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NOTE ON DOLBEAULT COHOMOLOGY AND HODGESTRUCTURES UP TO BIMEROMORPHISMS

DANIELE ANGELLA, TATSUO SUWA, NICOLETTA TARDINI,AND ADRIANO TOMASSINI

Abstract. We construct a simply-connected compact complex non-Kähler manifold satisfying the ∂∂-Lemma, and endowed with a balancedmetric. To this aim, we were initially aimed at investigating the stabil-ity of the property of satisfying the ∂∂-Lemma under modifications ofcompact complex manifolds and orbifolds. This question has been re-cently addressed and answered in [RYY17, YY17, Ste18b, Ste18c] withdifferent techniques. Here, we provide a different approach using Čechcohomology theory to study the Dolbeault cohomology of the blow-upXZ of a compact complex manifold X along a submanifold Z admittinga holomorphically contractible neighbourhood.

Introduction

The ∂∂-Lemma is a strong cohomological decomposition property definedfor complex manifolds, which is satisfied for example by algebraic projectivemanifolds and, more generally, by compact Kähler manifolds. The propertyis closely related to the fact that the Dolbeault cohomology provides a Hodgestructure on the de Rham cohomology (cf. Subsection 1.5 below).

This property yields also strong topological obstructions: the real ho-motopy type of a compact complex manifold satisfying the ∂∂-Lemma isa formal consequence of its cohomology ring [DGMS75]. Complex non-Kähler manifolds usually do not satisfy the ∂∂-Lemma: for example, itis never satisfied by compact non-tori nilmanifolds [Has89]. On the otherhand, some examples of compact complex non-Kähler manifolds satisfyingthe ∂∂-Lemma are provided by Moišhezon manifolds and manifolds in classC of Fujiki thanks to [DGMS75, Theorem 5.22], see [Hir62] for a concrete

Date: August 19, 2020.2010 Mathematics Subject Classification. 32Q99, 32C35, 32S45.Key words and phrases. complex manifold, non-Kähler geometry, ∂∂-Lemma, Hodge

decomposition, modification, blow-up, Dolbeault cohomology, orbifold.During the preparation of this work, the first-named author has been supported by

the Project SIR2014 “Analytic aspects in complex and hypercomplex geometry” (codeRBSI14DYEB), by the Projects PRIN “Varietà reali e complesse: geometria, topologia eanalisi armonica’ and PRIN2017 “Real and Complex Manifolds: Topology, Geometry andholomorphic dynamics” (code 2017JZ2SW5), and by GNSAGA of INdAM. The second-named author is supported by the JSPS grant no. 16K05116. The third-named authoris supported by Project PRIN “Varietà reali e complesse: geometria, topologia e analisiarmonica”, by SIR2014 project RBSI14DYEB “Analytic aspects in complex and hyper-complex geometry”, and by GNSAGA of INdAM. The fourth-named author is supportedby Project PRIN “Varietà reali e complesse: geometria, topologia e analisi armonica” andby GNSAGA of INdAM.

1

http://arxiv.org/abs/1712.08889v3

2 D. ANGELLA, T. SUWA, N. TARDINI, AND A. TOMASSINI

example. By the results contained in [Cam91, Corollary 3.13], [LP92, The-orem 1] and thanks to the stability property of the ∂∂-Lemma for smalldeformations [Voi02, Proposition 9.21], [Wu06, Theorem 5.12] one can pro-duce examples of compact complex manifolds satisfying the ∂∂-Lemma andnot bimeromorphic to Kähler manifolds. Other examples of this kind can befound among solvmanifolds [AK17a, AK17b, Kas13b]; moreover other exam-ples are provided by Clemens manifolds [Fri91, Fri17], which are constructedby combining modifications and deformations.

The main aim of this note is to construct a simply-connected compactcomplex non-Kähler manifold satisfying the ∂∂-Lemma.

The theorem in [DGMS75, Theorem 5.22] states that, for a modifica-tion X → X of compact complex manifolds, the property of ∂∂-Lemma ispreserved from X to X. So, it is natural to ask whether it is in fact aninvariant property by modifications. This is true, for example, for compactcomplex surfaces, thanks to the topological Lamari’s and Buchdahl’s crite-rion [Lam99, Buc99]. Note that, in higher dimension, the Kähler property isnot stable under modifications; but there are weaker metric properties thatare, 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 con-dition and the cohomological ∂∂-Lemma property are strictly related to eachother, 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 alongsubmanifolds. The strategy we follow is sheaf-theoretic, more precisely Čech-cohomological, in the spirit of [Suw09]. The de Rham case in the Kähler con-text is considered in [Voi02, Theorem 7.31]. For our argument, we need toassume that the centre admits a holomorphically contractible neighbourhood(this is clearly satisfied when blowing-up at a point, see also the explicit com-putations in Example 21) and another technical assumption (11) concerningthe kernel and images of certain morphisms. We can then deduce that:

Theorem 13. Let X be a compact complex manifold and Z a closed sub-manifold of X. If both X and the centre Z admits a Hodge structure (in thesense of Definition 4), then the same holds for the blow-up BlZX of X alongZ, provided that Z admits a holomorphically contractible neighbourhood andthe technical assumption (11) holds.

Along the way we give explicit expressions for the de Rham and Dolbeaultcohomologies of BlZX (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 defor-mation to the normal cone (see Question 23), along with the Weak Factor-ization Theorem for bimeromorphic maps in the complex-analytic category[AKMW02, Theorem 0.3.1], [Wlo03], may allow to use the above techniquesto 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 thesame problems appeared [RYY17, YY17, Ste18b], using different techniques.

DOLBEAULT COHOMOLOGY FOR BIMEROMORPHISMS 3

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 ∂∂-Lemmaproperty is a bimeromorphic invariant if and only if it is invariant by restric-tion.

Even if Stelzig’s theorem is clearly stronger than our Theorem 13, we thinkthat our argument may be interesting and useful in providing a broader pointof view for understanding (Čech-)Dolbeault cohomology.

The second and main aim of this note is to construct new explicit exam-ples of compact complex manifolds satisfying the ∂∂-Lemma: in particular,we provide a simply-connected example, see Example 26. To this aim, weneed to work with orbifolds in the sense of Satake [Sat56], and their desingu-larizations. 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 doesnot satisfy the ∂∂-Lemma; then we quotient it by a finite group of automor-phisms; and then we resolve its singularities. Finally, by Theorem 25, we getsimply-connected examples of complex manifolds satisfying the ∂∂-Lemma:

Theorem 27. There exist a simply-connected compact complex non-Kählermanifold, (not even in class C of Fujiki,) that satisfy the ∂∂-Lemma. Ourexample admits a balanced metric.

