Adjaratou Arame

J.-F Mattei

J Martinet

J.-P Ramis

R Moussu

PAIRS OF FOLIATIONS AND MATTEI-MOUSSU'S THEOREM

Keywords: 2010 Mathematics Subject Classification. 32M25, 32S65, 34C20, 34M35 Foliations, Web, Holonomy, Reduction of singularities. We

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Introduction

A singular holomorphic foliation by curves on a surface is locally given by a holomorphic vector field with isolated zeroes. Outside of the zero set, the complex integral curves of the vector field v define a regular holomorphic foliation by curves F. Another vector field will define the same foliation if, and only if, it takes the form f • v for a non vanishing holomorphic function f .

In [START_REF] Mattei | Holonomie et intégrales premieres[END_REF], Mattei and Moussu provide a topological characterization of singular points of holomorphic foliations that admit a non constant holomorphic first integral, i.e. such that the leaves are locally defined as the level curves of a holomorphic function h. For this, they use the reduction of singularities by blow-up, and they provide a careful study of the reduced singular points. In particular, they prove that the saddle singular points are determined by their eigenvalues and holonomy map (see Theorem 3.1, and also [START_REF] Elizarov | Remarks on orbital analytic classification of germs of vector fields[END_REF]). The proof of this famous result is in two parts. They consider two saddles with the same eigenvalues {λ 1 , λ 2 } and assume that the holomomies of the separatrices associated to λ 1 say are conjugated. First they use this conjugacy to construct a conjugacy of the foliations between neighborhoods of annuli contained into the separatrices; this is done by a simple and standard geometric argument. In a second step, they prove that the conjugacy extends on a neighborhood of the singular point minus the second separatrix, by controlling the boundedness by means of Gronwall's Inequality, and extends holomorphically along the separatrix by Riemann Extension Theorem. This second part of the proof is delicate and, in general, not written in full details. Here, we provide in Section 3 a complete and elementary proof avoiding Gronwall's Inequality. In fact, we compare the analytic conjugacy on the annulus with the formal conjugacy which is transversely formal on the disc: they differ by a transversely formal symmetry of the foliation on the annulus. It suffices to prove that all such symmetry extend as a transversely formal symmetry on the disc, and this allow to conclude the analytic extension of the conjugacy. Our proof remains valid in the saddle-node case when the central manifold (invariant curve tangent to the zero-eigendirection) is analytic (not only formal) although Gronwall's Inequality cannot be used to conclude in that case. We hope that this proof can be useful in other situations, like for higher dimensional saddles, for instance weakening assumptions of Reis' result in [START_REF] Reis | Equivalence and semi-completude of foliations[END_REF]. At least, this approach has been recently used by the first author to classify some natural pairs of singular foliations. Let us explain.

Mattei-Moussu's Theorem provide fibered conjugacy, that is to say, it can be thought as classification of pairs (F 1 , F 2 ) of a saddle F 2 with a regular foliation F 1 such that one of the separatrices of F 2 is a leaf of F 1 . We start a study of pairs of singular foliations by proving a reduction of singularities. In Theorem 4.2, we prove that, given a pair of singular foliations on a complex surface M , we can construct a proper map π : M → M obtained by a finite sequence of blowing-up such that the singular points of the lifted pair π * (F 1 , F 2 ) fits with a list of simple models. Reduced pairs of regular foliations can be easily handled and are well-known, let us mention [START_REF] Thom | Classification locale de bifeuilletages holomorphes sur les surfaces complexes[END_REF] for a recent contribution. Mattei-Moussu's Theorem and some results of Martinet and Ramis allows us to deal with the analytic classification of those models where only one of the two foliations is regular. In Section 5, we state a recent contribution of the first author to the case of two singular foliations: reduced pairs consist of pairs of reduced singular foliations sharing the same invariant curves. In [START_REF] Diaw | Géométrie de certains tissus holomorphes singuliers en dimension 2[END_REF][START_REF] Diaw | Pairs of foliations: reduction of singularities and local classification[END_REF], the first author provides a complete analytic classification in the case the two foliations have a reduced tangency divisor (i.e. without multiplicity) along the invariant curves (see Theorem 5.1). All details will be published in the forthcoming paper [START_REF] Diaw | Pairs of foliations: reduction of singularities and local classification[END_REF]. In the study of reduced pairs, only remains the case of two reduced singular foliations with higher tangency along the invariant curves. This seems feasible, but more technical and left to the future.

Singular holomorphic foliations in dimension 2

A singular holomorphic foliation by curves F on a complex surface M is defined by coherent analytic subsheaf T F ⊂ T M of rank 1 such that the quotient sheaf T M/T F is torsion free. The locus where T M/T F is not locally free is the singular locus, a discrete set. Locally, sections of T F take the form f • v where v is a holomorphic vector field with isolated zeroes, and f is any holomorphic function. Any other generating vector field takes the form f • v with f non vanishing. Equivalently, one can locally define T F as the kernel of a holomorphic 1-form ω with isolated zeroes, which is also defined up to a non vanishing factor, and globally by the conormal bundle N * F , a rank 1 coherent subsheaf of T * F with torsion-free cokernel (see [START_REF] Brunella | Birational Theory of Foliations[END_REF]Chapter 2]). Let v be a germ of holomorphic vector field at (C 2 , 0):

v = a(x, y)∂ x + b(x, y)∂ y , a, b ∈ C{x, y}
and we assume 0 is an isolated singular point of v, i.e. the common zero of a and b. The linear part of v is defined as the linear part of the map V : (x, y) → (a(x, y), b(x, y)), i.e. its differential D 0 V at 0; we denote by lin(v) this linear map. If Φ ∈ Diff(C, 0) is a coordinate change, then we have

lin (Φ * v)) = D 0 Φ -1 • lin(v) • D 0 Φ.
Therefore, the eigenvalues {λ 1 , λ 2 } of lin(v) are invariant under change of coordinates. If we are interested in the foliation F defined by v, then we note that

lin(f • v) = f (0) • lin(v)
so that only the ratio λ = λ 2 /λ 1 (when λ 1 = 0 say) is well defined by the foliation. When λ ∈ R ≤0 , then Poincaré and Dulac proved that we can reduce the foliation to the unique normal form [START_REF] Brunella | Birational Theory of Foliations[END_REF] x∂ x + λy∂ y , or x∂ x + (ny + x n )∂ y by an analytic change of coordinate; the latter case occur when λ or 1 λ = n ∈ Z >0 . When λ < 0, we are in the saddle case. Briot and Bouquet proved that there are two invariant curves, one in each eigendirection, so that after straigthening them on coordinates axis, we get a preliminary normalization (2)

x∂ x + (λ + f (x, y))y∂ y
where f is a function vanishing at 0. Poincaré and Dulac proved that we can reduce the foliation to the unique normal form by a formal change of coordinate:

(3)

x∂ x + λy∂ y , or x∂ x + - p q + u k + αu 2k y∂ y with    p, q, k ∈ Z >0 u = x p y q , α ∈ C
The latter case, called resonant saddle, occurs for generic f when λ ∈ Q <0 . We will provide below a proof of this formal reduction. It is known however that this formal normalization is divergent in general, and the analytic classification fails to be finite dimensional in this case. In fact, the classification of saddles up to analytic conjugacy is equivalent to the classification of germs of diffeomorphisms Diff(C, 0) up to analytic conjugacy. Before explaining this, let us just end our review of non degenerate singular points by the saddle-node case λ = 0 which admits a normal form similar to (3) with (p, q) = 0 and u = y:

(4) x∂ x + y k+1 + αy 2k+1 ∂ y with k ∈ Z >0 α ∈ C
In order to explain this, let us recall the construction of the holonomy. We claim that such a continuation exists because γ is contained in a regular leaf ∆ * of the foliation. Indeed, one can cover γ by small balls U i equipped with a local minimal first integral H i and choose finitely many of them by compacity: we enumerate such that γ successively intersect U 0 , U 1 , . . . , U n . Denote

V i = U i ∩ ∆ * . Near overlappings V i ∩ V i+1 , we have H i = ϕ i,i+1
• H i+1 for some ϕ i,i+1 ∈ Diff(C, 0) because they both define minimal first integrals; therefore, ϕ i,i+1 • H i+1 defines an analytic extension of H i near V i+1 . Starting from (U 0 , H 0 ) = (U, H), the analytic continuation H γ of H along γ is given by:

H = H 0 V 0 = ϕ 0,1 • H 1 V 1 = ϕ 0,1 • ϕ 1,2 • H 2 V 2 = • • • = ϕ 0,1 • ϕ 1,2 • • • • • ϕ n,0 • H 0 back to V 0 =: H γ .
By construction, H γ is a minimal first integral of F near V = U ∩ ∆ * , and we have

H γ = ϕ γ • H with ϕ γ ∈ Diff(C, 0).
It is easy to check that ϕ γ only depend on the homotopy type of γ; it is called the monodromy of H along γ. Moreover, we get a group morphism

Mon(H) : π 1 (∆ * , p) → Diff(C, 0) ; γ → ϕ γ .
Finally, if we start with another minimal first integral H = ϕ • H, then the monodromy is changed by

Hγ = ϕ • H γ = ϕ • ϕ γ • H = ϕ • ϕ γ • ϕ -1 φγ • H.
We call holonomy of F along γ the class of ϕ γ up to conjugacy in Diff(C, 0) (or a representative by abuse of notation); in the sequel, we will call holonomy of F along the leaf ∆ * the holonomy of F along a loop of index one, i.e. of the form γ(t) = (x 0 e 2iπt , 0). One can check that the holonomy of F of the form (2) along ∆ * is of the form ( 5) q exp(v) where v is the holomorphic vector field

ϕ(z) = e -2iπλ z + o(z).
v = -2iπ z kq+1 + αz 2kq+1 ∂ z .
Indeed, after introducing variables x = xq and z = xp y, we get u = z q and the foliation is defined by

F : x∂ x + q(z kq+1 + αz 2kq+1 )∂ z
which is the formal normal form for a saddle-node singular point. The holonomy of

F along ∆ * = {(x, 0) ; 0 < |x| < r} is the q th iterate of the holonomy of F along ∆ * , namely φ = ϕ •q = exp(q • v).
We conclude by noticing that any q th root takes the form ϕ = a • exp(v), and we must have the linear part a = e -2iπλ = e 2iπ p q (compare [10] and [START_REF] Martinet | Problèmes de modules pour des équations différentielles non linéaires du premier ordre[END_REF]). This latter computation is also valid for the saddle-node case (4) by setting (p, q) = (0, 1) and u = y.

