Note on Dolbeault cohomology and Hodge structures up to bimeromorphisms

We construct a simply-connected compact complex non-K\"ahler manifold satisfying the $\partial\bar\partial$-Lemma, and endowed with a balanced metric. To this aim, we were initially aimed at investigating the stability of the property of satisfying the $\partial\bar\partial$-Lemma under modifications of compact complex manifolds and orbifolds. This question has been recently addressed and answered in \cite{rao-yang-yang, yang-yang, stelzig-blowup, stelzig-doublecomplex} with different techniques. Here, we provide a different approach using \v{C}ech cohomology theory to study the Dolbeault cohomology of the blow-up $\tilde X_Z$ of a compact complex manifold $X$ along a submanifold $Z$ admitting a holomorphically contractible neighbourhood.


Introduction
The ∂∂-Lemma is a strong cohomological decomposition property defined for complex manifolds, which is satisfied for example by algebraic projective manifolds and, more generally, by compact Kähler manifolds. The property is closely related to the fact that the Dolbeault cohomology provides a Hodge structure on the de Rham cohomology (cf. Subsection 1.5 below).
This property yields also strong topological obstructions: the real homotopy type of a compact complex manifold satisfying the ∂∂-Lemma is a formal consequence of its cohomology ring [DGMS75]. Complex non-Kähler manifolds usually do not satisfy the ∂∂-Lemma: for example, it is never satisfied by compact non-tori nilmanifolds [Has89]. On the other hand, some examples of compact complex non-Kähler manifolds satisfying the ∂∂-Lemma are provided by Moišhezon manifolds and manifolds in class C of Fujiki thanks to [DGMS75,Theorem 5.22], see [Hir62] for a concrete example. By the results contained in [Cam91,Corollary 3.13], [LP92, Theorem 1] and thanks to the stability property of the ∂∂-Lemma for small deformations [Voi02,Proposition 9.21], [Wu06,Theorem 5.12] one can produce examples of compact complex manifolds satisfying the ∂∂-Lemma and not bimeromorphic to Kähler manifolds. Other examples of this kind can be found among solvmanifolds [AK17a,AK17b,Kas13b]; moreover other examples are provided by Clemens manifolds [Fri91,Fri17], which are constructed by combining modifications and deformations.
The main aim of this note is to construct a simply-connected compact complex non-Kähler manifold satisfying the ∂∂-Lemma.
The theorem in [DGMS75,Theorem 5.22] states that, for a modifica-tionX → X of compact complex manifolds, the property of ∂∂-Lemma is preserved fromX to X. So, it is natural to ask whether it is in fact an invariant property by modifications. This is true, for example, for compact complex surfaces, thanks to the topological Lamari's and Buchdahl's criterion [Lam99,Buc99]. Note that, in higher dimension, the Kähler property is not stable under modifications; but there are weaker metric properties that are, for example the balanced condition in the sense of Michelsohn [AB96, Corollary 5.7] or the strongly-Gauduchon condition in the sense of Popovici [Pop13,Theorem 1.3]. In fact, it is conjectured that the metric balanced condition and the cohomological ∂∂-Lemma property are strictly related to each other, see for example [Pop15b,Conjecture 6.1], see also [TW13,Pop15a]; and this provides another motivation for the above question.
In this note, we deal with the Dolbeault cohomology of the blow-up along submanifolds. The strategy we follow is sheaf-theoretic, more precisely Čechcohomological, in the spirit of [Suw09]. The de Rham case in the Kähler context is considered in [Voi02,Theorem 7.31]. For our argument, we need to assume that the centre admits a holomorphically contractible neighbourhood (this is clearly satisfied when blowing-up at a point, see also the explicit computations in Example 21) and another technical assumption (11) concerning the kernel and images of certain morphisms. We can then deduce that: Theorem 13. Let X be a compact complex manifold and Z a closed submanifold of X. If both X and the centre Z admits a Hodge structure (in the sense of Definition 4), then the same holds for the blow-up Bl Z X of X along Z, provided that Z admits a holomorphically contractible neighbourhood and the technical assumption (11) holds.
Along the way we give explicit expressions for the de Rham and Dolbeault cohomologies of Bl Z X (see Propositions 16 and 19).
Hopefully, a further study of the cohomological properties of submanifolds (see Question 22) and a deeper use of techniques as the MacPherson's deformation to the normal cone (see Question 23), along with the Weak Factorization Theorem for bimeromorphic maps in the complex-analytic category [AKMW02, Theorem 0.3.1], [Wlo03], may allow to use the above techniques to prove in full generality the stability of the ∂∂-Lemma under modifications, see Remark 24.
