Vestislav Apostolov
email: apostolov.vestislav@uqam.ca

Georges Dloussky

Locally Conformally Symplectic Structures on Compact Non-K ähler Complex Surfaces

We prove that every compact complex surface with odd first Betti number admits a locally conformally symplectic 2-form which tames the underlying almost complex structure.

Introduction

It is a well-known result [START_REF] Buchdahl | On compact K ähler surfaces[END_REF]30,[START_REF] Siu | Every K3 surface is K ähler[END_REF][START_REF] Todorov | Applications of the K ähler-Einstein-Calabi-Yau metric to moduli of K3 surfaces[END_REF]] that a compact complex surface S = (M, J) admits a K ähler metric if and only if its first Betti number b 1 (M) is even. A cornerstone for the proof of this result is the fact, proved independently in [START_REF] Gauduchon | La 1-forme de torsion d'une vari ét é hermitienne[END_REF]Lemme II.3;[START_REF] Harvey | An intrinsic characterization of K ähler manifolds[END_REF]p. 185] (see also [START_REF] Siu | Every K3 surface is K ähler[END_REF]p. 143,Proposition 1.6] for the case of a K3 surface), that b 1 (M) is even if and only if M admits a symplectic form ω which tames J, in the sense that the (1, 1) part of ω is positive-definite. This and the methods of proof in [START_REF] Buchdahl | On compact K ähler surfaces[END_REF]30] inspired the so-called "tamed to compatible" conjecture in symplectic geometry, which asks whether an almost complex structure on M which is tamed by a symplectic form admits a compatible symplectic form; see [START_REF] Donaldson | Two-Forms on Four-Manifolds and Elliptic Equations[END_REF][START_REF] Taubes | Tamed to compatible: symplectic forms via moduli space integration[END_REF].

A natural extension of the theory of K ähler manifolds to the non-K ählerian complex case can be obtained through the notion of locally conformally K ähler metrics, 2 V. Apostolov and G. Dloussky introduced and studied in foundational work by F. Tricerri and I. Vaisman; see for example, [START_REF] Dragomir | Locally Coformal K ähler Geometry[END_REF][START_REF] Ornea | A report on Locally Conformally K ähler Manifolds[END_REF] for an overview. Recall that a locally conformally K ähler (or lcK) metric on a complex manifold X = (M, J) is defined by a positive-definite (1, 1)-form F satisfying dF = θ ∧ F for a closed 1-form θ . The 1-form θ is uniquely determined and is referred to as the Lee form of F . The corresponding Hermitian metric g(•, •) = F (•, J•) defines a conformal class c on M. Changing the Hermitian metric g = e f g within c amounts to transforming the Lee form by θ = θ + df, so that the de Rham class [θ ] is an invariant of the conformal class c.

Of particular interest is the case of compact complex surfaces, where recent works [START_REF] Belgun | On the metric structure of non-K ähler complex surfaces[END_REF][START_REF] Brunella | Locally conformally K ähler metrics on certain non-K ählerian surfaces[END_REF][START_REF] Brunella | Locally conformally K ähler metrics on Kato surfaces[END_REF][START_REF] Gauduchon | Locally conformally K ähler metrics on Hopf surfaces[END_REF] showed that lcK metric exists for all known examples of compact complex surfaces with odd first Betti number, with the exception of the complex surfaces obtained by blowing-up points of certain Inoue surfaces with zero second Betti number, described in [START_REF] Belgun | On the metric structure of non-K ähler complex surfaces[END_REF]. However, a general existence result is still to come.

In this paper we study, on a compact complex surface S = (M, J) with odd first Betti number, the problem of existence of locally conformally symplectic forms ω which tame J, that is, 2-forms ω satisfying dω = θ ∧ ω for a closed 1-form θ (called Lee form of ω), and such that the (1, 1)-part of ω is positive-definite. This is, in general, a weaker condition than the existence of lcK metrics, which turns out to be related to the theory of bihermitian conformal structures [START_REF] Apostolov | Bihermitian structures on complex surfaces[END_REF][START_REF] Pontecorvo | Complex structures on Riemannian 4-manifolds[END_REF] in the case when the ω-conjugate of J determines another integrable almost-complex structure J ω on M, see [START_REF] Apostolov | From locally conformally K ähler to bihermitian structures on non-K ähler complex surfaces[END_REF].

We establish the following general existence result, which we believe is an important step towards the resolution of the existence problems for both lcK and bihermitian conformal structures on a non-K ähler complex surface, and which answers in the positive (in the case of complex surfaces) a question raised in [START_REF] Ornea | A report on Locally Conformally K ähler Manifolds[END_REF]Open Problem 1].

Theorem 1.1. Any compact complex surface S = (M, J) with odd first Betti number admits a locally conformally symplectic form ω which tames J.

The above theorem is derived from another existence result concerning a conformal class of Hermitian metrics on S = (M, J), which can be regarded as a twisted version of Gauduchon's celebrated theorem [START_REF] Gauduchon | Le th éor ème d'excentricit é nulle[END_REF], and can be stated as follows. Let a ∈ H 1 dR (M) be a de Rham cohomology class and α ∈ a a closed 1-form in a. Denote by d α := d -α ∧ .

the twisted differential operator defining the Lichnerowicz-Novikov complex, and let

d c α := JdJ -1 .
Theorem 1.2. Let S = (M, J) be a compact complex surface with odd first Betti number, and c a conformal class of Hermitian metrics on S. Then, there exists a non-zero de Rham class a ∈ H 1 dR (M) such that for any metric g ∈ c, there exists a representative α ∈ a such that the fundamental 2-form F of g satisfies

d α d c α F = 0.
The de Rham class a appearing in Theorem 1.2 determines, via the exponential map

H 1 dR (S, C) exp → H 1 (S, C * ) -→ Pic 0 (S), (1)
a flat holomorphic line bundle L a , and we derive Theorem 1.1 from Theorem 1.2 by showing that, when H 2 (S, L a ) = {0}, S also admits a locally conformally symplectic form with a Lee form α. The proofs of Theorems 1.1 and 1.2 are presented in Section 2, whereas the necessary analytical tools are collected in Appendix.

In the light of the "tamed to compatible" conjecture mentioned above, it is natural to compare the existence on S of lcK metrics and of locally conformally symplectic forms taming J, through the corresponding de Rham classes of their Lee forms. We thus introduce in Section 3 the subset C(S) (respectively, T (S)) in H 1 dR (M) of classes a for which there exists an lcK metric on S with Lee form θ ∈ a (respectively, for which there exists a locally conformally symplectic form which tames J, with Lee form in a). We obviously have the inclusion C(S) ⊆ T (S), and one may ask (see also [START_REF] Ornea | A report on Locally Conformally K ähler Manifolds[END_REF][START_REF] Brunella | Locally conformally K ähler metrics on certain non-K ählerian surfaces[END_REF]Remark 9]). Problem 1.3. Let S = (M, J) be a compact complex surface with odd first Betti number.

Determine the set T (S) ⊂ H 1

dR (M) of classes a for which there exists a locally conformally symplectic form ω which tames J and has a Lee form θ ∈ a. Is T (S) strictly bigger than C(S)?

Our initial motivation to study the above problem came from the theory of bihermitian conformal structures developed in [START_REF] Apostolov | From locally conformally K ähler to bihermitian structures on non-K ähler complex surfaces[END_REF], where the existence of the latter was reduced to answering the question of whether certain classes a ∈ H 1 dR (M) belong to T (S) and C(S).

A number of partial results concerning Problem 1.3 are obtained in Section 4, where we specialize to the case of a compact complex surface with first Betti number equal to 1.

In the last Section 5 of the paper, we consider some examples of non-K ähler complex surfaces in the Kodaira class VII (i.e. satisfying H 0 (S, K S ) = {0} for all ≥ 1, where K S stands for the canonical line bundle of S, see [START_REF] Barth | Compact Complex Surfaces[END_REF]), for which a complete answer to the above problem can be given. It is known that, in this case, the first Betti number equals 1 (see e.g. [START_REF] Barth | Compact Complex Surfaces[END_REF]), and that the degree with respect to some Gauduchon metric on S of the holomorphic line bundles determined via (1) induces an orientation on H 1 dR (S) ∼ = (R, >), which turns out to be independent of the choice of a Gauduchon metric (see [START_REF] Teleman | The pseudo-effective cone of a non-K ählerian surface and applications[END_REF]Remark 2.4] or Lemma 4.1). Thus, for any compact complex surface in the Kodaira class VII, one can naturally identify H 1 dR (S) with the oriented real line (-∞, +∞). In this notation, a combination of Propositions 5.1 and 5.2 gives the following theorem. We denote by L α the trivial real line bundle over M, endowed with a (non-trivial) flat connection ∇ α s := ds -α ⊗ s, where s is a smooth section of L. Similarly, ∇ α induces a holomorphic structure on the complex bundle In what follows, we shall tacitly identify the real flat bundles L α and L β (respectively, the flat holomorphic line bundles L α and L β ) associated to different representatives α, β ∈ a and (slightly abusively) denote by L a (respectively, L a ) either of them.

L α := L α ⊗ C. Writing α |U i = df i on an open covering U = (U i ) of M, s 0 := (U i , e f i )
Let L = L a be the flat real line bundle over M determined by α ∈ a, a ∈ H 1 dR (M) via the Convention 2.1, and denote by L * = L -a its dual. The differential operator which is isomorphic to the de Rham complex of differential forms with values in

d α = d -α ∧ . defines the Lichnerowicz-Novikov complex • • • d α → Ω k-1 (M) d α → Ω k (M) d α → • • • (2)
L * • • • d L * → Ω k-1 (M, L * ) d L * → Ω k (M, L * ) d L * → • • • . (3)
In particular, we have an isomorphism between the cohomology groups

H k α (M) H k d L * (M, L * ).
Considering the Dolbeault cohomology groups of S with values in the flat holomorphic line bundle L * , we have

d L * = ∂ L * + ∂L * , and d α = ∂ α + ∂α with ∂ α = ∂ -α 1,0 ∧ and ∂α = ∂ -α 0,1 ∧,
giving rise to the isomorphisms

H p,q ∂α (S) H p,q ∂L * (S, L * ). ( 4
)
Definition 2.2. Let X = (M, J) be a complex manifold. We shall say that a differentiable 2-form ω is a locally conformally symplectic form taming J if there exists a closed differentiable 1-form α such that d α ω = 0, and the (1, 1)-part ω 1,1 of ω is positive definite.

If, furthermore, ω is of type (1, 1), it defines a locally conformally K ähler structure on X.

The 1-form α is called the Lee form α of ω.

Remark 2.3. In terms of the isomorphism between (2) and (3), if we write

α |U i = df i on an open covering U = (U i ) of M, then ω i |U i := e -f i ω defines a d L * -closed 2-form with
values in L * , whose (1, 1)-part is positive definite (for the latter we use the fact that

L * = L -a is defined by the co-cycle (e f j -f i , U ij = U i ∩ U j )
which consists of positive constant real functions). Similarly, if ω is a locally conformally symplectic form taming J with d α ω = 0, then ω = e f ω is a locally conformally symplectic form taming J which satisfies d α ω = 0 with α = α + df. It follows that the existence of a locally conformally symplectic form taming J with Lee form α merely depends upon the de Rham

class a = [α].
Our first observation is the following lemma.

Lemma 2.4. Let ω be a locally conformally symplectic form on X = (M, J) with Lee form α. Denote by F := ω 1,1 the (1, 1)-part of ω. Then,

d α d c α F = 0,
where

d c α = Jd α J -1 = i( ∂α -∂ α ).

Proof.

Write

ω = F + ω 2,0 + ω 0,2
where ω 2,0 and ω 0,2 denote the (2, 0) and (0, 2)-part of ω, respectively. As

d α = ∂ α + ∂α , one has
d α ω = 0 ⇐⇒ ⎧ ⎨ ⎩ ∂ α F + ∂α ω 2,0 = 0, ∂α F + ∂ α ω 0,2 = 0, hence 2i∂ α ∂α F = d α d c α F = 0.
Lemma 2.5. Let S = (M, J) be a complex surface and g be a Hermitian metric with fun-

damental 2-form F (•, •) = g(J•, •). Then, d α d c α F = 0 ⇐⇒ δ(θ -α) + g(θ -α, α) = 0,
where δ is the co-differential with respect to g and θ = Jδ F is the Lee form of g.

Proof.

Using dF = θ ∧ F (which, as F is a self-dual 2-form with respect to g, is equivalent to the relation θ = Jδ F ), one gets

d α d c α F = [d(J(θ -α)) + (θ -α) ∧ J(θ -α)] ∧ F.
As F is self-dual, that is, * F = F , where * stands for the Hodge operator with respect to where D is the Riemannian connection of g, {e i , i = 1, . . . , 4} is any J-adapted orthonormal frame, and for passing from the third line to the fourth we have used the fact that D e i J is skew-symmetric with respect to g and anti-commutes with J, whereas for passing from the fourth line to the fifth, we have used the identity θ(X) = (δ F ) (-J X) = -4 i=1 g(J(D e i J)(e i ), X).

g, d α d c α F = 0 is equivalent to 0 = g(d(J(θ -α)) + (θ -α) ∧ J(θ -α), F ) = 4 i=1 (D e i (J(θ -α))(Je i )) + |θ -α| 2 g = 4 i=1 ((D e i J)(θ -α)(Je i ) + (D e i (θ -α))(e i )) + |θ -α| 2 g = (θ -α) J 4 i=1 (D e i J)(e i ) -δ(θ -α) + |θ -α| 2 g = -g(θ, θ -α) -δ(θ -α) + g(θ -α, θ -α) = -δ(θ -α) -g(θ -α, α),
Remark 2.6. In the light of Remark 2.3, it is easily seen that the condition

δ(θ -α) + g(θ -α, α) = 0 ( 5 )
is also conformally invariant. More precisely, it is straightforward to check that if g = e f g and α = α + df, then ( 5) is satisfied for (g, α) if and only if it is satisfied for ( g, α).

