Daniele Angella

AND Georges Dloussky

Adriano Tomassini

ON BOTT-CHERN COHOMOLOGY OF COMPACT COMPLEX SURFACES

Keywords: 2010 Mathematics Subject Classification. 32C35, 57T15, 32J15 compact complex surfaces, Bott-Chern cohomology, class VII, solvmanifold

We study Bott-Chern cohomology on compact complex non-Kähler surfaces. In particular, we compute such a cohomology for compact complex surfaces in class VII and for compact complex surfaces diffeomorphic to solvmanifolds.

Introduction

For a given complex manifold X, many cohomological invariants can be defined, and many are known for compact complex surfaces.

Among these, one can consider Bott-Chern and Aeppli cohomologies. They are defined as follows:

H •,• BC (X) := ker ∂ ∩ ker ∂ im ∂∂ and H •,• A (X) := ker ∂∂ im ∂ + im ∂ .
Note that the identity induces natural maps

H •,• BC (X) x x & & H •,• ∂ (X) & & H • dR (X; C) H •,• ∂ (X)
x x H •,• A (X) where H •,• ∂ (X) denotes the Dolbeault cohomology and H •,• ∂ (X) its conjugate, and the maps are morphisms of (graded or bi-graded) vector spaces. For compact Kähler manifolds, the natural map p+q=• H p,q BC (X) → H • dR (X; C) is an isomorphism. Assume that X is compact. The Bott-Chern and Aeppli cohomologies are isomorphic to the kernel of suitable 4th-order differential elliptic operators, see [19, §2.b, §2.c]. In particular, they are finitedimensional vector spaces. In fact, fixed a Hermitian metric g, its associated C-linear Hodge- * -operator induces the isomorphism H p,q BC (X) → H n-q,n-p A (X) , for any p, q ∈ {0, . . . , n}, where n denotes the complex dimension of X. In particular, for any p, q ∈ {0, . . . , n}, one has

dim C H p,q BC (X) = dim C H q,p BC (X) = dim C H n-p,n-q A (X) = dim C H n-q,n-p A (X) .
For the Dolbeault cohomology, the Frölicher inequality relates the Hodge numbers and the Betti numbers: for any k ∈ {0, . . . , 2n},

p+q=k dim C H p,q ∂ (X) ≥ dim C H k dR (X; C) .
Similarly, for Bott-Chern cohomology, the following inequality à la Frölicher has been proven in [3, Theorem A]: for any k ∈ {0, . . . , n}, p+q=k

(dim C H p,q BC (X) + dim C H p,q A (X)) ≥ 2 dim C H k dR (X; C) .
The equality in the Frölicher inequality characterizes the degeneration of the Frölicher spectral sequence at the first level. This always happens for compact complex surfaces. On the other side, in [START_REF] Angella | On the ∂∂-Lemma and Bott-Chern cohomology[END_REF]Theorem B], it is proven that the equality in the inequality à la Frölicher for the Bott-Chern cohomology characterizes the validity of the ∂∂-Lemma, namely, the property that every ∂-closed ∂-closed d-exact form is ∂∂-exact too, [START_REF] Deligne | Real homotopy theory of Kähler manifolds[END_REF]. The validity of the ∂∂-Lemma implies that the first Betti number is even, which is equivalent to Kählerness for compact complex surfaces. Therefore the positive integer numbers

∆ k := p+q=k (dim C H p,q BC (X) + dim C H p,q A (X)) -2 b k ∈ N ,
varying k ∈ {1, 2}, measure the non-Kählerness of compact complex surfaces X.

Compact complex surfaces are divided in seven classes, according to the Kodaira and Enriques classification, see, e.g., [START_REF] Barth | Compact complex surfaces[END_REF]. In this note, we compute Bott-Chern cohomology for some classes of compact complex (non-Kähler) surfaces. In particular, we are interested in studying the relations between Bott-Chern cohomology and de Rham cohomology, looking at the injectivity of the natural map H 2,1 BC (X) → H 3 dR (X; C). This can be intended as a weak version of the ∂∂-Lemma, compare also [START_REF] Fu | A note on small deformations of balanced manifolds[END_REF]. More precisely, we start by proving that the non-Kählerness for compact complex surfaces is encoded only in ∆ 2 , namely, ∆ 1 is always zero. This gives a partial answer to a question by T. C. Dinh to the third author. Theorem 1.1. Let X be a compact complex surface. Then:

(i) the natural map H 2,1 BC (X) → H 2,1 ∂ (X) induced by the identity is injective; (ii) ∆ 1 = 0.
In particular, the non-Kählerness of X is measured by just ∆ 2 ∈ N.