During the preparation of this work, several other attempts to solve the same problems appeared [RYY17,YY17,Ste18b], using different techniques.
In particular, the work by Jonas Stelzig [Ste18a] finally ties up the problem, as far as now: Theorem 1 ([Ste18b, Theorem 8], [Ste18c,Corollary 25]). The ∂∂-Lemma property is a bimeromorphic invariant if and only if it is invariant by restriction.
Even if Stelzig's theorem is clearly stronger than our Theorem 13, we think that our argument may be interesting and useful in providing a broader point of view for understanding (Čech-)Dolbeault cohomology.
The second and main aim of this note is to construct new explicit examples of compact complex manifolds satisfying the ∂∂-Lemma: in particular, we provide a simply-connected example, see Example 26. To this aim, we need to work with orbifolds in the sense of Satake [Sat56], and their desingularizations. We take advantage of Stelzig's general results, see Theorem 25. The construction of Example 26 goes as follows, see e.g. [FM08,BFM14]: we start from a manifold isomorphic to the Iwasawa manifold, which does not satisfy the ∂∂-Lemma; then we quotient it by a finite group of automorphisms; and then we resolve its singularities. Finally, by Theorem 25, we get simply-connected examples of complex manifolds satisfying the ∂∂-Lemma: Theorem 27. There exist a simply-connected compact complex non-Kähler manifold, (not even in class C of Fujiki,) that satisfy the ∂∂-Lemma. Our example admits a balanced metric.
As far as we know, these are the first explicit examples of simply-connected compact complex non-Kähler manifolds satisfying the ∂∂-Lemma in the literature.
Given a closed set S in X, we can take U 0 := X \ S and U 1 an open neighbourhood of S in X, and the open covering where the first map is the inclusion and the second map is the projection on the first element, we obtain a long exact sequence in cohomology (1) From this we see that H h D (U, U 0 ) is determined uniquely modulo canonical isomorphisms, independently of the choice of U 1 . We denote it also by H h D (X, X \ S) and call it the relative Čech-de Rham cohomology. We recall that excision holds: for any neighbourhood U of S in X, it holds In fact we have, [Suw08], H h D (X, X \ S) ≃ H h (X, X \ S; C), the relative cohomology of the pair (X, X \ S).
Consider now a smooth complex vector bundle π : E → M of rank k on a smooth manifold M . Consider the bundle ̟ : π * E → E defined by the fibre product / / M and its diagonal section s ∆ . The zero-set of s ∆ is the image of the zerosection of E, which is identified with M . In this situation, the Thom class of E is given as the localization of the top Chern class c k (π * E) by s ∆ . That is: consider the covering W = {W 0 := E \ M, W 1 } of E, where W 1 is a neighbourhood of M in E; consider ∇ 0 a connection on W 0 such that ∇ 0 s ∆ = 0, and ∇ 1 a connection on W 1 ; then the Chern class c k (π * E) is represented by where c k (∇ 0 , ∇ 1 ) is the Bott difference form of ∇ 0 and ∇ 1 ; in fact, since c k (∇ 0 ) = 0, this defines a class Ψ E ∈ H 2k D (E, E \ M ) represented by (ψ 1 , ψ 01 ) := (c k (∇ 1 ), c k (∇ 0 , ∇ 1 )). It turns out that the map is an isomorphisms, [Suw98, Theorem 5.3], called the Thom isomorphism, where the cup product Ψ E π * [θ] is represented by (ψ 1 ∧ π * θ, ψ 01 ∧ π * θ). Its inverse is the integration along the fibres: where π 1 is the restriction of π to a bundle T 1 of disks of complex dimension k in W 1 , and π 01 is the restriction of π to the bundle T 01 = −∂T 1 of spheres of real dimension 2k − 1 with opposite orientation. In particular, the Thom class Ψ E is characterized in H 2k D (E, E \ M ) by the property π * Ψ E = 1. Finally, we recall the projection formula, [Suw98, Ch.II, Proposition 5.1]: for Given a closed complex submanifold Z, of complex codimension k, of a complex manifold X, of complex dimension n, we can define the Thom isomorphism and the Thom class of Z as follows. Consider the normal bundle π : N Z|X → Z, of complex rank k. By the Tubular Neighbourhood Theorem, there exist neighbourhoods U of Z in X, and W of Z as zero section in N Z|X , and a smooth diffeomorphism ϕ : U → W such that ϕ| Z = id. Then, setting N = N Z|X , we get isomorphisms Define the Thom class Ψ Z ∈ H 2k D (X, X \ Z) of Z as the image of Ψ N Z|X via the above isomorphisms, and the Thom isomorphism T Z : Čech-Dolbeault cohomology. Let X be a complex manifold and let A p,q (U ) be the space of smooth (p, q)-forms on an open set U in X. Let U = {U 0 , U 1 } be an open covering of X and consider The differential operatorD : The Čech-Dolbeault cohomology associated to the covering U is then defined by H •,• D (U) = kerD/imD (see [Suw09] where this definition is given for an arbitrary open covering of the manifold X). The morphism A p,q (X) → A p,q (U) given by ω → (ω| U 0 ω| U 1 , 0) induces an isomorphism in cohomology ∂ (X) denotes the Dolbeault cohomology of X, [Suw09, Theorem 1.2]. In particular, the definition is independent of the choice of the covering of X. Moreover, the inverse map is given by assigning to the class of ξ = (ξ 0 , ξ 1 , ξ 01 ) the class of the global ∂-closed form ρ 0 ξ 0 +ρ 1 ξ 1 −∂ρ 0 ∧ξ 01 , where (ρ 0 , ρ 1 ) is a partition of unity subordinate to U.
