse statement is trivially true: if F is transversely projective, transversely affine, or virtually transversely Euclidean, then the same holds true for G. Indeed, it suffices to pull-back the 1-forms ω i defining the structure by F . 2.4. Uniqueness of tranversely homogeneous structures. Transversely homogeneous structures for codimension one foliations are unique except for very special exceptions which are described by the next result. Lemma 2.2. Let F be a codimension one foliation on a projective manifold. The following assertions hold true.

(1) If F admits two non-equivalent transversely projective structures then F admits a virtually transversely Euclidean structure.

(2) If F admits two non-equivalent transversely affine structures then F is defined by a closed rational 1-form, i.e. F admits a transversely Euclidean structure.

(3) If F admits two non-equivalent virtually transversely Euclidean structure then F admits a rational first integral.

Proof. The first two possibilities are described in [START_REF] Cerveau | Complex codimension one singular foliations and Godbillon-Vey sequences[END_REF]Lemma 2.20]. For describing the last possibility, assume that F is defined a rational 1-form ω 0 and observe that the existence of two non-equivalent virtually transversely Euclidean structures is equivalent to the existence of two linearly independent logarithmic 1-forms ω 1 , ω ′ 1 , both with periods commensurable to πi and satisfying

dω 0 = ω 0 ∧ ω 1 = ω 0 ∧ ω ′ 1 . Their difference ω 1 -ω ′
1 is non-zero, proportional to ω 0 , and a suitable complex multiple of it has all its periods in 2πiZ. In other words, there exists a constant λ ∈ C * such that

exp λ(ω 1 -ω ′ 1 )
is a rational function constant along the leaves of F .

EXTENSION OF TRANSVERSE STRUCTURES

The purpose of this section is to prove the following result.

Theorem 3.1. Let F be a codimension one foliation on a projective manifold X. Suppose there exists a projective variety B and a morphism f : X → B with general fiber irreducible such that the F is transversely projective/transversely affine/virtually transversely Euclidean when restricted to a very general fiber of f . If the restriction of F to the very general fiber does not admit a rational first integral then F is, respectively, transversely projective/transversely affine/virtually transversely Euclidean.

In the statement of Theorem 3.1 one can replace the existence of the morphism f by the existence of a covering family Z of subvarieties of X such that F is transversely projective/transversely affine/virtually transversely Euclidean when restricted to a very general element of the family in order to achieve the same conclusion. One reduces to the statement above by pulling back the foliation to the total space of the family of cycles, applying Theorem 3.1 to this pull-back foliation, and descending the conclusion to X using Lemma 2.1.

Before proceeding to the proof of Theorem 3.1 we point out that the projectiveness of X is essential for the validity of the Lemma 3.5 as the example below shows.

Example 3.2. Let F = E × E be the square of an elliptic curve and consider the automorphism of it defined by ϕ(x, y) = (2x + y, x + y). This is the standard example of an hyperbolic automorphism. It leaves invariant two linear foliations on E × E, say F + , F -, defined by closed holomorphic 1-forms ω + , ω -respectively. As can be easily seen ϕ * ω ± = λ ± ω ± where λ ± are the eigenvalues of the matrix 2 1 1 1 .

If we now consider a projective curve B and a suspension of a representation π 1 (B) → Aut(F ± ), containing ϕ in its image, we get a genuinely transversely affine foliation on compact complex manifold X fibering over B with fibers F such that the restriction to any fiber is transversely Euclidean and without rational first integral. The ambient manifold X is not Kähler. Indeed, aiming at a contradicition, suppose that X is Kähler. Then ϕ has to preserve a Kähler class and [7, Proposition 2.2] implies that some non trivial power of ϕ is a translation. This gives the sought contradiction.

3.1. Proof of Theorem 3.1. Theorem 3.1 follows from the next three results combined.

Lemma 3.3. Let F be a codimension one foliation on a projective manifold X. Assume that X is endowed with a fibration f : X → B such that the restriction of F to the very general fiber of f is transversely projective. Then there exists rational 1-forms ω 0 , ω 1 , ω 2 and ω 3 with the following properties.