As far as we know, these are the first explicit examples of simply-connectedcompact complex non-Kähler manifolds satisfying the ∂∂-Lemma in the lit-erature.

Acknowledgments. The authors warmly thank Giovanni Bazzoni, Carlo Collari,

Hisashi Kasuya, Sheng Rao, Jonas Stelzig, Valentino Tosatti, Claire Voisin, Victor

Vuletescu, Song Yang, Xiangdong Yang, Weiyi Zhang for interesting and useful

discussions on many occasions. Thanks also to the anonymous Referees for some

suggestions and comments that improve the presentation of the paper. Parts of this

work have been written during the visit of the first-named author at the Hokkaido

University and that of the second-named author at Università di Firenze: they

would like to thank the both Departments for the warm hospitality.

1. Preliminaries on Čech-Dolbeault cohomology and ∂∂-Lemma

In this Section, we recall the main definitions and results about relativeČech-de Rham and Čech-Dolbeault cohomologies; for more details and ap-plications we refer to [Suw98] and [Suw09]. We also recall the ∂∂-Lemmaand some of its characterizations.

1.1. Čech-de Rham cohomology and relative de Rham cohomology.Let X be a smooth manifold. Let U = U0, U1 be an open covering of Xand set U01 := U0 ∩U1. Denoting by Ah(U) the space of (C-valued) smoothh-forms on an open set U in X, we set

Ah(U) := Ah(U0)⊕Ah(U1)⊕Ah−1(U01).


The differential operator D : Ah(U)→ Ah+1(U) defined by D (σ0, σ1, σ01) =(dσ0, dσ1, σ1 − σ0 − dσ01) yields a differential complex (A•(U),D): theČech-de Rham cohomology associated to the covering U is then definedby H•

D(U) = kerD/imD. The morphism Ah(X) → Ah(U) given byω 7→ (ω|U0

, ω|U1, 0) induces an isomorphism in cohomology, [Suw98, The-

orem 3.3],H•

dR(X)∼−→ H•

D(U),

whose inverse is given by assigning to the class of σ = (σ0, σ1, σ01) theclass of the global d-closed form ρ0σ0 + ρ1σ1 − dρ0 ∧ σ01, where (ρ0, ρ1) isa partition of unity subordinate to U. In the above H•

dR(X) denotes thede Rham cohomology of X. The de Rham theorem says it is isomorphic toH•(X;C), the simplicial, singular or sheaf cohomology of X with coefficentsin C. See [Suw98] for further results, including cup product, integration ontop-degree cohomology, duality.

Given a closed set S in X, we can take U0 := X \ S and U1 an openneighbourhood of S in X, and the open covering U = U0, U1. In this case,define Ap(U, U0) := σ ∈ Ah(U) | σ0 = 0 = Ah(U1) ⊕ Ah−1(U01). Then(A•(U, U0),D) is a differential sub-complex of (A•(U),D). Let Hh

D(U, U0)denote the associated cohomology. From the short exact sequence

0 −→ A•(U, U0) −→ A•(U) −→ A•(U0) −→ 0,

where the first map is the inclusion and the second map is the projection onthe first element, we obtain a long exact sequence in cohomology

· · · −→ Hh−1dR (U0)

δ−→ HhD(U, U0)

j∗−→ HhD(U)

i∗−→ HhdR(U0) −→ · · · . (1)

From this we see that HhD(U, U0) is determined uniquely modulo canoni-

cal isomorphisms, independently of the choice of U1. We denote it alsoby Hh

D(X,X \ S) and call it the relative Čech-de Rham cohomology. Werecall that excision holds: for any neighbourhood U of S in X, it holdsHh

D(X,X \ S) ≃ HhD(U,U \ S). In fact we have, [Suw08],

HhD(X,X \ S) ≃ Hh(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 fibreproduct

π∗E //

E

π

Eπ

// M

and its diagonal section s∆. The zero-set of s∆ is the image of the zero-section of E, which is identified with M . In this situation, the Thom classΨE ∈ H2k

D (E,E \M) of E is given as the localization of the top Chern classck(π∗E) by s∆. That is: consider the covering W = W0 := E \M,W1of E, where W1 is a neighbourhood of M in E; consider ∇0 a connectionon W0 such that ∇0s∆ = 0, and ∇1 a connection on W1; then the Chernclass ck(π∗E) is represented by

(

ck(∇0), ck(∇1), c

k(∇0,∇1))

∈ H2kD (W) ≃

H2kdR(E), where ck(∇0,∇1) is the Bott difference form of ∇0 and ∇1; in fact,


since ck(∇0) = 0, this defines a class ΨE ∈ H2kD (E,E \M) represented by

(ψ1, ψ01) := (ck(∇1), ck(∇0,∇1)). It turns out that the map

TE : H•−2kdR (M)

∼−→ H•D(E,E \M), [θ] 7→ ΨE ` π∗[θ]

is an isomorphisms, [Suw98, Theorem 5.3], called the Thom isomorphism,where the cup product ΨE ` π∗[θ] is represented by (ψ1 ∧ π∗θ, ψ01 ∧ π∗θ).Its inverse is the integration along the fibres:

π∗ : H•D(E,E \M) −→ H•−2k

dR (M), π∗ (σ1, σ01) = (π1)∗σ1 + (π01)∗σ01,

where π1 is the restriction of π to a bundle T1 of disks of complex dimensionk in W1, and π01 is the restriction of π to the bundle T01 = −∂T1 of spheresof real dimension 2k − 1 with opposite orientation. In particular, the Thomclass ΨE is characterized in H2k

D (E,E \ M) by the property π∗ΨE = 1.Finally, we recall the projection formula, [Suw98, Ch.II, Proposition 5.1]: forσ ∈ Ap(W,W0), θ ∈ Aq(M),

π∗(σ ` π∗θ) = π∗σ ∧ θ.Given a closed complex submanifold Z, of complex codimension k, of

a complex manifold X, of complex dimension n, we can define the Thomisomorphism and the Thom class of Z as follows. Consider the normal bundleπ : NZ|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 NZ|X ,and a smooth diffeomorphism ϕ : U →W such that ϕ|Z = id. Then, settingN = NZ|X , we get isomorphisms

H•D(X,X \ Z) ≃ H•

D(U,U \ Z)∼←−ϕ∗

H•D(W,W \ Z) ≃ H•

D(N,N \ Z).