One can define cup product, integration on top-degree cohomology and Kodaira-Serre duality and they turn out to be compatible with the above isomorphism (cf. [Suw09] for more details).
1.3. Relative Čech-Dolbeault cohomology. Let S be a closed set in X. We set U 0 = X \ S and U 1 to be an open neighbourhood of S in X, and we consider the associated covering U = {U 0 , U 1 } of X. For any p, q, we set where the first map is the inclusion and the second map is the projection on the first element, we obtain a long exact sequence in cohomology (2) Therefore, H •,• D (U, U 0 ) is determined uniquely modulo canonical isomorphism, independently of the choice of U 1 . We denote it also by H •,• D (X, X \S) and we call it the relative Čech-Dolbeault cohomology of X, see [Suw09, Section 2], where it is denoted by H •,• ∂ (X, X \ S). We recall that excision holds: for any neighbourhood U of S in X, it holds H •,• D (X, X \S) ≃ H •,• D (U, U \S). In fact we have, [Suw19], H p,q D (X, X \ S) ≃ H q (X, X \ S; Ω p ), the relative cohomology of the pair (X, X \ S) with coefficients in the sheaf Ω p of holomorphic p-forms.
Together with integration theory, the relative Čech-Dolbeault cohomology has been used to study the localization of characteristic classes, see [Suw09,ABST13], and has found more recent applications to hyperfunction theory, see [HIS18].
Notice that if X andX are complex manifolds, S andS are closed sets in X andX respectively and f :X → X is a holomorphic map such that f (S) ⊂ S and f (X \S) ⊂ f (X \ S), then f induces a natural map in relative cohomology. More precisely, let U 0 := X \ S,Ũ 0 :=X \S and let U 1 ,Ũ 1 be open neighborhoods of S andS in X andX respectively, chosen in such a way that f (Ũ 1 ) ⊂ U 1 . Let U := {U 0 , U 1 } andŨ := Ũ 0 ,Ũ 1 be open coverings of X andX respectively, then we have a morphism which induces a morphism in relative cohomology We consider a holomorphic vector bundle π : E → X of rank k on a complex manifold X and we identify X with the image of the zero section. In this situation we have the Dolbeault-Thom class, ∂-Thom class for short,Ψ E ∈ H k,k D (E, E \ X) and the Dolbeault-Thom morphism, ∂-Thom morphism for short,T E : . They are given as follows, see [ABST13,Suw09]. Consider the fibre product The Dolbeault-Thom morphism The inverse ofT E is given by the ∂-integration along the fibres of π: It is defined as follows. Let T 1 denote a bundle of discs of complex dimension k in W 1 and set T 01 = −∂T 1 , which is a bundle of spheres of real dimension 2k − 1 endowed with the orientation opposite to that of the boundary ∂T 1 of T 1 . Set π 1 = π| T 1 and π 01 = π| T 01 . Then we have the usual integration along the fibres The map (π 1 ) * sends a (p, q)-form to a (p − k, q − k)-form, while, if ξ 01 is a (p, q − 1)-form on W 01 , (π 01 ) * (ξ 01 ) consists of (p − k, q − k) and In this situation,π * •T E = 1. Thusπ * is surjective andT E gives a splitting of 1.5. ∂∂-Lemma and Hodge structures. Although these may be wellknown to experts, we recall what the ∂∂-Lemma means and some alternative ways of saying that for later use. Let X be a complex manifold. (3) We describe the above property in terms of filtrations. Note that A • (X) has two natural filtrations. The first filtration on A h (X) is given by

a Hodge filtration of weight h if and only if
Moreover, if this is the case, there is a canonical isomorphism It is rather straightforward to show the equivalence of two expressions for Hodge filtrations. We only indicate a proof of the last statement for later use. In the sequel we denote The spectral sequence associated with the first filtration of A • (X) is the Frölicher spectral sequence [Frö55], for which we have . A complex manifold X satisfies the ∂∂-Lemma if and only if the following two conditions hold: (1) the Frölicher spectral sequence degenerates at E 1 , (2) the filtration ( ′ F p H h dR (X)) is a Hodge filtration of weight h for every h ≥ 0.