Recall the fundamental result of Gauduchon [START_REF] Gauduchon | Le th éor ème d'excentricit é nulle[END_REF] which affirms that if X = (M, J)

is an m-dimensional compact complex manifold endowed with a Hermitian metric g, then there exists (a unique up to scale) conformal metric g = e f g whose fundamental form F satisfies dd c F m-1 = 0, or equivalently, for which δ J δ F = 0. Such Hermitian metric is referred to as Gauduchon metric. By Lemma 2.5 and Remark 2.6, we then obtain Proposition 2.7. Let S = (M, J) be a compact complex surface, a ∈ H 1 dR (M) be a de Rham class, and c = [g] be a conformal class of Hermitian metrics on M, with g a Gauduchon metric in c. Then the following conditions are equivalent:

(i) for any g ∈ c, the fundamental 2-form F satisfies d α d c α F = 0 for some α ∈ a; (ii) there exists a positive smooth function ψ on M which satisfies the equation

L g,a (ψ) = Δ g (ψ) -g(θ g -2a g h , dψ) + g(θ g -a g h , a g h )ψ = 0, (6)
where Δ g is the Riemannian Laplace operator of g, θ g is the corresponding co-closed Lee form, and a g h is the harmonic representative for a with respect to g. Proof. By Remark 2.6, if, for some metric g ∈ c, the corresponding fundamental form F satisfies d α d c α F = 0, for any other metric g = e -f g in c, the fundamental 2-form F satisfies d α d c α F = 0 with α = αdf. It follows that the condition (i) is equivalent to the existence of a 1-form α ∈ a such that the fundamental 2-form of a Gauduchon metric g ∈ c satisfies

d α d c α F = 0. Writing α = a g h -dlog ψ with ψ > 0, Lemma 2.5 reads 0 = δ(θ g -α) + g(θ g -α, α) = -δα + g(θ g -α, α) = -δ(a g h -dlog ψ) + g(θ g -a g h + dlog ψ, a g h -dlog ψ) = Δ g (log ψ) -g(θ g -2a g h , dlog ψ) + g(θ g -a g h , a g h ) -g(dlog ψ, dlog ψ) = 1 ψ (Δ g ψ -g(θ g -2a g h , dψ) + g(θ g -a g h , a g h )ψ), (7)
and the claim follows.

The following elementary observation regarding the linear operator L g,a will be used throughout.

Lemma 2.8. For any everywhere positive smooth function

ψ on M M L g,a (ψ) ψ v g = M g(θ g , a g h ) - M ||a g h || 2 g + 1 ψ 2 ||dψ|| 2 g v g . (8)
In particular, if ( 6) admits a positive solution with respect to some

a = 0 ∈ H 1 dR (M), then M g(θ g , a g h )v g = M g(θ g h , a g h )v g = M ||a g h || 2 g + 1 ψ 2 ||dψ|| 2 g v g > 0. ( 9
)
Proof. The equality [START_REF] Bogomolov | Classification of surfaces of class VII 0 with b 2 = 0[END_REF] follows by integration by parts and using that g is Gauduchon,

that is, δθ g = 0; the inequality ( 9) is an immediate consequence.

The general theory for the existence of positive solutions of the elliptic linear second-order PDE L g,a (ψ) = 0 is reviewed in Appendix to this paper. We recollect below the following variational characterization.

Proposition 2.9. The PDE (6) has a positive solution ψ if and only if

λ a (g) := sup u∈C ∞ (M),u>0 min M L g,a (u) u = 0. ( 10
)
Furthermore, λ a (g) is a finite number that varies analytically with respect to linear vari- Proof of Theorem 1.2. By Propositions 2.7 and 2.9, it is enough to fix a Gauduchon metric g ∈ c and show that λ a (g) = 0 for a suitable choice of a = 0. Let θ g h denote the harmonic part of the Lee form of g. It is well known (see e.g. [3, Proposition 1] or Proposition 3.5) that [θ g h ] = 0 under the assumption that the first Betti number is odd. We let a t = t 2 [θ g h ] with t > 0 and denote by a t h = t 2 θ g h the harmonic representative of a t with respect to g. We are going to show that λ a t (g) < 0 when t ≥ 2 and λ a t (g) > 0 for t close to 0; by the second part of Proposition 2.9, this would imply that λ a t (g) = 0 for some t ∈ (0, 2). Let u t > 0 be the eigenfunction of L g,a t , corresponding to the principal eigenvalue

ations g t = (1 -t)g + t g of
λ(t) := λ a t (g), normalized by M u 2 t v g = 1 (see Theorem A.1)
. By Lemma 2.8 we have

λ(t)vol g (M) = M L g,a t (u t ) u t v g = - M 1 u 2 t ||du t || 2 g v g + M g(θ g h -a t h , a t h )v g = - M 1 u 2 t ||du t || 2 g v g + t(2 -t) 4 M ||θ g h || 2 g v g . ( 11
)
Taking t ≥ 2 in (11) yields λ(t) ≤ 0. We are going to show that λ(t) > 0 for positive t close to 0. Indeed, as for each t λ(t) is a simple eigenvalue (see Theorem A.1), by the Kato-

Rellich theory [START_REF] Kato | Perturbation Theory for Linear Operators[END_REF][START_REF] Rellich | Perturbation Theory of Eigenvalue Problems[END_REF] (see Theorem A.3), λ(t) and u t vary analytically with respect to t.

As L g,0 does not have a zero-order term, by the Hopf maximum principle (see e. v g has a global minimum 0 at t = 0, so differentiating [START_REF] Buchdahl | On compact K ähler surfaces[END_REF] at t = 0 we get

λ (0)vol g (M) = 1 2 M ||θ g h || 2 g v g > 0.
In order to obtain a converse of Lemma 2.4, we first note the following lemma.

Lemma 2.10. Let L be a flat holomorphic line bundle over a compact complex surface

S, such that H 0 (S, L) = H 2 (S, L) = {0}. Then, H 2,1 ∂L * (S, L * ) = {0}. Proof. As H 2,1 ∂L * (S, L * ) ∼ = H 1 (S, K S ⊗ L * ), by Serre duality dim H 2,1 ∂L * (S, L * ) = h 1 (S, K S ⊗ L * ) = h 1 (L).
Using the vanishing of the cohomology groups from the hypothesis of the lemma and the Riemann-Roch formula, one concludes dim

H 2,1 ∂L * (S, L * ) = 0.
Let L a be the flat holomorphic line bundle corresponding to a = [α]; see Convention 2.1. Its degree with respect to the Gauduchon metric g is defined to be [START_REF] Gauduchon | La 1-forme de torsion d'une vari ét é hermitienne[END_REF] deg

g (L a ) = 1 2π M ρ L a ∧ F,
where 1 2π ρ L a is a pluriharmonic representative of the first Chern class of L a . Writing

α |U i = df i on an open covering U = (U i ) of M,
L a is the topologically trivial complex line bundle over M with s 0 = (U i , e f i ) being a nowhere vanishing smooth section, whereas

h |U i = e -2 f i (•, •) introduces a Hermitian metric on L a with h(s 0 , s 0 ) = 1. It follows that ρ = -1 2 dd c log e -2 f i = -d c α is the curvature of the Chern connection of (L a , h), so that deg g (L a ) = - 1 2π M d c α ∧ F = - 1 2π M g(d c α, F )v g = - 1 2π M g(θ g , α)v g = - 1 2π M g(θ g h , a g h )v g , ( 12
)
where for the last line we have used that dα = 0, δθ g = 0. It then follows from Proposition 2.7, the inequality (9) of Lemma 2.8 and the properties of the degree (see e.g. [START_REF] Gauduchon | La 1-forme de torsion d'une vari ét é hermitienne[END_REF]).

Lemma 2.11. Let g be a Gauduchon Hermitian metric on a compact complex surface S, whose fundamental 2-form F satisfies d α d c α F = 0 for some closed but not exact 1-form α. Then, the flat holomorphic line bundle L a determined by a = [α] ∈ H 1 dR (M) via (1) and Convention 2.1 satisfies

deg g (L a ) = - 1 2 M ||a g h || 2 g + 1 ψ 2 ||dψ|| 2 g v g < 0, so that H 0 (S, L a ) = {0} for all ≥ 1.
We thus obtain the following Proposition 2.12. Let g be a Gauduchon Hermitian metric on a compact complex sur-

∂L * a (S, L * a ) ∼ = H 2,1 ∂α (S) = {0}. As ∂α ∂ α F = i 2 d α d c α F = 0, it fol- lows that ∂ α F + ∂α β = 0,
d α ω = (∂ α + ∂α )(F + β + β) = ∂ α F + ∂α β + ∂α F + ∂ α β = 0,
which completes the proof.

Proof of Theorem 1.1. Let S = (M, J) be a compact complex surface with odd first Betti number b 1 (M). If the Kodaira dimension of S is non-negative, it follows from [START_REF] Belgun | On the metric structure of non-K ähler complex surfaces[END_REF][START_REF] Tricerri | Some examples of locally conformal K ähler manifolds[END_REF] that S admits a locally conformally K ähler metric. We therefore suppose that the Kodaira dimension of S is negative, that is, S belongs to class VII of the Kodaira list; see for example, [START_REF] Barth | Compact Complex Surfaces[END_REF]. Denote by S 0 the minimal model of S.

We first suppose that the second Betti number of S 0 is greater than or equal to 1, that is, S 0 is in the Kodaira class VII + 0 (for which a complete classification is still to come). In this case, Theorem 1.1 follows from Theorem 1.2, Proposition 2.12, and the following vanishing result. Lemma 2.13. Let S be a compact complex surface with b 1 (S) = 1 and negative Kodaira dimension. Suppose that the second Betti number of its minimal model S 0 is > 0. Then, for any topologically trivial line bundle L ∈ Pic 0 (S), we have

H 2 (S, L) = 0.
Proof. By Serre duality, we have to show that, for any L ∈ Pic 0 (S), H 0 (S, K S ⊗ L) = 0.

Suppose that there exists a non-trivial section σ ∈ H 0 (S, K S ⊗ L). The line bundle K S ⊗ L is not trivial as otherwise K -1 S must be flat, and therefore 0 = c 2 1 (S) = -b 2 (S) = -c 2 (S) (see [START_REF] Barth | Compact Complex Surfaces[END_REF]), which is a contradiction. It follows that σ must vanish along an effective divisor

D with [D] = K S ⊗ L. Therefore, in H 2 (S, Z), 0 = (K S + L -D) • K S = K 2 S -K S • D. ( 13
)
We show the assertion by induction on the number p of blowing ups.

If p = 0, the surface is minimal and by [34, p. 399], an irreducible curve C of D is either

• a rational curve such that C 2 ≤ -2, and therefore, by the adjunction formula,

0 = π(C ) = 1 + K S •C +C 2 2
and K S • C ≥ 0, or

• an elliptic or a rational curve C with a double point. By the same formula,

K S • C + C 2 = 0, hence K S • C ≥ 0,
We deduce that, for any effective divisor D, K S • D ≥ 0. Moreover, as b 2 (S) = -c 1 (S) 2 = -K 2 S > 0, we have a contradiction in [START_REF] Demailly | Numerical characterization of the K ähler cone of a compact K ähler manifold[END_REF].

If p ≥ 1, there is an exceptional curve of the first kind E and let B x : S → Š be the blowing down of E to a point x ∈ Š. Let Ǔ be a ball centred at x and U = B x -1 ( Ǔ ) be a simply connected neighborhood of E. The line bundle L is holomorphically trivial on U and, therefore, the coherent sheaf Ľ = (B x ) * L is in fact a topologically trivial line bundle on Š.

As (B x ) * K S = K Š, any non-trivial section σ ∈ H 0 (S, K S ⊗ L) gives, via the biholomorphism

B x : S \ E ∼ = Š \ {x} a non-trivial section σ ∈ H 0 ( Š \ {x}, Ǩ Š ⊗ Ľ)
. By Hartogs' theorem, this section extends to Š. As b 2 ( Š) ≥ b 2 (S 0 ) > 0, by the induction hypothesis σ = 0, which is a contradiction.

Let us now consider the case when b 2 (S 0 ) = 0. According to [START_REF] Bogomolov | Classification of surfaces of class VII 0 with b 2 = 0[END_REF][START_REF] Li | Hermitian Yang-Mills Connections on non-K ähler Manifolds[END_REF][START_REF] Teleman | Projectively flat surfaces and Bogomolov's theorem on class VII 0 -surfaces[END_REF], S 0 is either an Inoue surface (see [START_REF] Inoue | On surfaces of class VII 0[END_REF]) or a Hopf surface (see [START_REF] Kato | Topology of Hopf surfaces[END_REF]). The arguments in [START_REF] Tricerri | Some examples of locally conformal K ähler manifolds[END_REF][START_REF] Vuletescu | Blowing-up points on locally conformally K ähler manifolds[END_REF] can be used without any change to show that the blow-up S of S 0 admits a locally conformally symplectic form taming J if S 0 does. We thus consider the case S = S 0 .