(1) The 1-forms ω 0 , . . . , ω 3 satisfy the system of equations

(3.1)    dω 0 = ω 0 ∧ ω 1 dω 1 = ω 0 ∧ ω 2 dω 2 = ω 0 ∧ ω 3 + ω 1 ∧ ω 2 (2)
The foliation F is defined by ω 0 .

(3) The restriction of 1-form ω 3 to a general fiber of f is zero. (4) If we further assume that the restriction of F to a very general fiber of f is transversely affine then we have that both ω 2 and ω 3 restrict to zero at a general fiber of f .

Proof. We start by choosing α 0 , α 1 , α 2 , α 3 rational 1-forms on X such that α 0 defines F and which satisfy the system equations (3.1). The existence of such 1-forms is well-known and can be traced back to Godbillon-Vey, see [5, Section 2.1].

Let G be the restriction of F to the (schematic) generic fiber X of f . Hence X is a projective manifold defined over the function field C(B) of the basis of the fibration and G is a codimension one foliation on X . The algebraic nature of transversely projective structures implies that G is transversely projective. If we denote by β 0 , . . . , β 3 the restrictions of α 0 , . . . , α 3 to X then, according to [5, Section 2.3], we can replace β 2 by β 2 + f β 0 where f is a rational function on X in such a way that they now satisfy

   dβ 0 = β 0 ∧ β 1 dβ 1 = β 0 ∧ β 2 dβ 2 = β 1 ∧ β 2
Consider a lift of f to a rational function on X (still denoted by f ) and set ω 0 = α 0 , [START_REF] Cerveau | Complex codimension one singular foliations and Godbillon-Vey sequences[END_REF]Corollary 2.4] there exists a unique ω 3 such that the system of equations (3.1) is satisfied. Moreover, the uniqueness of ω 3 implies that its restriction to the generic fiber of f must be zero. This is sufficient to prove item (3) of the lemma. The proof if item (4) is completely similar. Proposition 3.4. Let F be a codimension one foliation on a projective manifold X. Assume that X is endowed with a fibration f : X → B and that the restriction of F to a general fiber of f does not admit a rational first integral. If there exists 1-forms ω 0 , ω 1 , ω 2 , ω 3 satisfying item (3) of Lemma 3.3 then F is transversely projective. Similarly, if there exists 1-forms ω 0 , ω 1 , ω 2 , ω 3 satisfying item (4) of Lemma 3.3 then F is transversely affine.

ω 1 = α 1 , ω 2 = α 2 + f α 0 . According to
Proof. Assume that the dimension of B is q and let f 1 , . . . , f q be algebraically independent rational functions in f * C(B) ⊂ C(X). Notice that item (3) of Lemma 3.3 guarantees that ω 3 ∧df 1 ∧. . .∧df q = 0. Assume that the foliation ω 0 ∧df 1 ∧• • •∧df q has some transcendental leaf. We want to prove that F is transversely projective. The proof is recursive on q: we first prove that, on each fixed codimension q -1 subvariety defined by (an irreducible component of ) f 2 , . . . , f q = constant, the restriction of F is transversely projective. Note that this later foliation must have transcendental leaves also so that we can go on with codimension q -2 subvarieties defined by f 3 , . . . , f q = constant. It is thus enough to consider the case q = 1.

We start with the easier case where F restricts as a transversely Euclidean foliation on the level sets. In other words, assume that

ω 1 ∧ df = 0.
Then we can write ω 1 = gdf for some rational function g and derivating dω 0 = ω 0 ∧ ω 1 , we get

ω 0 ∧ df ∧ dg = 0. If df ∧ dg = 0, then dω 1 = 0 so that F is transversely affine. If df ∧ dg = 0, then g is a rational first integral for F | f =constant .
Before proceeding to the next case, note that our assumption in the previous case is equivalent to dω 0 ∧ df = 0. Indeed, this means that ω 0 ∧ ω 1 ∧ df = 0 and by division (recall that ω 0 ∧df = 0) ω 1 = gdf +hω 0 for rational functions g and h. But (see [5, Section 2.1]) we can then replace ω 1 by ω 1hω 0 without changing the equality dω 0 = ω 0 ∧ ω 1 .