Note that every element of
. The condition dω = 0 implies that ∂ω p,q = 0 when we write ω = p+q i=p ω i,p+q−i . The condition (1) above is equivalent to saying that, for every (p, q), the assignment We ask when these relations carry on to the cohomologies.
Definition 4. 1. We say that X admits a Hodge structure of weight h, if there exist isomorphisms 2. A Hodge structure as above is said to be natural, if the following conditions hold: (H1) Every class in H p,q ∂ (X), p + q = h, admits a representative ω with ∂ω = 0 and ∂ω = 0, i.e., dω = 0. Moreover, the assignment ω →ω induces the first isomorphism above. (H2) Every class in H h dR (X) admits a representative ω which may be written ω = p+q=h ω p,q , where ω p,q is a (p, q)-form with dω p,q = 0. Moreover, the assignment ω → (ω p,q ) p+q=h induces the second isomorphism above.
Proposition 7. A complex manifold X admits a natural Hodge structure of weight h if and only if the following conditions hold : Suppose X admits the natural Hodge structure of weight h. We claim that there is an isomorphism compatible with the one in (H2) in the sense that the following is commutative: compatible with the isomorphism of (H2). It is obviously injective. The surjectivity follows from (H1) and it is the desired isomorphism.
(X) and the correspondence is the one as given in (i).

Dolbeault cohomology of the projectivization of a holomorphic vector bundle
Let X be a smooth manifold. Also let π : V → X be a complex vector bundle of rank k and denote by ρ : P(V ) → X its projectivization. We may regard H • dR (P(V )) as an H • dR (X)-module (in fact, H • dR (X)-algebra). Here we regard it as a right module by our convention and the module structure is given by c · a = c ρ * (a) for c ∈ H • dR (P(V )) and a ∈ H • dR (X), where denotes the cup product. In the sequel it will be simply denoted by ·, if there is no fear of confusion.
In the above situation we have the tautological bundle T on P(V ), which is a rank one subbundle of ρ * V with the universal bundle Q as the quotient so that we have an exact sequence of vector bundles on P(V ): Recall that We recall the following, which is a direct consequence of the Leray-Hirsch theorem (cf. The essential point in the above is that the restriction of γ to each fibre, which is the projective space P k−1 , is the first Chern of the tautological bundle (dual of the hyperplane bundle) on P k−1 and that their powers up to the (k − 1)-st form a C-basis of H • dR (P k−1 ). As an H • dR (X)-algebra, where c k−i (V ) is the (k − i)-th Chern class of V . The relation can be seen from c(T ) · c(Q) = ρ * c(V ), the relation among the total Chern classes, which follows from (7). If we take a metric connection for T , its curvature form κ is of type (1, 1) and is simultaneously d-and ∂-closed. We also haveκ = −κ. The class of √ −1 2π κ in H 2 dR (P(V )) is the first Chern class γ = c 1 (T ) and its class in H 1,1 ∂ (P(V )) is the first Atiyah class a 1 (T ). Note that they cannot be compared directly on the cohomology level, in general. However, their restrictions to each fibre of P(V ) → X may be identified, as the fibre is P k−1 and it satisfies the ∂∂-Lemma. dual to {u p,q i } via the Kodaira-Serre duality, h = h p,q = h n−p,n−q : Obviously {α r · ρ * u p,q i } 0≤r≤k−1, p, q, i span the C-vector space H •,• ∂ (P(V )). We show that they are linearly independent over C, which will prove the claim and the proposition. For this we introduce a relation > in the set Λ of indices λ = (r, p, q, i) by saying that (r 1 , p 1 , q 1 , i 1 ) > (r 2 , p 2 , q 2 , i 2 ) if one of the following holds: (1) 2r 1 + p 1 + q 1 > 2r 2 + p 2 + q 2 , (2) 2r 1 + p 1 + q 1 = 2r 2 + p 2 + q 2 and p 1 + q 1 > p 2 + q 2 , (3) r 1 = r 2 , p 1 + q 1 = p 2 + q 2 and p 1 > p 2 , (4) r 1 = r 2 , p 1 = p 2 , q 1 = q 2 and i 1 > i 2 .