The Inoue surfaces with second Betti number equal to zero are classified by Inoue [START_REF] Inoue | On surfaces of class VII 0[END_REF] who shows that they do not admit curves. Thus, if S is such a surface, the condition H 2 (S, L) ∼ = H 0 (S, K S ⊗ L * ) = {0} is equivalent to L = K S , where K S denotes the canonical bundle of S. Let g be a Gauduchon metric whose fundamental 2-form F satisfies d α d c α F = 0 for a 1-form α with a = [α] = 0, given by Theorem 1.2. The corresponding line bundle L = L a has negative degree with respect to g by ( 12), whereas it is shown in [START_REF] Teleman | The pseudo-effective cone of a non-K ählerian surface and applications[END_REF]Remark 4.2] that the degree of K S is positive with respect to any Gauduchon metric. Thus, L = K S , showing that H 2 (S, L) = {0}, and therefore S admits a locally conformally symplectic 2-form taming J with Lee form α by Proposition 2.12.

If S is a Hopf surface, Lemma 2.13 generally fails, so we cannot directly use Theorem 1.2 in this case. However, according to [START_REF] Belgun | On the metric structure of non-K ähler complex surfaces[END_REF][START_REF] Gauduchon | Locally conformally K ähler metrics on Hopf surfaces[END_REF], S admits an lcK metric.

Cohomological Invariants of a non-K ähler Complex Manifold

An important and well-studied invariant associated to a compact complex manifold X = (M, J) which admits K ähler metrics is its K ähler cone, K(X), defined to be the subset of classes Ω ∈ H 1,1 dR (X, R) for which there exists a K ähler metric on X whose fundamental 2-form belongs to Ω. A characterization of K(X) in terms of the intersection form of the cohomology ring of X, the Hodge structure, and homology of analytic cycles has been obtained by Buchdahl [START_REF] Buchdahl | On compact K ähler surfaces[END_REF][START_REF] Buchdahl | A Nakai-Moishezon criterion for non-K ähler surfaces[END_REF] and Lamari [30,[START_REF] Lamari | Le cone k ähl érien dune surface[END_REF] when X is a compact complex surface, and Demailly-Paun [START_REF] Demailly | Numerical characterization of the K ähler cone of a compact K ähler manifold[END_REF] in general. These results imply in particular that the classes in

H 1,1
dR (X) which contain symplectic forms taming J coincide with K(X).

In order to introduce similarly designed cohomological invariants in the non-K ähler lcK case, one can consider the sets (see [START_REF] Brunella | Locally conformally K ähler metrics on certain non-K ählerian surfaces[END_REF][START_REF] Ornea | A report on Locally Conformally K ähler Manifolds[END_REF]) as follows.

Definition 3.1. Let X = (M, J) be a compact complex manifold. We introduce the following subsets of H 1 dR (M):

• The subset of classes of Lee forms of lcK metrics:

C(X) = {[α] | ∃F ∈ Ω 1,1 (X), F > 0, d α F = 0}.
• The subset of classes of Lee forms of locally conformally symplectic forms taming J:

T (X) = {[α] | ∃ω ∈ Ω 2 (X), ω 1,1 > 0, d α ω = 0}.
• The subset of classes of the harmonic parts of the Lee forms of Gauduchon metrics

G(X) = {[θ g h ] | ∃F ∈ Ω 1,1 (X), F > 0, dd c F m-1 = 0, d θ F m-1 = 0},
where θ g h denotes the harmonic part of θ with respect to the Riemannian metric g(•, •) = F (•, J•).

Remark 3.2. It follows from the definition that

C(X) ⊆ T (X), C(X) ⊆ G(X),
G(X) is connected while T (X) is invariant under small deformations of X. Using similar techniques as in [START_REF] Goto | On the stability of locally conformal K ähler structures[END_REF], one can show that a ∈ T (X) is an interior point, provided that

H 3 d L * (M, L * ) = {0}
, where L = L a is the real flat line bundle determined by a.

We now consider the blow-up X of X at a point x and denote by B x X → X the blow-down map which is a biholomorphism between X \ E → X \ {x}, where E ∼ = CP m-1

is the exceptional divisor. The following lemma is well-known.

Lemma 3.3. Let B x : X → X be the blow-down map which contracts a divisor E ∼ = CP m-1 ⊂ X with normal bundle N E ∼ = O(-1) to a point x ∈ X. For any a ∈ H 1 dR (X) with generator α ∈ a, denote by α = B * x (α). Then B * x : H k α (X) → H k α ( X) is surjective for any positive k = 2(m -1), and is injective for any positive k = 2m -1.

Proof. As H k

α (X) does not depend on the choice of α ∈ a (see the discussion in Section 2), we can choose α such that it identically vanishes on a open ball U centred at x. It follows that α = B *

x (α) vanishes on Û = B -1 x (U ). We will first prove that B *

x :

H k α (X) → H k α ( X) is surjective. With our choice for α, any d α -closed k-form φ on X is closed over Û . As H k dR ( Û ) ∼ = H k dR (CP m-1 ) = {0} for k = 2 (m -1)
, we can write φ| Û = d( ξ| Û ). Multiplying ξ| Û by the pull-back via B x of a bump function centred at x and support in U , we can assume that ξ is globally defined on X and φ = φd α ξ is another form representing [ φ] ∈ H k α ( X), which vanishes identically on a tubular neighborhood of E. Then, the diffeomorphism (B -1

x ) : X \ E → X \ {x} allows one to define a smooth k-form φ = (B -1

x ) * ( φ) on X with d α φ = 0 and B * x (φ) = φ. We now prove that B *

x :

H k α (X) → H k α ( X) is injective. Suppose that ϕ is a d α -closed k-form on X, such that φ = B * x (ϕ) = d α ξ . As H k dR (U ) = {0}
, we can modify ϕ with a d α -exact form (as we did above with φ) and assume without loss that ϕ |U ≡ 0. It follows that the (k -1)-form ξ satisfies d ξ| Û ≡ 0. If k = 1, ξ is a smooth function on X, which is constant on Û and, therefore, is the pull-back to X of a smooth function ξ on X (which is constant

on U ). It follows that ϕ = d α ξ . If k > 1, H k-1 dR ( Û ) ∼ = H k-1 (CP m-1 ) = {0}
, so we conclude that ξ| Û is exact, that is, ξ Û = dη Û . Multiplying η by a bump-function, we obtain that ξd α η is identically zero in a neighborhood of E, so that it descends to X to define a (k -1)-form

ξ with d α ξ = ϕ.
We recall the following result established in [START_REF] Tricerri | Some examples of locally conformal K ähler manifolds[END_REF][START_REF] Vuletescu | Blowing-up points on locally conformally K ähler manifolds[END_REF]. The argument in [START_REF] Vuletescu | Blowing-up points on locally conformally K ähler manifolds[END_REF] applies without change to the case of locally conformally symplectic structures taming J. Proposition 3.4. Let X be the blow-up of X at a point x.

Then (a) if a ∈ T (X), then â = B * x (a) ∈ T ( X); (b) a ∈ C(X) if and only if â = B * x (a) ∈ C( X).
When X = S is a compact complex surface, the following result is well known.

Proposition 3.5. On a compact complex surface S = (M, J) the following conditions are equivalent:

(i) 0 ∈ C(S), that is, S is K ähler;

(ii) 0 ∈ T (S);

(iii) 0 ∈ G(S); Proof. "(i) ⇒ (ii)" and "(iv) ⇒ (i)" are obvious. In order to prove "(ii) ⇒ (iii)", let ω be a symplectic 2-form on M which tames J. Letting F = ω 1,1 be the positive-definite

(iv) C(S) = T (S) = G(S) = {0}.
(1, 1)-part, it defines a Hermitian metric g(•, •) = F (•, J•) which is Gauduchon (see e.g. Lemma 2.4 with α = 0). Furthermore, ω is a closed self-dual 2-form with respect to g, and is therefore co-closed. Writing ω = F + Ψ , where Ψ is of type (2, 0) + (0, 2), we get

0 = δω = δ F + δΨ = -Jθ g -J(δJ Ψ ),
where J stands for the natural action of the almost-complex structure J on the bundle of real 2-forms of type (2, 0) + (0, 2) by J Ψ (X, Y) := -Ψ (J X, Y). This shows that the Lee form θ of g is co-exact and, therefore, θ [START_REF] Buchdahl | On compact K ähler surfaces[END_REF]30,[START_REF] Siu | Every K3 surface is K ähler[END_REF][START_REF] Todorov | Applications of the K ähler-Einstein-Calabi-Yau metric to moduli of K3 surfaces[END_REF].

In order to prove T (S) = {0}, we use Lemma 2.4, Proposition 2.7, and Lemma 2.8:

as θ g h = 0 in our case, one gets 0 ≤ -M ||a g h || 2 g v g , showing a g h = 0.
Recall the following definition from [START_REF] Ornea | Topology of Locally Conformally K ähler Manifolds with Potential[END_REF].

Definition 3.6. An lcK metric with potential g on X = (M, J) is an lcK metric such that the pull-back F of its fundamental 2-form F to the universal covering space X of X is of the form F = dd c f f , where f > 0 is a positive plurisubharmonic function on X which satisfies γ * f = e c γ f (c γ ∈ R) for any deck-transform γ ∈ π 1 (X).

Examples of lcK metrics with potential include the Vaisman lcK metrics (i.e. lcK metrics for which the Lee form θ is parallel) or more generally, pluricanonical lcK metrics, introduced and studied by Kokarev [START_REF] Kokarev | On pseudo-harmonic maps in conformal geometry[END_REF], for which the covariant derivative Dθ of the Lee form is of type (2, 0) + (0, 2) with respect to J; see [START_REF] Ornea | Topology of Locally Conformally K ähler Manifolds with Potential[END_REF]Claim 3.3;[START_REF] Ornea | Compact pluricanonical manifolds are Vaisman[END_REF]. (In [START_REF] Ornea | Topology of Locally Conformally K ähler Manifolds with Potential[END_REF], the authors claim that an lcK metric admits a potential if and only if it is pluricanonical, but a proof is given only in one direction; see [START_REF] Ornea | Compact pluricanonical manifolds are Vaisman[END_REF] for the precise link between the two notions.) The following observation is taken from [START_REF] Ornea | A report on Locally Conformally K ähler Manifolds[END_REF]. Lemma 3.7. Let X = (M, J) be a compact complex manifold endowed with an lcK metric with potential, g, and Lee form θ . Then, for any t ≥ 1, tθ is the Lee form of an lcK metric with potential on X. If, furthermore, g is a pluricanonical lcK metric, then tθ is the Lee form of a pluricanonical lcK metric for any t > 0.

Proof. Writing F = dd c f f on X, the pull-back of the Lee form is θ = -d f f . For any t ≥ 1 put

Ft := dd c ft ft = t dd c f f + t(t -1) d f ∧ d c f f2 .
For t ≥ 1, Ft defines a positive-definite (1, 1)-form satisfying d Ft = θt ∧ Ft with θt = -t d f f = t θ. As Ft is invariant under any deck transformation, it defines an lcK metric with potential, g t , on X, whose Lee form is θ t = tθ .

It is not hard to see that the pluricanonical condition (D

g X θ)(Y) = -(D g J X θ)(JY) is equivalent to dJθ = -|θ | 2 g F + θ ∧ Jθ. ( 14
)
This is essentially the formula appearing in [36, p. 724], by noting that there is a sign error in [START_REF] Ornea | Topology of Locally Conformally K ähler Manifolds with Potential[END_REF] in deriving the formula for d(I θ) from the previous one, and an omission of a factor |θ | 2 g before g in the formula expressing ∇θ -Dθ ; the precise statement appears in [START_REF] Ornea | Compact pluricanonical manifolds are Vaisman[END_REF]. For convenience of the reader we supply here a brief argument for ( 14): it is well known (see e.g. [27, II, Proposition 4.2]) that when J is integrable

(D X F )(Y, Z ) = -1 2 (dF (X, JY, J Z) -dF (X, Y, Z )).
The lcK condition implies dF = θ ∧ F , which allows one to rewrite the above equality as

D X F = 1 2 (X ∧ Jθ + J X ∧ θ), ( 15
)
where X denotes the g-dual 1-form to X. Thus, using [START_REF] Donaldson | Two-Forms on Four-Manifolds and Elliptic Equations[END_REF],

(D X Jθ)(Y) = -θ((D X J)(Y)) -(D X θ)(JY) = 1 2 (θ (X)Jθ(Y) -Jθ(X)θ (Y) -|θ | 2 g F (X, Y)) -(D X θ)(JY).
It then follows

dJθ = -|θ | 2 g F + θ ∧ Jθ + (J Dθ) anti ,
where (J Dθ) anti denotes the anti-symmetrization of the (2, 0)-tensor (J Dθ)(X, Y) := -Dθ(X, JY); as dθ = 0 (i.e. Dθ is symmetric), J Dθ anti (X, Y) = -(Dθ) 1,1 (X, JY) where

Dθ 1,1 (X, Y) := (D X θ)(Y) + (D J X θ)(JY)
. Thus, for an lcK metric g, equation ( 14) is equivalent to Dθ 1,1 = 0, that is, to g being pluricanonical. Differentiating ( 14) one more time yields

d|θ | 2 g ∧ F = 0,
showing that |θ | 2 g is a constant. Thus

F t := F + t |θ | 2 g θ ∧ Jθ, t > -1
defines a family of positive-definite (1, 1)-forms with dF t = (1 + t)θ ∧ F t . Clearly, F t give rise to a family of lcK metrics which verify the pluricanonical condition [START_REF] Dragomir | Locally Coformal K ähler Geometry[END_REF].