Define the Thom class ΨZ ∈ H2kD (X,X \ Z) of Z as the image of ΨNZ|X

via the above isomorphisms, and the Thom isomorphism TZ : H•−2kD (Z)

∼→H•

D(X,X \ Z) as TZ(z) = ΨZ ` r∗z, where r = π ϕ : U → Z.

1.2. Čech-Dolbeault cohomology. Let X be a complex manifold and letAp,q(U) be the space of smooth (p, q)-forms on an open set U in X. LetU = U0, U1 be an open covering of X and consider

Ap,q(U) := Ap,q(U0)⊕Ap,q(U1)⊕Ap,q−1(U01).

The differential operator D : Ap,q(U) → Ap,q+1(U) is defined on every ele-ment (ξ0, ξ1, ξ01) ∈ Ap,q(U) by

D (ξ0, ξ1, ξ01) =(

∂ξ0, ∂ξ1, ξ1 − ξ0 − ∂ξ01)

.

The Čech-Dolbeault cohomology associated to the covering U is then definedby H•,•

D(U) = ker D/im D (see [Suw09] where this definition is given for

an arbitrary open covering of the manifold X). The morphism Ap,q(X) →Ap,q(U) given by ω 7→ (ω|U0

ω|U1, 0) induces an isomorphism in cohomology

H•,•∂

(X)∼−→ H•,•

D(U),

where H•,•∂

(X) denotes the Dolbeault cohomology of X, [Suw09, Theo-rem 1.2]. In particular, the definition is independent of the choice of thecovering of X. Moreover, the inverse map is given by assigning to the class


of ξ = (ξ0, ξ1, ξ01) the class of the global ∂-closed form ρ0ξ0+ρ1ξ1−∂ρ0∧ξ01,where (ρ0, ρ1) is a partition of unity subordinate to U.

One can define cup product, integration on top-degree cohomology andKodaira-Serre duality and they turn out to be compatible with the aboveisomorphism (cf. [Suw09] for more details).

1.3. Relative Čech-Dolbeault cohomology. Let S be a closed set in X.We set U0 = X \ S and U1 to be an open neighbourhood of S in X, and weconsider the associated covering U = U0, U1 of X. For any p, q, we set

Ap,q(U, U0) := ξ ∈ Ap,q(U) | ξ0 = 0 = Ap,q(U1)⊕Ap,q−1(U01).

Then(

Ap,•(U, U0), D)

is a subcomplex of(

Ap,•(U), D)

. Let Hp,q

D(U, U0)

be the cohomology associated to(

Ap,•(U, U0), D)

. From the short exactsequence

0 −→ Ap,•(U, U0) −→ Ap,•(U) −→ Ap,•(U0) −→ 0,

where the first map is the inclusion and the second map is the projection onthe first element, we obtain a long exact sequence in cohomology

· · · −→ Hp,q−1

∂(U0)

δ−→ Hp,q

D(U, U0)

j∗−→ Hp,q

D(U)

i∗−→ Hp,q

∂(U0) −→ · · · .

(2)Therefore, H•,•

D(U, U0) is determined uniquely modulo canonical isomor-

phism, independently of the choice of U1. We denote it also by H•,•D

(X,X\S)and we call it the relative Čech-Dolbeault cohomology of X, see [Suw09, Sec-tion 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],

Hp,q

D(X,X \ S) ≃ Hq(X,X \ S; Ωp),

the relative cohomology of the pair (X,X \ S) with coefficients in the sheafΩp of holomorphic p-forms.

Together with integration theory, the relative Čech-Dolbeault cohomologyhas 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 and X are complex manifolds, S and S are closed setsin X and X respectively and f : X → X is a holomorphic map such thatf(S) ⊂ S and f(X \ S) ⊂ f(X \S), then f induces a natural map in relativecohomology. More precisely, let U0 := X \ S, U0 := X \ S and let U1, U1

be open neighborhoods of S and S in X and X respectively, chosen in sucha way that f(U1) ⊂ U1. Let U := U0, U1 and U :=

U0, U1

be open

coverings of X and X respectively, then we have a morphism

f∗ : A•,•(U, U0) −→ A•,•(U, U0)

defined on every element (ξ1, ξ01) ∈ A•,•(U, U0) as

f∗(ξ1, ξ01) := (f∗ξ1, f∗ξ01)


which induces a morphism in relative cohomology

f∗ : H•,•D

(X,X \ S) −→ H•,•D

(X, X \ S) .

1.4. Dolbeault-Thom morphism. We consider a holomorphic vector bun-dle π : E → X of rank k on a complex manifold X and we identify X with theimage of the zero section. In this situation we have the Dolbeault-Thom class,∂-Thom class for short, ΨE ∈ Hk,k

D(E,E \X) and the Dolbeault-Thom mor-

phism, ∂-Thom morphism for short, TE : Hp−k,q−k

∂(X)→ Hp,q

D(E,E \X).

They are given as follows, see [ABST13, Suw09]. Consider the fibre prod-uct

π∗E //

E

π

Eπ

// X.

The bundle : π∗E → E admits the diagonal section s∆, whose zero set isX ⊂ E. The Dolbeault-Thom class ΨE is the localization of the top Atiyahclass ak(π∗E) of π∗E by s∆. More precisely, let W0 = E \X and let W1 bea neighbourhood of X in E, and consider the covering W = W0,W1 of E.For a (1, 0)-connection ∇ for π∗E, we denote by ak(∇) the k-th Atiyah formof ∇, namely, ak(∇) =

(

√−12π

)kσk(K

1,1), where K1,1 is the (1, 1)-componentof the curvature seen as a 2-form with values in Hom(E,E) and σk denotesthe k-th elementary symmetric polynomial, see [Suw09, Section 5] for moredetails. The class ak(π∗E) is represented inHk,k

∂(E) ≃ Hk,k

D(W) by the triple

ak(∇∗) = (ak(∇0), ak(∇1), a

k(∇0,∇1)), where ∇i is a (1, 0)-connection forπ∗E on Wi, i = 0, 1, and ak(∇0,∇1) is the difference form of ∇0 and ∇1. Ifwe take ∇0 to be s∆-trivial, we have the vanishing ak(∇0) = 0 and ak(∇∗)defines a class in Hk,k

D(W,W0) = Hk,k

D(E,E \X) that is the Dolbeault-Thom

class ΨE of E.The Dolbeault-Thom morphism

TE : Hp−k,q−k

∂(X) −→ Hp,q

D(E,E \X).

is given by the cup product with ΨE , i.e. if ΨE is represented by (ψ1, ψ01),it is induced by

θ 7→ (ψ1 ∧ π∗θ, ψ01 ∧ π∗θ).The inverse of TE is given by the ∂-integration along the fibres of π:

π∗ : Hp,q

D(E,E \X) −→ Hp−k,q−k

∂(X).