Recalling that dim X = n and dim P(V ) = n + k − 1, we have, in the case (1), (2) or (3), In the case (4), P(V ) v λ · v λ ′ = 0 by a similar computation as above. Thus the Kodaira-Serre dual of the matrix (v λ · v λ ′ ) is triangular with ±1's along the diagonal, which shows that the α r · ρ * u p,q i 's are linearly independent over C.
Corollary 12. Let X be a compact complex manifold and V → X a holomorphic fibre bundle on X. If X satisfies the ∂∂-lemma, so does P(V ).
Proof. The statement follows from Corollary 8 and Propositions 10 and 11, noting that γ and α are both represented by the same form √ −1 2π κ as above.

Hodge structures under blow-ups
We can now prove explcit expressions for the de Rham (Proposition 16) and Dolbeault (Proposition 19) cohomologies of the blow-up and then Theorem 13. Compare also [YY17, Theorem 1.3] for similar results using Bott-Chern cohomology, and [Ste18c, Corollary 25] for a clear statement and argument.
Let X be a compact complex manifold of dimension n and Z a closed complex submanifold of codimension k. Also let τ :X :=X Z → X be the blow-up of X along Z with exceptional divisor E = P(N Z|X ). Here we assume that Z admits a holomorphically contractible neighbourhood (9) that is, there exists U ⊃ Z with r : U → Z holomorphic and r| Z = id. In this case E also admits a holomorphically contractible neighbourhoodŨ ⊃ E with r :Ũ → E holomorphic andr| E = id. Thus we have the following diagram: where the horizontal arrows are the ∂-integrations along the fibres, τ * is the morphism induced by τ andχ is given by z → a k−1 (Q) · τ * E z, see the proof below for details. Here we spend some words to clarify the heavy notation: accordingly with [Suw09], the bar refers to the holomorphic aspects of the theory, while the tilde concerns to the level of the blow-up.
We do not know whether or not the diagram (10) is commutative. The first condition in (11) below is apparently weaker than the commutativity (cf. Remark 20. (5) below).
Theorem 13. Let X be a compact complex manifold and Z a closed submanifold of X. Also let τ :X Z → X be the blow-up of X along Z. Assume that the conditions (9) above and hold. Then, if both X and Z admit a Hodge structure, so doesX Z .
Proof. Algebraic preliminaries. We quote the following lemma, see for instance [Bla56, Lemme II.6]: Lemma 14. Let R be a commutative ring with unity and let be a commutative diagram of R-modules with exact rows such that f 1 is surjective, f 2 and f 5 are injective and f 4 is an isomorphism. Then f 3 is injective and g induces an isomorphism In the above situation, we have the diagram with an exact row: where π is the canonical surjection. If there is a splitting η : B 2 /f 2 A 2 → B 3 , i.e., a morphism with (g) −1 • π • η = id, we have an isomorphism Note that the isomorphism depends on the splitting. In the sequel, we try to express the cohomology ofX in terms of those of X and Z using the above.
de Rham cohomology. Let us start with the de Rham case. Note that, for this case, the assumption (9) (or (11)) is not necessary; for the map r, simply take the one given by the Tubular Neighbourhood Theorem, although it is only smooth that is sufficient.
Considering the exact sequence (1) for the pairs (X, X \Z) and (X,X \E), we have the commutative diagram with exact rows: We study the morphism τ * : We try to express the left hand side in terms of the cohomologies of Z and E and along the way we reprove the injectivity of τ * on the relative cohomology (cf. Remark 20. (1) below). Let π : N := N Z|X → Z be the normal bundle of Z in X. Recall that E is the projectivization P(N ) of N and that τ E := τ | E : E = P(N ) → Z is the projection of the bundle. The normal bundle of E inX is the tautological bundleπ : T → E = P(N ). It is a subbundle of τ * E N with the universal bundle Q as the quotient so that we have an exact sequence of vector bundles on E (cf. (7)): where p and ̟ denote the restrictions of the projections onto the first and the second factors, respectively. Let ϕ : U ∼ → W be a diffeomorphism as given by the Tubular Neighbourhood Theorem, with U and W neighbourhoods of Z in X and N , respectively. We set r = π • ϕ : U → Z. We may choose neighbourhoodsŨ andW of E inX and T and a diffeomorphismφ :Ũ In the above situation, we have where c k−1 (Q) is the top Chern class of Q.