For compact complex surfaces in class VII, we show the following result, where we denote h p,q BC := dim C H p,q BC (X) for p, q ∈ {0, 1, 2}. Theorem 2.2. The Bott-Chern numbers of compact complex surfaces in class VII are:

h 0,0 BC = 1 h 1,0 BC = 0 h 0,1 BC = 0 h 2,0 BC = 0 h 1,1 BC = b 2 + 1 h 0,2 BC = 0 h 2,1 BC = 1 h 1,2 BC = 1 h 2,2 BC = 1 .
According to Theorem 1.1, the natural map H 2,1 BC (X) → H 2,1 ∂ (X) is injective for any compact complex surface. One is then interested in studying the injectivity of the natural map H 2,1 BC (X) → H 3 dR (X; C) induced by the identity, at least for compact complex surfaces diffeomorphic to solvmanifolds. In fact, by definition, the property of satisfying the ∂∂-Lemma, [START_REF] Deligne | Real homotopy theory of Kähler manifolds[END_REF], is equivalent to the natural map p+q=• H p,q BC (X) → H • dR (X; C) being injective. Note that, for a compact complex manifold of complex dimension n, the injectivity of the map H n,n-1 BC (X) → H 2n-1 dR (X; C) implies the (n -1, n)-th weak ∂∂-Lemma in the sense of J. Fu and S.-T. Yau, [START_REF] Fu | A note on small deformations of balanced manifolds[END_REF]Definition 5].

We then compute the Bott-Chern cohomology for compact complex surfaces diffeomorphic to solvmanifolds, according to the list given by K. Hasegawa in [START_REF] Hasegawa | Complex and Kähler structures on compact solvmanifolds[END_REF], see Theorem 4.1. More precisely, we prove that the cohomologies can be computed by using just left-invariant forms. Furthermore, for such complex surfaces, we note that the natural map H 2,1 BC (X) → H 3 dR (X; C) is injective, see Theorem 4.2. We note that the above classes do not exhaust the set of compact complex non-Kähler surfaces, the cohomologies of elliptic surfaces being still unknown.

Acknowledgments. The first and third authors would like to thank the Aix-Marseille University for its warm hospitality. Many thanks are due to the referee for her/his suggestions that improved the presentation.

Non-Kählerness of compact complex surfaces and Bott-Chern cohomology

We recall that, for a compact complex manifold of complex dimension n, for k ∈ {0, . . . , 2n}, we define the "non-Kählerness" degrees, [3, Theorem A],

∆ k := p+q=k h p,q BC + h n-q,n-p BC -2 b k ∈ N , .
where we use the duality in [19, §2.c] giving h p,q BC := dim C H p,q BC (X) = dim C H n-q,n-p A (X). According to [3, Theorem B], ∆ k = 0 for any k ∈ {0, . . . , 2n} if and only if X satisfies the ∂∂-Lemma, namely, every ∂-closed ∂-closed d-exact form is ∂∂-exact too. In particular, for a compact complex surface X, the condition ∆ 1 = ∆ 2 = 0 is equivalent to X being Kähler, the first Betti number being even, [START_REF] Kodaira | On the structure of compact complex analytic surfaces. I[END_REF][START_REF] Miyaoka | Kähler metrics on elliptic surfaces[END_REF][START_REF] Siu | Every K3 surface is Kähler[END_REF], see also [START_REF] Lamari | Courants kählériens et surfaces compactes[END_REF]Corollaire 5.7], and [START_REF] Buchdahl | On compact Kähler surfaces[END_REF]Theorem 11].

We prove that ∆ 1 is always zero for any compact complex surface. In particular, a sufficient and necessary condition for compact complex surfaces to be Kähler is ∆ 2 = 0. Theorem 1.1. Let X be a compact complex surface. Then:

(i) the natural map H 2,1 BC (X) → H 2,1 ∂ (X) induced by the identity is injective; (ii) ∆ 1 = 0.
In particular, the non-Kählerness of X is measured by just ∆ 2 ∈ N.

Proof. (i) Let α ∈ ∧ 2,1 X be such that [α] = 0 ∈ H 2,1 ∂ (X). Let β ∈ ∧ 2,
0 X be such that α = ∂β. Fix a Hermitian metric g on X, and consider the Hodge decomposition of β with respect to the Dolbeault Laplacian : let β = β h + ∂ * λ where β h ∈ ∧ 2,0 X ∩ ker , and λ ∈ ∧ 2,1 X. Therefore we have

α = ∂β = ∂∂ * λ = -∂ * (∂ * λ) ∈∧ 2,0 X = -∂ (∂ * λ) = ∂∂ ( * λ) ,
where we have used that any (2, 0)-form is primitive and hence, by the Weil identity, is self-dual. In particular, α is ∂∂-exact, so it induces a zero class in H 2,1 BC (X). (ii) On the one hand, note that

H 1,0 BC (X) = ker ∂ ∩ ker ∂ ∩ ∧ 1,0 X im ∂∂ = ker ∂ ∩ ker ∂ ∩ ∧ 1,0 X ⊆ ker ∂ ∩ ∧ 1,0 X = ker ∂ ∩ ∧ 1,0 X im ∂ = H 1,0 ∂ (X) . It follows that dim C H 0,1 BC (X) = dim C H 1,0 BC (X) ≤ dim C H 1,0 ∂ (X) = b 1 -dim C H 0,1 ∂ (X) ,
where we use that the Frölicher spectral sequence degenerates, hence in particular b