It is defined as follows. Let T1 denote a bundle of discs of complex dimensionk in W1 and set T01 = −∂T1, which is a bundle of spheres of real dimension2k − 1 endowed with the orientation opposite to that of the boundary ∂T1of T1. Set π1 = π|T1

and π01 = π|T01. Then we have the usual integration

along the fibres

(π1)∗ : Ar(W1) −→ Ar−2k(X) and (π01)∗ : Ar−1(W01) −→ Ar−2k(X).

The map (π1)∗ sends a (p, q)-form to a (p − k, q − k)-form, while, if ξ01 isa (p, q − 1)-form on W01, (π01)∗(ξ01) consists of (p − k, q − k) and (p − k +


1, q − k − 1)-components. We define

(π01)∗ : Ap,q−1(W01) −→ Ap−k,q−k(X)

by taking the (p− k, q − k)-component of (π01)∗(ξ01), then

π∗ξ = (π1)∗ξ1 + (π01)∗ξ01.

In this situation,

π∗ TE = 1.

Thus π∗ is surjective and TE gives a splitting of

0 −→ kerπ∗ −→ Hp,q

D(E,E \X)

π∗−→ Hp−k,q−k

∂(X) −→ 0.

For the ∂-Thom class ΨE ∈ Hk,k

D(E,E \X), we have π∗ΨE = [1] ∈ H0,0

∂(X).

1.5. ∂∂-Lemma and Hodge structures. Although these may be well-known to experts, we recall what the ∂∂-Lemma means and some alternativeways of saying that for later use.

Let X be a complex manifold. The de Rham complex (A•(X), d) of Xis the single complex associated with the double complex (A•,•(X), ∂, ∂),d = ∂ + ∂. Recall that [DGMS75] X satisfies the ∂∂-Lemma if

ker ∂ ∩ ker ∂ ∩ im d = im ∂∂. (3)

We describe the above property in terms of filtrations. Note that A•(X)has two natural filtrations. The first filtration on Ah(X) is given by

′F pAh(X) =h

⊕

i=p

Ai,h−i(X).

It induces a filtration on HhdR(X) by

′F pHhdR(X) = ker dh ∩ ′F pAh(X)/ im dh−1 ∩ ′F pAh(X).

The second filtration on Ah(X) is given by

′′F qAh(X) =

h⊕

j=q

Ah−j,j(X)

and it induces a filtration (′′F qHhdR(X)) on Hh

dR(X).Since Aq,p(X) = Ap,q(X), we may identify the filtration (′F qAh(X)) con-

jugate to (′F qAh(X)) with the second filtration: ′F qAh(X) = ′′F qAh(X),which leads to the identification

′F qHhdR(X) = ′′F qHh

dR(X).

We say that the filtration (′F pHhdR(X)) is a Hodge filtration of weight h if

HhdR(X) =

⊕

p+q=h

′F pHhdR(X) ∩ ′F qHh

dR(X).


Lemma 2. The filtration (′F pHhdR(X)) is a Hodge filtration of weight h if

and only if

HhdR(X) = ′F pHh

dR(X)⊕ ′F qHhdR(X) for every (p, q) with p+ q = h+ 1.

Moreover, if this is the case, there is a canonical isomorphism′F pHh

dR(X) ∩ ′F qHhdR(X) ≃ ′GpHh

dR(X) for every (p, q) with p+ q = h,

where ′GpHhdR(X) = ′F pHh

dR(X)/′F p+1HhdR(X).

Proof. It is rather straightforward to show the equivalence of two expressionsfor Hodge filtrations. We only indicate a proof of the last statement for lateruse. In the sequel we denote Hh

dR(X) by Hh.For c ∈ ′F pHh we denote by [c]p its class in ′GpHh. We define a morphism

′F pHh ∩ ′F qHh −→ ′GpHh by c 7→ [c]p

and show that it is an isomorphism. For the surjectivity, take [c]p ∈′GpHh, c ∈ ′F pHh. Then we may write uniquely c =

∑hi=0 c

i,h−i withci,h−i ∈ ′F iHh ∩ ′F h−iHh. We have [c]p = [c′]p, c′ =

∑pi=0 c

i,h−i. Since∑h

i=p+1 ci,h−i ∈ ′F p+1Hh ⊂ ′F pHh, we have c′ ∈ ′F pHh. On the other hand,

c′ is also in ′F qHh and c′ 7→ [c]p. For the injectivity, take c ∈ ′F pHh ∩ ′F qHh

such that [c]p = 0. This means that c ∈ ′F p+1Hh ∩ ′F qHh = 0.

The spectral sequence associated with the first filtration of A•(X) is theFrölicher spectral sequence [Frö55], for which we have

Ep,q1 ≃ Hp,q

∂(X), Ep,q

∞ ≃ ′GpHp+qdR (X),

Proposition 3 ([DGMS75]). A complex manifold X satisfies the ∂∂-Lemmaif and only if the following two conditions hold:

(1) the Frölicher spectral sequence degenerates at E1,(2) the filtration (′F pHh

dR(X)) is a Hodge filtration of weight h for everyh ≥ 0.

Note that every element of ′GpHp+qdR (X) is expressed as [[ω]]p, where ω

is a d-closed form in ′F pAp+q(X), [ω] is the class of ω in ′F pHp+qdR (X) and

[[ω]]p is the class of [ω] in ′GpHp+qdR (X). The condition dω = 0 implies that

∂ωp,q = 0 when we write ω =∑p+q

i=p ωi,p+q−i.

The condition (1) above is equivalent to saying that, for every (p, q), theassignment [[ω]]p 7→ [ωp,q] is well-defined and induces an isomorphism

′GpHp+qdR (X)

∼−→ Hp,q

∂(X).

Recall that Ap,q(X) = Aq,p(X) and Ah(X) =⊕

p+q=hAp,q(X). We ask

when these relations carry on to the cohomologies.

Definition 4. 1. We say that X admits a Hodge structure of weight h, ifthere exist isomorphisms

Hp,q

∂(X) ≃ Hq,p

∂(X), p+ q = h, and Hh

dR(X) ≃⊕

p+q=h

Hp,q

∂(X).