Mattei-Moussu's Theorem tells us that any two saddles of the form (2) are conjugated by an analytic diffeomorphism preserving the two separatrices (i.e. not permuting the coordinate axis) if, and only if, they share the same invariant λ, and the corresponding holonomies along ∆ * are analytically conjugated. On the other hand, Pérez-Marco and Yoccoz proved in [START_REF] Marco | Germes de feuilletages holomorphes à holonomie prescrite. Complex analytic methods in dynamical systems (Rio de Janeiro[END_REF] that any diffeomorphism germ ϕ(z) = e -2iπλ z + o(z) is the holonomy of a saddle of the form [START_REF] Diaw | Géométrie de certains tissus holomorphes singuliers en dimension 2[END_REF]. Therefore, for a fixed λ < 0, analytic classification of saddles and one-dimensional diffeomorphisms are equivalent. A result of Siegel shows that for generic (with respect to Lebesgue measure) λ, any diffeomorphism, and therefore any saddle is analytically linearizable. But for special "diophantine" λ, the moduli space is infinite dimensional as shown by the works of Yoccoz in the irrational case, and Écalle-Malgrange-Voronin in the rational case. Martinet and Ramis gave a complete analytic classification of resonant saddles (i.e. non linear case of (3)) in [START_REF] Martinet | Classification analytique des équations différentielles non linéaires résonnantes du premier ordre[END_REF], and of saddle-nodes λ = 0 in [START_REF] Martinet | Problèmes de modules pour des équations différentielles non linéaires du premier ordre[END_REF]. i.e. where all a n are analytic on D, and nothing is asked about convergence in the yvariable. A transversely formal diffeomorphism is a "map" Φ(x, y) = (x+yf (x, y), yg(x, y)) where f, g ∈ O(D)[[y]] and g(x, 0) does not vanish on D. We denote by Diff(D × (C, 0)) the group of transversely formal diffeomorphisms. Proposition 2.2. Let ∆ = {x ; |x| < r} be a disc and let F be the foliation defined by ( 2) with f transversely formal on ∆ × (C, 0) ∋ (x, y) (for instance, analytic on the neighborhood of ∆ × {0}). Then there exists a transversely formal diffeomorphism Φ ∈ Diff(D × (C, 0)) of the form Φ(x, y) = (x, yg(x, y)) such that Φ * F is defined by (3).

Remark 2.3. The resonant case of (1) does not occur since we have imposed in expression (2) that F has two invariant curves. Also, in the saddle-node case λ = 0, the central manifold, tangent to the 0-eigendirection, is convergent in expression (2).

Proof. It is more convenient to define F as the kernel of the 1-form

ω = xdy -(λ + f )ydx, or better dy y -(λ + f ) dx
x as coordinate change will be easier to handle on 1-forms. Our strategy is to make successive changes of y-coordinate to kill (or reduce) step-by-step the coefficients a n (x) in f = n a n (x)y n . Let us first consider a linear change y → ϕ 0 (x)y. Then Φ * ω writes

dy y -λ + a 0 (x) - xϕ ′ 0 (x) ϕ 0 (x) y + o(y) dx x ;
as a 0 vanishes at x = 0, we can integrate and find ϕ 0 = exp( a 0 x ) ∈ O * (∆). Now we can assume a 0 ≡ 0, i.e. f (x, 0) ≡ 0, and all further changes of coordinate will be tangent to the identity, i.e. with g(x, 0) ≡ 1. Consider now a change of the form y → y(1 + ϕ n (x)y n ), n > 0. Then Φ * ω writes after normalization (i.e. multiplication by a non vanishing analytic function)

dy y -λ + f -nλϕ n + xϕ ′ n y n + o(y n ) dx x .
Since linear operator

O(∆) → O(∆) ; ϕ → nλϕ + xϕ ′ writes m≥0 b m x m → m≥0 (nλ + m)b m x m
we deduce, when λ ∈ Q ≤0 , that the operator is onto, so that we can find at each step a ϕ n killing the coefficient a n of f . The composition

Φ = • • • • Φ n • • • • • Φ 2 • Φ 1 • Φ 0 converges in Diff(D × (C, 0)) providing a linearization of F, i.e. Φ * ω is colinear to xdy -λydx. Indeed, for each n > 0, the jet of order n of Φ in y-variable is determined by Φ n • • • • • Φ 2 • Φ 1 • Φ 0 since
all other terms in the composition are tangent to the identity up to order n. Assume now that λ ∈ Q <0 , i.e. λ = -p q with p, q ∈ Z >0 relatively prime. Then we see that we can kill successively all the terms expect powers of u = x p y q in f . Therefore, we are led by a transversely formal diffeomorphism to the preliminary normal form

(p -f (u)) dx x + q dy y = f (u) du uf (u)
dx x where f is now a formal power series, with f (0) = 0. One easily check that the freedom in this normalization is up to a change of the form Φ(x, y) = (x, yψ(u)), ψ(0) = 0; moreover, this induces a change u → ϕ(u) = u(ψ(x)) q , and therefore Φ * du uf (u)

- dx x = ϕ * du uf (u) - dx x
and we are led to a normalization of a (formal) meromorphic 1-form in one variable.

It is well known (see proof of [10, Proposition 2.1] or [7, Proposition 1.1.3]) that there exists a unique change of coordinate ϕ tangent to the identity such that

ϕ * du uf (u) = du u k+1 + α du u
which is analytic (resp. formal) if f is analytic (resp. formal). The only invariants are the pole order k + 1 (here, k > 0 is the vanishing order of f ), and the residue α. The last normalization is therefore given by the choice of a determination of ψ(u) = ϕ(u) u 1/q .

In the normal form (3), we have choosen another normalization, namely

du qu k+1 (1+αu k )
where the residue is -α q . Finally, in the saddle-node case λ = 0, we note that everything works the same with u = y (i.e. p = 0 and q = 1).

We deduce a formal version of Mattei-Moussu's Theorem: Corollary 2.4. Let F 1 and F 2 be two foliations of the form

F i = ker(ω i ), ω i = xdy -(λ + o(y))ydx, λ ∈ C
and assume they are both analytic at the neighborhood of the disc ∆. Then, there exists a transversally formal diffeomorphism Φ(x, y) = (x, y + o(y)) along ∆ conjugating the foliations Φ * F 2 ∧ F 1 , i.e. Φ * ω 2 ∧ ω 1 = 0, if and only if, the respective holonomies along the punctured disc ∆ * are formally conjugated.

Proof. If we have a transversally formal conjugacy between foliations along even the punctured disc is enough to conclude that the holonomies are conjugated. For the converse, we can assume that F 1 and F 2 are into normal form (3), with the same λ (resp. ( 4) when λ = 0. Only the resonant saddle case λ = -p q and saddle-node case λ = 0 need some arguments, as we have to distinguish between the linear case, and the several possible non-linear cases of (3), i.e how to retrieve k and α from the holonomy. But, as shown in Example 2.1, the holonomy of the non-linear saddle (3) or saddle-node (4) takes the form

ϕ •q (z) = exp z kq+1 - α 2iπq z 2kq+1 ∂ z = z+z kq+1 + kq + 1 2 - α 2iπq z 2kq+1 +o z 2kq+1 .
It is well-known (see [10, p. 581] or [START_REF] Loray | Pseudo-groupe d'une singularité de feuilletage holomorphe en dimension 2[END_REF]Section1.3]) that the formal conjugacy class of the tangent-to-identity diffeomorphism ϕ •q is characterized by two invariants, namely the maximal contact kq to the identity, and the coefficient of the monomial z 2kq+1 once we have killed intermediate coefficients; the first one gives us k, and we deduce α from the latter one.

Mattei-Moussu's Theorem

Theorem 3.1 (Mattei-Moussu). Let F 1 and F 2 be two foliations of the form

F i = ker(ω i ), ω i = xdy -(λ + o(y))ydx, λ ∈ R <0 .
Assume that they are well defined near ∆ * = {(x, 0) ; 0 < |x| < r} and have same holonomy along this leaf. Then, there exists an analytic diffeomorphism Φ(x, y) = (x, y + o(y)) conjugating the foliations:

Φ * F 2 ∧ F 1 , i.e. Φ * ω 2 ∧ ω 1 = 0.
Proof. The first part of the proof follows the paper [START_REF] Mattei | Holonomie et intégrales premieres[END_REF]. Let p 0 ∈ ∆ * and H 1 , H 2 be local first integrals of F 1 , F 2 respectively at p 0 . Assume that H 1 and H 2 have same monodromy ϕ = ϕ γ ∈ Diff(C, 0) along a generating loop γ of π 1 (∆ * ). Notice that the local diffeomorphism Φ i (x, y) = (x, H i (x, y)) at p 0 is conjugating F i to the horizontal foliation {y = constant}, i.e. Φ * i dy ∧ ω i = 0. Therefore, the local diffeomorphism Φ := (Φ 2 ) -1 • Φ 1 is conjugating F 1 to F 2 as in the statement, near p 0 . By analytic continuation of H 1 , H 2 along paths in ∆ * (see Section 2.1) one can deduce the analytic continuation of Φ by the same formula. Since H 1 and H 2 have the same monodromy ϕ, we see that Φ is uniform: after analytic continuation along the generating loop γ, we get