1 = dim C H 1,0 ∂ (X) + dim C H 0,1 ∂ (X).
On the other hand, by part (i), we have

dim C H 1,2 BC (X) = dim C H 2,1 BC (X) ≤ dim C H 2,1 ∂ (X) = dim C H 0,1 ∂ (X) ,
where we use the Kodaira and Serre duality

H 2,1 ∂ (X) H 1 (X; Ω 2 X ) H 1 (X; O X ) H 0,1 ∂ (X)
. By summing up, we get

∆ 1 = dim C H 0,1 BC (X) + dim C H 1,0 BC (X) + dim C H 1,2 BC (X) + dim C H 2,1 BC (X) -2 b 1 ≤ 2 b 1 -dim C H 0,1 ∂ (X) + dim C H 0,1 ∂ (X) -b 1 = 0 ,
concluding the proof.

Class VII surfaces

In this section, we compute Bott-Chern cohomology for compact complex surfaces in class VII.

Let X be a compact complex surface. By Theorem 1.1, the natural map

H 2,1 BC (X) → H 2,1
∂ (X) is always injective. Consider now the case when X is in class VII. If X is minimal, we prove that the same holds for cohomology with values in a line bundle. We will also prove that the natural map

H 1,2 BC (X) → H 1,2 ∂ (X) is not injective. Proposition 2.1. Let X be a compact complex surface in class VII 0 . Let L ∈ H 1 (X; C * ) = Pic 0 (X). The natural map H 2,1 BC (X; L) → H 2,1 ∂ (X; L) induced by the identity is injective. Proof. Let α ∈ ∧ 2,1 X ⊗ L be a ∂ L -exact (2, 1)-form. We need to prove that α is ∂ L ∂ L -exact too. Consider α = ∂ L ϑ, where ϑ ∈ ∧ 2,0 X ⊗ L. In particular, ∂ L ϑ = 0, hence θ defines a class in H 0,2 ∂ (X; L). Note that H 0,2 ∂ (X; L) H 2 (X; O X (L)) H 0 (X; K X ⊗ L -1 ) = {0} for surfaces of class VII 0 , [9, Remark 2.21]. It follows that θ = -∂ L η for some η ∈ ∧ 1,0 X ⊗ L. Hence α = ∂ L ∂ L η, that is, α is ∂ L ∂ L -exact.
We now compute the Bott-Chern cohomology of class VII surfaces.

Theorem 2.2. The Bott-Chern numbers of compact complex surfaces in class VII are:

h 0,0 BC = 1 h 1,0 BC = 0 h 0,1 BC = 0 h 2,0 BC = 0 h 1,1 BC = b 2 + 1 h 0,2 BC = 0 h 2,1 BC = 1 h 1,2 BC = 1 h 2,2 BC = 1 . Proof. It holds H 1,0 BC (X) = ker ∂∩ker ∂∩∧ 1,0 X im ∂∂ = ker ∂ ∩ ker ∂ ∩ ∧ 1,0 X ⊆ ker ∂ ∩ ∧ 1,0 X = ker ∂∩∧ 1,0 X im ∂ = H 1,0 ∂ (X) = {0} hence h 1,0 BC = h 0,1 BC = 0. On the other side, by Theorem 1.1, 0 = ∆ 1 = 2 h 1,0 BC + h 2,1 BC -b 1 = 2 h 2,1 BC -1 hence h 2,1 BC = h 1,2 BC = 1.
Similarly, it holds

H 2,0 BC (X) = ker ∂∩ker ∂∩∧ 2,0 X im ∂∂ = ker ∂ ∩ ker ∂ ∩ ∧ 2,0 X ⊆ ker ∂ ∩ ∧ 2,0 X = ker ∂∩∧ 2,0 X im ∂ = H 2,0 ∂ (X) = {0} hence h 2,0 BC = h 0,2 BC = 0. Note that, from [3, Theorem A], we have 0 ≤ ∆ 2 = 2 h 2,0 BC + h 1,1 BC + h 0,2 BC -b 2 = 2 h 1,1 BC -b 2 hence h 1,1 BC ≥ b 2 .
→ im d ∩ ∧ 1,1 X im ∂∂ → H 1,1 BC (X) → im H 1,1 BC (X) → H 2 dR (X; C) → 0
is clearly exact. Furthermore, fix a Gauduchon metric g. Denote by ω := g(J•, ••) the (1, 1)-form associated to g, where J denotes the integrable almost-complex structure. By definition of g being Gauduchon, we have ∂∂ω = 0. The sequence

0 → im d ∩ ∧ 1,1 X im ∂∂ •|ω → C is exact. Indeed, firstly note that for η = ∂∂f ∈ im ∂∂ ∩ ∧ 1,1 X, we have η|ω = X ∂∂f ∧ * ω = X ∂∂f ∧ ω = X f ∂∂ω = 0
by applying twice the Stokes theorem. Then, we recall the argument in [21, Lemma 2.3(ii)] for proving that the map