2. A Hodge structure as above is said to be natural, if the following conditionshold:


(H1) Every class in Hp,q

∂(X), p + q = h, admits a representative ω with

∂ω = 0 and ∂ω = 0, i.e., dω = 0. Moreover, the assignment ω 7→ ωinduces the first isomorphism above.

(H2) Every class in HhdR(X) admits a representative ω which may be

written ω =∑

p+q=h ωp,q, where ωp,q is a (p, q)-form with dωp,q =

0. Moreover, the assignment ω 7→ (ωp,q)p+q=h induces the secondisomorphism above.

Remark 5. In (H1) above, Hq,p

∂(X) denotes the vector space conjugate to

Hq,p

∂(X), i.e., the vector space with underlying set Hq,p

∂(X) and the complex

multiplication given by c · ω = c ω. We may rephrase (H1) as:(H1)′ Every class in Hp,q

∂(X), p + q = h, admits a representative ω with

∂ω = 0 and ∂ω = 0, i.e., dω = 0. Moreover, the assignment ω 7→ ωinduces an isomorphism Hp,q

∂(X) ≃ Hp,q

∂ (X).

Remark 6. See [COUV16, Proposition 4.3] for an example of a compactcomplex manifold with a non-natural Hodge structure.

Proposition 7. A complex manifold X admits a natural Hodge structure ofweight h if and only if the following conditions hold :

(i) the morphism ker d ∩ ′F pAh(X) → Ap,h−p(X), ω 7→ ωp,h−p, inducesan isomorphism ′GpHh

dR(X) ≃ Hp,h−p

∂(X) for every p,

(ii) (′F pHhdR(X)) is a Hodge filtration on Hh

dR(X) of weight h.

Proof. Suppose X admits the natural Hodge structure of weight h. We claimthat there is an isomorphism

′F pHhdR(X) ≃

h⊕

i=p

H i,h−i

∂(X) (4)

compatible with the one in (H2) in the sense that the following is commuta-tive:

′F pHhdR(X)

∼⋂

⊕hi=pH

i,h−i

∂(X)

⋂

HhdR(X)

∼ ⊕hi=0H

i,h−i

∂(X).

For this, take θ ∈ ker dh ∩ ′F pAh(X) and write θ =∑h

i=p θi,h−i with θi,h−i ∈

Ai,h−i(X). From dθ = 0 and (H1), we see that there exist ωi,h−i and αi,h−i

in Ai,h−i(X), p ≤ i ≤ h, such that dωi,h−i = 0 and that

θi,h−i = ωi,h−i + ∂αi−1,h−i + ∂αi,h−i−1,

where we set αp−1,h−p = 0. Then we have

θ =

h∑

i=p

ωi,h−i + d

h∑

i=p

αi,h−i−1.

By (H2), the assignment θ 7→ (ωi,h−i)hi=p induces a well-defined morphism′F pHh

dR(X)→⊕hi=pH

i,h−i

∂(X) compatible with the isomorphism of (H2). It

is obviously injective. The surjectivity follows from (H1) and it is the desiredisomorphism.


From (4), we have ′GpHhdR(X) ≃ Hp,h−p

∂(X) and the correspondence is

the one as given in (i).We also have an isomorphism ′F qHh

dR(X) ≃⊕hj=qH

h−j,j

∂(X) compatible

with the one in (H2). Thus for (p, q) with p + q = h + 1, HhdR(X) =

′F pHhdR(X)⊕ ′F qHh

dR(X) and we have (ii).Now we prove the converse. The condition (ii) implies (cf. Lemma 2)

HhdR(X) =

⊕

p+q=r


dR(X) and (5)


dR(X) ≃ ′GpHhdR(X), h = p+ q. (6)

From the condition (i) and (6), we have

Hp,q

∂(X) ≃ ′GpHh

dR(X) ≃ ′F pHhdR(X) ∩ ′F qHh

dR(X)

≃ ′GqHhdR(X) ≃ Hq,p

∂(X), h = p+ q.

We look at the correspondence above. Take c ∈ ′F pHhdR(X) ∩ ′F qHh

dR(X),we may write c = [ω1] = [ω2], where ω1 =

∑hi=p ω

i,h−i1 ∈ ′F pAh(X), dω1 = 0

(thus ∂ωp,q1 = 0) and ω2 =

∑hj=q ω

h−j,j1 ∈ ′F qAh(X), dω2 = 0 (thus ∂ωp,q

2 =

0). Then the correspondence is given by [ωp,q1 ] ↔ [ωp,q

2 ]. From [ω1] = [ω2],we see that there exist θp,q−1 ∈ Ap,q−1(X) and θp−1,q ∈ Ap−1,q(X) such that

ωp,q1 − ω

p,q2 = ∂θp−1,q + ∂θp,q−1.

Then ω = ωp,q1 −∂θp,q−1 = ωp,q

2 +∂θp−1,q is a representative as in (H1). From(5), (6) and the condition (i), we have (H2).

Corollary 8. A complex manifold satisfies the ∂∂-Lemma if and only if itadmits a natural Hodge structure of weight h for every h.

Remark 9. If we use the Bott-Chern cohomology H•,•BC(X) = ker ∂∩ker ∂

im ∂∂, the

condition (3) means that the morphism

H•,•BC(X) −→ H•

dR(X)

induced by the identity is injective. A numerical characterization of the ∂∂-Lemma in terms of the dimension of the Bott-Chern cohomology and theBetti numbers is provided in [AT13] and in [AT17] using only Bott-Chernnumbers.

2. Dolbeault cohomology of the projectivization of aholomorphic vector bundle

Let X be a smooth manifold. Also let π : V → X be a complex vectorbundle of rank k and denote by ρ : P(V )→ X its projectivization. We mayregard H•

dR(P(V )) as an H•dR(X)-module (in fact, H•

dR(X)-algebra). Herewe regard it as a right module by our convention and the module structureis 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 ·, ifthere is no fear of confusion.


In the above situation we have the tautological bundle T on P(V ), whichis a rank one subbundle of ρ∗V with the universal bundle Q as the quotientso that we have an exact sequence of vector bundles on P(V ):

0 −→ T −→ ρ∗V −→ Q −→ 0. (7)

Recall that ρ∗V = (v, l) ∈ V × P(V ) | π(v) = ρ(l) . We may think of apoint l in P(V ) as a line in Vx = C

k, x = π(v), and we have T = (v, l) ∈ρ∗V | v ∈ l .