Remark 3.8. If g is a pluricanonical lcK (non-K ähler) metric on X = (M, J) with Lee form θ , writing the pull-back metric as g = e φ gK where gK is a K ähler metric on the universal cover X conformal to g (or, equivalently, writing the pull-back θ of the Lee form as θ = d φ)

gives rise to a potential function f := e -φ for the fundamental 2-form F of g, that is, F = dd c f f by ( 14).

4 Compact Complex Surfaces with b 1 (S) = 1

We start with the following easy consequence of Proposition 3.5. Proof. Let g be a Gauduchon metric on S. By [START_REF] Buchdahl | A Nakai-Moishezon criterion for non-K ähler surfaces[END_REF], deg g (L a ) = 0 for any Gauduchon metric if a = 0. On the other hand, if a = 0, θ g h = μ a (g)a g h for a constant μ a (g), so that by [START_REF] Buchdahl | A Nakai-Moishezon criterion for non-K ähler surfaces[END_REF] again, the sign of deg g (L a ) is equal to the sign of -μ a (g). We know by Proposition 3.5 that μ a (g) = 0. As the space of Gauduchon metrics is convex (and therefore connected), it follows that the sign of μ a (g) is non-zero and is independent of g.

The above lemma suggests the following definition.

(M) = 1. For a, b ∈ H 1 dR (M) we will say that a > b if deg g (L a ⊗ L * b ) = deg g (L (a-b) ) = deg g (L a ) -deg g (L b ) > 0,
for some (and hence any) Gauduchon metric g.

We will thus identify the ordered set H 1 dR (S) ∼ = (R, >).

Proposition 4.3. Let S = (M, J) be a compact complex surface with b 1 (M) = 1. Then G(S) ⊆ (-∞, 0) and T (S) ⊆ (-∞, 0). Furthermore, for each a ∈ T (S), there exists a class b ≤ a which belongs to G(S).

Proof. Let g be Gauduchon metric on S and θ g h be the harmonic part of the corresponding Lee form θ g . Applying [START_REF] Buchdahl | A Nakai-Moishezon criterion for non-K ähler surfaces[END_REF] for a = [θ g h ] (and using Proposition 3.5) yields deg

g (L a ) = -1 2π M ||θ g h || 2 g v g < 0,
thus showing the first inclusion. The second inclusion follows from Proposition 3.5, and Lemmas 2.4 and 2.11. The inequality M g(θ

g h -a g h , a g h )v g = M ( 1 ψ ||dψ|| 2 g )v g ≥ 0 (see Lemma 2.8
) and the fact that a and b = [θ g h ] are both negative (so that θ

g h = μa g h with μ > 0) show deg g (L a-b ) ≥ 0, that is, the de Rham class b ≤ a. Proposition 4.4. Suppose S = (M, J) is a compact complex surface with b 1 (M) = 1. If b ∈ C(S), c ∈ G(S) with b ≤ c, then any a ∈ [b, c] ⊂ H 1 dR (S)
, such that the corresponding flat line bundle L a satisfies H 2 (S, L a ) = {0}, belongs to T (S).

Proof. Let g be an lcK metric on S whose closed Lee form θ g belongs to b, and suppose (without loss) that g is a Gauduchon metric: thus, the Lee form of g is harmonic, that is,

θ g = b g h . For any a ∈ H 1 dR (M)
, let a g h denote the harmonic representative of a with respect to g: thus, writing

a g h = μb g h for a constant μ, we have a ≥ b iff deg g (L b-a ) = - 1 2π M g(b g h , b g h -a g h )v g = (μ -1) 2π M ||b g h || 2 g v g ≤ 0, that is, iff μ ∈ [0, 1]. Thus, for any a ∈ [b, c], L g,a (ψ) = Δ g ψ + (2μ -1)g(b g h , dψ) + (1 -μ)μ||b g h || 2 g ψ. ( 16
)
We claim that λ a (g) ≥ 0. Indeed,

λ a (g) ≥ min M L g,a (1) = (1 -μ)μ min M ||b g h || 2 g ≥ 0.
Similarly, let g be a Gauduchon metric on S for which the de Rham class c = [ θ g h ] of the harmonic part of its the Lee form satisfies c ≥ b. For any a ≤ c, we have

θ g h = νa g h for a real constant ν ∈ [0, 1]. It then follows that M g( θ g -a g h , a g h )v g = (ν -1) M ||a g h || 2 gv g ≤ 0.
As we saw in the proof of Theorem 1.2, this implies λ a ( g) ≤ 0. Considering the linear path g t = (1t)g + t g, t ∈ [0, 1] of Gauduchon metrics and using the continuity of λ a (g t ) (see Proposition 2.9), one concludes that there exists a Gauduchon metric g on S with λ a (g ) = 0. Our claim then follows from Proposition 2.12.

Examples

In this section, we illustrate the previous discussion on the known compact complex surfaces S in the Kodaira class VII [START_REF] Barth | Compact Complex Surfaces[END_REF]. Note that in this case, b 1 (S) = 1.

Hopf surfaces

These are, by definition, compact complex surfaces with universal covering space C 2 \ {(0, 0)}. It is shown by Kodaira [28] that the fundamental group Γ of such a surface is a finite extension of the infinite cyclic group Z. The list of concrete realizations of Γ as a group of automorphisms of C 2 can be found in [START_REF] Kato | Topology of Hopf surfaces[END_REF], and we summarize this classification in the following rough form: Γ = H γ 0 , where γ 0 denotes the infinite cyclic group generated by the contraction

γ 0 of C 2 γ 0 (z 1 , z 2 ) = (αz 1 + λz m 2 , βz 2 ),
where the complex numbers α, β, λ satisfy 0 < |α| ≤ |β| < 1, λ(α -β m ) = 0 for an integer m ∈ N * . Furthermore, it follows by the classification in [START_REF] Kato | Topology of Hopf surfaces[END_REF] that when λ = 0, H ⊂ U(1) × U( 1) is abelian and commutes with γ 0 . We denote by S α,β,λ = C 2 {(0, 0)}/ γ 0 the corresponding primary Hopf surface and by S α,β,λ;H the further quotient by H , called secondary Hopf surface.

Hopf surfaces with λ = 0 are called diagonal (or, confusingly, of class 1).

Belgun has shown [6, Theorem 1] that any such surface admits a Vaisman lcK metric.

By Lemma 3.7 and Proposition 4.3, in this case we have

T (S) = C(S) = (-∞, 0).
Hopf surfaces with λ = 0 are called resonant (also called, even more confusingly, of class 0). Let S β m ,β,λ;H be a resonant Hopf surface. The analytic family S λ := S β m ,β,λ;H , λ ∈ C has as central fibre the diagonal Hopf surface S 0 = S β m ,β,0;H , while for λ = 0 the surfaces S λ are isomorphic (see [START_REF] Gauduchon | Locally conformally K ähler metrics on Hopf surfaces[END_REF]). As S 0 admits a taming conformally symplectic form with Lee form in any a ∈ (-∞, 0) so does S λ , just by continuity using that the taming condition is open. It follows that T (S λ ) = (-∞, 0).

Similarly, as S 0 admits a Vaisman metric by [START_REF] Belgun | On the metric structure of non-K ähler complex surfaces[END_REF][START_REF] Gauduchon | Locally conformally K ähler metrics on Hopf surfaces[END_REF], it has a Vaisman lcK metric g a with fundamental 2-form F a and Lee form in a for any a ∈ (-∞, 0), by Lemma 3.7.

(Recall that any Vaisman lcK metric is pluricanonical.) As the corresponding H commutes with γ 0 in this case, the potential f for F a can be chosen to be H -invariant (by averaging over H ). By the argument in [START_REF] Gauduchon | Locally conformally K ähler metrics on Hopf surfaces[END_REF] , the f can be deformed to define an Hinvariant potential for an lcK metric on S λ , for λ small enough, with the same constant The Inoue surfaces S 0 with second Betti number equal to zero are classified by Inoue [START_REF] Inoue | On surfaces of class VII 0[END_REF] into three types, S M , S - N; p,q,r , and S + N; p,q,r;u , where the parameters M, N are matrices with integer coefficients, p, q, r are integers, and u is a complex number. Any such surface is the quotient of C × H (where H denotes the upper half-plane in C) under a discrete subgroup Γ of the group A(2, C) of affine transformations of C 2 , leaving C × H invariant.

The specific description of Γ in each case is given in [START_REF] Inoue | On surfaces of class VII 0[END_REF], but we shall not make use of this. The relevant information for the discussion below is the fact, shown by Tricerri [START_REF] Tricerri | Some examples of locally conformal K ähler manifolds[END_REF], that all Inoue surfaces admit lcK metrics, except the surfaces of the type S u := S + N; p,q,r;u for which the complex parameter u is not real. In the latter case, Belgun [START_REF] Belgun | On the metric structure of non-K ähler complex surfaces[END_REF] shows that there are no lcK metrics at all. Nevertheless, we proved in Theorem 1.1 that S u always admits timing locally conformally symplectic structures.

Let us now consider in a little more detail the analytic family S u , u∈ C \ R of Inoue surfaces of the third type. We will show, by using an argument from [START_REF] Belgun | On the metric structure of non-K ähler complex surfaces[END_REF], that in this case T (S u ) is a single point.

It is known [START_REF] Belgun | On the metric structure of non-K ähler complex surfaces[END_REF]p. 35;[START_REF] Hasegawa | Complex and K ähler structures on compact solvmanifolds[END_REF]Theorem 1;[START_REF] Wall | Geometric structures on compact complex analytic surfaces[END_REF] that S u = (C × H)/Γ u where Γ u is a lattice in the solvable Lie group

Sol 4 1 = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ⎛ ⎜ ⎜ ⎝ 1 a b+ i log γ 0 γ c 0 0 1 ⎞ ⎟ ⎟ ⎠ , γ > 0, a, b, c ∈ R ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ ⊂ GL(3, C).
The group Sol 4 1 acts itself simply transitively (and holomorphically) on C × H. This allows one to identify S u with the quotient Sol 4 1 , endowed with a (fixed) left-invariant

Y = ⎛ ⎜ ⎜ ⎝ 0 1 0 0 0 0 0 0 0 ⎞ ⎟ ⎟ ⎠ Z = ⎛ ⎜ ⎜ ⎝ 0 0 1 0 0 0 0 0 0 ⎞ ⎟ ⎟ ⎠ T = ⎛ ⎜ ⎜ ⎝ 0 0 i 0 1 0 0 0 0 ⎞ ⎟ ⎟ ⎠ U = ⎛ ⎜ ⎜ ⎝ 0 0 0 0 0 1 0 0 0 ⎞ ⎟ ⎟ ⎠
be the generators of the Lie algebra of Sol 4 1 with

Z central, [Y, T] = Y, [T, U ] = U, [Y, U ] = Z
and denote with the same letters the induced left-invariant vector fields on Sol 4 1 . Then the left-invariant complex structure J on Sol 4 1 is given by [6, (35)]

JY = -Z , J Z = Y, JT = -U -Z , JU = T -Y.
Furthermore, Sol 4 1 admits an ad-invariant 1-form α 0 , defined by α 0 (T) = 1, α 0 (U ) = α 0 (Y) = α 0 (Z ) = 0, which descends to define a closed but not exact 1-form (still denoted byα 0 ) on S u . As b 1 (S u ) = 1, by Remark 2.3 we can assume without loss that S u admits a locally conformally symplectic 2-form F which tames J, and whose Lee form equals kα 0 for a non-zero real constant k. As Sol 4 1 has a bi-invariant volume form v [20, Lemma 4] (which defines a volume form on the quotient S u , still denoted by v), for any left-invariant vector fields U and V on Sol 4 1 (which define vector fields on the quotient S u , still denoted by U and V), one can consider the average of F over S u :

F 0 (U, V) := S u F (U, V)v.
It can be shown, as in the proof of [START_REF] Belgun | On the metric structure of non-K ähler complex surfaces[END_REF]Theorem 7], that F 0 defines a left-invariant 2-form on (Sol 4 1 , J) which tames J and satisfies dF 0 = kα 0 ∧ F 0 . Evaluating the later equality over Y, Z , T yields (see also [6, (36) [START_REF] Goto | On the stability of locally conformal K ähler structures[END_REF]Sect. 7.1]), using the explicit description of S u of [START_REF] Inoue | On surfaces of class VII 0[END_REF] and the isomorphism C * = H 1 (S, C * ) ∼ = Pic 0 (S) of [28, I, p. 756], that the corresponding holomorphic line bundle L a 0 is isomorphic to the anticanonical line bundle K * S u . Noting finally that the existence of a locally conformally symplectic structure taming J is an open condition (with respect to J) and that the Inoue surfaces of the type S - N, p,q,r,s are quotients of Inoue surfaces of the type S + N, p,q,r;0 by an involution [START_REF] Inoue | On surfaces of class VII 0[END_REF], we obtain the following Proposition 5.2. Let S be an Inoue surface with b 2 (S) = 0 in one of the types S + N, p,q,r;u or S - N; p,q,r . Then C(S) and T (S) are given as follows:

]) F 0 (Y, Z ) = kF (Y, Z ); as Y = J Z and the (1, 1)- part of F 0 is positive-definite, it follows that k = 1. This shows that T (S u ) = {a 0 } with a 0 = [α 0 ]. It is easy to check (see
• C(S) = T (S) = {a 0 } iff S is of the type S - N; p,q,r or S + N, p,q,r;u with u∈ R; • C(S) = ∅, T (S) = {a 0 } iff S is of the type S + N, p,q,r;u with u∈ C \ R,

where L a 0 = K * S u .