•|ω : im d ∩ ∧ 1,1 X im ∂∂ → C is injective. Take α = d β ∈ im d ∩ ∧ 1,1 X ∩ ker •|ω . Then Λα|1 = α|ω = 0 ,
where Λ is the adjoint operator of ω ∧ • with respect to

:= α -∂∂u, we have [α ] = [α] ∈ im d ∩∧ 1,1 X im ∂∂
, and Λα = 0, and α = d β where β := β -∂u. In particular, α is primitive. Since α is primitive and of type (1, 1), then it is anti-self-dual by the Weil identity. Then

α 2 = α |α = X α ∧ * α = - X α ∧ α = - X d β ∧ d β = - X d β ∧ d β = 0
and hence α = 0, and therefore [α] = 0. Since the space im d ∩∧ 1,1 X im ∂∂ is finite-dimensional, being a sub-space of H 1,1 BC (X), and since the space im

H 1,1 BC (X) → H 2 dR (X; C) is finite-dimensional, being a sub-space of H 2 dR (X; C), we get that dim C im d ∩ ∧ 1,1 X im ∂∂ ≤ dim C C = 1 ,
and hence

b 2 < dim C H 1,1 BC (X) = dim C im H 1,1 BC (X) → H 2 dR (X; C) + dim C im d ∩ ∧ 1,1 X im ∂∂ ≤ b 2 + 1 . We get that dim C H 1,1 BC (X) = b 2 + 1.
Finally, we prove that the natural map

H 1,2 BC (X) → H 1,2 ∂ (X) is not injective.
Proposition 2.3. Let X be a compact complex surface in class VII. Then the natural map H 1,2 BC (X) → H 1,2 ∂ (X) induced by the identity is the zero map and not an isomorphism.

Proof. Note that, for class VII surfaces, the pluri-genera are zero. In particular,

H 1,2 ∂ (X) H 1,0 ∂ (X) = {0},
by Kodaira and Serre duality. By Theorem 2.2, one has H 1,2 BC (X) = {0}. 2.1. Cohomologies of Calabi-Eckmann surface. In this section, as an explicit example, we list the representatives of the cohomologies of a compact complex surface in class VII: namely, we consider the Calabi-Eckmann structure on the differentiable manifolds underlying the Hopf surfaces.

Consider the differentiable manifold X := S 1 × S 3 . As a Lie group, S 3 = SU (2) has a global leftinvariant co-frame e 1 , e 2 , e 3 such that d e 1 = -2e 2 ∧ e 3 and d e 2 = 2e 1 ∧ e 3 and d e 3 = -2e 1 ∧ e 2 . Hence, we consider a global left-invariant co-frame f, e 1 , e 2 , e 3 on X with structure equations

           d f = 0 d e 1 = -2 e 2 ∧ e 3 d e 2 = 2 e 1 ∧ e 3 d e 3 = -2 e 1 ∧ e 2 .
Consider the left-invariant almost-complex structure defined by the (1, 0)-forms

ϕ 1 := e 1 + i e 2
ϕ 2 := e 3 + i f .

By computing the complex structure equations, we get

∂ϕ 1 = i ϕ 1 ∧ ϕ 2 ∂ϕ 2 = 0 and ∂ϕ 1 = i ϕ 1 ∧ φ2 ∂ϕ 2 = -i ϕ 1 ∧ φ1 .
We note that the almost-complex structure is in fact integrable.

The manifold X is a compact complex manifold not admitting Kähler metrics. It is bi-holomorphic to the complex manifold M 0,1 considered by Calabi and Eckmann, [START_REF] Calabi | A class of compact, complex manifolds which are not algebraic[END_REF], see [START_REF] Parton | Explicit parallelizations on products of spheres and Calabi-Eckmann structures[END_REF]Theorem 4.1].

Consider the Hermitian metric g whose associated (1, 1)-form is

ω := i 2 2 j=1 ϕ j ∧ φj .
As for the de Rham cohomology, from the Künneth formula we get 12 12 , (where, here and hereafter, we shorten, e.g.,

H • dR (X; C) = C 1 ⊕ C ϕ 2 -φ2 ⊕ C ϕ 12 1 -ϕ 1 12 ⊕ C ϕ
ϕ 12 1 := ϕ 1 ∧ ϕ 2 ∧ φ1 ).
By [12, Appendix II, Theorem 9.5], one has that a model for the Dolbeault cohomology is given by

H •,• ∂ (X) x 2,1 , x 0,1 ,
where x i,j is an element of bi-degree (i, j). In particular, we recover that the Hodge numbers

h p,q ∂ := dim C H p,q ∂ (X) p,q∈{0,1,2} are h 0,0 ∂ = 1 h 1,0 ∂ = 0 h 0,1 ∂ = 1 h 2,0 ∂ = 0 h 1,1 ∂ = 0 h 0,2 ∂ = 0 h 2,1 ∂ = 1 h 1,2 ∂ = 0 h 2,2 ∂ = 1 .
We note that the sub-complex