We will treat the affine case, now assuming

ω 2 ∧ df = 0.
One can write ω 2 = gdf as before and we get dω 1 = gω 0 ∧ df . If g = 0 then dω 1 = 0 and F is transversely affine. If g = 0, by derivation we get

ω 1 - dg g ∧ ω 0 ∧ df = 0.
Substituting ω 0 = g • ω0 and ω 1 = ω1 + dg g , we get (the same conclusion with g = 1) ω1 ∧ ω0 ∧ df = 0 and thus dω 0 ∧ df = 0 and we are back to the Euclidean case.

We finally end with the general projective case, assuming

ω 3 ∧ df = 0.
Writing ω 3 = gdf , we get dω 2 = gω 0 ∧ df + ω 1 ∧ ω 2 . If g = 0, then F is transversely projective; if not, after derivation we get

(ω 1 - 1 2 dg g ) ∧ ω 0 ∧ df = 0.
After division, we find

ω 1 = 1 2 dg g + hdf + kω 0
for rational functions h and k. In fact, we can replace ω 1 by ω 1kω 0 thus assuming k = 0 (we now forget ω 2 ). If dh ∧ df = 0, then ω 1 = 1 2 dg g + hdf is closed and F is transversely affine. If dh ∧ df = 0, then

ω 0 ∧ dh ∧ df = ω 0 ∧ dω 1 = 0 and h is a first integral for F | f =constant .
Lemma 3.5. Let F be a transversely affine foliation of codimension one on a projective manifold X. Assume there exists a projective surface Σ ⊂ X for which the restriction of F to Σ is virtually transversely Euclidean and admits no rational first integral then the same holds true for F .

Proof. This is a consequence of the structure theorem for transversely affine foliations, see [START_REF] Cousin | Transversely affine foliations on projective manifolds[END_REF] and [START_REF] Loray | Representations of quasi-projective groups, flat connections and transversely projective foliations[END_REF]Theorem D], which says that transversely affine foliations on projective manifolds which are not virtually transversely Euclidean are pull-backs of Riccati foliations on surfaces. Aiming at a contradiction, assume that F is transversely affine but not transversely Euclidean. Thus there exists a rational map ϕ : X S to a ruled surface, and a Riccati foliation on S such that ϕ * R = F . The restriction of ϕ to Σ must be dominant as otherwise F |Σ would admit a rational first integral. Thus, if we assume that F |Σ is virtually transversely Euclidean then Lemma 2.1 implies that the same holds true for R. But this would imply that F = ϕ * R is also virtually transversely Euclidean contrary to our assumptions. This gives the sought contradiction which proves the lemma.

TANGENTIAL FOLIATION

This section is devoted to the proof of Theorem B. We start by recalling the definition of the tangential foliation. For a more detailed account, we invite the reader to consult [8, Section 6].

4.1. Tangential foliation. Let X be a uniruled projective manifold. These are characterized by the existence of non-constant morphisms f : P 1 → X such that f * T F is generated by global sections, the so called free morphisms. At a neighborhood of a free morphism the scheme Mor(P 1 , X) is smooth and the tangent space of Mor(P 1 , X) at the point f is naturally isomorphic to H 0 (P 1 , f * T X ).

Let us fix a foliation F on X and an irreducible component M ⊂ Mor(P 1 , X) of the space of morphism from P 1 to X containing a free morphism f . Recall from [START_REF] Loray | Singular foliations with trivial canonical class[END_REF]Section 6] that F induces a natural foliation on M , called the tangential foliation of F and denoted by F tang , with tangent space at a general free morphism f given by H 0 (P 1 , T F ) ⊂ H 0 (P 1 , T X ).

The tangential foliation admits the following alternative description. If ev : M × P 1 → X is the evaluation morphism, p : M × P 1 → M is the natural projection, and H is the foliation on M × P 1 defined by the projection to P 1 then F tang coincides with p * (H ∩ ev * F ).

4.2.