Kato surfaces

These are minimal complex surfaces in the Kodaira class VII whose second Betti number is strictly greater than 0, and which have a global spherical shell (GSS). Conjecturally, any minimal surface in the class VII should be either a Hopf surface, an Inoue surface, or a Kato surface. However, this conjecture is still far from being solved.

For Kato complex surfaces, Brunella [START_REF] Brunella | Locally conformally K ähler metrics on certain non-K ählerian surfaces[END_REF][START_REF] Brunella | Locally conformally K ähler metrics on Kato surfaces[END_REF] has shown that C(S) = ∅ and that C(S) has -∞ as an accumulation point. Note that any Kato surface S is diffeomorphic to (S 1 × S 3 ) kCP 2 (where k := b 2 (S)); see for example, [START_REF] Nakamura | On surfaces of class V I I 0 with curves[END_REF]. As S 1 × S 3 , with a complex structure of a Hopf surface of class 1, admits a Vaisman metric, it follows by [START_REF] De L Éon | On the computation of the Lichnerowicz-Jacobi cohomology[END_REF][START_REF] Ornea | Morse-Novikov cohomology of locally conformally K ähler manifolds[END_REF] that Further progress in this case seems to depend on a better understanding of the subset G(S), as the following result suggests. the Perron-type Theorem mentioned above to the case of a compact manifold without boundary (as we failed to find a reference to this result in the literature).

H 3 d L (S 1 × S 3 , L) = {0}
Theorem A.1. Let L(u) = Δu + g(α, du) + cu be a linear strongly elliptic linear differential operator of order 2 on a compact Riemannian manifold (M, g). Then, there exists a real eigenvalue λ 0 for L which admits a smooth everywhere positive eigenfunction u 0 . Furthermore, λ 0 satisfies the following properties: Proof of Theorem A.1. The proof will be divided into four steps, corresponding to the statements in the Theorem A.1 as follows.

Step 1. We shall establish in this Step the existence of a real eigenvalue λ 0 corresponding to an everywhere positive smooth eigenfunction u 0 . To this end, we shall work by [START_REF] Hormander | The Analysis of Linear Partial Differential Operators III[END_REF]Cor. 19.1.8] and, as Ker L = {0}, L is invertible and bounded (by the standard L p -estimates, see e.g. [7, p. 463]). Using the compactness of W 2 k+1 (M) ⊂ W 2 k (M) again, we conclude that the composition We are now going to show that, for any > 0, the closed subset of W 2 k (M) S := {u∈ C | ∃λ, 0 ≤ λ ≤ 2μ, u= λA(u + w)}

A : W 2 k (M) L -1 -→ W 2 k+2 ⊂ W 2 k (M)

Theorem 1 . 4 .2Convention 2 . 1 .

 1421 Let S be a compact complex surface with a minimal model S 0 .(i) If S 0 is a Hopf surface, then T (S) = C(S) = (-∞, 0). (ii) If S isan Inoue surface of the type S + N, p,q,r;u with u∈ C \ R, then C(S) = ∅ and T (S) = {a 0 }. (iii) If S is an Inoue surface of the type S + N, p,q,r;u with u∈ R, then C(S) = T (S) = {a 0 }, where a 0 ∈ H 1 dR (S) denotes the de Rham class for which the holomorphic line bundle determined by (1) is isomorphic to the anti-canonical line bundle K * S . Existence of Locally Conformally Symplectic Forms Taming the Complex Structure Throughout the paper, we shall use the following convention. Let α be a closed 1-form on M, representing a de Rham class a = [α].

  defines a nowhere vanishing smooth parallel (respectively, holomorphic) section of L α (respectively, L α ). In terms of (1), L α represents the isomorphism class of flat holomorphic line bundles determined by the deRham class a = [α].

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2 t

 2 g. [4, III, Section 8, 3.71]) λ(0) = 0 and u 0 = 1/vol g (M). It thus follows that the function t → M du t || 2 g u

  face S, whose fundamental 2-form F satisfies d α d c α F = 0 for some closed but not exact 1-form α. If the holomorphic flat bundle L a corresponding to the deRham class a = [α] via (1) and Convention 2.1 satisfies H 2 (S, L a ) = {0}, then there exists a locally conformally symplectic 2-form ω on S with Lee form α and whose (1, 1)-part is F . Proof. As H 0 (S, L a ) = {0} by Lemma 2.11, the hypothesis H 2 (S, L a ) = {0}, Lemma 2.10 and the isomorphism (4) imply H 2,1

  by guest on November 10, 2015 http://imrn.oxfordjournals.org/ Downloaded from Locally Conformally Symplectic Structures 11 for a (2, 0)-form β. Letting ω = F + β + β, one has

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g h = 0 .

 0 It remains to establish "(iii) ⇒ (iv)". To this end, by [3, Proposition 1], b 1 (M) is even and G(S) = {0}. It follows that C(S) = {0} as C(S) ⊆ G(S) and C(S) = ∅ by the characterization of K ähler surfaces

Lemma 4 . 1 .

 41 Let S = (M, J) be a compact complex surface with b 1 (M) = 1. For any a ∈ H 1 dR (M), denote by L a the corresponding flat holomorphic line bundle defined via (1) and Convention 2.1. Then, the sign of deg g (L a ) does not depend on the choice of a Gauduchon metric on S and is zero if and only if a = 0.

Definition 4 . 2 .

 42 Let S = (M, J) be a compact complex surface with b 1

c γ 0 Proposition 5 . 1 . 5 . 2

 05152 (see Definition 3.6). Using the isomorphism C * = H 1 (S, C * ) ∼ = Pic 0 (S) established for class VII surfaces in [28, I, p. 756], this shows that the induced lcK metrics on S λ will have Lee forms in the same de Rham class a. We thus see that C(S λ ) = (-∞, 0) too. Using Proposition 3.4 (and Lemma 3.3 with a = 0 and k = 1), we conclude the following proposition. For any compact complex surface S whose minimal model is a Hopf surface, T (S) = C(S) = (-∞, 0). Inoue surfaces with b 2 = 0

  for any non-trivial flat real line bundle L. By Lemma 3.3, H 3 d L (S, L) = {0} and, therefore, by Remark 3.2, T (S) must be an open subset of (-∞, 0). Similar conclusion holds true for C(S) ⊆ T (S), by [20] and Lemma 2.10, together with the vanishing of H 2 (S, L) and H 0 (S, L) established in Lemmas 2.13 and 2.4 and 2.11, respectively.

Proposition 5 . 3 .

 53 Let S be a Kato surface. Then,G(S) ⊆ T (S).Proof. Let c ∈ G(S).Brunella shows (see [9, Remark 9]) that there exists an lcK metric whose Lee form defines a de Rham class b with b < c. By Proposition 4.4 and Lemma 2.13, any a ∈ [b, c] (in particular a = c) belongs to T (S).

Funding

  V.A. was supported in part by an NSERC Discovery Grant; V.A. and G.D. were supported in part by the ANT Grant MNGNK-ANR-10-BLAN-0118. by guest on November 10, 2015 http://imrn.oxfordjournals.org/ Downloaded from Locally Conformally Symplectic Structures 23

Definition A. 2 .

 2 (i) λ 0 is of multiplicity one. (ii) If λ is another eigenvalue, then Re(λ) > λ 0 . (iii) λ 0 = sup u∈A (inf x∈M L(u) u ), where A = {u∈ C ∞ (M) | u> 0}.The eigenvalue λ 0 is called the principal eigenvalue of L.

2 ,

 2 with the Sobolov spaces W 2 k (M) corresponding to the norm where |D j f| is the point-wise norm of the jth covariant derivative and v g is the Riemannian volume form. In what follows, we shall choose k sufficiently large so that we have a continuous embeddingW 2 k (M) ⊂ C 2 (M).Because M is compact, we also have the continuous embeddingsW 2 k (M) ⊂ W 2 l (M),for any integers 0 ≤ l ≤ k.We shall assume (without loss) in what follows that the smooth function c ≥ 0 (otherwise we consider the operator L -(inf M c)Id instead of L). Then, by the maximum principle [4, III, Section 8, 3.71] (which in the sequel we shall always apply to -L and -u), 0 is not an eigenvalue of L, that is, KerL = {0}. It is a standard fact that L : W 2 k+2 (M) → W 2 k (M) is then invertible, and L -1 : W 2 k (M) → W 2 k+2 (M) is bounded. Indeed, as L = Δ + T with Δ being the self-adjoint Riemannian Laplacian and T of order ≤ 1, the composition T : W 2 k+2 (M) → W 2 k+1 (M) ⊂ W 2 k (M) is a compact operator by the Rellich-Kondrachov theorem (see [7, p. 458]); it follows that Index(L) = Index(Δ) = 0 by guest on November 10, 2015 http://imrn.oxfordjournals.org/ Downloaded from Locally Conformally Symplectic Structures 25

  is a compact operator. We define the cone C := {u∈ W 2 k (M) | u≥ 0}. (A.1) By the maximum principle [4, III, Section 8, 3.71] again, we have that A(C ) ⊂ C and if w ∈ C with w ≡ 0, then Aw > 0. Using standard elliptic regularity (see e.g. [7, p. 467]), it will be enough to show that A has a non-trivial eigenfunction w 0 ∈ C , corresponding to a real positive eigenvalue 1/λ 0 (we will have then that u 0 := λ 0 A(w 0 ) is a smooth strictly positive eigenfunction of L, corresponding to the real positive eigenvalue λ 0 ). Let us fix a function w ∈ C , w ≡ 0, and let v := Aw. By continuity, there exists μ > 0 such that μv ≥ w on M. We claim that if the equation u= λA(u + w) with > 0 and λ > 0 has an everywhere positive solution u, then necessarily λ ≤ μ. (A.2) Indeed, u= λA(u) + λA( w) > λA( w) = λ v ≥ only if λ ≤ μ.

by guest on November 10, 2015 http://imrn.oxfordjournals.org/ Downloaded from

Acknowledgments

V.A. is grateful to the Universit é Aix-Marseille and the Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences where a part of this project was realized. G.D. is grateful to the UQAM for their hospitality during the preparation of the work. The authors are very grateful to Dimiter Vassilev for his invaluable help with the spectral theory of strongly elliptic operators, and to Dimitry Jakobson, Franc ¸ois Hamel, and Fr éderic Rochon for sharing with us their expertise on spectral analysis and the theory of elliptic PDE's. The authors are also grateful to Liviu Ornea and Misha Verbitsky for clarifying their results and giving them an access to the unpublished work [39]. They also thank the referee for her/his valuable suggestions on improving the presentation.

Appendix A Perron-Type Theorem for Second-Order Strongly Elliptic Linear

Operators on a Compact Riemannian Manifold

We review here some spectral properties of the second-order strongly elliptic linear operators in the form

where g is a Riemannian metric on a compact manifold M, Δ g = δ g d is the corresponding Riemannian Laplacian (which we will consequently denote by Δ), α is a given smooth 1-form and c a given smooth function on M. Note that L need not to be self-adjoint in general, a case where the spectral theory is well-established.

In a local chart U , L takes the form

where

) is uniformly positive-definite on U . In this case, the Perron-type Theorem established in [16, p. 360; 35] states that if Ω ⊂ U is compactly supported domain with smooth boundary, the operator L U taken on smooth functions on Ω with zero boundary value has real eigenvalue λ 0 (called principal eigenvalue) such that, for any other eigenvalue λ, Re(λ) ≥ λ 0 ; furthermore, λ 0 is simple and the corresponding eigenspace is generated by a nowhere vanishing smooth function u 0 on Ω.

Recall that, for any strongly elliptic linear operator L : C ∞ (M) → C ∞ (M), the set of (complex) eigenvalues is discrete, having a limit point only at infinity; see for example, [7, p. 465] or [4, p. 126]. We want to establish the following adaptation of by guest on November 10, 2015 http://imrn.oxfordjournals.org/ Downloaded from is unbounded. To this end, we shall use (A.2) in conjunction with the well-known Schaefer theorem (see [16, p. 540]), which states that if à : W → W is a continuous compact (not necessarily linear) mapping of a Banach space W and C ⊂ W is a convex subset stable by Ã, then à has a fixed point in C provided that the subset

which is continuous compact because A is, and

is the convex subset introduced in (A.1). Furthermore, if S were bounded, à would satisfy the hypothesis of the Schaefer theorem with respect to C , so à would have a fixed point u in C , that is,

Since S is not bounded, there exists u ∈ S with u k ≥ 1 . Denote by λ (0 ≤ λ ≤ 2μ) the corresponding real number such that

For any sequence m → 0 let λ m := λ m and u m :=

Since A is compact, we can take a subsequence if necessary and suppose that (λ m ) and

(u m ) are convergent. Let λ 0 := lim m→∞ λ m and u 0 := lim m→∞ u m ≥ 0. Taking limit in (A.3), we get

The condition u m k = 1 prevents λ 0 and u 0 from vanishing. Thus, u 0 = λ 0 A(u 0 ) > 0, and applying L to the above equality, we obtain L(u 0 ) = λ 0 u 0 with λ 0 > 0.