Φ γ = (Φ γ 2 ) -1 • Φ γ 1 = (φ • Φ 2 ) -1 • (φ • Φ 1 ) = (Φ 2 ) -1 • φ -1 • φ • Φ 1 = (Φ 2 ) -1
• Φ 1 = Φ where φ(x, y) = (x, ϕ). We therefore obtain an analytic diffeomorphism Φ :

U 1 → U 2 between neighborhoods U i of ∆ * conjugating the restrictions F 1 to F 2 .
In the second part of the proof, we want to extend Φ analytically on the neighborhood of 0. Our idea is to compare Φ with the transversely formal diffeomorphism Φ along ∆ provided by Corollary 2.4. As they are both conjugating F 1 to F 2 , φ := Φ-1 • Φ defines a transversely formal diffeomorphism along ∆ * that preserves the foliation F 1 . We are going to show, in Lemma 3.2, that this forces φ to extends as a transversely formal diffeomorphism on the whole disc ∆. Therefore, the diffeomorphism Φ • φ = Φ is both analytic near ∆ * and transversely formal on ∆. Equivalently, the Laurent power series decomposition of its coefficients are convergent and with only positive exponents: they define holomorphic germs at 0, therefore extending Φ. Lemma 3.2. Let F be a foliation of the form

F = ker(ω), ω = xdy -(λ + o(y)) ydx, λ ∈ C \ Q >0
analytic near the disc ∆ = {(x, 0) ; |x| < r}. Then, any transversely formal diffeomorphism Φ(x, y) = (x, φ(x, y)) along the punctured disc ∆ * commuting with F, i.e. Φ * ω ∧ ω = 0, extends as a transversely formal diffeomorphism along the whole disc ∆.

Proof. By Proposition 2.2, we can assume that F is in formal normal form (3) and consider first the linear case: we can equivalently define F with ω = dy yλ dx x . If we write Φ(x, y) = (x, y • g(x, y)), then the condition Φ * ω ∧ ω = 0 writes

dg ∧ dy y -λ dx x = 0 (or (x∂ x + λy∂ y ) • g = 0)
i.e. g is a first integral for F. If we decompose g in Laurent power series, then we get

g = m∈Z, n≥0 a m,n x m y n (x∂ x + λy∂ y ) • g = m∈Z, n≥0
(m + λn)a m,n x m y n = 0.

The obstruction to extend at 0 comes from non zero coefficients a m,n with negative m; but this can only occur when λ ∈ Q >0 , what we have excluded.

Consider now the resonant case, and assume F is defined by

ω = du u k+1 + α du u - dx x = 1 u k + α du u - dx x
(see proof of Proposition 2.2). Then we have

Φ * ω = 1 u k g k + α du u + dg g - dx x .
The ratio of the closed one-forms ω and Φ * ω must be a first integral of F and therefore be constant. Moreover, since they have the same residu along ∆ * (mind that g is not vanishing), then they should be equal, and therefore dg ∧ du = 0, i.e. g = g(u). But non zero coefficients a m,n of g occur only for n ≥ 0, and therefore only for m ≥ 0 since u = x p y q . So g extends on the whole of ∆.

Remark 3.3. Our proof of Mattei-Moussu's Theorem works not only for saddles λ < 0, but also for saddle-node λ = 0 provided that the central manifold is convergent (i.e. the formal invariant curve along the 0-eigendirection). It was proved in [START_REF] Martinet | Problèmes de modules pour des équations différentielles non linéaires du premier ordre[END_REF] that the holonomy of the strong manifold characterize the foliation, but the proof was completely indirect, by comparing the two moduli spaces. The original approach of Mattei and Moussu fails to conclude in that case. Here, we get a direct and simple proof when the central manifold is convergent. We will explain later how to adapt to the case of a divergent central manifold (the generic case).

Remark 3.4. In the saddle-node case λ = 0, it is well-known that the holonomy of the central manifold, when it is convergent, fails to characterize the saddle-node, even formally. In fact, if we want to normalize a saddle-node of the form x 2 dy -(1 + o(y))ydx by a transversly formal change of coordinates Φ(x, y) = (x, φ(x, y)) along the disc ∆ or even along the punctured disc ∆ * , we find obstructions at each step. Indeed, if it takes the form dy y -(1 + f (x)y n + o(y n ) dx x and we apply a change of the form y → y(1 + g(x)y n ), then we find

dy y -(1 + f (x)y n + o(y n ) dx x with f = f -x 2 g ′ (x) -ng(x).
If we want to make f = 0 as it should be for n >> 0, we see by integration that we must set g = e -n/x e n/x f (x)

x 2
which is multiform on ∆ * in general. Moreover, there are many transversely formal symmetries along ∆ * that do not extend: we can take any transformation of the form y → yg(ye 1/x ) with g(0) = 0. Remark 3.5. If we want to compare our proof to the original one of Mattei and Moussu, the present one has the disadvantage that we have to discuss between the different formal types, and deal with existence of first integrals, closed one-forms; only in the non resonant part λ ∈ Q it is very short and easy. On the other hand, this approach can be used to classify pairs of singular foliations as we shall see in the next section.

Remark 3.6. Mattei-Moussu's Theorem can be considered as a classification of pairs of foliations. Indeed, consider a pair (F 1 , F 2 ) of foliations germs such that F 1 is regular and F 2 is a saddle, one invariant curve of which is a leaf of F 1 . If we have two such pairs, (F 1 , F 2 ) and (G 1 , G 2 ), one can easily find coordinates in which F 1 , G 1 are vertical, defined by ker(dx), and the two invariant curves of F 2 , G 2 are on coordinate axis. Then, the two pairs are conjugated if and only if they share the same λ and have conjugated holonomies. Indeed, the conjugacy between the foliations F 2 and G 2 provided by Theorem 3.1 preserves the variable x and therefore F 1 = G 1 . This was one of the motivation to us for looking at classifications of pairs in a more general setting.

Reduction of singularities for a pair of foliations

Consider a pair (F 1 , F 2 ) of holomorphic singular foliations on a complex surface M . Denote by Tang(F 1 , F 2 ) the tangency divisor between the two foliations: locally, if F i = ker(ω i ) where ω i is a holomorphic one-form with isolated zeroes, then Tang(F 1 , F 2 ) is class of F 2 is characterized by λ and the analytic class of its holonomy along the invariant curve y = 0 in (5.2) (Mattei-Moussu's Theorem), or x = 0 in (5.3) (see [START_REF] Martinet | Problèmes de modules pour des équations différentielles non linéaires du premier ordre[END_REF]Corollaire 3.3]).

So far, only the classification of type ( 6) was not known. In this direction, the first author recently obtained (see [START_REF] Diaw | Géométrie de certains tissus holomorphes singuliers en dimension 2[END_REF][START_REF] Diaw | Pairs of foliations: reduction of singularities and local classification[END_REF]) the complete classification in the case Tang(F 1 , F 2 ) is reduced (i.e. T = (xy), without multiplicity). We will state the results in Section 5. We note that in case [START_REF] Loray | Versal deformation of the analytic saddle-node. Analyse complexe, systèmes dynamiques, sommabilité des séries divergentes et théories galoisiennes[END_REF], any saddle occuring among F i must have analytic invariant curves, i.e. not formal divergent component, since they are contained in T which is analytic.

The list of Theorem 4.2 is not stable by arbitrary blowing-up: if we blow-up items (2) or (3) for instance, then we have to blow-up more in order to recover a pair with reduced singular points as in the statement.

Proof. Let M , (F 1 , F 2 ) and T as in the statement. First of all, there exists a proper map π : M → M obtained by a finite sequence of punctual blowing-up such that the pull-back π * F i have both only reduced singular points. This is done by applying twice Seidenberg's Theorem [START_REF] Seidenberg | Reduction of singularities of the differential equation Ady=Bdx[END_REF], successively for F 1 and F 2 . These properties are stable by additional blowing-up and the final map π : M → M of the statement will be obtained by additional blowing-ups. Therefore, we can assume without lost of generality that both F i have reduced singular points from the beginning.

Recall that T is passing through all singular points of F 1 or F 2 . Outside the support of T , the pair is regular, of type [START_REF] Brunella | Birational Theory of Foliations[END_REF]. Along each irreducible component of T , the pair is generically of type ( 2) or (3), depending whether that component is generically transversal to the F i 's, or is F i -invariant; moreover, this generic feature is for a Zariski open set of T . We therefore conclude that, apart from a finite set of points in M , the situation is as in (1), ( 2) and (3) of the list. So the problem is local: let us consider a point p where (F 1 , F 2 , T ) is not as (1), ( 2) or (3), and prove that after finitely many blowing-up infinitesimally close to p, we get only points in the list (1)-( 6); as we shall see, we will get only local models of type (3)-( 6) along the exceptional divisor.

In order to show this, let us consider the local formal invariant curve Γ i of F i at p: it consists of one or two smooth and transversal branches depending on whether the foliation is regular or singular. Then consider the germ Γ = Γ 1 ∪ Γ 2 ∪ T at p; mind that they can share common branches. Then we can blow-up until Γ = π * Γ has only normal crossing singular points. We note that all local invariant curves for Fi = π * F i are contained in Γ; otherwise, any extra invariant curve would descend as an additional invariant curve for F i outside Γ, providing a contradiction. We also note that Γ contains the support of Tang( F1 , F2 ) for a similar reason. We claim that all points along Γ are of type (3)-( 6). Let us check this.