We recall the following, which is a direct consequence of the Leray-Hirschtheorem (cf. [GH78, Proposition page 606], [Voi02, Lemma 7.32]):

Proposition 10. In the above situation, H•dR(P(V )) is a free H•

dR(X)-module with basis 1, γ, . . . , γk−1, where γ = c1(T ) is the first Chern classof T .

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 tautologicalbundle (dual of the hyperplane bundle) on P

k−1 and that their powers upto the (k − 1)-st form a C-basis of H•

dR(Pk−1). As an H•

dR(X)-algebra,H•

dR(P(V )) is generated by γ with the single relation

k∑

i=0

(−1)iγi · ρ∗ck−i(V ) = 0, (8)

where ck−i(V ) is the (k − i)-th Chern class of V . The relation can be seenfrom c(T ) ·c(Q) = ρ∗c(V ), the relation among the total Chern classes, whichfollows 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 have κ = −κ. The classof

√−12π κ in H2

dR(P(V )) is the first Chern class γ = c1(T ) and its class inH1,1

∂(P(V )) is the first Atiyah class a1(T ). Note that they cannot be com-

pared directly on the cohomology level, in general. However, their restric-tions to each fibre of P(V ) → X may be identified, as the fibre is P

k−1 andit satisfies the ∂∂-Lemma.

Proposition 11. Let V → X be a holomorphic vector bundle of rank k ona compact complex manifold X. Then H•,•

∂(P(V )) is a free H•,•

∂(X)-module

with basis 1, α, . . . , αk−1, where α = a1(T ) is the first Atiyah class of T .

Proof. By [CFGU00, Lemma 18], we see that H•,•∂

(P(V )) is generated by αas an H•,•

∂(X)-algebra. We have a relation as (8), replacing γ and ck−i(V )

with α and ak−i(V ), the (k− i)-th Atiyah class of V , from which we see that1, α, . . . , αk−1 generate H•,•

∂(P(V )) as an H•,•

∂(X)-module. The proposition

follows from the following:Claim. 1, α, . . . , αk−1 are linearly independent over H•,•

∂(X).

To prove this, we look at the C-algebra structure of H•,•∂

(P(V )). Let n =

dimX and hp,q = dimHp,q

∂(X). For each (p, q) with hp,q 6= 0, we take a basis

up,qi 1≤i≤h of Hp,q

∂(X) so that un−p,n−q

i′ 1≤i′≤h is the basis of Hn−p,n−q

∂(X)


dual to up,qi via the Kodaira-Serre duality, h = hp,q = hn−p,n−q:∫

X

up,qi · un−p,n−qi′ = ±δii′ .

Obviously αr · ρ∗up,qi 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 theclaim and the proposition. For this we introduce a relation > in the set Λof indices λ = (r, p, q, i) by saying that (r1, p1, q1, i1) > (r2, p2, q2, i2) if oneof the following holds:

(1) 2r1 + p1 + q1 > 2r2 + p2 + q2,(2) 2r1 + p1 + q1 = 2r2 + p2 + q2 and p1 + q1 > p2 + q2,(3) r1 = r2, p1 + q1 = p2 + q2 and p1 > p2,(4) r1 = r2, p1 = p2, q1 = q2 and i1 > i2.

With this, Λ becomes a totally ordered set. Let Λ′ denote the set Λ with theorder defined by reversing the inequalities in (1), (2) and (3) and keepingthat in (4) above. We consider the matrix (vλ · vλ′)(λ,λ′)∈Λ×Λ′ , where, forλ = (r, p, q, i), vλ = αr ·ρ∗up,qi and similarly for vλ′ . On the diagonal, we havevλ · vλ′ for which r + r′ = k − 1, p+ p′ = n, q + q′ = n and i = i′ (note thatthis makes sense as p+ p′ = n and q + q′ = n), when we write λ = (r, p, q, i)and λ′ = (r′, p′, q′, i′). In this case, noting that ρ∗αk−1 = (−1)k−1, as therestriction of α to each fibre is the first Atiyah class of the dual of thehyperplane bundle, by the projection formula,

∫

P(V )(αr · ρ∗up,qi ) · (αr′ · ρ∗up′,q′i′ ) = ρ∗α

k−1 ·∫

X

up,qi · un−p,n−qi = ±1.

On the upper triangle, but off the diagonal, we have vλ · vλ′ for which oneof the following holds:

(1) 2r + 2r′ + p+ p′ + q + q′ > 2(n+ k − 1),(2) 2r + 2r′ + p+ p′ + q + q′ = 2(n+ k − 1) and p+ p′ + q + q′ > 2n,(3) r + r′ = k − 1, p+ p′ + q + q′ = 2n and p+ p′ > n,(4) r + r′ = k − 1, p+ p′ = n, q + q′ = n and i < i′.

Recalling that dimX = n and dimP(V ) = n + k − 1, we have, in the case(1), (2) or (3),

(αr · ρ∗up,qi ) · (αr′ · ρ∗up′,q′i′ ) = 0,

by dimension reason. In the case (4),∫

P(V ) vλ · vλ′ = 0 by a similar com-putation as above. Thus the Kodaira-Serre dual of the matrix (vλ · vλ′) istriangular with ±1’s along the diagonal, which shows that the αr · ρ∗up,qi ’sare linearly independent over C.

Corollary 12. Let X be a compact complex manifold and V → X a holo-morphic 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

√−12π κ as above.


3. 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 The-orem 13. Compare also [YY17, Theorem 1.3] for similar results using Bott-Chern cohomology, and [Ste18c, Corollary 25] for a clear statement andargument.

Let X be a compact complex manifold of dimension n and Z a closedcomplex submanifold of codimension k. Also let τ : X := XZ → X be theblow-up ofX along Z with exceptional divisor E = P(NZ|X). Here we assumethat

Z admits a holomorphically contractible neighbourhood (9)that is, there exists U ⊃ Z with r : U → Z holomorphic and r|Z = id. In thiscase E also admits a holomorphically contractible neighbourhood U ⊃ E withr : U → E holomorphic and r|E = id. Thus we have the following diagram:

Hp−k,q−k

∂(Z)

χ

Hp,q

D(X,X \ Z)

τ∗

r∗oo

Hp−1,q−1

∂(E) Hp,q

D(X, X \ E),

¯r∗oo

(10)

where the horizontal arrows are the ∂-integrations along the fibres, τ∗ is themorphism induced by τ and χ is given by z 7→ ak−1(Q) · τ∗

Ez, 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 thetheory, while the tilde concerns to the level of the blow-up.