Step 2. We will now prove that the multiplicity of λ 0 is one.

As L is a real operator, it is enough to consider a smooth non-identically zero real valued function u on M, such that Lu= λ 0 u. Furthermore, replacing u by -u if necessary, we shall assume that u is somewhere positive. Thus, letting

we then have χ > 0, v := u 0 -χ u≥ 0 on M, and hence Lv = λ 0 v ≥ 0. By the Hopf maximum principle [4, III, Section 8, 3.71], we have that either v > 0 on M or v ≡ 0. By the definition of χ we conclude that v ≡ 0.

Step 3. We now show that, for any other (complex) eigenvalue λ of L, Re(λ) > λ 0 .

Let u ≡ 0 be a complex-valued smooth function on M with Lu= λu and let v := u/u 0 . We then have

where we have set

Dividing the last equality by u 0 , we have

and, as K is a real operator,

It follows

Suppose for a contradiction that Re(λ) -λ 0 ≤ 0. Then, the Hopf maximum principle [4, III, Section 8, 3.71] implies |v| = const, and therefore

showing that Re(λ) = λ 0 (as v = u/u 0 ≡ 0). Going back to (A.4) (and using again K(|v| 2 ) = 0), we have

that is, v = u/u 0 is a constant. As λ = λ 0 by assumption, this is a contradiction.

Step 4. We finally have to prove that To this end, we will show first that the formal conjugate operator

of L (with respect to the global L 2 product on (M, g)) has the same principal eigenvalue as L. Indeed, let λ * 0 be the principal eigenvalue of L * and u * 0 > 0 be an eigenfunction. Then

Since u 0 > 0 and u * 0 > 0, it follows that λ 0 = λ * 0 . Now, let

We clearly have μ ≥ inf x∈M L(u 0 ) u 0 = λ 0 . To establish the converse inequality, let (w m ) be a maximizing sequence in A such that

The following result follows from the general Rellich-Kato theory, see for example, [26, II, Section 1.8; IV, Theorem 3.16, and VII, Theorem 1.7] applied to the bounded operator A constructed in the proof of Theorem A.1, by noting that the principal eigenvalue λ 0 of L is simple and can be separated from the remainder of the spectrum.

Theorem A.3. Let L(t) be an analytic family of linear strongly elliptic operators as in Theorem A.1. For each t, denote by λ 0 (t) the principal eigenvalue of L(t) with corresponding eigenfunction u t > 0, normalized by M u 2 t v g = 1. Then λ 0 (t) and u t vary analytically with respect to t.


Introduction: Polynomial approximations are widely used in digital systems for function evaluation (e.g. 1/x, √ x, sin, cos, exp, log). In some digital signal processing applications, such as frequency demodulation, low-degree polynomials are often used for evaluating reciprocals. In hardware implementation of polynomial approximations, the size of the multipliers is a major concern. Several solutions have been investigated to limit their size: argument reduction and series expansions in [START_REF] Ercegovac | Reciprocation, square root, inverse square root, and some elementary functions using small multipliers[END_REF], small table and a modified multiplication in [START_REF] Takagi | Powering by a table look-up and a multiplication with operand modification[END_REF], or the multipartite tables method [START_REF] Schulte | Approximating elementary functions with symmetric bipartite tables[END_REF][START_REF] De Dinechin | Multipartite table methods[END_REF].

In this work we focus on polynomial approximations with sparse coefficients (i.e.

the multiplier operand can be written so that it contains predefined strings of bits stuck at 0). Sparse coefficients allow us to replace the complete reduction tree of the multipliers by smaller ones. This letter is an improved and extended version of the paper presented in [START_REF] Brisebarre | Sparse-coefficient polynomial approximations for hardware implementations[END_REF] and uses a new recursive coefficient filtering.

Background: When evaluating a function f on a real interval [a, b], one usually uses polynomial approximations, such as minimax approximations which minimize the distance

||p -f || ∞,[a,b] = sup a≤x≤b |p(x) -f (x)|,
where p ∈ R d [X], the set of polynomials with real coefficients and degree at most d. Minimax approximations, that can be computed thanks to an algorithm due to Remez have a major drawback: in most cases, their coefficients are not exactly representable using a finite number of bits. In [START_REF] Brisebarre | Computing machine-efficient polynomial approximations[END_REF], the authors propose an efficient method for computing a polynomial which minimizes the distance

||p -f || ∞,[a,b]
among the polynomials p ∈ R d [X] that fulfill some given constraints on the format of the coefficients. The result polynomials q are of the form:

q(x) = q 0 2 m 0 + q 1 2 m 1 x + q 2 2 m 2 x 2 + . . . + q d 2 m d x d .
The degree d and the integer sequence m 0 , . . . , m d are input parameters of the method. The coefficients are such that q i ∈ Z. The method presented in [START_REF] Brisebarre | Computing machine-efficient polynomial approximations[END_REF] provides

result polynomials q such that ||q -f || ∞,[a,b]
is minimal among the polynomials that fulfill the constraints. Those polynomials can be represented as the integer points of a polytope (cf. [START_REF] Brisebarre | Computing machine-efficient polynomial approximations[END_REF]) and efficient scanning of all the polytope points is performed using linear programming tools.

Here, we look for polynomials with sparse coefficients, i.e. there are several bits predefined to 0 as illustrated in Figure 1. Each coefficient q i /2 m i is decomposed into k chunks c i,j with j ∈ {1, . . . , k}. The chunks are small signed s j -bit integers. The number of useful bits in coefficients is S = k j=1 s j . The weight of the chunk c i,j is the value 2 w j as illustrated on Figure 1. Notice that the size and the weight of the chunks are the same for all the coefficients. The value of a coefficient numerator is then q i = k j=1 c i,j × 2 w j . For a given number of chunks k, there are several possible sparse decompositions (the values s j and w j ) with non-overlapping chunks.

Polynomials with sparse coefficients are interesting for circuit implementation as soon as the number of nonzero bits is much smaller than the total format width. This corresponds to:

d i=0 k j=1 s j = (d + 1) × S << d i=0 m i .
Finding Sparse-Coefficient Polynomials: In [START_REF] Brisebarre | Sparse-coefficient polynomial approximations for hardware implementations[END_REF], we proposed two methods to find such sparse-coefficient polynomials. The first method is a very simple filter applied to results from [START_REF] Brisebarre | Computing machine-efficient polynomial approximations[END_REF]. For each candidate polynomial, the filter verifies that all its coefficients can be represented using the target sparse format. This solution is limited to degree-2 polynomials in practice (huge memory requirements and computation times).

The second proposed solution uses a new polytope formulation. Instead of having a d + 1-dimensional polytope, each coefficient introduces k chunks, then the dimension of the polytope is k × (d + 1). Using this solution, up to degree-3 polynomials can be reached.

Our new solution is based on a recursive coefficient filtering strategy. As in the first solution proposed in [START_REF] Brisebarre | Sparse-coefficient polynomial approximations for hardware implementations[END_REF], we build a d + 1-dimensional polytope but we do not scan all its integer points. For each possible sparse decomposition (i.e. the values s j and w j ), we scan only the integer points that fit this decomposition. This last scanning is recursively performed: if coefficient q i can be decomposed into the sparse format then the next dimension corresponding to q i+1 is checked. If not, we scan another decomposition. This new solution leads to efficient programs up to degree-5 polynomials and 6 chunks.

FPGA Implementation Results: Implementations have been done on Xilinx XC3S400-5 FPGAs using ISE7.1i tools. The reported area (number of slices) and delay (ns) are post-P&R values with standard effort.

Approximations to the cosine function on [0, π/4] with 24-bit format for input and output, with a degree-4 polynomial have been implemented. The number of chunks is 2 ≤ k ≤ 6 and number of useful bits in the coefficients is 7 ≤ S ≤ 12.

The obtained operators are compared to the reference implementation with 24-bit full-width standard multiplication. Figure 2 summarizes the results while Table 1 presents the decompositions for which we get an absolute accuracy larger than 18.5 bits. The column named "acc." is the accuracy expressed in number of correct bits.

The last line of this table presents the result for the reference solution.

The results show that significant area and speed improvements are possible using our method. For instance, a 19-bit accuracy for the cosine function on [0, π/4] can be reached using 6 chunks and only 12 useful bits (S) compared to the 24-bit format. The corresponding decomposition is defined by s 6...1 = (3, 2, 2, 2, 1, 2) and w 6...1 = [START_REF] De Dinechin | Multipartite table methods[END_REF]9,12,15,18,22). It corresponds to an operator that requires 39 slices and a delay of 7.3 ns while the standard solution (last line of Table 1) requires 332 slices and 16.2 ns. Hence, our method provides an 8× area reduction and a 2× speed improvement. Notice that due to the huge impact of multipliers on the circuit area, a 2× reduction on the number of useful bits leads to an 8× area reduction.

Conclusion:

This letter presents a new solution for the design of hardware operators dedicated to function evaluation. It uses polynomial approximations with sparse coefficients. The proposed solution improves the results from [START_REF] Brisebarre | Sparse-coefficient polynomial approximations for hardware implementations[END_REF] using an efficient recursive filtering of the polynomial coefficients that are viewed as points of rational polytopes. The results obtained using this solution lead to up to 2× faster and 8× smaller results have been reported compared to standard implementations.

  HARDWARE OPERATORS FOR FUNCTION EVALUATION USING SPARSE-COEFFICIENT POLYNOMIALS N. BRISEBARRE, J.-M. MULLER, A. TISSERAND AND S. TORRES Abstract: This article presents dedicated hardware arithmetic operators for function evaluation. The proposed solution uses polynomial approximations with sparse coefficients which leads to efficient hardware implementations. Up to 2× faster and 8× smaller operators are reported compared to standard implementations.

Fig. 2 :

 2 Figure captions: Fig. 1: 2 and 3-Chunk Decompositions Fig. 2: FPGA Implementation results for cos on [0, π/4]

Figure 2

 2 Figure 1

Table captions :

 captions Tab. 1: Results for cos on [0, π/4]

			m i					m i
	s 2				s 1	s 3		s 2		s 1
	c i,2	0	...	0	c i,1	c i,3	0...0	c i,2	0...0	c i,1
		w 2			w 1	w 3		w 2		w 1

Acknowledgment: This work was partly supported by the French Ministry Grant ACI "New interfaces of mathematics".


Delphine Bresch-Pietri
email: delphine.bresch-pietri@gipsa-lab.fr

Nicolas Petit
email: nicolas.petit@mines-paristech.fr

Prediction-Based Control of Linear Systems by Compensating Input-Dependent Input Delay of Integral-Type

published or not. The documents may come

Introduction

Numerous control systems involve a physical dead-time which proves to be troublesome in the design and tuning of feedback control laws. The dead-time results from the fact that sensors and actuators are rarely co-located, for example in processes that involve the transport of materials, such as mixing processes for liquid or gaseous fluids, chemical reactors [START_REF] Harmand | The optimal design of two interconnected (bio) chemical reactors revisited[END_REF], automotive engine and exhaust lines [START_REF] Depcik | One-dimensional automotive catalyst modeling[END_REF], heat collector plants [START_REF] Sbarciog | Nonlinear predictive control of processes with variable time delay. A temperature control case study[END_REF], networks for blending liquids and solids [START_REF] Chebre | Feedback control and optimization for the production of commercial fuels by blending[END_REF], and batch processes [START_REF] Petit | Motion planning for two classes of nonlinear systems with delays depending on the control[END_REF]. In all of these, and also in the crushing mill system described by [START_REF] Richard | Time-delay systems: an overview of some recent advances and open problems[END_REF], the lag directly depends on the control variable and is inherently input-dependent.

It seems that the stabilization of such processes with input-dependent time-delay in the input D(u) or D(u t ), where u t denotes past values over a finite horizon, has seldom been theoretically studied. Rather, a widely considered approach is to recast the delay dependence on the control, e.g., by modeling this dependence as D(u) ≈ D(t) (or even by a constant average value D) and expecting the controller to deal with a certain level of unstructured model variability. The advantage of such an approach is that it enables the use of a wide range of technical tools previously developed to deal with disturbances. However, recasting this input-dependency into the more general class of input-disturbance reduces performance potential. For this reason we propose to explicitly take this dependency into account and to employ a prediction-based controller, in an effort of improving transient performances.