ι : ϕ 1 , ϕ 2 , φ1 , φ2 → ∧ •,• X
is such that H ∂ (ι) is an isomorphism. More precisely, we get 12 12 , where we have listed the harmonic representatives with respect to the Dolbeault Laplacian of g. By [2, Theorem 1.3, Proposition 2.2], we have also H BC (ι) isomorphism. In particular, we get 12 12 , where we have listed the harmonic representatives with respect to the Bott-Chern Laplacian of g. By [19, §2.c], we have 12 12 , where we have listed the harmonic representatives with respect to the Aeppli Laplacian of g. Note in particular that the natural map

H •,• ∂ (X) = C 1 ⊕ C ϕ 2 ⊕ C ϕ 12 1 ⊕ C ϕ
H •,• BC (X) = C 1 ⊕ C ϕ 1 1 ⊕ C ϕ 12 1 ⊕ C ϕ 1 12 ⊕ C ϕ
H •,• A (X) = C 1 ⊕ C ϕ 2 ⊕ C ϕ 2 ⊕ C ϕ 2 2 ⊕ C ϕ
H 2,1 BC (X) → H 2,1 ∂ (X)
induced by the identity is an isomorphism, and that the natural map H 2,1 BC (X) → H 3 dR (X; C) induced by the identity is injective.

Complex surfaces diffeomorphic to solvmanifolds

Let X be a compact complex surface diffeomorphic to a solvmanifold Γ\G . By [11, Theorem 1], X is (A) either a complex torus, (B) or a hyperelliptic surface, (C) or a Inoue surface of type S M , (D) or a primary Kodaira surface, (E) or a secondary Kodaira surface, (F) or a Inoue surface of type S ± , and, as such, it is endowed with a left-invariant complex structure.

In each case, we recall the structure equations of the group G, see [START_REF] Hasegawa | Complex and Kähler structures on compact solvmanifolds[END_REF]. More precisely, take a basis {e 1 , e 2 , e 3 , e 4 } of the Lie algebra g naturally associated to G. We have the following commutation relations, according to [START_REF] Hasegawa | Complex and Kähler structures on compact solvmanifolds[END_REF]: (A) differentiable structure underlying a complex torus: Denote by e 1 , e 2 , e 3 , e 4 the dual basis of {e 1 , e 2 , e 3 , e 4 }. We recall that, for any α ∈ g * , for any x, y ∈ g, it holds d α(x, y) = -α ([x, y]). Hence we get the following structure equations: (A) differentiable structure underlying a complex torus: Consider the G-left-invariant (1, 0)-forms

           d e 1 =
ϕ 1 := e 1 + i e 2 ϕ 2 := e 3 + i e 4 .
In case (F), consider the G-left-invariant almost-complex structure J on X defined by Je 1 := e 2 and Je 2 := -e 1 and Je 3 := e 4 -q e 2 and Je 4 := -e 3 -q e 1 , where q ∈ R. Consider the G-left-invariant (1, 0)-forms ϕ 1 := e 1 + i e 2 + i q e 4 ϕ 2 := e 3 + i e 4 .

With respect to the G-left-invariant coframe ϕ 1 , ϕ 2 for the holomorphic tangent bundle T 1,0 Γ\G , we have the following structure equations. (As for notation, we shorten, e.g., ϕ 1 2 := ϕ 1 ∧ φ2 .) (A) torus:

d ϕ 1 = 0 d ϕ 2 = 0 (B) hyperelliptic surface: d ϕ 1 = -1 2 ϕ 12 + 1 2 ϕ 1 2 d ϕ 2 = 0 (C) Inoue surface S M : d ϕ 1 = α-i β 2 i ϕ 12 -α-i β 2 i ϕ 1 2 d ϕ 2 = -i α ϕ 2 2
(where α ∈ R \ {0} and β ∈ R); (D) primary Kodaira surface:

d ϕ 1 = 0 d ϕ 2 = i 2 ϕ 1 1
(E) secondary Kodaira surface:

d ϕ 1 = -1 2 ϕ 12 + 1 2 ϕ 1 2 d ϕ 2 = i 2 ϕ 1 1 (F) Inoue surface S ± : d ϕ 1 = 1 2 i ϕ 12 + 1 2 i ϕ 2 1 + q i 2 ϕ 2 2 d ϕ 2 = 1 2 i ϕ 2 2 .

Cohomologies of complex surfaces diffeomorphic to solvmanifolds

In this section, we compute the Dolbeault and Bott-Chern cohomologies of the compact complex surfaces diffeomorphic to a solvmanifold.

We prove the following theorem. Theorem 4.1. Let X be a compact complex surface diffeomorphic to a solvmanifold Γ\G ; denote the Lie algebra of G by g. Then the inclusion ∧ •,• g * , ∂, ∂ → ∧ •,• X, ∂, ∂ induces an isomorphism both in Dolbeault and in Bott-Chern cohomologies. In particular, the dimensions of the de Rham, Dolbeault, and Bott-Chern cohomologies and the degrees of non-Kählerness are summarized in Table 5.