We do not know whether or not the diagram (10) is commutative. Thefirst 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 sub-manifold of X. Also let τ : XZ → X be the blow-up of X along Z. Assumethat the conditions (9) above and

im ¯r∗ τ∗ ⊂ im χ, ker ¯r∗ ⊂ im τ∗ (11)

hold. Then, if both X and Z admit a Hodge structure, so does XZ .

Proof. Algebraic preliminaries. We quote the following lemma, see for in-stance [Bla56, Lemme II.6]:

Lemma 14. Let R be a commutative ring with unity and let

A1//

f1

A2//

f2

A3//

f3

A4//

f4≀

A5

f5

B1// B2

g// B3

// B4// B5

be a commutative diagram of R-modules with exact rows such that f1 issurjective, f2 and f5 are injective and f4 is an isomorphism. Then f3 isinjective and g induces an isomorphism

g : B2/f2A2∼−→ B3/f3A3.


In the above situation, we have the diagram with an exact row:

0 // A3f3

// B3π

// B3/f3A3// 0

B2/f2A2,

≀ g

OO

η

ee

where π is the canonical surjection. If there is a splitting η : B2/f2A2 → B3,i.e., a morphism with (g)−1 π η = id, we have an isomorphism

A3 ⊕ (B2/f2A2)∼−→ B3, (a, [b]) 7→ f3(a) + η([b]).

Note that the isomorphism depends on the splitting.In the sequel, we try to express the cohomology of X in terms of those of

X and Z using the above.

de Rham cohomology. Let us start with the de Rham case. Note that, forthis case, the assumption (9) (or (11)) is not necessary; for the map r, simplytake the one given by the Tubular Neighbourhood Theorem, although it isonly 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:

Hh−1dR (X \ Z) δ

//

τ∗≀

HhD(X,X \ Z)

j∗//

τ∗

HhdR(X)

i∗//

τ∗

HhdR(X \ Z)

δ//

τ∗≀

Hh+1D (X,X \ Z)

τ∗

Hh−1dR (X \ E) δ

// HhD(X, X \ E)

j∗// Hh

dR(X)i∗

// HhdR(X \ E)

δ// Hh+1

D (X, X \ E).

We study the morphism τ∗ : H•D(X,X \Z)→ H•

D(X, X \E) more closely.First, it is injective by [Tar19, Theorem 3.2] and Lemma 14 shows thatτ∗ : Hh

dR(X)→ HhdR(X) is injective (in fact this is already implied by [Wel74,

Theorem 3.1]) and that j∗ in the second row induces an isomorphism

HhD(X, X \ E)/τ∗Hh

D(X,X \ Z)∼−→ Hh

dR(X)/τ∗HhdR(X). (12)

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 := NZ|X → Z be the normal bundle of Z in X. Recall that E

is the projectivization P(N) of N and that τE := τ |E : E = P(N)→ Z is theprojection of the bundle. The normal bundle of E in X is the tautologicalbundle π : T → E = P(N). It is a subbundle of τ∗

EN with the universal bundle

Q as the quotient so that we have an exact sequence of vector bundles on E

(cf. (7)):

0 −→ Tι−→ τ∗

EN −→ Q −→ 0. (13)

Recall that τ∗EN = (ν, e) ∈ N × E | π(ν) = τE(e) so that we have the

commutative diagram

E

τE

τ∗EN

oo

p

Z N,π

oo


where p and denote the restrictions of the projections onto the first andthe second factors, respectively.

Let ϕ : U∼→ W be a diffeomorphism as given by the Tubular Neighbour-

hood Theorem, with U andW neighbourhoods of Z inX andN , respectively.We set r = π ϕ : U → Z. We may choose neighbourhoods U and W ofE in X and T and a diffeomorphism ϕ : U

∼→ W so that U = τ−1U andϕ τ (ϕ)−1 : W →W is equal to p ι|W . We set r = π ϕ : U → E so thatwe have the commutative diagram

E

τE

Ur

oo

τ |U

Z U.r

oo

We have the Thom class ΨZ ∈ H2kD (X,X \ Z) = H2k

D (U,U \ Z) of Z andthat ΨE ∈ H2

D(X, X \ E) = H2D(U , U \ E) of E.

Lemma 15. In the above situation, we have

τ∗ΨZ = ΨE ` r∗ck−1(Q),

where ck−1(Q) is the top Chern class of Q.

Proof of Lemma 15. Noting that r τ = τE r, we have the exact sequenceof vector bundles on U :

0 −→ r∗T −→ τ∗r∗N −→ r∗Q −→ 0. (14)

Let s∆ and s∆ denote the diagonal sections of π∗N on N and of π∗T on T ,respectively. We denote the corresponding sections of r∗N on U and of r∗Ton U by s and s. We claim that s is mapped to τ∗s by the first morphismabove. To see this, first note that s∆(ν) = (ν, ν), where we think of the firstcomponent as the fibre component. The section s is given by, for x ∈ U ,s(x) = ϕ(x) ∈ (r∗N)x = Nz, z = r(x) = π ϕ(x). On the other hands∆(t) = (t, t) and s is given by, for x ∈ U , s(x) = ϕ(x) ∈ (r∗T )x = Te,e = r(x) = π ϕ(x). We have τ∗s(x) = s(τ(x)) = ϕ τ(x) = p ι ϕ(x),which proves the claim.

Recall that ΨZ is the localization of ck(r∗N) by s so that τ∗ΨZ is thelocalization of ck(τ∗r∗N) by τ∗s. The latter can be described as follows. Let∇0 be an s-trivial connection for r∗T on U0 and let ∇Q be a connection forQ on E. Then there exists a τ∗s-trivial connection ∇0 for τ∗r∗N on U0 suchthat (∇0,∇0, r

∗∇Q) is compatible with (14) on U0. Let ∇1 be an arbitraryconnection for r∗T on U . Then there exists a connection ∇1 for τ∗r∗N onU such that (∇1,∇1, r

∗∇Q) is compatible with (14) on U . Then τ∗ΨZ isrepresented by

(ck(∇1), ck(∇0,∇1)) = (c1(∇1) · r∗ck−1(∇Q), c1(∇0, ∇1) · r∗ck−1(∇Q).

Since (c1(∇1), c1(∇0, ∇1)) represents ΨE, we have the lemma.