Let us consider first the case where Γ is smooth. Then the two foliations are smooth (only one invariant curve) with Γ as a common leaf and no other tangency: we are in type [START_REF] Diaw | Pairs of foliations: reduction of singularities and local classification[END_REF]. Now, in order to consider the case Γ has two components, note that any common component between two among the three curves Γi = π * Γ i , i = 1, 2, and T = π * T must be also a component of the third curve. Indeed, if Γ1 and Γ2 have a common component, then F1 and F2 must be tangent along this branch and it must be a component of T ; in a similar way, if T and Γ1 , say, have a common component, then that branch of Γ1 must be invariant by F2 as well, and be a component of Γ2 . We conclude that, each branch of Γ consists of

• either a branch of T only,

• or a branch of Γi only,

• or a common branch of Γ1 , Γ2 and T . In the first case above, then F1 and F2 are regular (only one invariant curve contined in the other branch of Γ) and transversal to that branch of T only: we are in case (4). In the second case above, say i = 1, then F2 is regular (only one invariant curve) and transversal to that branch of Γ1 , and tangent to the other branch: we are in case [START_REF] Loray | Sur les Théorèmes I et II de Painlevé. Geometry and dynamics[END_REF]. Finally, if all branches are shared by T , Γ1 and Γ2 , then we are in case [START_REF] Loray | Versal deformation of the analytic saddle-node. Analyse complexe, systèmes dynamiques, sommabilité des séries divergentes et théories galoisiennes[END_REF]. Finally, in case ( 6), if ever F1 and F2 are saddle-nodes oriented in opposite way, i.e. each branch of T is the strong manifold (non zero eigendirection) of one and the central manifold (zero eigendirection) of the other, then one check that after one additional blowing-up, we get two singular pairs along the exceptional divisor with only one saddle-node in each pair.

To conclude, applying the above strategy of resolution by blowing-up at each point p in M where the pair is not locally of type ( 1)-( 3) yields a global proper map π : M → M such that the lifted pair is locally of type ( 1)-( 6) everywhere.

Reduced pairs of foliations: a partial classification

This section is devoted to announcement of results of the first author in her thesis [START_REF] Diaw | Géométrie de certains tissus holomorphes singuliers en dimension 2[END_REF]. All details will appear in a forthcoming paper [START_REF] Diaw | Pairs of foliations: reduction of singularities and local classification[END_REF].

We consider a local pair (F 1 , F 2 ) of type ( 6) in the list of Theorem 4.2 and assume that Tang(F 1 , F 2 ) = (xy) is reduced, i.e. without multiplicity. One easily check that this implies that we can write

F i = ker( xdy -(λ i + o(1))ydx)
with λ i ∈ C \ Q ≥0 , and with λ 1 = λ 2 (otherwise Tang(F 1 , F 2 ) cannot be reduced). Conversely, any pair (F 1 , F 2 ) as above with λ 1 = λ 2 is of type [START_REF] Loray | Versal deformation of the analytic saddle-node. Analyse complexe, systèmes dynamiques, sommabilité des séries divergentes et théories galoisiennes[END_REF] with reduced tangency along axis. Denote

Diff × (C 2 , 0) = {Φ(x, y) = (xa(x, y), yb(x, y)) ; a, b ∈ O × (C 2 , 0)} ⊂ Diff(C 2 , 0)
the subgroup of those diffeomorphisms preserving the axis (without permutation) and

F ∼ G or (F 1 , F 2 ) ∼ (G 1 , G 2 )
when two foliations or two pairs are conjugated by a diffeomorphism Φ ∈ Diff × (C 2 , 0). with λ 1 = λ 2 , λ i ∈ C \ Q ≥0 , and f i , g i ∈ O(C, 0) vanishing at 0. Then we have

(F 1 , G 1 ) ∼ (F 2 , G 2 ) ⇔ F 1 ∼ F 2 and G 1 ∼ G 2 .
In other words, if there are Φ i ∈ Diff × (C 2 , 0) such that F i = Φ * i G i for i = 1, 2, then there exists a single Φ ∈ Diff × (C 2 , 0) such that F i = Φ * G i . An equivalent statement is given by Theorem 5.2 (A. A. Diaw [START_REF] Diaw | Géométrie de certains tissus holomorphes singuliers en dimension 2[END_REF][START_REF] Diaw | Pairs of foliations: reduction of singularities and local classification[END_REF]). Consider a pair (F 1 , F 2 ) as in Theorem 5.1. Then, for any Ψ ∈ Diff × (C 2 , 0), there exist Ψ 1 , Ψ 2 ∈ Diff × (C 2 , 0) such that Ψ = Ψ 1 • Ψ 2 and Ψ * 1 F i = F i , i = 1, 2. Proof of the equivalence between the two statements. Assume we are two pairs (F 1 , F 2 ) and (G 1 , G 2 ) as in Theorem 5.1 and Φ i ∈ Diff × (C 2 , 0) such that F i = Φ * i G i . Then we can apply Theorem 5.2 to decompose Ψ := Φ -1

1 • Φ 2 = Ψ 1 • Ψ 2 and check that Φ := Φ 1 • Ψ 1 = Φ 2 • Ψ -1
2 is conjugating the pairs (F 1 , F 2 ) to (G 1 , G 2 ). Conversely, given Ψ ∈ Diff × (C 2 , 0), we can apply Theorem 5.1 to produce conjugacy of pairs

(F 1 , Ψ * F 2 ) Φ 1 -→ (F 1 , F 2 )
so that we have a conjugacy (F 1 , F 2 )

Ψ•Φ -1 1 -→ (Ψ * F 1 , F 2 ).
Then we have decomposition Ψ = Ψ 1 •Ψ 2 where Ψ 1 = Φ 1 preserves F 1 , and

Ψ 2 = Ψ•Φ -1 1 preserves F 2 .
The technics involved to prove Theorem 5.1 use local study of a pair of regular foliations along a common leaf, as was already done in [START_REF] Loray | Versal deformation of the analytic saddle-node. Analyse complexe, systèmes dynamiques, sommabilité des séries divergentes et théories galoisiennes[END_REF][START_REF] Thom | Classification locale de bifeuilletages holomorphes sur les surfaces complexes[END_REF][START_REF] Loray | Two-dimensional neighborhoods of elliptic curves: formal classification and foliations[END_REF][START_REF] Thom | Formal classication of two-dimensional neighborhoods of genus g ≥ 2 curves with trivial normal bundle[END_REF]; this allows to construct conjugacy between of pairs of foliations at the neighborhood of a punctured disc ∆ * contained in the invariant curve. On the other hand, one can prove a version of the decomposition in Theorem 5.2 in the transversely formal setting along the complete disc ∆, implying a transversely formal conjugacy between the pairs of foliations. Then a careful study of transversely formal symmetries of a pair along ∆ * allow the first author to conclude to the extension of the conjugacy at 0, likely as in our proof of Mattei-Moussu's Theorem.

We expect that it will be possible to obtain the complete classification even for non reduced tangency divisors, but there will be additional invariants in that case, and more complicated statement. The reduced case is motivated by the study of Hilbert Modular Surfaces which admit a pair of foliations with reduced tangency divisor (see [START_REF] Rousseau | Curves in Hilbert modular varieties[END_REF]).

2. 1 .

 1 Holonomy. Fix a disc ∆ = {(x, 0) ; |x| < r} ⊂ C 2 such that the foliation is of the form (2), with f analytic at the neighborhood of ∆, λ + f non vanishing on ∆. Denote by ∆ * the punctured disc, where we have deleted the singular point of F. Fix a point p ∈ ∆ * and choose a minimal first H(x, y) ∈ O(C 2 , p): locally at p, the 1-form dH is non vanishing and its kernel defines the foliation, i.e. dH(v) = v • H = 0. Consider a loop γ : [0, 1] → ∆ * based at p = γ(0) = γ(1), and consider the analytic continuation of H along γ.

Example 2 . 1 .

 21 The holonomy of the linear model in 3 is the monodromy of the local first integral H(x, y) = x -λ y (after choosing a local determination of x -λ = exp(-λ log(x))), namely ϕ(z) = e -2iπλ z. Equivalently, one can integrate the vector field v = x∂ x + λy∂ y and deduce a global symmetry φ(x, y) = exp(-2iπv) = (x, e -2iπλ y) of the foliation; and then restrict to a transversal x = x 0 .The holonomy of the non-linear model in 3 is ϕ(z) = e 2iπ p

2. 2 .

 2 Normalization in the transversely formal setting. Let D ⊂ C be a domain. A transversely formal function on D × (C, 0) ∋ (x, y) is a power series f = n≥0 a n (x)y n ∈ O(D)[[y]]

Theorem 5 . 1 (

 51 A. A. Diaw[START_REF] Diaw | Géométrie de certains tissus holomorphes singuliers en dimension 2[END_REF][START_REF] Diaw | Pairs of foliations: reduction of singularities and local classification[END_REF]). Consider two pairs (F 1 , F 2 ) and (G 1 , G 2 ) of the formF i = ker(xdy -(λ i + f i (x, y))ydx) and G i = ker(xdy -(λ i + g i (x, y))ydx)

grant, CAPES-COFECUB Ma 932/19 project, Université de Rennes 1 and Centre Henri Lebesgue CHL.

defined by the ideal (f ) given by ω 1 ∧ ω 2 = f dx ∧ dy. We note that Tang(F 1 , F 2 ) is passing through all singular points of each F i . Recall Seidenberg's Theorem: Theorem 4.1 (Seidenberg [START_REF] Seidenberg | Reduction of singularities of the differential equation Ady=Bdx[END_REF]). Let F be a holomorphic singular foliations on a complex surface M . Then there is a proper map π : M → M obtained by a finite sequence of punctual blowing-up such that the lifted foliation F = π * F has only reduced singular points, i.e. of the type

Moreover, this property is stable under additional blowing-up.

This allows to reduce the study of singular points to non degenerate models. We note that after reduction of singularities, all singular points have exactly two invariant curves (one of them might be formal divergent in the saddle-node case λ = 0) which are smooth and transversal. We want to prove now a similar result for pairs of foliations. Theorem 4.2. Let (F 1 , F 2 ) be a pair of holomorphic singular foliations on a compact complex surface M , and T be the support of Tang(F 1 , F 2 ). Then there is a proper map π : M → M obtained by a finite sequence of punctual blowing-up such that the pair of lifted foliations ( F1 , F2 ) = π * (F 1 F 2 ) is, at each point of M , one of the following types:

(1) T = ∅, both F i are regular and transversal to each other;

(2) T is smooth, both F i are regular, transversal to T ;

(3) T is smooth, both F i are regular, tangent to T ;

(4) T has a normal crossing, both F i are regular, tangent to one component of T , and transversal to the other one; (5) T is smooth, F 1 is regular, F 2 has a reduced singular point, and T is a common leaf/invariant curve of F i ; (6) T has a normal crossing, both F i have a reduced singular point, and T is the common set of invariant curves of F i ; moreover, in case both F i are saddle-node, they share the non zero invariant curve.