Proof. Firstly, we compute the cohomologies of the sub-complex of G-left-invariant forms. The computations are straightforward from the structure equations.

(A) torus (B) hyperelliptic (C) Inoue SM (p, q) H p,q ∂ dimC H p,q ∂ H p,q BC dimC H p,q BC H p,q ∂ dimC H p,q ∂ H p,q BC dimC H p,q BC H p,q ∂ dimC H p,q ∂ H p,q BC dimC H p,q BC (0, 0) 1 1 1 1 1 1 1 1 1 1 1 1 (1, 0) ϕ 1 , ϕ 2 2 ϕ 1 , ϕ 2 2 ϕ 2 1 ϕ 2 1 0 0 0 0 (0, 1) ϕ 1, ϕ 2 2 ϕ 1, ϕ 2 2 ϕ 2 1 ϕ 2 1 ϕ 2 1 0 0 (2, 0) ϕ 12 1 ϕ 12 1 0 0 0 0 0 0 0 0 (1, 1) ϕ 11 , ϕ 12 , ϕ 21 , ϕ 22 4 ϕ 11 , ϕ 12 , ϕ 21 , ϕ 22 4 ϕ 11 , ϕ 22 2 ϕ 11 , ϕ 22 2 0 0 ϕ 22 1 (0, 2) ϕ 12 1 ϕ 12 1 0 0 0 0 0 0 0 0 (2, 1) ϕ 121 , ϕ 122 2 ϕ 121 , ϕ 122 2 ϕ 121 1 ϕ 121 1 ϕ 121 1 ϕ 121 1 (1, 2) ϕ 112 , ϕ 212 2 ϕ 112 , ϕ 212 2 ϕ 112 1 ϕ 112 1 0 0 ϕ 112 1 (2, 2) ϕ 1212 1 ϕ 1212 1 ϕ 1212 1 ϕ 1212 1 ϕ 1212 1 ϕ 1212 1
Table 1. Dolbeault and Bott-Chern cohomologies of compact complex surfaces diffeomorphic to solvmanifolds, part 1.

(D) primary Kodaira In Tables 1 and2 and in Tables 3 and4, we summarize the results of the computations. The subcomplexes of left-invariant forms are depicted in Figure 1 (each dot represents a generator, vertical arrows depict the ∂-operator, horizontal arrows depict the ∂-operator, and trivial arrows are not shown.) The dimensions are listed in Table 5.

(E) secondary Kodaira (F) Inoue S± (p, q) H p,q ∂ dimC H p,q ∂ H p,q BC dimC H p,q BC H p,q ∂ dimC H p,q ∂ H p,q BC dimC H p,q BC H p,q ∂ dimC H p,q ∂ H p,q BC dimC H p,q (1, 0) ϕ 1 1 ϕ 1 1 0 0 0 0 0 0 0 0 (0, 1) ϕ 1, ϕ 2 2 ϕ 1 1 ϕ 2 1 0 0 ϕ 2 1 0 0 (2, 0) ϕ 12 1 ϕ 12 1 0 0 0 0 0 0 0 0 (1,
k H k dR dim C H k dR H k dR dim C H k dR H k dR dim C H k dR 0 1 1 1 1 1 1 1 ϕ 1 , ϕ 2 , ϕ 1, ϕ 2 4 ϕ 2 , ϕ 2 2 ϕ 2 -ϕ 2 1 2 ϕ 12 , ϕ 1 1, ϕ 1 2, ϕ 2 1, ϕ
H k dR dim C H k dR H k dR dim C H k dR 0 1 1 1 1 1 1 1 ϕ 1 , ϕ 1, ϕ 2 -ϕ 2 3 ϕ 2 -ϕ 2 1 ϕ 2 -ϕ 2 1 2 ϕ 12 , ϕ
On the one side, recall that the inclusion of left-invariant forms into the space of forms induces an injective map in Dolbeault and Bott-Chern cohomologies, see, e.g., [START_REF] Console | Dolbeault cohomology of compact nilmanifolds[END_REF]Lemma 9], [START_REF] Angella | The cohomologies of the Iwasawa manifold and of its small deformations[END_REF]Lemma 3.6]. On the other side, recall that the Frölicher spectral sequence of a compact complex surface X degenerates at the first level, equivalently, the equalities

dim C H 1,0 ∂ (X) + dim C H 0,1 ∂ (X) = dim C H 1 dR (X; C) and dim C H 2,0 ∂ (X) + dim C H 1,1 ∂ (X) + dim C H 0,2 ∂ (X) = dim C H 2 dR (X; C)
hold. By comparing the dimensions in Table 5 with the Betti numbers case by case, we find that the left-invariant forms suffice in computing the Dolbeault cohomology for each case. Then, by [1, Theorem 3.7], see also [2, Theorem 1.3, Theorem 1.6], it follows that also the Bott-Chern cohomology is computed using just left-invariant forms.