Proof of Lemma 15. Noting that r • τ = τ E •r, we have the exact sequence of vector bundles onŨ : Let s ∆ ands ∆ denote the diagonal sections of π * N on N and ofπ * T on T , respectively. We denote the corresponding sections of r * N on U and ofr * T onŨ by s ands. We claim thats is mapped to τ * s by the first morphism above. To see this, first note that s ∆ (ν) = (ν, ν), where we think of the first component as the fibre component. The section s is given by, for x ∈ U , On the other hand s ∆ (t) = (t, t) ands is given by, forx ∈Ũ ,s( which proves the claim. Recall that Ψ Z is the localization of c k (r * N ) by s so that τ * Ψ Z is the localization of c k (τ * r * N ) by τ * s. The latter can be described as follows. Let ∇ 0 be ans-trivial connection forr * T onŨ 0 and let ∇ Q be a connection for Q on E. Then there exists a τ * s-trivial connection ∇ 0 for τ * r * N onŨ 0 such that (∇ 0 , ∇ 0 ,r * ∇ Q ) is compatible with (14) onŨ 0 . Let∇ 1 be an arbitrary connection forr * T onŨ . Then there exists a connection ∇ 1 for τ * r * N oñ U such that (∇ 1 , ∇ 1 ,r * ∇ Q ) is compatible with (14) onŨ . Then τ * Ψ Z is represented by Since (c 1 (∇ 1 ), c 1 (∇ 0 ,∇ 1 )) represents Ψ E , we have the lemma.
From the above lemma, we see that the following diagram is commutative: where χ is the morphism given by z → c k−1 (Q) τ * E z. In the above T Z and r * are isomorphisms and the inverses of each other, similarly for T E andr * . Thus χ is injective and T E induces an isomorphism Now we study the left hand side. We claim that H • dR (E) is a free H • dR (Z)module with basis 1, γ, . . . , γ k−2 , c k−1 (Q), γ = c 1 (T ). To see this, from (13) we have the relation c(T ) · c(Q) = τ * E c(N ) among the total Chern classes. Thus c(Q) = c(T ) −1 · τ * E c(N ) and we have which proves the claim in view of Proposition 10. Thus we have By (12), (16) and (18), we have the diagram: The restriction of the Gysin morphism (i E ) * = j * • T E : H h−2 dR (E) → H h dR (X) gives a splitting η and we have: Dolbeault cohomology. Considering the exact sequence (2) for the pairs (X, X \ Z) and (X,X \ E), we have the commutative diagram with exact rows: The essential difference from the de Rham case occurs for the relative cohomology and the morphism τ * : H p,q D (X, X \ Z) → H p,q D (X,X \ E), which we are going to analyze. First, it is injective by [Tar19, Theorem 3.1] and Lemma 14 shows that τ * : H p,q ∂ (X) → H p,q ∂ (X) is injective (again this is already implied by [Wel74, Theorem 3.1]) and that j * in the second row induces an isomorphism We try to express the left hand side in terms of cohomologies of Z and E.
Recall that the normal bundle π : N → Z of Z is a holomorphic vector bundle of rank k on Z. By the assumption (9), we see that there exist neighbourhoods U and W of Z in X and N , respectively, and a biholomorphic map ϕ : U → W so that r = π • ϕ : U → Z. Thus we have isomorphisms where the first and the last isomorphisms are excisions. The ∂-Thom class Ψ Z of Z is, by definition, the class in H k,k D (X, X \ Z) that corresponds tō Ψ N by the above isomorphism. We have the ∂-Thom morphism , which is given by z →Ψ Z r * z. It gives a splitting of Under the assumption (9), E also admits a holomorphic retractionr :Ũ → E,Ũ = τ −1 U , such that the following diagram is commutative: , which is given by a →Ψ E r * a. It gives a splitting of We have the following lemma, which is the holomorphic analogue of Lemma 15 and is proven by the same argument with de Rham cohomology and Chern classes are replaced by Dolbeault cohomology and Atiyah classes, respectively: Lemma 17. We have: , where a k−1 (Q) denotes the top Atiyah class of Q.