Predictor-based control strategies [START_REF] Artstein | Linear systems with delayed controls: a reduction[END_REF][START_REF] Manitius | Finite spectrum assignment problem for systems with delays[END_REF][START_REF] Smith | A controller to overcome dead time[END_REF], which are state-of-the-art for systems with constant input time-delays (as studied in [START_REF] Bresch-Pietri | Adaptive control scheme for uncertain time-delay systems[END_REF][START_REF] Gu | Survey on recent results in the stability and control of time-delay systems[END_REF][START_REF] Jankovic | Recursive predictor design for linear systems with time delay[END_REF][START_REF] Michiels | Stability and Stabilization of Time-Delay Systems[END_REF][START_REF] Moon | Robust stabilization of uncertain input-delayed systems using reduction method[END_REF] or [START_REF] Richard | Time-delay systems: an overview of some recent advances and open problems[END_REF] and the references therein) are much more complex to apply for time-varying delays. As shown in [START_REF] Nihtila | Finite pole assignment for systems with time-varying input delays[END_REF] or, more recently, in [START_REF] Krstic | Delay Compensation for Nonlinear, Adaptive, and PDE Systems[END_REF], to compensate a time-varying input delay, the prediction has to be calculated over a time window whose length matches the value of the future delay. In other words, future variations in the delay must be predicted. This can be done, for example, when one has access to a given delay model as in [START_REF] Witrant | Stabilisation des Systemes Commandes par Rseaux[END_REF] or when the delay is state-dependent, by carefully predicting the future system state, as proposed in [START_REF] Bekiaris-Liberis | Compensation of state-dependent input delay for nonlinear systems[END_REF].

Yet, when the delay depends on the input, things become very involved. Determining the required prediction horizon is then an implicit issue, which in practice may not be resolvable or even well-posed. This implicit nature is caused by the reciprocal interactions between the (current and past) control values and the delay, yielding a closed-loop dependency.

In order to design such a prediction-based control law, we advocate a two-step method of disrupting the implicit loop, as proposed in our recent works on the topic [START_REF] Bresch-Pietri | Invoking Halanay inequality to conclude on closedloop stability of a process with input-varying delay[END_REF][START_REF] Bresch-Pietri | Sufficient condition for prediction-based stabilization of linear system subject to input-dependent input-delay[END_REF] and establish sufficient conditions for asymptotic stabilization.

In a first step, we considered the input-dependency as a particular form of timevariation. It enables us to then use a robust compensation result for linear systems with time-varying input delay, using the backstepping tools proposed in [START_REF] Krstic | Boundary Control of PDEs: a Course on Backstepping Designs[END_REF] for the analysis of input-delay system stability. Technically, this guarantees stabilization, provided the delay variations are sufficiently small. Second, a sufficient bound on the delay variations was obtained by relating them to the control tracking error, which is analyzed using the asymptotic convergence of delay differential equations (DDE) (Halanay-type inequalities [START_REF] Halanay | Differential Equations: Stability, Oscillations, Time Lags[END_REF]).

The result holds for potentially unstable linear systems of any arbitrary order n ∈ N. Assuming that the kernel defining the integral is a positive polynomial function of the input, our previous result was extended to such a framework. This is the main novelty of this study. As shown, this delay model encompasses a large class of transport processes. For this class of systems, robust compensation is shown to be achievable, provided that the initial conditions are sufficiently close to equilibrium and that the feedback gain is chosen accordingly.

This chapter is organized as follows. In Section 2, the problem is illustrated by a rocket engine system as an motivating example. A prediction-based controller is designed in Section 3. Sufficient conditions for stabilization are derived through a proof of convergence, calling on Halanay-like inequalities arguments in Section 4.

Notation and Definitions. In the following, |.| stands for the usual Euclidean norm, and C 0 (S 1 , S 2 ) denotes the set of all continuous functions on a set S 1 with values into a set S 2 . For a given symmetric matrix, λ (.) and λ (.) stand respectively for minimum and maximum eigenvalues.

Classically, x t refers to the function x t : [-D, 0] → x(t + s) defined by x t (s) = x(t + s) for a given function x and D > 0 and max|x t | = max{|x t (s)| : s ∈ [-D, 0]}.

For any bounded function k defined on [-D, 0] and any polynomial function π, we write π(x t ) = x(t 1 ), . . . , x(t n-2 ),

t n t n-1 k(t -s)x(s)ds (1)
for (t 1 , . . . ,t n ) ∈ [t -D,t] n . We also refer to π(x t ) as a polynomial function in x t .

A polynomial function π in the variables (x 1 , . . . , x n , x n+1 ) is said to be at least quadratic in x 1 , . . . , x n if, for any given x n+1 , the corresponding polynomial function π x n+1 defined as

π x n+1 (x 1 , . . . , x n ) =π(x 1 , . . . , x n , x n+1 )
has no terms of order 0 or 1, e.g., π = x 2 1 + x 1 x 2 x 3 and π = x 2 x 1 + x 3 x 2 1 are both at least quadratic in (x 1 , x 2 ) while π = x 3 + x 3 x 2 2 is not.

Problem Statement

Consider the following potentially unstable linear time-invariant plant sketched in Fig. 1:

x (n) (t) + a n-1 x (n-1) (t) + . . . + a 1 ẋ(t) + a 0 x(t) = b 0 u(t -D(t)) (2)
driven by a delayed input, where the varying delay D(t) is implicitly defined in terms of the input history by

t t-D(t) ϕ(u(s))ds = 1 , (3)
in which ϕ : R → [ϕ, ∞) is a polynomial function with positive values (i.e., ϕ > 0). Physically, the integral equation ( 3) corresponds to a plug-flow assumption [START_REF] Perry | Perry's Chemical Engineers' Handbook[END_REF] in a transport phenomenon and appears in a large class of applications. The variable ϕ is a (normalized) flow rate, which depends polynomially on the system input. Since ϕ ≥ ϕ > 0, this transport delay is well defined 1 and is upper-bounded, as follows:

1

The delay is positive and, besides, its derivative can be expressed as

Ḋ(t) = 1 -ϕ(u(t))/(ϕ(u(t -D))) < 1 which guarantees strict causality. Process Transport delay t t-D(t) ϕ(u(s))ds = 1 u(t) u(t -D)
Fig. 1 The addressed problem, where the input is delayed by a transport delay which is inputvarying.

D(t) ≤ D = 1/ϕ for all t ≥ 0. (4)
In particular, as

D → t t-D ϕ(u(s))ds (5)
is strictly increasing, it is invertible and the delay can be calculated and so is assumed to be known in the following.

Motivating Example Consider a liquid non-hypergolic propellant rocket such as the one pictured on Fig. 2. The fuel and oxidizer are stored in separate tanks, blended and then fed through a system of pipes, valves, and turbopumps to a com- bustion chamber, where they are burned to generate thrust. As the mixture is nonhypergolic, combustion is started by an ignition system in the chamber. Compared to more classical setups (with dual pumps and pipes convecting the tanks directly to the combustion chamber), this architecture has been chosen for cost reasons, as it uses only one pipe, one turbopump and one choke.

After being premixed downstream of the tanks, the mixture flows to the combustion chamber, which generates a transport delay satisfying

t t-D(t) Q(s) V P ds =1 , (6)
in which V P is the pipe holdup and

Q(s) = Q 1 + Q 2 (s)
is the total volume flow rate after mixing, i.e., the sum of the (constant) outlet fuel flow rate and the comburant flow rate. Following Newton's law, the (longitudinal) dynamics of the rocket can therefore be written as

ẍ(t) =η(t)F 0 ,
in which F 0 is the optimal thrust, corresponding to a stoichiometric mixture, and η is the combustion efficiency. The latter is significantly nonlinear at high flow-rate ratios, which vanishes for low ratios [START_REF] Sutton | Rocket Propulsion Elements[END_REF]. Hence, it can reasonably be modeled as

η(t) = η 0 Q 2 (t -D(t))
for example (without loss of generality), with the oxygen volume flow rate Q 2 (t -D(t) entering the combustion chamber. Therefore, accounting for (6) and setting

u(t) = η 0 F 0 Q 2 (t), (7)
one obtains

     ẍ(t) = u(t -D(t)) t t-D(t) 1 V P Q 1 + u(s) 2 η 2 0 F 2 0 ds = 1 which is indeed in the form (2)-(3) with ϕ(u(s)) = 1 V P Q 1 + u(s) 2 η 2 0 F 2 0 ( 8
)
being a second-order polynomial function with strictly positive values. More complex polynomial expressions can be considered to account for more detailed fit of experimental data and thrust maps.

Control Design Objective

The control task is to stabilize the plant at any equilibrium point x r such that x r = b 0 /a 0 u r , where u r is the corresponding input. For this purpose, a predictor-based feedback law will be employed here.

With this aim in view, we first formulate a state-space representation of this system as

   Ẋ(t) = AX(t) + Bu(t -D(t)) t t-D(t)
ϕ(u(s))ds = 1

where

A =      0 1 0 . . . . . . 0 0 1 -a 0 -a 1 . . . -a n-1      and B =      0 . . . 0 b 0      . (9)
For the sake of clarity, we assume in the following that the system state X is fully measured. Extending this to output feedback for observable systems is straightforward, following the methodology of [START_REF] Bresch-Pietri | Prediction-based stabilization of linear systems subject to input-dependent input delay of integral-type[END_REF]. To construct a prediction-based control law, we use the following theorem, a proof of which is given in [START_REF] Bresch-Pietri | Invoking Halanay inequality to conclude on closedloop stability of a process with input-varying delay[END_REF] and [START_REF] Bresch-Pietri | Prediction-based feedback control of a class of processes with input-varying delay[END_REF].

Theorem 1 (Step 1: Preliminary result). Consider the closed-loop single input system

Ẋ(t) =AX(t) + Bu(t -D(t)) (10a) u(t) =K e AD(t) X(t) + t t-D(t)
e A(t-s) Bu(s)ds ,

where X ∈ R n , u ∈ R, K is chosen such that A + BK is Hurwitz, and D : R

+ → [0, D] is a time-differentiable function. Define ϒ 0 (t) =|X(t)| 2 + t t-D(t) u(s) 2 ds + D(t) 2 t t-D(t)
u(s)2 ds .

There exists ∆ * (K) ∈ (0, 1) such that, if

| Ḋ(t)| < ∆ * (K) , t ≥ 0 , (11)
then the plant (10a) exponentially converges to the origin, in the sense that there exist R > 0 and ρ > 0 such that ϒ 0 (t) ≤ Rϒ 0 (0)e -ρt for all t ≥ 0.

The prediction controller (10b) is a natural extension from the case of constant delay. It forecasts values of the state over a time window of varying length D(t). Of course, exact compensation of the delay is not achieved with this controller. To do so, one would need to consider a time window whose length exactly matched the value of the future delay, as done in [START_REF] Nihtila | Finite pole assignment for systems with time-varying input delays[END_REF] and [START_REF] Krstic | Delay Compensation for Nonlinear, Adaptive, and PDE Systems[END_REF]. In detail, defining η(t) = t -D(t) and assuming that its inverse exists 2 , exact delay-compensation is obtained form the feedback law U(t) = KX(η -1 (t)). Yet, implementing this relation requires the future variation of the delay to be predicted via η -1 (t), which is not achievable in practice for an input-varying delay.

Equation ( 11) can be interpreted as a condition for achieving robust delay compensation 3 . The essence of this condition is that, if the delay were to vary sufficiently slowly, its current value D(t) used for prediction would remain sufficiently close to its future values, and the corresponding prediction would be accurate enough to guarantee the stabilization of the plant through the feedback loop.

We now focus on an alternative sufficient condition guaranteeing [START_REF] Halanay | Differential Equations: Stability, Oscillations, Time Lags[END_REF] when the delay varies in accordance with the integral relation [START_REF] Bresch-Pietri | Adaptive control scheme for uncertain time-delay systems[END_REF]. For practical control design, this new condition involves the control gain and the initial conditions.

Control Design

In this section, we now establish a sufficient condition guaranteeing [START_REF] Halanay | Differential Equations: Stability, Oscillations, Time Lags[END_REF] when the delay varies in accordance with the integral relation [START_REF] Bresch-Pietri | Adaptive control scheme for uncertain time-delay systems[END_REF]. This new condition is relevant to the initial conditions (and thus, to the choice of control gain).

Theorem 2. Consider the closed-loop system ϕ((u(s))ds = 1 ( 13)

Ẋ(t) = AX(t) + Bu(t -D(t)) (12)
u(t) = u r + K e AD(t) X(t) + t t-D(t) e A(t-s) Bu(s)ds -X r . ( 14
)
where ϕ : R → [ϕ, ∞) is a polynomial function, with ϕ > 0 constant, A and B are defined in [START_REF] Depcik | One-dimensional automotive catalyst modeling[END_REF], K is such that A + BK is Hurwitz, u is scalar, X r is the state equilibrium corresponding to the original equilibrium x r of the plant (2) and u r is the corresponding (constant) reference control. Consider the functionals

Θ (t) =|X(t) -X r | + max s∈[t-D,t] [u(s) -u r u(s) . . . u (n-1) (s)] T (15)
ϒ (t) =|X(t) -X r | 2 + t t-D(t) (u(s) -u r ) 2 ds + D(t) 2 t t-D(t) u(s) 2 ds . ( 16
)
Provided that u 0 ∈ C n ([-D, 0], R), there exists θ : R n → R + such that, if Θ (0) < θ (K), then condition (11) is fulfilled and the plant exponentially converges to X r in the sense that there exist R > 0 and ρ > 0 such that ϒ (t) ≤ Rϒ (0)e -ρt for all t ≥ 0.

This result has a relatively direct interpretation: the previously presented Theorem 1 requires the delay to vary sufficiently slowly, while on the other hand, the delay variations implicitly depend on the control input through the integral equation (3), with variations whose rapid swings are scaled by the gain K. Therefore, it would seem to be a natural requirement to restrict input variations by choosing initial conditions sufficiently near the desired equilibrium and which comply with the magnitude of the feedback gain.