Foliations on B × P 1 . Let L be a leaf of F tang and L its Zariski closure. We recall below [START_REF] Loray | Singular foliations with trivial canonical class[END_REF]Theorem 6.5] which describes the foliation G = ev * F |L×P 1 .

Theorem 4.1. Let B be an algebraic manifold, let G be a codimension one foliation on B × P 1 , let π : B × P 1 → B be the natural projection, and let H be the codimension one foliation defined by the fibers of the other natural projection ρ : B × P 1 → P 1 . If the general fiber of π is generically transverse to G and the general leaf of the direct image T = π * (G ∩ H) is Zariski dense then the codimension of T is at most three. Moreover,

(1) If codim T = 1 then G is defined by a closed rational 1-form; (2) If codim T = 2 then G is transversely affine; (3) If codim T = 3 then G is transversely projective.
Remark 4.2. The proof presented in [START_REF] Loray | Singular foliations with trivial canonical class[END_REF] gives more information as we now proceed to recall. It starts by showing that the foliation G is defined by a rational 1-form which can be written as ω = dz + a(z)α + b(z)β + c(z)γ where α, β, γ are rational 1-forms on B; a, b, c are rational functions on P 1 ; and z is a coordinate on P 1 . To avoid overburdening the notation, we identify 1-forms on B with their pull-backs through π to B × P 1 , and we also identify 1-forms on P 1 with their pullbacks through ρ : B × P 1 → P 1 . Furthermore, the codimension of T coincides with the dimension of the C-vector space generated by α, β, γ.

When codim T = 1 then the foliation G is induced by the sum of pull-backs under the natural projections of closed rational 1-form on B and on P 1 . To wit, G is defined by dz f (z) + η, where f is rational function on P 1 and η is closed rational 1-form on B. Notice that the zero set of this closed rational 1-form ω has no divisorial components.

If instead codim T ≥ 2 then the proof of [8, Theorem 6.5] establishes the existence of a rational map ϕ : P 1 → P 1 and a Riccati foliation R on the manifold B × P 1 such that G = (id B ×ϕ) * R. The case where codim T = 3 is distinguished from the case where codim T = 2 by the existence of fibers of the projection ρ : B × P 1 → P 1 invariant by G. When codim T = 3 none of the ρ-fibers are invariant. When codim T = 2 the Riccati foliation R has exactly one invariant ρ-fiber and, consequently, the foliation G has at least one invariant ρ-fiber.

We proceed to investigate the converse of Theorem 4.1. We start with a simple observation. Proof. By the definition of F tang the restriction of Y to L × P 1 is horizontal. Therefore the restriction to Y ∩ (L × P 1 ) of the projection L × P 1 → P 1 is locally constant. Since L is Zariski dense L = B the same holds true over Y itself. Therefore Y is horizontal. Proof. Aiming at a contradiction, let us assume that G is virtually transversely Euclidean but δ = codim T ≥ 2. According to Remark 4.2 we can assume that G is the pull-back of a Riccati foliation R on B × P 1 through a rational map of the form id B ×ϕ : B × P 1 B × P 1 . In particular, R is defined by a rational 1-form which can be written as

ω = dz + α + zβ + z 2 γ
where α, β, γ are rational 1-forms on B which define the foliation T and z is a coordinate on P 1 . The integrability condition ω ∧ dω = 0 implies that

dα = α ∧ β dβ = 2α ∧ γ dγ = β ∧ γ .
In particular, the assumption codim T ≥ 2 implies dω = 0.

Since we are assuming that G is virtually transversely Euclidean, the same holds true for R according to Lemma 2.1. Therefore there exists a non-zero closed logarithmic 1-form ω 1 on B × P 1 with periods commensurable to √ -1π such that dω = ω ∧ ω 1 . Since any holomorphic function on P 1 is constant, the canonical multi-valued first integral

F = exp ω 1 ω 0
must have a hypersurface contained in its singular set dominating the basis of the fibration π : B × P 1 → B. As this hypersurface is clearly invariant by R, we can apply Lemma 4.3 to guarantee that, after a change of coordinates on P 1 , the section {z = ∞} is invariant by R. In these new coordinates, γ = 0, i.e. R is defined by ω = dz + α + zβ. This already shows that codim T ≤ 2, since T is the foliation defined by α and β.