Before proving the theorem, let us comment on the possible types of reduced singular points. First of all, types (1) -( 4) admit simple local normal form:

(1) (F 1 , F 2 ) ∼ (dy, dx) (the generic regular case);

(

(F 1 , F 2 ) ∼ (dy, d(y + x k+1 y l )) and T = (x k y l ), k, l ∈ Z >0 . Generic points of each branch of T are either of type [START_REF] Diaw | Géométrie de certains tissus holomorphes singuliers en dimension 2[END_REF], or of type [START_REF] Diaw | Pairs of foliations: reduction of singularities and local classification[END_REF].

Type (5) splits into 3 cases:

Moreover, the classification of the pair up to analytic diffeomorphism is equivalent to the analytic classification of F 2 in each case. Poincaré Linearization Theorem provides linearization in (5.1). The saddle case (5.2) with λ < 0 corresponds to Mattei-Moussu's Theorem (see Remark 3.6). The saddle-node case splits into (5.2) with λ = 0 (see Remark 3.3) and (5.3) (see [START_REF] Martinet | Problèmes de modules pour des équations différentielles non linéaires du premier ordre[END_REF]Proposition 2.2.3]). Moreover, for fixed λ, the analytic


Jacky Potier

Dolley-Hitze Thibault

Didier Hamel

Isabelle Landru

Erick Cardineau

Guillaume Queffeulou

Elie Zagdoun

Eric Renaudineau

Nicolas Molinari

Lucie Gamon

Long-term effects of citric acid-based bicarbonate haemodialysis on patient outcomes: a survival propensity score-matched study in western France

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Pascal Lavieille
email: pascal.lavieille@laplace.univ-tlse.fr

Marco Azzolin

Arianna Berto

Patrick Queeckers

Andrey Gluschuk

Benoit Schlegel

Stefano Bortolin

Marc Miscevic

Carlo Saverio Iorio

Davide Del Col
email: davide.delcol@unipd.it

New measurement technique of liquid film thickness distribution with high spatial and temporal resolution: application to convective condensation in microgravity

Introduction

Prediction of condensing two phase flow (flow pattern, heat transfer and pressure drop) needs an accurate knowledge of liquid film thickness distribution. Indeed liquid film thickness is a key parameter to understand the main mechanisms inducing the instability and governing pressure drop and heat transfer. Unfortunately this liquid film is often very thin and unsteady. This thinness coupled with a highly dynamic change of slope of the interface make challenging the measurements of the liquid thickness. This lack of experimental data limits the validation and improvements of numerical or analytical model and so development of the laws that manage two-phase flow. The present work proposes a technique to determine accurately with a high spatial and temporal resolution the liquid film thickness distribution inside a tube. This technique combines a high resolution punctual measurement system and high speed image acquisition of the flow inside a tube owning an external specific shape designed in order to magnify small liquid film thickness.

Limitations of standard visualizations

In the 62 nd ESA parabolic flight campaign realized in June 2015 a shadowgraph method with a parallel light source has been used in order to determine temporal evolution of liquid film distribution of condensation flows in microgravity. The advantage of this measurement technique is to observe with a high temporal resolution (1500 Hz) the axial film thickness distribution at the top and at the bottom of the tube. In order to obtain quantitative measurements, distorsions induced by the cylindrical shape of the tube have been corrected using a ray-tracing model. This allows to convert the seeming thickness of the image into the real liquid film thickness present inside the tube. Two main limitations reduce the performances of this technique: the resolution of the measurement is limited by the spatial resolution of the camera and the liquid film can not be detected below a threshold value due to light deviation (figure 1). Another limitation induced by the cylindricity of the external shape of the tube is to produce a defocus of light which make ineffective or less effective point measurement techniques such as confocal chromatic or interferometric.

In order to overcome these limitations in the 70 th ESA parabolic flight campaign realized in Nov. 2018 a glass tube with a specifc external shape has been designed and realized.

Theoretical optical performances of the specific external tube shape

From the optical properties of the materials and the fluid, the shape of the external surface of the glass tube has been calculated in order to make detectable liquid film down to one micrometer, to produce a magnification of small liquid film thickness and to allow the use of punctual measuring sensors. The theroretical performances deduced from the ray tracing model are presented in figure 2. Magnification of small liquid film is obtained. Indeed from this calibration curve it can be deducted that when the liquid film thickness inside the tube is 50 µm, the camera will see the liquid film as if its thickness is equal to 267 µm. In that case even if the spatial resolution of the camera is 10 µm/pixel the liquid film thickness will be determined with an accuracy of 6% (knowing that when the seeming thickness is 277 µm the real thickness is 53 µm) instead of 20% when there is no magnifying effect. Figure 3 shows this relative accuracy according to the liquid film thickness inside the tube. With such a camera resolution the relative accuracy fall down below 10% when liquid film thickness is higher than 20 µm and below 5% for liquid film thickness greater than 100 µm. Such magnification (figure 4) allows to combine high accuracy and large field of view with a high temporal resolution. For the 70 th ESA parabolic flight campaign this optical configuration has been used in order to measure liquid film structures at the top and at the bottom of the tube on a length of 13 mm with a camera frame rate of 1000 Hz.

Punctual measurement techniques

One of the main difficulties of this optical technique is the realization of the calculated external tube shape. Indeed it requires a high accuracy of the shape of the inner and outer surfaces but also of their relative position and roughness. Error in one of these parameters creates a modification of the calibration curve and reduces the capability to determine precisely the liquid film thickness. In order to overcome these difficulties and also to keep the possibility to determine more accurately liquid film thickness, punctual measurement sensors have been added. The external surface of the tube has two flat surfaces that allow to use techniques analysing the reflected light properties to determine liquid film thickness. A white light interferometer and a confocal chromatic sensor have been placed respectively at the top and at the bottom of the tube. The white light interferometer used, allows to determine liquid film thickness between 0.5 µm to 100 µm with an accuracy of 10 nm. It is particularly well suited for small liquid film thickness. The main limitation is that the angle between the liquid-vapor interface and the tube surface must be small (roughly below 15°). The confocal sensor used can accept slope of the interface up to 28° and can measure liquid film thickness between 20 µm and 300 µm.

Thanks to these sensors and this optical configuration, the liquid film thickness can be measured simultaneously with the camera and one of these sensors at 1000 Hz. This accurate measurement allows to check and eventually correct the theoretical calibration curve of the machinning tube and then create a way to determine accurately the liquid film thickness distribution taking into account the distorsion induced by machining precision. Such optical measurement set-up was used to study the gravity effect on the condensation of HFE-7000 inside a 3.4 mm internal tube diameter for mass flow rate between 30 and 55 kg m -2 s -1 .

Experimental set-up

The experimental facility is similar to the one used for the 62 nd ESA Parabolic flight campaign [START_REF] Azzolin | Experimental investigation of in-tube condensation in microgravity[END_REF] with two main loops: the refrigerant (HFE-7000) loop and the coolant water loop. The optical measurement section is placed between two heat exchangers (figure 5). The first heat exchanger is divided in three sub-sectors, the second one in two sub-sectors. These sub-sectors allow to determine quasilocal heat flux and heat transfer coefficient. The heat flux is obtained thanks to the determination of the water mass flow rate and the temperature difference measured with a copperconstantan three-junction thermopile. For each sub-sector, six T-type thermocouples embedded on the wall allow to deduce the heat transfer coefficient. Effect of gravity, mass flow rate and quality on liquid film thickness distribution will be presented in the workshop.

Conclusions

A new technique to accurately measure the liquid film thickness distribution has been proposed and applied during the 70 th ESA parabolic flight campaign. A specific designed external tube shape has been realized to magnify the liquid film thickness and to easily detect, with a standard camera resolution, liquid film thickness down to few micrometer. This technique was coupled with confocal and white light interferometric sensors to reach submicrometric precision at a frequency rate of 1000 Hz.

Figure 1 :

 1 Figure 1: Ray tracing of parallel light inside a tube where a thin liquid film is distributed around the internal wall. The light deviation angle is too high to be collected by the CCD sensor of the camera.

Figure 2 :

 2 Figure 2: Theoretical calibration curve of the tube with the calculated specific external shape.

Figure 3 :

 3 Figure 3: Relative accuracy in measuring film thickness as a function of film thickness inside the tube, using a camera with a resolution of 10 µm/pixel.

Figure 4 :

 4 Figure 4: Example of image showing the magnification of liquid film thickness during microgravity inside a tube of 3.4 mm internal diameter.

Figure 5 :

 5 Figure 5: Layout of the experimental test section.

Acknowledgements

The authors acknowledge the financial support of the European Space Agency through the MAP Condensation program ENCOM-3 (AO-2004-096).


Laurent Mermet

Knowledge that is actionable by whom? Underlying models of organized action for conservation

Introduction

Many ecologists and conservation biologists feel a need for their research to make a difference in the face of the biodiversity crisis. As this need has been taken up by the field's literature and professional societies, the prevalence of a "knowing-doing gap" [START_REF] Knight | Knowing but not doing: selecting priority conservation areas and the research-implementation gap[END_REF] has emerged as a salient point of such reflection. More recently, the focus is moving towards responsibility of the researchers themselves. Attention is then drawn to the connexions that scientists should reinforce or create with actors in the practical world, whether these are defined as decision-makers, practitioners, stakeholders, or local people [START_REF] Cook | Achieving conservation science that bridges the knowledge-action boundary[END_REF][START_REF] Toomey | Navigating the space between research and implementation in conservation[END_REF]. Many authors also point to the content of research, and to the responsibility of scientists to produce research that is relevant for action. Margaret Palmer describes such research as actionable "because it has the potential to inform decisions (in government, business, and the household), to improve the design of implementation of public policies, or to influence public-or private-sector strategies, planning and behaviours that affect the environment." (Palmer, 2012, p.5).