A) torus (B) hyperell (C) Inoue S M (D) prim Kod (E) sec Kod (F) Inoue S ± (p, q) h p,q ∂ h p,q BC b k ∆ k h p,q ∂ h p,q BC b k ∆ k h p,q ∂ h p,q BC b k ∆ k h p,q ∂ h p,q BC b k ∆ k h p,q ∂ h p,q BC b k ∆ k h p,q ∂ h p,q BC b k ∆ k (0, 0) 1 (1, 0) 2 2 4 0 1 1 2 0 0 0 1 0 1 1 3 0 0 0 1 0 0 0 1 0 (0, 1) 2 2 1 1 1 0 2 1 1 0 1 0 (2, 0) 1 1 6 0 0 0 2 0 0 0 0 2 1 1 4 2 0 0 0 2 0 0 0 2 (1, 1) 4 4 2 2 0 1 2 3 0 1 0 1 (0, 2) 1 1 0 0 0 0 1 1 0 0 0 0 (2, 1) 2 2 4 0 1 1 2 0 1 1 1 0 2 2 3 0
Table 5. Summary of the dimensions of de Rham, Dolbeault, and Bott-Chern cohomologies and of the degree of non-Kählerness for compact complex surfaces diffeomorphic to solvmanifolds.

We prove the following result.

Theorem 4.2. Let X be a compact complex surface diffeomorphic to a solvmanifold. Then the natural map H 2,1 BC (X) → H 2,1 ∂ (X) induced by the identity is an isomorphism, and the natural map H 2,1 BC (X) → H 3 dR (X; C) induced by the identity is injective.

Proof. By the general result in Theorem 1.1, the natural map H 2,1 BC (X) → H 2,1 ∂ (X) is injective. In fact, it is an isomorphism as follows from the computations summarized in Tables 1 and2. As for the injectivity of the natural map H 2,1 BC (X) → H 3 dR (X; C), it is a straightforward computation from Tables 1 and2 and Tables 3 and4.

As an example, we offer an explicit calculation of the injectivity of the map H 2,1 BC (X) → H 3 dR (X; C) for the Inoue surfaces of type 0, see [START_REF] Inoue | On surfaces of Class V II 0[END_REF], see also [START_REF] Tricerri | Some examples of locally conformal Kähler manifolds[END_REF]. We will change a little bit the notation. Recall the construction of Inoue surfaces: let M ∈ SL(3; Z) be a unimodular matrix having a real eigenvalue λ > 1 and two complex eigenvalues µ = µ. Take a real eigenvector (α 1 , α 2 , α 3 ) and an eigenvector (β 1 , β 2 , β 3 ) of M . Let H = {z ∈ C | m z > 0}; on the product H × C consider the following transformations defined as f 0 (z, w) := (λz, µw) f j (z, w) := (z + α j , w + β j ) for j ∈ {1, 2, 3} .

Denote by Γ M the group generated by f 0 , . . . , f 3 ; then Γ M acts in a properly discontinuous way and without fixed points on H × C, and S M := H × C/Γ M is an Inoue surface of type 0, as in case (C) in [START_REF] Hasegawa | Complex and Kähler structures on compact solvmanifolds[END_REF]. Denoting by z = x + i y and w = u + i v, consider the following differential forms on H × C:

[

  e j , e k ] = 0 for any j, k ∈ {1, 2, 3, 4} ; (hereafter, we write only the non-trivial commutators); (B) differentiable structure underlying a hyperelliptic surface: [e 1 , e 4 ] = e 2 , [e 2 , e 4 ] = -e 1 ; (C) differentiable structure underlying a Inoue surface of type S M : [e 1 , e 4 ] = -α e 1 + β e 2 , [e 2 , e 4 ] = -β e 1 -α e 2 , [e 3 , e 4 ] = 2α e 3 , where α ∈ R \ {0} and β ∈ R; (D) differentiable structure underlying a primary Kodaira surface: [e 1 , e 2 ] = -e 3 ; (E) differentiable structure underlying a secondary Kodaira surface: [e 1 , e 2 ] = -e 3 , [e 1 , e 4 ] = e 2 , [e 2 , e 4 ] = -e 1 ; (F) differentiable structure underlying a Inoue surface of type S ± : [e 2 , e 3 ] = -e 1 , [e 2 , e 4 ] = -e 2 , [e 3 , e 4 ] = e 3 .

0 d e 2 d e 1 = e 2 ∧ e 4 d e 2 = -e 1 ∧ e 4 d e 3 d e 1 = 4 d e 2 = 4 d e 3 =d e 1 = e 2 ∧ e 4 d e 2 = -e 1 ∧ e 4 d e 3 = e 1 ∧ e 2 d e 4 d e 1 = e 2 ∧ e 3 d e 2 = e 2 ∧ e 4 d e 3 = -e 3 ∧ e 4 d e 4 = 0 .