The behavior of θ with respect to K must be investigated in future work. From the expression (31), this would involve a study of the solution of the Lyapunov equation and the behavior of its eigenvalues with respect to K. One would expect that θ (K) → 0 while K → -∞ (since r(K) is expected in this case to tend to ∞). For this reason, for given initial conditions, the magnitude of the feedback gain should be chosen accordingly. For relatively large initial conditions, this would imply the use of a small feedback gain; therefore, this result can be interpreted as a small-gain condition. We now detail the proof of Theorem 2 .

Proof of Theorem -Halanay-like Inequalities

We prove Theorem 2. Taking a time-derivative of ( 13) and defining the error variable ε = uu r , one gets

Ḋ(t) = ϕ(u(t -D(t))) -ϕ(u(t)) ϕ(u(t -D(t))) = ϕ(ε(t -D(t)) + u r ) -ϕ(ε(t) + u r ) ϕ(u(t -D(t))) .
As ϕ is a polynomial function, it is locally Lipschitz with a constant M(κ) on the interval [-κ, κ] for any positive scalar κ. Then, since ϕ ≥ ϕ, we obtain

Ḋ(t) ≤ M(κ)|ε(t-D(t))+u r -(ε(t)+u r )| ϕ = M(κ)|ε(t-D(t))-ε(t)| ϕ ≤ 2M(κ)max|ε t | ϕ , if max|ε t | ≤ κ.
Hence, for any positive parameter κ > 0, condition ( 11) is satisfied if

∀t ≥ 0 , max|ε t | < min ϕ∆ * (K) 2M(κ) , κ . (17)
This is the condition we now focus on. It yields the analysis of the dynamics of the variable ε. Before beginning this analysis in detail, we recall some well-known stability results for DDE and their extensions.

Halanay Inequality for Delay Equations of Order n ≥ 1

We first recall the following result ( [START_REF] Halanay | Differential Equations: Stability, Oscillations, Time Lags[END_REF], [START_REF] Ivanov | Halanay inequality, yorke 3/2 stability criterion, and differential equations with maxima[END_REF]) 4 .

Lemma 1. (Halanay inequality) Consider a continuous positive-and real-valued function x such that, for some t 0 ∈ R,

ẋ(t) ≤ -ax(t) + bmax x t , t ≥ t 0 with a ≥ b ≥ 0. Then, there exists γ ≥ 0 such that ∀t ≥ t 0 , x(t) ≤ max x t 0 e -γ(t-t 0 )
Straightforward extensions of this lemma are stated below.

Corollary 1. Consider a positive-valued continuous function such that

ẋ(t) ≤ -ax(t) + bh(t, x t ) , t ≥ t 0 x t 0 = ψ ∈ C 0 ([-D, 0], R + ) , (18)
where h is a continuous functional satisfying the sup-norm relation

h(t, x t ) ≤ max|x t | , for max|x t | < ω , (19)
for a given ω > 0. If the initial condition is such that maxψ < ω and if a ≥ b ≥ 0, then there exists γ ≥ 0 (where γ = 0 if a = b and γ > 0 otherwise) such that

∀t ≥ t 0 , x(t) ≤ max x t 0 e -γ(t-t 0 ) . (20)
Proof. Consider ψ such that x is a non-trivial continuous solution5 of (18) which, using [START_REF] Moon | Robust stabilization of uncertain input-delayed systems using reduction method[END_REF], does satisfy the inequality

ẋ(t) ≤ -ax(t) + bmax x t if maxx t < ω .
Following the seminal proof of [START_REF] Halanay | Differential Equations: Stability, Oscillations, Time Lags[END_REF], define y(t) = ke -γ(t-t 0 ) , with k > 0 and γ chosen such that y satisfies the corresponding differential equation6 ẏ(t) =ay(t) + bmax y t , t ≥ t 0 and y t 0 = k . Now, we define the difference z = yx, which is a continuous function, and we are interested in its sign change. We choose k ∈ (max x t 0 , ω), which is a not empty set as max x t 0 = max ψ < ω. This guarantees that z(t) > 0 for t ∈ [t 0 -D,t 0 ]. The function z being continuous, we define

t 1 = inf {t > t 0 |z(t) = 0} ∈ R ∪ {∞} .
Assume that t 1 < ∞. From the definition of t 1 , z(t) > 0 for t ∈ [t 0 ,t 1 ] and, from the analytical expression of y and as both x and y are continuous, x(t) < y(t) < k < ω for t ∈ [t 0 ,t 1 ). Therefore, the following inequality holds ∀t ≤ t 1 , ż(t) ≥ -az(t) + b (max y tmax x t ) .

Then, ż(t 1 ) ≥ max y t 1max x t 1 > 0, by definition of t 1 . Yet, one has

ż(t 1 ) = lim t→t - 1 z(t) -z(t 1 ) t -t 1 = lim t→t - 1 z(t) t -t 1 ≤ 0 as z(t) ≥ 0 on [t 0 ,t 1 ] .
We finally conclude that t 1 = ∞. Then, for t ≥ t 0 and all z(t) > 0 and, for any ε = kmax x t 0 > 0, we get ∀t ≥ t 0 x(t) < (max x t 0 + ε)e -γ(t-t 0 ) , which gives the result.

Lemma 2. [stability of a n th order DDE] Let x be a solution of the n th order DDE

x (n) (t) + α n-1 x (n-1) (t) + . . . + α 0 x(t) = c (t, x t , . . . x (n-1) t ) , t ≥ t 0 X t 0 = ψ ∈ C 0 ([-D, 0], R) with max|ψ| < ω , (21)
where the left-hand side of the differential equation defines a polynomial whose roots have only strictly negative real parts, c > 0, ω > 0 and is a continuous functional. Then, there exist c * > 0 and r * > 0 (r * = 1 and c * = α 0 if n = 1) such that, if

• c ≤ c * ;
• satisfies the following sup norm relation, with X = [x ẋ . . . x (n-1) ] T ,

| (t, x t , . . . , x (n-1) t )| ≤ max|X t | , for t ≥ t 0 , max|X t | < rω with r ≥ r * ,
then, there exists γ ≥ 0 such that |X(t)| ≤ r * max|X t 0 |e -γ(t-t 0 ) holds for all t ≥ 0.

Proof. Using the scalar result of Corollary 1, define the scalar positive valued function m(t) = X(t) T PX(t) where P is, as defined in the statement of the Lemma, the symmetric positive definite matrix solution of the Lyapunov equation A T 0 P + PA 0 = -Q, for some given symmetric positive definite matrix Q and

A 0 =      0 1 . . . . . . 0 1 -α 0 -α 1 . . . -α n-1 .     
is the companion matrix. Taking a time-derivative of m, we obtain

ṁ(t) = -X(t) T QX(t) + 2X(t) T P      0 . . . 0 c (t, x t , . . . , x (n-1) t ) .      Therefore, defining a ∆ = λ (Q) λ (P) and b ∆ = 2c λ (P) λ (P) , one obtains ṁ(t) ≤ -am(t) + bh(t), where (22)
h(t) = λ (P)

λ (P) X(t) T P       0 . . . 0 (t, x t , . . . , x (n-1) t )       ≤ λ (P)|X(t)|| (t, x t , . . . , x (n-1) t )| ≤ λ (P) m(t)| (t, x t , . . . , x (n-1) t )|, (23)
which satisfies the following for max|X t | < rω (and therefore, in particular, for max m t < r 2 ω 2 λ (P)):

|h(t)| ≤ m(t)max √ m t ≤ max m t . Define the condition max m t 0 < r 2 ω 2 λ (P) (24)
If ( 24) holds then we conclude by applying Corollary 1 to [START_REF] Petit | Motion planning for two classes of nonlinear systems with delays depending on the control[END_REF], that, if a > b, then there exists γ > 0 such that ∀t ≥ t 0 , m(t) ≤ max m t 0 e -2γ(t-t 0 ) .

A sufficient condition for max m t 0 < r 2 ω 2 λ (P) is that

max|X t 0 | 2 < r 2 ω 2 λ (P)/λ (P). (25)
By assumption |maxX t 0 | < ω. Hence, to guarantee that (24) holds, we simply have to pick r ≥ r * λ (P)/λ (P).

Finally, the condition a > b can be reformulated as

c < λ (P)λ (Q) 2λ (P) 2 c * (27)
and one obtains ∀t ≥ t 0 , |X(t)| ≤ λ (P) λ (P) max|X t 0 |e -γ(t-t 0 ) , which concludes the proof.

Application to Dynamical Equation Defined through the Predictor-Based Control Law

We now focus on the DDE governing ε, which is given in the following lemma, the proof of which follows exactly the same lines as the one provided in [START_REF] Bresch-Pietri | Sufficient condition for prediction-based stabilization of linear system subject to input-dependent input-delay[END_REF]. This lemma can be extended without difficulty to the case of a piecewise continuous initial condition u 0 . However, we prefer to restrict our exposition for the sake of simplicity.

in which c * is introduced in Lemma 2. This neighborhood depends on the functional and on the constant c * and therefore on the feedback gain K. Finally, define c (t, ε t , . . . , ε

(n-1) t

) =π 1 Ḋ, . . . , D (n) , ε t , . . . , ε

we ensure that this condition is fulfilled for any t ≥ 0, that the initial condition lies in the neighborhood Ω and that

|E(t)| ≤ min u∆ * (|K|) 2M(κ) , κ , t ≥ 0. (32)
In particular, condition ( 17) is also fulfilled. Finally, the choice max|E 0 | ≤ θ (K) can be expressed in terms of Θ . This gives the conclusion.

The proof of Theorem 2 directly follows from Lemma 4.

Conclusion

In this study we have extended some of our recent work to the compensation of input-dependent delay input defined through an integral equation and considered a wide class of kernels (positive polynomial functions of the input). This class of systems is representative of a large number of processes involving transport of material, such as the low-cost rocket design presented above. We prove that robust compensation is achieved by using the current value of the delay as prediction horizon, provided that the initial conditions are in accordance with the magnitude of the feedback gain. The proposed approach is based on a two-step methodology which, first, requires the delay variations to be bounded and, second, relates these variations to input fluctuations. Quantitative comparison of the practical performances of the proposed approach with those of a prediction-based controller using the (constant) final reference delay D = 1/ϕ(u r ) is the natural next step in this approach. Exten-sion to explicitly time-dependent delay-integral kernels is also a direction of future work.

1 Fig. 2

 12 Fig. 2 Left: Schematic view of a low cost designed (single choke) bipropellant rocket engine. Zoom in on the transport phenomenon in the top right corner. Right: Schematic view of the evolution of the combustion efficiency with respect to the comburant/fuel ratio entering the chamber.

  max|E t ([-D, 0])| < ω * (K), we obtain c (t, ε t , . . . , ε (n-1) t ) ≤ c * max|E t |.Therefore, Lemma 2 guarantees the existence of r * (K) > 0 and γ ≥ 0 such that∀t ≥ 0 , |E(t)| ≤ r * (K)max|E 0 |e -γt .Hence, by choosingmax|E 0 | ≤ 1 r * (K) min u∆ * (K) 2M(κ) , κ, ω * (K) ∆ = θ (K)

This is the case if Ḋ < 1.

Interestingly, a similar condition is often stated in Linear Matrix Inequality approaches, such as[START_REF] Yue | Delayed feedback control of uncertain systems with time-varying input delay[END_REF] for example, where the delay is also assumed to be time-differentiable.

More precisely, in[START_REF] Halanay | Differential Equations: Stability, Oscillations, Time Lags[END_REF], this result is stated for a > b > 0.

The case when x is identically 0 is trivial. The continuity (and even more) is obtained by assuming ψ is smooth enough.

γ ≥ 0 is the unique solution on [0, ∞[ of aγ = bexp(γD).

 Lemma 3. Provided that u 0 ∈ C n ([-D, 0], R), the error variable ε = uu r with u defined in [START_REF] Jankovic | Recursive predictor design for linear systems with time delay[END_REF] satisfies the following differential equation for t ≥ 0

where [-k 0 . . .k n-1 ] ∆ = K and π 1 is a polynomial function which is at least quadratic in the variables Ḋ, . . . , D (n) , ε t , . . . , ε (n-1) t . Now that we are equipped with this last lemma, it is possible to use Lemma 2 to guarantee that the stability condition (17) holds.

Lemma 4. Consider the functional Θ defined in [START_REF] Krstic | Boundary Control of PDEs: a Course on Backstepping Designs[END_REF]. Then, there exists a function θ :

is fulfilled, which implies that (11) holds.

Proof. The input error ε satisfies dynamics [START_REF] Yue | Delayed feedback control of uncertain systems with time-varying input delay[END_REF], which is compliant with the assumptions of Lemma 2. In detail, first, the left-hand side of ( 28) is stable, as it represents the last line of the Hurwitz companion matrix A + BK. Second, by observing that ϕ and hence its derivatives are polynomial functions and that

we obtain by induction that, for m ≥ 1, D (m) is a polynomial function in

without terms of order 0 or 1. Therefore, π 1 is directly a polynomial function of the variables ε t , . . . , ε

which is at least quadratic in the variables ε t , . . . , ε

(n-1) t

. Observing that

we conclude that this term is continuous and therefore bounded on a given neighborhood of the origin. Hence, as π 1 is at least quadratic, it is possible to properly define a scalar ω * > 0 such that π 1 ε t , . . . , ε