The integrability condition implies that (ω, β) is a transverse affine structure for R. Therefore both (ω, β) and (ω, ω 1 ) are transverse structures for R. If they are the same then ω 1 = β and exp(n β) is a rational function constant along the leaves of T for any sufficiently divisible integer n (ω 1 is logarithmic with periods commensurable to √ -1π). As we are assuming that the leaves of T are Zariski dense, we deduce that β = 0 and, consequently, codim T = 1 contrary to our assumptions.

We can assume that ω 1 and β are distinct closed rational 1-forms. Their difference is a non-zero closed rational 1-form defining R. Therefore we can write βω 1 = hω for a certain non-constant rational h ∈ C(B × P 1 ). The irreducible components of the divisor of zeros and poles of h, according to Lemma 4.3, are either fibers of π : B × P 1 → B or fibers of ρ : B × P 1 → P 1 . Therefore we can write h as a product of a ∈ C(B) with b ∈ C(P 1 ).

We claim that the function a ∈ C(B) is constant. Indeed, from the equalities d(hω) = 0 and dz ∧ db = 0 we deduce that dz ∧ d(aω) = 0 =⇒ dz ∧ da ∧ ω + adz ∧ dω = 0 .

Using that dβ = 0 we can writing dz ∧ da ∧ (α + zβ) + adz ∧ dα = 0 .

Finally, taking the wedge product of this last identity with β we conclude that dz ∧ da ∧ α ∧ β = 0. It follows that a is a first integral for T , and as such must be constant. Now from d(bω) = 0 for b ∈ C(P 1 ), we deduce the identity db ∧ α + bdα + d(zb) ∧ β + zhdβ = 0. After taking the wedge product with α and using the vanishing of dβ and of α ∧ dα, we conclude that α ∧ β = 0. This implies codim T ≤ 1. The proposition follows.

Arguing as in the beginning of the proof above, one also obtains the following result.

Synthesis (Proof of Theorem B).

Let X be a uniruled projective manifold and F be a codimension one foliation on X. We fix an irreducible component M of the space of morphisms from P 1 to X containing a free morphism and let F tang stand for the tangential foliation of F defined on this irreducible component M . We denote by F tang the foliation on M with general leaf given by the Zariski closure of a leaf of F tang . The existence of a foliation with these properties follows from [START_REF] Bonnet | Minimal invariant varieties and first integrals for algebraic foliations[END_REF].

First notice that F must have some algebraic leaf. Otherwise the existence of a transverse structure for F given by Theorem 4.6 and the simple connectedness of X would give rise to the existence of a rational first integral for F . It follows that the leaves of F tang are algebraic contradicting our initial assumption. Therefore we can apply Lemma 4.7 to deduce that F is a transversely affine foliation. Fix rational 1-forms ω 0 and ω 1 on X such that dω 0 = ω 0 ∧ ω 1 , dω 1 = 0, and ω 0 defines F .

If F is virtually transversely Euclidean then the result follows from Theorem 4.6. Assume from now on that F is not virtually transversely Euclidean. Since X is simply connected, this implies that ω 1 is not a logarithmic differential with rational residues. Notice that ω 0 and ω 1 are not uniquely defined, but any different pair (ω ′ 0 , ω ′ 1 ) satisfying the conditions above will satisfy ω 0 = hω ′ 0 and ω 1 = ω ′ 1 + d log h. In particular, the fact that ω 1 is not a logarithmic 1-form with rational residues does not depend on the choice of the pair.

Aiming at a contradiction assume that dim L -dim L ≥ 2. According to Remark 4.2, the foliation G is the pull-back of a Riccati foliation on U × P 1 leaving invariant the section at infinity. A simple computation shows that this implies the existence of a transverse affine structure for G defined by a pair (α 0 , α 1 ) such that the restriction of α 1 at a general fiber P 1 is logarithmic with integral residues. But the pull-back of ω 1 under the evaluation morphism is not of this form since the image of f intersects all components of the polar divisor of ω 1 . Hence G admits two non-equivalent transversely affine structures. Lemma 2.2 implies that G is transversely Euclidean. The lemma follows from Theorem 3.1. Theorem 4.9 (Theorem C of the Introduction). If X is simply connected, the general leaf of F tang is not algebraic, and the general morphism f ∈ M intersects non-trivially and transversely every algebraic hypersurface invariant by F then F is defined by a closed rational 1-form without divisorial components in its zero set.