From the above lemma, we see that the following diagrams are commutative: whereχ is the morphism given by z → a k−1 (Q) τ * E z. From the first commutative diagram above,TZ induces a well-defined morphism . Proposition 18. Under the assumption (11), ψ is an isomorphism.
Remark 20. (1) Even if X and Z admit a natural Hodge structure, i.e., satisfy the ∂∂-Lemma, it is not clear, from the above arguments, whether or notX has the same property. The problem is that the cohomology of E contributes to the cohomology ofX through the Gysin morphisms and it is not clear if these morphisms send good representatives to good ones as in (H1) and (H2) of Definition 4. 2.
(2) In view of the commutative diagram (15), which is a consequence of Lemma 15, the injectivity of τ * on the relative cohomology is equivalent to that of χ. From the definition of χ, we see that this is also equivalent to the injectivity of τ * E . The injectivity of χ can also be proven as follows, independently of the injectivity of τ * .
Recalling that τ E : E = P(N ) → Z is a P k−1 -bundle, we have the integration along the fibres (τ E ) * : This can be seen from (17), the projection formula and the facts that (τ E ) * γ i = 0, for i = 0, . . . , k − 2, by dimension reason, and (τ E ) * γ k−1 = (−1) k−1 , as γ restricted to each fibre is the first Chern class of the tautological bundle on P k−1 . Then by the projection formula and (21), (3) The statement of Proposition 16 is proven in the Kähler context, e.g. in [Voi02,Theorem 7.31] by excision and by the Thom isomorphism in cohomology with Z-coefficients. Presumably, the Kähler condition is necessary there to show that χ or τ * E is injective using the above-mentioned theorem [Wel74, Theorem 4.1]. The novelty here is the elimination of this restriction by a result of [Tar19] or Lemma 15, which also gives a precise relation between the Thom classes of Z and E and this in turn gives a precise relation between τ * and χ.
This subject is treated in the algebraic category in [Ful84,§ 6.7]. (4) In the Dolbeault case, we can show the injectivity ofχ similarly as for χ (cf. (1) above). However this does not directly imply the injectivity of τ * on the relative cohomology. The injectivity ofχ is equivalent to that of . If E is Kähler, the latter again follows from [Wel74, Theorem 4.1]. (5) The condition regarding the holomorphically-contractible neighbourhood of Z in Theorem 13 holds, for example, if Z is a point (see Example 21), or if X is a fibration (e.g. a Hopf manifold) with Z a fibre. (6) The first condition in (11) is implied by the commutativity of the diagram (10), which may be verified for the top degree cohomology using the projection formula.
Example 21 (Blow-up in a point; see also [YY17,Proposition 3.6]). The very particular case when Z is a point is easier, and follows by the description of the Dolbeault cohomology in [GH78]. For completeness we outline the proof in this situation. Let X be a compact complex manifold and consider τ :X → X the blow-up of X on a point p. If X admits a Hodge structure, then alsoX does.
We denote by E = P n−1 = τ −1 (p) the exceptional divisor of the blowup. We recall that the de Rham and Dolbeault cohomologies of X andX are related as follows (see [GH78,): for k / ∈ {0 , 2n}, for (p, q) / ∈ {(0, 0) , (n, n)}, . In particular, h p,p (X) = h p,p (X) + 1 and h p,q (X) = h p,q (X) for p = q. Since, by hypothesis, X satisfies the ∂∂-lemma and E clearly does, we have that and . Question 22. We ask whether a submanifold of a manifold satisfying the ∂∂-Lemma, still satisfies the ∂∂-Lemma. Note that, in general, existence of Hodge structures is not preserved by blow-ups: Claire Voisin suggested to us an example that appears in [Vul12] by Victor Vuletescu: take the blow-up of a Hopf surface inside S 3 × S 3 × P 1 . (Compare also [YY17,Concluding Remarks].) Question 23. We ask whether if X and Z satisfy the ∂∂-Lemma, then we can perform constructions like the deformation to the normal cone for (X, Z) that still satisfies the ∂∂-Lemma. We recall that the deformation to the normal cone by MacPherson [Ful84, Chapter 5] allows to modify the pair (X, Z) to the pair (N Z|X , Z) as deformation, where clearly Z has the property of admitting a holomorphically contractible neighbourhood in its normal bundle N Z|X . We briefly recall the construction, see also [Suw09, Section 8]: consider a 1-dimensional disc D; define X * := Bl Z×{0} (X × D) \ Bl Z×{0} (X × {0}) that provides a deformation path through X * t = X to X * 0 = N Z|X . We notice that N Z|X is clearly non-compact. We also recall that satisfying the  [Wlo03]. It states that f can be functorially factored as a sequence of blow-ups and blow-downs with non-singular centres. For a blowup ϕ :

The orbifold case
We now consider the orbifold case, applying the Stelzig arguments to the orbifold Dolbeault cohomology studied in [Bai54,Bai56]. Recall that an orbifold, also called V-manifold [Sat56], is a singular complex space whose singularities are locally isomorphic to quotient singularities C n /G, where G ⊂ GL(n, C) is a finite subgroup. Tensors on an orbifold are defined to be locally G-invariant. In particular, this yields the notions of orbifold de Rham cohomology and orbifold Dolbeault cohomology, for which we have both a sheaf-theoretic and an analytic interpretation [Sat56,Bai54,Bai56], and Hodge decomposition in cohomology defines the orbifold ∂∂-Lemma property.