 21424314243142432413243440 = 0 d e 3 = 0 d e 4 = 0 ; (B) differentiable structure underlying a hyperelliptic surface: differentiable structure underlying a Inoue surface of type S M : α e 1 ∧ e 4 + β e 2 ∧ e -β e 1 ∧ e 4 + α e 2 ∧ e -2α e 3 ∧ e 4 d e 4 = 0 ; (D) differentiable structure underlying a primary Kodaira surface: e 3 = e 1 ∧ e 2 d e 4 = 0 ; (E) differentiable structure underlying a secondary Kodaira surface: differentiable structure underlying a Inoue surface of type S ± : In cases (A), (B), (C), (D), (E), consider the G-left-invariant almost-complex structure J on X defined by Je 1 := e 2 and Je 2 := -e 1 and Je 3 := e 4 and Je 4 := -e 3 .

Figure 1 .

 1 Figure 1. The double-complexes of left-invariant forms over 4-dimensional solvmanifolds.

(

2 e 2 ∧ e 4 . 3 .

 2243 e 3 := √ y d u , e 4 := √ y d v . (Note that e 1 and e 2 , and e 3 ∧ e 4 are Γ M -invariant, and consequently they induce global differential forms on S M .) We obtain d e 1 = e 1 ∧ e 2 , d e 2 = 0 , d e 3 = 1 2 e 2 ∧ e 3 , d e 4 = 1 Consider the natural complex structure on S M induced by H × C. Locally, we have Je 1 = -e 2 and Je 2 = e 1 and Je 3 = -e 4 and Je 4 = e Considering the Γ M -invariant (2, 1)-Bott-Chern cohomology of S M , we obtain that H 2,1 BC (S M ) = C e 1 ∧ e 3 ∧ e 4 + i e 2 ∧ e 3 ∧ e 4 . Clearly ∂ e 1 ∧ e 3 ∧ e 4 + i e 2 ∧ e 3 ∧ e 4 = 0 and e 1 ∧ e 3 ∧ e 4 + i e 2 ∧ e 3 ∧ e 4 = e 1 ∧ e 3 ∧ e 4 + i d e 3 ∧ e 4 , therefore the de Rham cohomology class e 1 ∧ e 3 4 + i e 2 ∧ e 3 ∧ e 4 = e 1 ∧ e 3 ∧ e 4 ∈ H 3 dR (S M ) is non-zero.

  More precisely, from[START_REF] Angella | On the ∂∂-Lemma and Bott-Chern cohomology[END_REF] Theorem B] and Theorem 1.1, we have that h 1,1 BC = b 2 if and only if ∆ 2 = 0 if and only if X satisfies the ∂∂-Lemma, in fact X is Kähler, which is not the case.Finally, we prove that h 1,1 BC = b 2 + 1. Consider the following exact sequences from [21, Lemma 2.3].

	More precisely, the sequence
	0

Table 2 .

 2 Dolbeault and Bott-Chern cohomologies of compact complex surfaces diffeomorphic to solvmanifolds, part 2.

	1)	ϕ 12 , ϕ 21	2	ϕ 11 , ϕ 12 , ϕ 21	3	0	0	ϕ 11	1	0	0	ϕ 22	1
	(0, 2)	ϕ 12	1	ϕ 12	1	0	0	0	0	0	0	0	0
	(2, 1)	ϕ 121 , ϕ 122	2	ϕ 121 , ϕ 122	2	ϕ 121	1	ϕ 121	1	ϕ 121	1	ϕ 121	1
	(1, 2)	ϕ 212	1	ϕ 112 , ϕ 212	2	0	0	ϕ 112	1	0	0	ϕ 112	1
	(2, 2)	ϕ 1212	1	ϕ 1212	1	ϕ 1212	1	ϕ 1212	1	ϕ 1212	1	ϕ 1212	1
				(A) torus			(B) hyperelliptic		(C) Inoue S M

Table 3 .

 3 2 2, ϕ de Rham cohomology of compact complex surfaces diffeomorphic to solvmanifolds, part 1.

			12	6	ϕ 1 1, ϕ 2 2	2	0	0
	3	ϕ 12 1, ϕ 12 2, ϕ 1 12 , ϕ 2 12		4	ϕ 12 1, ϕ 1 12	2	ϕ 12 1 -ϕ 1 12	1
	4	ϕ 12 12		1	ϕ 12 12	1	ϕ 12 12	1
		(D) primary Kodaira	(E) secondary Kodaira		(F) Inoue S ±
	k H k dR	dim C H k dR

Table 4 .

 4 1 2, ϕ 2 1, ϕ de Rham cohomology of compact complex surfaces diffeomorphic to solvmanifolds, part 2.

		12	4	0	0	0	0
	3	ϕ 12 2, ϕ 2 12 , ϕ 12 1 -ϕ 1 12	3	ϕ 12 1 -ϕ 1 12	1	ϕ 12 1 -q ϕ 12 2 -ϕ 1 12 + q ϕ 2 12	1
	4	ϕ 12 12	1	ϕ 12 12	1	ϕ 12 12	1

This work was supported by the Project PRIN "Varietà reali e complesse: geometria, topologia e analisi armonica", by the Project FIRB "Geometria Differenziale e Teoria Geometrica delle Funzioni", by GNSAGA of INdAM, and by ANR "Méthodes nouvelles en géométrie non kählerienne".