Proof. We start by showing that F is defined by a closed rational 1-form. Lemma 4.8 implies that F is virtually transversely Euclidean. Therefore the transverse structure for F is defined by a pair (ω 0 , ω 1 ) where ω 1 is a closed logarithmic 1-form with periods commensurable to π √ -1. If the periods of ω 1 are integral multiples of 2π √ -1 then exp ω 1 ω 0 is the sought closed rational 1-form. Assume from now on that the periods of ω 1 are not integral multiples of 2π √ -1. Let L be a general leaf of F tang and U ⊂ L be the smooth locus of its Zariski closure. Let G = ev * F |U×P 1 be the pull-back of F to M × P 1 under the evaluation morphism. On the one hand, according to Remark 4.2, G is defined by a closed rational 1-form. On the other hand, the transversality of the general f ∈ M with the F -invariant algebraic hypersurfaces imply that the periods of ev * ω 1 are also not integral multiples of 2π √ -1. Hence G admits two non-equivalent virtually transversely Euclidean structures. Lemma 2.2 implies that all the leaves of G are algebraic. It follows that all leaves of F tang are algebraic contrary to our assumptions. This concludes the proof that F is defined by a closed rational 1-form ω.

To verify that ω does not have divisorial components in its zero set, we proceed similarly. Pull-back ω to U × P 1 using the evaluation morphism. Any irreducible divisorial component of the zero set of ω is a F -invariant algebraic hypersurface. Therefore, by assumption, the restriction of the 1-form ev * ω on U × P 1 will have an irreducible component in its zero set which dominates U . This is only possible if ev * ω does not depend on the variables of U , i.e. ev * ω = a(z)dz where z is a coordinate on P 1 and a ∈ C(P 1 ) is a rational function. It follows that L and L have the same dimension, contrary to our assumptions. Theorem C follows. 4.5. Tangential foliation with algebraic leaves. When F tang is a foliation by algebraic leaves, the original foliation F inherits a subfoliation by algebraic leaves. We keep the notation settled at the beginning of Section 4.3. Proposition 4.10. If all the leaves of F tang are algebraic, then there exists a foliation by algebraic leaves A contained in F such that H 0 (P 1 , f * T A ) = H 0 (P 1 , f * T F ) for any sufficiently general morphism f ∈ M . In particular, F is the pull-back under a rational map of a foliation on a lower dimensional manifold.

Proof. Let A be the maximal foliation by algebraic leaves contained in F . The existence of such A is assured by [START_REF] Loray | Singular foliations with trivial canonical class[END_REF]Lemma 2.4]. If L is a general leaf of F tang then the image of L × {z} (for any fixed z ∈ P 1 ) under the evaluation morphism ev : M × P 1 → X is contained in a leaf of F . Moreover, it is also contained in a leaf of A. This makes clear that A tang contains F tang . But since A is contained in F , we must have A tang contained in F tang . Hence F tang = A tang and the equality H 0 (P 1 , f * T A ) = H 0 (P 1 , f * T F ) holds true for any sufficiently general morphism f ∈ M .

Lemma 4 . 3 .

 43 Notation and assumptions as in Theorem 4.1. If Y is an algebraic leaf of G generically transverse to the fibers of π then Y is contained in a fiber of ρ.

Proposition 4 . 4 .

 44 Notation and assumptions as in Theorem 4.1. If G is virtually transversely Euclidean then codim T = 1.

Corollary 4 . 5 .

 45 Notation and assumptions as in Theorem 4.1. If G is transversely affine then codim T ≤ 2.

FOLIATIONS OF DEGREE THREE

5.1. Proof of Theorem A. Let F be a codimension one foliation of degree three on P 3 . The canonical bundle of

If the general leaf of F tang is not algebraic then Theorem C guarantees that F is defined by a closed rational 1-form without divisorial components in its zero set. Thus F is described by item (1).