But by whom exactly should such research be actionable? And with whom should scientists engage in "research-implementation spaces", as [START_REF] Toomey | Navigating the space between research and implementation in conservation[END_REF] encourage them to do? Doing our own work in management, a discipline that puts much emphasis on who exactly is supposed to do what for goals to be met, on who actually does what and on who is accountable to whom in the process, we find ourselves regularly puzzled as we read and listen to conservation scientists discussing their connexions with the world of conservation action. We are quite attentive to whom they see as taking up science (or participating in it) and acting on research findings and we often find their answers to be more or less incomplete. As we shall elaborate, sometimes the answer is vague (e.g. "humanity must act now to curb the loss of tropical forests", or "it is essential to improve governance of biodiversity conservation mechanisms on the ground"), sometimes it is glossed over e.g. by using the passive form ("the regulations will have to be changed"), or it points to only some actors ("ecosystem management agencies", or "local people must be involved alongside with protected-area managers") and is not structured (as in Palmer's citation above, which offers a rather comprehensive enumeration, but doesn't locate potential interlocutors in an organized scheme or a pattern that helps hierarchize between them). This hitch with the identification of immersed-in-action-partners points to the wider issue of organized action. If relevant science is going to make a difference through action, that action will surely be collective action of one sort or another. It could be much enhanced, and the debate enriched, by a wider and clearer general view of the issues of agency (who is supposed to take action?) and of organized action (what are the structural links and power dynamics between the main actors) that underlie conservation action in general, and the way it uses science in particular.

We propose here an explanation and analysis of the alternative conceptual models of collective action that inform thought and practice in the field of action on conservation issues. We have come to this choice and explanation of conceptual models gradually through years of research on decision-making processes on environmental issues (Mermet and Leménager, 2015), on conservation and the use of various policy tools [START_REF] Mermet | Tools for What Trade? Analysing the Utilisation of Economic Instruments and Valuations in Biodiversity Management[END_REF], on environmental policy evaluation [START_REF] Mermet | Concern-focused evaluation for ambiguous and conflicting policies : an approach from the environmental field[END_REF]. Admittedly this explanation of underlying models is a work in progress and the reader will find their presentation here more complete (more models considered) and with further conceptualization than in previous tentative expositions [START_REF] Mermet | Five paradigms of collective action underlying the human dimension of conservation[END_REF].

We shall start by a critique of one often encountered but highly anti-pragmatic implicit model of action, i.e. that we are all responsible for biodiversity and affected by it, and thus, should act all together. As we cannot literally act all together, but only some of us can or want to act on collective problems even if they are shared by all, we then turn to a systematic review of the main possible answers to the question: "who is to act on our collective problems, and what is the main thrust of such action?" We will review six underlying conceptual models of organized action, each of which offers a distinctive answer to this question. Each of them outlines a very different context for the place and role of research in conservation action. Each of them frames quite differently the meaning and value of conservation research. Each of them suggests different (and not always compatible) avenues for improving how conservation scientists connect with conservation actors and action. In the final discussion we shall return to the current state of the debate on bridging research and action in biodiversity conservation, hoping to show that the proposed clarification enhances existing contributions and actively invites and contributes to new ones.

Dispelling the hathroologic illusion and spelling out explicit models of agency and organized action

In 2002 at the Rio + 10 conference in Johannesburg, French president Jacques Chirac opened his speech with a sentence that struck a chord: "our house is on fire, but we are looking the other way". This expresses the most basic model of collective action on environmental problems: we are one humanity, we have one problem (climate change, biodiversity crisis, etc.) so we must act as one, exactly as would do one person or one family that discovers that their house is on fire. Fifteen years later, as philosopher Clive Hamilton denounces "the great climate silence" and warns that "we are on the edge of the abyss but we ignore it" (Hamilton, 2017), he expresses exactly the same underlying model of collective action. It is, again, the same model that the "knowing but not doing" phrase expresses in the conservation debate. We propose to name that model "hathroologic", based on greek hathroos that qualifies large groups (e.g. the people of a city) acting en masse, as one. It is important to name that model because we find it to be widespread in biodiversity conservation (and more generally, environmental) debates. It needs to be explained and criticized because it sounds grand, but obfuscates crucial issues of agency.

Let us use the "climate silence" as an example for critical analysis of hatroology. Is everybody "silent" on climate issues? Far from it: it is abundantly discussed publicly, including by the author of the phrase (Hamilton, 2017). Does everybody ignore it? Far from it: it is a salient point on the international agenda, in all major news media and a lot of practical action is being taken by many people, at many levels. Similarly in biodiversity conservation, it would be hard to argue seriously that no meaningful action is taking place, that not many people are involved in it, or that what action is taking place makes no use of knowledge from research.

The hatroologic discourse disassembles quickly under scrutiny for two reasons. (1) It calls to action by entities (society, humanity, "us all", etc.) that exist as aggregates but have no concrete existence as actual pragmatic operators, operators that could actually act on knowledge provided by scientists; this encourages unrealistic reasoning about moving from knowledge to action. (2) It exacerbates feelings of powerlessness as what is a relative lack of power (some of us are acting a lot for conservation, but against powerful opposition and difficult odds) is turned into absolute powerlessness (compared to what would be "real" action -i.e. unanimous and completely effective -we do nothing). Indeed, assigning action to "humanity" or "society" can be a figurative manner of speech, shorthand for complex series of actions that leads to results than can be assessed as if "humanity" or "society" as a whole had acted. And leaving open the issue of who might do the action can be a way to hand over an important agenda point or technical tool without having to presuppose who exactly is to act on it. Such moves amount to conservation scientists addressing their findings and recommendations "to whomever they may concern". This may be appropriate in a variety of contexts. In many cases however, it is neither appropriate nor useful. For instance, it is incoherent to address one's findings to "whomever" and then lament that no one seems to express concern and act on it. Also, if one is looking for relevant partners to take up knowledge about biodiversity and act on it, reasoning on the basis of "humanity should" or "one could" will provide poor guidance.

As they participate in knowledge-to-action initiatives, conservation scientists and their social science allies have to steer away from hatroologic reasoning and passive voice statements and enter the fray where concrete actors, the structural links and power dynamics between them shape biodiversity outcomes. Many of them do that through a case by case approach by engaging specific field situations and reflecting in a practical manner on the actors that may be relevant for conservation action in that situation and on the relations between them. Others advocate more general approaches to improve the paths that lead from conservation knowledge to action, e.g. to increase stakeholder involvement in research, or include a critique of large scale power structures in the analysis of specific field situations.

A further step to sustain such efforts is to examine in a systematic manner the various possible perspectives that one can adopt to analyse the kind of operators and the kind of action that they envision as potentially leading from knowledge to action. The following set of seven questions can guide that exercise.

-Who are we expecting to act on whom in order to produce the changes we recommend? Who do we expect to lead the action? -Who gets to define the issues and prioritize them (e.g. between conservation and other concerns)?

-Who is accountable to whom (for biodiversity, and for other issues that may be affected by conservation)?

-What is the place and role of actors who are specialized in conservation (NGOs, conservation scientists, etc.)? -What (or who) causes the biodiversity problem that conservation action intends to address? -What course of action could effectively address this (or these) biodiversity degradation inducing actions or processes? -What privileged forms of action could be envisaged?

The first four are focusing on actors and organisational relations between them, the last three focus on problem identification and problem solving. Realistic action for conservation is a combination of power relations and problem solving: dealing with power relations in a way that leads to effectively halting a biodiversity degrading process.

Precise answers to such questions are highly context-specific and this set of questions can be taken as a guide to quick and dirty diagnostic of specific conservation knowledge-to-action situations.

However on a more general level, the kind of answers that one tends to privilege defines a repeatable perspective and amounts to a broad underlying conceptual model of organized action.

In the conservation literature and debate, as well as in concrete conservation initiatives, we have found six such models to be particularly salient and useful to consider. As we now set out to describe each of them in turn, let us just keep in mind that they are ways to structure reasoning about organized action for conservation, conceptual models that may underlie both the design of conservation action and its analysis in research about conservation action. For each model, we propose a new technical term, alongside with a lay phrase that may alternatively designate it. The former may seem arcane at first, but they have the advantage (1) of being more concise, (2) of building on fundamental conceptual synergies (e.g. the combination of initiative and power in the "archic" model, or of dispersion and planting seeds in the "speiric"model), (3) of preventing misunderstandings (e.g. "this is not how I see governance, or government") and ( 4) of associating explicitly each model with the overall framework that is proposed here.

Government-as-the-operator, or the archic model

In this model it is established public authority that is expected to act on behalf of society.

Government (e.g. national or local) is seen as the main operator in charge of taking action on public problems like conservation and of prioritizing issues. Here, all stakeholders, all members of the public are accountable to government for their behaviour and it's impacts; and government is accountable in turn for its choices and for its effectiveness in solving conservation problems. In this perspective, the major path from knowledge to action, is for knowledge to inform government decision-makers at all stages of policy: agenda setting, prioritizing conservation or amongst conservation issues, policy design as it has to rest on sound knowledge, and policy evaluation. This sets roles for conservation scientists as advisers to the prince, as experts participating in public policy, as a public voice speaking directly to the authorities about what they could or should do for conservation.

We call this model archic based on greek archos, that combines the idea of the leader, the chief, the source of initiative. In archic reasoning the initiative and the power of acting for conservation both rest on the authorities. This paradigm plays a major role in conservation research, as conservation scientists strive to influence the agenda of public policy, to inspire regulation, to provide technical tools (maps, models, inventories, etc.) for the implementation of policy, and as it (often implicitly) posits conservation researchers who reflect on action in a form of advisory dialogue with the authorities.