The following result generalizes the contents of Theorem 1 to orbifolds of global-quotient type, namely, X/G, where X is a complex manifold and G is a finite group of biholomorphisms of X. We can interpret this case as the smooth case with the further action of a group G: for example, an orbifold morphism Z/H → X/G is just an equivariant map Z → X. The orbifold Dolbeault cohomology of X/G is the cohomology of the complex of Ginvariant forms, (∧ •,• X) G , ∂ . The notion of ∂∂-Lemma for orbifolds refers to the cohomological decomposition for the double complex (∧ •,• X) G , ∂, ∂ . This result follows directly by the work of Jonas Stelzig and it will let us construct new examples of compact complex manifolds satisfying the ∂∂-Lemma, as resolutions of orbifolds obtained starting from compact quotients of solvable Lie groups. (Here, by asking that j o : Z o = Z/G → X o = X/G is a suborbifold, we mean that Z is a G-invariant submanifold of X, and the embedding j : Z → X is G-equivariant.) Theorem 25 (see [Ste18c]). Let X o = X/G be a compact complex orbifold of complex dimension n, and j o : Proof. We first notice thatX Z o itself is a (possibly smooth) orbifold of globalquotient type. Indeed, by the universal property of blow-up, see e.g. [GH78, page 604], the action G X yields the action G X Z the blow-up of X along Z. The proof then follows by considering the E 1 -quasi-isomorphism ∧ •,•X Z ≃ 1 ∧ •,• X ⊕ k−1 j=1 ∧ •−j,•−j Z. This means that there is a morphism of double complexes that induces an isomorphism at the first page E 1 of the Frölicher spectral sequence, that is, the Dolbeault cohomology, see [Ste18c,Definition D]. The fact that there is an E 1 -quasi-isomorphism as above is [Ste18c,Theorem 23], see also [Ste18b]. Since the action of G is compatible with the above morphism, we get also an E 1 -quasi-isomorphism (∧ •,•X Z ) G ≃ 1 (∧ •,• X) G ⊕ k−1 j=1 (∧ •−j,•−j Z) G . Recall that the Dolbeault and the de Rham cohomologies of the orbifold are computed as the cohomologies of the complex of G-invariant forms, as said above. Therefore the properties of Hodge decomposition for X and Z reflects on the property of Hodge decomposition forX Z by means of the above quasi-isomorphism.
Example 26 (resolution of an orbifold covered by the Iwasawa manifold). In this example, starting from a smooth compact complex manifold which does not satisfy the ∂∂-lemma, we construct a simply-connected smooth compact complex manifold that does.
In fact, the same argument as [FM08, Proposition 2.3] adapted to our manifold M , which is a principal 2-torus bundle over a 4-torus, yields that M is simply-connected. Moreover, the metric ω := √ −1 2 3 j=1 ϕ j ∧φ j on M is σ-invariant and so it descends to the orbifold M o . We can also obtainM by blowing-up M and then by quotienting by Z 3 . Therefore, ω yields a balanced metric onM thanks to [AB96]. Finally, we notice thatM is not in class C of Fujiki, since M is not.
Summarizing the contents of the last example: Theorem 27. There exists a simply-connected compact complex non-Kähler manifoldM such that: it is non-Kähler, in fact it does not belong to class C of Fujiki; it satisfies the ∂∂-Lemma; and it is endowed with a balanced metric.