If the general leaf of F tang is algebraic then let A ⊂ F be the foliation by algebraic leaves given by Proposition 4.10. In particular, h 0 (P 1 , f * T A) = h 0 (P 1 , f * T F ) = 0.

If dim A = 2 then all the leaves of F are algebraic and Darboux-Jouanolou Theorem guarantees that F admits a rational first integral. The foliation F fits into the description given by item (4).

If instead dim A = 1 then T A is a line bundle and h 0 (P 1 , f * T A) = 0 implies that T A = O P 3 or T A = O P 3 (1). In the first case, A has degree one and F is described by item (2). In the second case, A has degree zero and its leaves are lines through a unique point of P 3 . The foliation F is described by item (3).

Theorem 4.6 (Theorem B of the Introduction). Let δ = dim F tang -dim F tang . If the general leaf of F tang is not algebraic (i.e. δ > 0) then δ ≤ 3. Furthermore the following assertions hold true.

(1) δ = 3 if and only if F is transversely projective but not transversely affine; and

(2) δ = 2 if and only if F is transversely affine but not virtually transversely Euclidean; and (3) δ = 1 if and only if F is virtually transversely Euclidean.

Proof. Let L be a general leaf of F tang which we will assume not algebraic. If L is the Zariski closure of L and U is the smooth locus of L then we are in position to apply Theorem 4.1 to G = (ev * F ) |U×P 1 and deduce that G is transversely projective and dim L -dim L = δ ∈ {1, 2, 3}. Theorem 3.1 implies that ev * F is also transversely projective.

If the general leaf of F tang is not algebraic then combining Theorem 4.1 with Theorem 3.1 one deduces that F is transversely projective.

Let us first prove assertion (3). If δ = 1 then G is defined by a closed rational 1-form, see Remark 4.2. Theorem 3.1 implies that ev * F is virtually transversely Euclidean. Lemma 2.1 implies that the same holds true for F . Reciprocally, if F is virtually transversely Euclidean then the same holds true for G. We apply Proposition 4.4 to deduce that δ = 1. Assertion (3) follows.

The proof of assertion ( 2) is similar. If δ = 2 then G is transversely affine. Theorem 3.1 implies that ev * F is transversely affine and Lemma 2.1 implies that the same holds true for F . Reciprocally, if F is transversely affine then Corollary 4.5 implies δ ≤ 2. Assertion (3) implies δ ≥ 2 and assertion (2) follows.

As before, Theorem 3.1 combined with Lemma 2.1 imply that F is transversely projective. Reciprocally, as we already know that δ ≤ 3, assertions (2) and (3) imply assertion (1).

Bootstrapping (Proof of Theorem C).

We proceed to prove Theorem C. We keep the notations from Section 4.3.

Lemma 4.7. Assume that the general leaf of F tang is not algebraic. If F has an algebraic leaf Y ⊂ X which intersects the image of a general morphism f ∈ M then F is transversely affine.

Proof. Let L be a general leaf of F tang ; L be its Zariski closure; and U ⊂ L be the smooth locus of the Zariski closure. Lemma 4.3 implies that G = ev * F |U×P 1 has a horizontal leaf. Since G = (id U ×ϕ) * R it follows that the Riccati foliation R also has a horizontal leaf. Thus, in a suitable coordinate system where the horizontal is at {z = ∞}, the Riccati foliation R is defined by

Hence the restriction of F tang to L is defined by ω 0 and ω 1 and therefore dim F tangdim F tang ≤ 2. We apply Theorem 4.6 to conclude that F is transversely affine. Lemma 4.8. Assume X is simply connected and that the general leaf of F tang is not algebraic. If the image of a general morphism f ∈ M intersects non-trivially every algebraic leaf of F then F is virtually transversely Euclidean.

Proof. We keep the notation from Lemma 4.7. According to Theorem 4.6 it suffices to show that G = ev * F |U×P 1 is defined by a closed rational 1-form.