A focus on legal official goals and legal norms, on the kind of instruments (laws, subsidies, etc.) that are governmental in nature, on the authorities as legitimate to act and having a duty to do so, on policy implementation, its difficulties and shortcomings, are all indicators that one is immersed in archic (government-as-operator) reasoning.

Coordination-as-resolution, or the agoric model

The agoric model assigns the responsibility for acting on the problem to the stakeholders themselves, without the involvement of authorities. It is negotiation between them that should prioritize issues, decide on what action to take, implement and evaluate such action. In this perspective, stakeholders are accountable directly to one another. In agoric reasoning the main vector of action is through approaches like mediation and other tools that enhance communication, coordination and collaboration, and focus on sitting all the actors around the proverbial table .   Elinor Ostrom's Governing the Commons (1990) is a very influential example of agoric reasoning, as it posits direct negotiation of resource-use rules between resource users themselves as the crux of managing common-pool resources. Another example is the classic definition of payments for ecosystem services [START_REF] Wunder | Payments for Environmental Services: Some Nuts and Bolts[END_REF], that insists that ecosystem services users directly negotiate and effect payments to providers. It is worth underlining that from an organisational and power relations standpoint, the important trait of the agoric model is not whether one negotiates about rules, about money, about technical options or about other issues: the fundamental trait is that stakeholders negotiate directly solutions to environmental problems with one another. The greek word agora reflects that trait as it means both the assembly and the market: on the city's central square, people negotiated all their dealings, practical, political, social or commercial.

In the agoric (coordination) paradigm, conservation scientists' role can be either as support to mediation, or as support to one of the stakeholders negotiating for better biodiversity management.

Forms of applied conservation research that promote direct engagement with stakeholders as they negotiate between themselves, co-production with them of knowledge and action, not involving the authorities, refer to that paradigm of collective action.

Governance-process: the diagogic model

The third model is a hybrid combination of the previous two as it sits somewhere between government and direct negotiation of solutions between stakeholders. It finds its source in two converging pressures. On the one hand, officials conducting government policies often meet difficulties in design and implementation, to the point that they endeavour to involve stakeholders at various stages of policy. On the other hand, social innovators who experiment with agoric solutions -i.e. direct negotiation -are often confronted with the limitations of what can be achieved without government involvement (e.g. without the support of law enforcement, or without legal changes, or without support from public funds) and so, they may seek collaboration of public authorities to bottom-up negotiated solutions. The resulting model of organized action is governance: "multi-level interactions […] among, but not limited to, three main actors, i.e., state, market, and civil society, which interact with one another […] in formulating and implementing policies in response to environment-related demands and inputs from the society, bound by rules, procedures, processes, and widely-accepted behavior, […] for the purpose of attaining environmentally-sustainable development." (Floranoy, http://ecogov. blogspot.com/2007/04/ -for a discussion of issues in defining governance, see Mermet et al., 2013, p.52). This model is identified by two main traits. (1) It insists on forms of collaboration between actors with, and without, public mandates: public participation in decision-making, public private partnership, joint committees and other forms of deliberative forums that downplay the heterogeneity in the institutional standing of members. (2) It assigns to decision-making procedures the central role in treating collective action problems. This contrasts both with archic reasoning (the central point is substantive and instrumental: a decision-maker choosing the right tool for action) and with agoric reasoning (the central point is agreement between peers, whatever the ways in which the agreement has been reached). Explicit, visible, often complex procedures reduce the tension inherent in situations where operators with a clear institutional mandate (e.g. elected officials, civil servants) collaborate with others who do not have any (e.g. NGO activists, inhabitants who have been invited to a round table or who have simply shown up at a public participation meeting). Thus the choice of the term diagogic for this model: procedure is expected to support action by leading through (diagoo) situations involving heterogeneous actors and issues.

The diagogic (governance-process) paradigm is familiar to conservation operators in many forms like action plans designed by committees and public participation events, biodiversity strategies involving government, NGOs and private firms, or community-based protected area management.

For conservation scientists, this creates environments where the way from knowledge to action is through active involvement in procedures of biodiversity governance. This can take many forms, e.g. involving the public in the research itself, involving stakeholders in the research agenda, participating as experts in conservation planning procedures, etc.

Revolution-required-for-change: the stasic model

In the stasic model, the authorities -and more widely the political-economic regime -are seen not as the source of solutions, nor as part of the solution, but as the main source of the problem as they act to perpetuate dysfunctional activities and power relations that are seen as being the root cause of biodiversity loss. In this perspective, the main course of action has to be to stand up in the face of the system and overthrow the government that keeps it in place. Everyone is accountable, in a Manichean way: either they participate in the activities of the system and they are responsible for the problems, or they oppose squarely the system and they are part of the solution. Greek stasisstanding up in revolt, attempts at overthrowing the government, civil war -reflects both the structure of this confrontation of powers and the spirit of standing up to the powers that be that pervades much of the environmental literature based on that model. The title of Joel Kovel's book "The Enemy of Nature-the End of Capitalism or the End of the World" (2002) summarizes in a nutshell one of the possible diagnostics of the biodiversity crisis based on that underlying model.

Who is then to act? To effect a revolution, a mobilisation of the masses is required because only they can accumulate the strength to overturn the political regime: they (and the revolutionaries who mobilise and lead them) are the decisive operators. Such mobilisation cannot rely on one very specific issue (like biodiversity). To mobilize masses, it is required to present a large set of public concerns as being jointly the consequences of the existing system, and revolution as the joint solution to that whole set of problems. The issue is then not to prioritize issues but on the contrary (in line with the Manichean structure of that model) to amalgamate on the one hand all the issues that one sees as rightful and suffering from the system (from biodiversity to poverty, from discrimination to educational problems, etc.) and on the other hand all the issues that are deemed illegitimate and part of the system (large scale logistics for food production and distribution, automobile traffic issues, etc.).

The stasic (revolution) paradigm is troubling for conservation scientists. Should they join wider oppositional mobilizations or should they participate in governments' efforts towards sustainable development? On April 29th, 2015, protesters opposing the construction of an airport at Notre-Dame des Landes (France) attacked scientists from an ecology lab who were doing field work because they saw their studies as a contribution to decision-making about the project and thus as part of the system that justifies and supports such ecosystem-destroying projects [START_REF] Certain | Vigneux-de-Bretagne: des biologists agressés sur la ZAD[END_REF].

Minority-actor-for-change: the zelic model

The zelic model shares one premise with the previous one (stasic). The initiative of change cannot rest primarily on the government, nor on the circle of major existing stakeholders, since states of affairs detrimental to biodiversity are precisely the result of existing balances of power and interests that government (in the archic paradigm) and negotiation between major stakeholders (in the agoric paradigm) act to maintain. So the onus of acting in favour of biodiversity has to rest on those actors for whom it is a priority concern to the point that they are ready to struggle against other stakeholders and established balances of power and interest. In the zelic paradigm however, the scope, the aims and the means of this action for change differ from stasic reasoning.

In terms of scope, it rests on pluralism, i.e. on recognizing that there is a multiplicity of legitimate public concerns, and that they are not necessarily always compatible or convergent (e.g. a dam project that alleviates rural poverty in some valley can be detrimental to biodiversity). As a result, each public concern can be defended on its own grounds in the face of all the other (public or private) concerns. It also rests on the diagnostic that biodiversity issues are caused not so much by "the system" as a whole as by specific actors, techniques and projects. In this perspective, the massive impact of European agriculture on terrestrial biodiversity is the result of a finite series of inappropriate choices in terms of crops (e.g. irrigated corn), farming techniques (e.g. pesticides), location (e.g. replace high nature value agricultural systems by intensive ones), etc. And behind such choices are identifiable operators and it is them that action for change should target. The aim of action for change is thus to transform the system, so that specific biodiversity issues are better managed, not to throw down the economic and political system wholesale (as in the stasic paradigm). And the means for this are all the possible means for a struggle to increase the level of priority given to specific biodiversity issues, and to identify and overcome specific actors resisting change.

The zelic paradigm essentially attribute to minority set of actors the onus to change large systems of practice and power that negatively affect biodiversity, the main spring of action being the particularly high priority they assign to that issue (thus the choice of the word zelos, greek for passion, zeal for something, somebody, or against somebody). The idea of a minority actor reflects the feeling of an uphill battle, of strategy from the weak to the strong, that defenders of biodiversity often express. On a deeper theoretical level, it refers not to the amount of power available to such actors, but to the exact opposite of the hatroologic discourse: the fact that by opting for one cause that is just one amongst many in a deeply pluralistic world, one becomes logically a minority [START_REF] Deleuze | Mille Plateaux[END_REF]. Rachel Carson's book "Silent spring" (1962) is an emblematic example: one singular voice confronting a host of interests that converge against biodiversity. A striking point of the model is that seen from the minority-actor's perspectives, other actors are structurally seen as having too much power. For instance as Chris Sandbrook (2017) critically examines the power positions of conservation groups, he finds them at first sight to be too weak when seen from their own perspective, and too strong in relation to marginalised people that may be affected by conservation -i.e. when seen from the perspective of other minority-interest actors, such as those who defend the rural poor or indigenous people. It is not a matter of uneven power, as seen from an overlooking moral standpoint, but of asymmetry of prupose and perspective.

Theory-wise, in the environmental field there are several approaches that analyse how this can result in actual change, despite apparently impossible odds. Sabbatier and Jenkins' advocacy coalition theory (1993) shows how the gradual construction of coalitions around an issue can (sometimes) result in changing the course of public policies. The concept of policy windows [START_REF] Rose | Policy windows for the environment: tips for improving the uptake of scientific knowledge[END_REF] -discrete periods of time during which the policy making process presents opportunities for conservation groups to successfully promote their agenda and solutions -is inherently based on the premise that steps forward in conservation are pushed by conservation interest groups on the authorities as they adopt policies. Michel Callon's translation theory (1986) dissects how an initially powerless innovator can (sometimes) succeed in making major stakeholders change course. Our own work on strategic environmental management analysis ( 2011) adapts strategic management reasoning to environmental situations where a minority actor confronts resistance to environmental change.

Practice-wise, the zelic (minority actor) paradigm is familiar to conservation operators. It is the model of an NGO mobilizing on a specific biodiversity issue and managing after a struggle to impose major changes of policy. It is the model of a grassroots movement that mobilizes against a biodiversity-damaging development project and eventually succeeds in halting that project, after obtaining support from various allies.

In this paradigm, knowledge leads to action mainly by providing resources to minority actors or alliances defending biodiversity. An ecology professor who participates in the creation of an NGO to mobilize biodiversity issues linked to his work, conservation biologists doing research with results that they know (but maybe don't say) will be instrumental as it will be used by biodiversity advocates, a committee of ecology experts that obtains an official recognition of a biodiversity issue that they know will not lead directly to government action, but will provide leverage for biodiversity activists and advocates, are some of the examples of knowledge-to-action paths in that model of action.

8. Drop-out-and-seed: the speiric model Speiric reasoning shares with the stasic paradigm the diagnostic that "the system", including government policy is the cause of the problem, not the solution, but differs deeply in the formula for action. Instead of trying to amalgamate all those who have discontents, it relies on dispersing, i.e. on dropping out of the system individually or in small groups, and instead of confronting the system, on embarking on innovative experiments while eluding the system and offering as little hold as possible to its enforcers. The underlying theory of action for change is two steps: I can change my circumstances and biodiversity impacts individually now -provided I drop out of the system and innovate in my lifestyle -and if that works, and if other people do it successfully too, our initiatives will sow the seeds for large scale change. In the words of Guy [START_REF] Mcpherson | Going back to the land in the age of entitlement[END_REF] in a vivid expression of his own drop out and seed reasoning and experience: "We make every effort to live outside the industrial economy […]. By our example, we are demonstrating how society can be restructured so that children and other humans will understand and value the origins of food and life". Thus the choice of speiric, derived from greek verb speiroo that expresses both the act of dispersing, and the act of sowing.

How conservation science can find its place in that paradigm remains an open question. The tension here stems from the fact that science is by definition a large scale, highly institutionalized system for establishing, validating and disseminating knowledge, whereas speiric ("drop out & seed") reasoning promotes small-scale, non-institutionalized forms of organization. Here knowledge is important, but it is expected to be independent from institutions, originated in personal experience, and (actively) exchanged in informal, non-normalized, non-institutionalized forms of validation and exchange.

Deeply engrained and contradictory models

Table 1 An overview of six underlying conceptual models of collective action in conservation research. The six models structure a system of crossed critiques that underlies debates on what may constitute significant action for biodiversity conservation (each box summarizes critiques from model on top of column against model on left of line). x Table 1 gives an overview of the six models and how they each give a different meaning to conservation research being relevant for action. This overview can help clarify debates on bridging conservation science and action. For this, four points need to be discussed.

Model

(1) The six paradigms of collective action are at least partly incompatible with one another. The most obvious examples are the strong oppositions between the stasic model and the archic, diagogic, zelic and speiric models, respectively on the role of existing authorities, on the fruitfulness of improving governance with existing partners, on the desirability of improving just some biodiversity issues at the possible expense of hindering a more comprehensive revolution, and on the respective merits of individual versus mass action. Less obvious examples are just as important. For instance direct negotiation between stakeholders (agoric) does mean that there has to be no or little interference from government [START_REF] Ostrom | Governing the Commons: The Evolution of Institutions for Collective Action[END_REF] which contradicts the strong sense, in the archic paradigm of the legitimacy of authorities to supervise the resolution of public problems.

As a result diagogic arrangements for governance procedures have to manage deep tensions between the two models as they try to conciliate them. Beyond these examples, such tensions can be systematized as each model has specific critiques against each of the five others. Table 2 presents this system of crossed critiques which we think is a useful contribution to analysing controversies about conservation actions, and especially discussing contrasting views about knowledge to action paths in biodiversity conservation.

(2) There is not a line of progress that would lead over time from one model to the next. All six models have coexisted in parallel for a long time in the field of conservation. Waves of fashion can bring one or another in turn to the fore of academic and public discourse. A good example is the steady rise to a quasi-hegemony (in discourse if not in practice) of the diagogic (governanceprocess) model in the 1990s and 2000s, under the pressures for integrating private interests in the funding and implementation of conservation, and for accommodating a wide array of local or traditional interest groups and concerns. Another is the rise in the current decade of the stasic (revolution) and speiric (drop out and seed) models. But beyond striking variations in their front page presence in debates -and their quite real but more moderately fluctuating influence in practice -all play an important role today, both in guiding action and in analysing knowledge-to-action problems (for a detailed example of the first five models playing out against one another over time in conservation field practice, see [START_REF] Mermet | Five paradigms of collective action underlying the human dimension of conservation[END_REF]. Claims that one models comes to embrace the others or to succeed them-e.g. that stakeholder involvement and governance procedures are a progress that displaces the "old" ways of government policy and of environmentalist pressure groups -are simply one of the rhetoric devices by which proponents of each model may try to disqualify the others.

(3) These models are not superficial framings that can be interchanged easily. On the part of the persons involved, they are most often deeply held belief and value systems, such as the (archic) sense of public service in a managing agency's staff versus the profound (stasic) indignation felt by some activists at the very thought of the compromises that such an agency makes and of the authority it exercises to the detriment of some people, contrasting again with the (zelic) relentless zeal deployed by some environmental activists as they strive to move mountains of antienvironmental inertia. In terms of theory, each of the model has deeply different roots and developments that again make it difficult to jump easily from one model to another.

(4) Finally there is no overarching model that would integrate all six perspectives -that is, there is no actionable overarching model of that sort. Indeed, it is often easy to see how different models of collective action can relate or lead to one another, as when (zelic) pressure by an NGO leads to (diagogic) prolonged debates and policy-defining procedures until (archic) authorities establish and enforce a new law. But the crucial point is not that an external observer or a historian can step back and assemble them into a wider picture, it is that in order to act, each scientist (or team of scientists) has to step forward and become an actor or associate with one or another actor, and so become situated within the wider social system and situation and adopt by necessity a partial and specific perspective on it, as they see the action system not from the outside but from within it. According to where conservation scientists happen or prefer to step into action, they will have to choose and develop a view of action and the respective roles of various actors in it that is actionable, i.e. able to guide (and to justify) their actions and that of their partners. In other words, in contrast with [START_REF] Watts | Should social science be more solution-oriented?[END_REF] we do not consider desirable nor possible the possibility of more integration of different social science approaches to situations through open testing on action problems, but rather that engaging in action does imply adopting a distinct perspective, and that social scientists, inasmuch as their research aims to be actionable -and thus in line with a position engaged in actionhave to do likewise, at the cost of the fragmentation of the theoretical landscape for which we propose an overview here.

Clarifying debates on knowledge to action options

The six models approach proposed here is a typology of the possible perspectives one can adopt about where and how to act on the social system from within it, in the name of biodiversity. This typology can help clarify debates within the conservation community on knowledge-to-action options. At the most basic level, alongside with the critique of the hatroologic model of action and other imprecise treatments of agency, it can help conservation scientists realize the importance of the organizational reasoning that underlies all action-related proposals, and to identify the organizational models underlying various options. Then, it can help navigate the debates, controversies and confrontations between various proposals and claims about how to overcome the knowing-doing gap and create useful connexions. Clashes between promoters of different options can be examined in a colder spirit once they are situated in the system of crossed critiques between the six models.

For instance when looking at the diversity of reactions of groups with different political or analytical orientations as they consider the possibility of systemic collapse of our economic model and its ecological basis (see for instance [START_REF] Pollard | Preparing for Collapse[END_REF][START_REF] Servigne | Comment tout peut s'effondrer. Petit manuel de col-lapsologie à l'usage des générations présentes[END_REF], one can see how they map out on the models of collective action. An archic reaction may be for authorities to claim control, or act to organize control of the situation (and for citizens to demand control); an agoric response may be to promote a return to community organization; for a stasic analyst or militant, collapse will be viewed as an ally and welcome opportunity in overthrowing the system; a zelic response might be concern that painfully acquired conservation guarantees (e.g. protected areas) might be wiped away in the turmoil, while a speiric-oriented person will advance the unstoppable advance of collapse as the perfect justification for going off-grid to dodge the catastrophy while making sure the seeds for a new world are in place when the old one will have proposed actions pragmatically promise tangible positive biodiversity outcomes versus other outcomes (political, social, economic, etc.) that may be of concern to the authors of the analysis or proposal?

As the conservation science community continues its mobilization, its strategic reflections and its soul-searching on knowledge-to-action paths, it becomes increasingly clear that several very different and even contradictory models of action will continue to coexist and develop in parallel. A clearer perception of underlying models of organised action will help put in perspective recurrent clashes based on premises and principles, and focus on deepening what each perspective can bring in terms of practical and context-relevant understanding, depending on the particulars of each biodiversity field situation and context for action. It is important to realise that work based on each of the models has both some potential to help conservation scientists find relevant paths for action and some serious limitations; both are best explored in reflexive action, confronting actual biodiversity issues.

  ceased to function. And what about the diagogic model, one may ask? Well, as it requires everyone to agree on the issue and the solutions… it may hardly be able to consider seriously and in time the possibility of collapse.Positions in such debates (one may think also of the debate on evidence-based conservation, on community-based conservation, on biodiversity offsets and many others) can be broken down between on the one hand controversies of principle between premises, perspectives, schools of thought or general political positions, and on the other hand different tangible and specific aspects of actual field situations, of conservation problems that one is considering before taking action. The first bring only very generic information to the table, and information on the general beliefs and affiliations of the authors. The second reflect the complexity of actual conservation situations on the ground. An acid test to tell them apart could consist in asking systematically three questions. What is the underlying paradigm of organized action in this analysis or proposal? What part of the analysis or proposal would have resulted from the premises regardless of field investigation, and what part is rooted in case-specific field investigation? To what extent do the (implicit or explicit)