From the above lemma, we see that the following diagram is commutative:

Hh−2kdR (Z)

χ

TZ// Hh

D(X,X \ Z)r∗oo

τ∗

Hh−2dR (E)

TE// Hh

D(X, X \ E),r∗

oo

(15)

where χ is the morphism given by z 7→ ck−1(Q) ` τ∗Ez. In the above TZ and

r∗ are isomorphisms and the inverses of each other, similarly for TE and r∗.Thus χ is injective and TE induces an isomorphism

Hh−2dR (E)/χHh−2k

dR (Z)∼−→ Hh

D(X, X \ E)/τ∗HhD(X,X \ Z). (16)

Now we study the left hand side. We claim that H•dR(E) is a free H•

dR(Z)-module with basis 1, γ, . . . , γk−2, ck−1(Q), γ = c1(T ). To see this, from (13)we have the relation c(T ) · c(Q) = τ∗

Ec(N) among the total Chern classes.

Thus c(Q) = c(T )−1 · τ∗Ec(N) and we have

ck−1(Q) =k−2∑

i=0

(−1)iγi · τ∗Eck−1−i(N) + (−1)k−1γk−1, (17)

which proves the claim in view of Proposition 10. Thus we have

Hh−2dR (E)/χHh−2k

dR (Z) ≃k−2⊕

i=0

γi · τ∗EHh−2i−2

dR (Z) ⊂ Hh−2dR (E). (18)

By (12), (16) and (18), we have the diagram:

0 // HhdR(X)

τ∗// Hh

dR(X)π

// HhdR(X)/τ∗Hh

dR(X) // 0

⊕k−2i=0 γ

i · τ∗EHh−2i−2

dR (Z).

≀OO

ηii

The restriction of the Gysin morphism (iE)∗ = j∗ TE : Hh−2dR (E)→ Hh

dR(X)gives a splitting η and we have:

Proposition 16. There is an isomorphism

HhdR(X)⊕

k−2⊕

i=0

Hh−2i−2dR (Z)

∼−→ HhdR(X),

which is given by (x, (zi)k−2i=0 ) 7→ τ∗x+

∑k−2i=0 (iE)∗(γ

i · τ∗Ezi) for x ∈ Hh

dR(X)

and zi ∈ Hh−2i−2dR (Z).

Dolbeault cohomology. Considering the exact sequence (2) for the pairs(X,X \ Z) and (X, X \ E), we have the commutative diagram with exactrows:

Hp,q−1

∂(X \ Z) δ

//

τ∗≀

Hp,q

D(X,X \ Z) j∗

//

τ∗

Hp,q

∂(X)

i∗//

τ∗

Hp,q

∂(X \ Z) δ

//

τ∗≀

Hp,q+1D

(X,X \ Z)

τ∗

Hp,q−1

∂(X \ E) δ

// Hp,q

D(X, X \ E) j∗

// Hp,q

∂(X)

i∗// Hp,q

∂(X \ E) δ

// Hp,q+1D

(X, X \ E).


The essential difference from the de Rham case occurs for the relativecohomology and the morphism τ∗ : Hp,q

D(X,X \Z)→ Hp,q

D(X, X \E), which

we are going to analyze. First, it is injective by [Tar19, Theorem 3.1] andLemma 14 shows that τ∗ : Hp,q

∂(X) → Hp,q

∂(X) is injective (again this is

already implied by [Wel74, Theorem 3.1]) and that j∗ in the second rowinduces an isomorphism

Hp,q

D(X, X \ E)/τ∗Hp,q

D(X,X \ Z) ∼−→ Hp,q

∂(X)/τ∗Hp,q

∂(X). (19)

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 neigh-bourhoods U and W of Z in X and N , respectively, and a biholomorphicmap ϕ : U →W so that r = π ϕ : U → Z. Thus we have isomorphisms

Hp,q

D(X,X \ Z) ≃ Hp,q

D(U,U \ Z) ∼←−

ϕ∗Hp,q

D(W,W \ Z) ≃ Hp,q

D(N,N \ Z),

where the first and the last isomorphisms are excisions. The ∂-Thom classΨZ of Z is, by definition, the class in Hk,k

D(X,X \ Z) that corresponds to

ΨN by the above isomorphism. We have the ∂-Thom morphism

TZ : Hp−k,q−k

∂(Z) −→ Hp,q

D(U,U \ Z) = Hp,q

D(X,X \ Z),

which is given by z 7→ ΨZ ` r∗z. It gives a splitting of

0 −→ ker r∗ −→ Hp,q

D(X,X \ Z) r∗−→ Hp−k,q−k

∂(Z) −→ 0.

Under the assumption (9), E also admits a holomorphic retraction r : U →E, U = τ−1U , such that the following diagram is commutative:

E

τE

Ur

oo

τ |U

Z U.r

oo

Thus we have the ∂-Thom class ΨE ∈ H1,1D

(X, X \ E) = H1,1D

(U , U \ E) of Eand the ∂-Thom morphism

TE : Hp−1,q−1

∂(E) −→ Hp,q

D(U , U \ Z) = Hp,q

D(X, X \ E),

which is given by a 7→ ΨE ` r∗a. It gives a splitting of

0 −→ ker ¯r∗ −→ Hp,q

D(X, X \ E) ¯r∗−→ Hp−1,q−1

∂(E) −→ 0.

We have the following lemma, which is the holomorphic analogue ofLemma 15 and is proven by the same argument with de Rham cohomol-ogy and Chern classes are replaced by Dolbeault cohomology and Atiyahclasses, respectively:

Lemma 17. We have:

τ∗ΨZ = ΨE ` r∗ak−1(Q),

where ak−1(Q) denotes the top Atiyah class of Q.


From the above lemma, we see that the following diagrams are commuta-tive:

Hp−k,q−k

∂(Z)

TZ//

χ

Hp,q

D(X,X \ Z)

τ∗

Hp−1,q−1

∂(E)

TE// Hp,q

D(X, X \ E),

Hp−k,q−k

∂(Z)

TZ//

χ

Hp,q

D(X,X \ Z)

τ∗

Hp−1,q−1

∂(E) Hp,q

D(X, X \ E),

¯r∗oo

where χ is the morphism given by z 7→ ak−1(Q) ` τ∗Ez.

From the first commutative diagram above, TZ induces a well-definedmorphism

ψ : Hp−1,q−1

∂(E)/χHp−k,q−k

∂(Z) −→ Hp,q

D(X, X \ E)/τ∗Hp,q

D(X,X \ Z).

Proposition 18. Under the assumption (11), ψ is an isomorphism.

Proof of Proposition 18. For the surjectivity, take [c], c ∈ Hp,q

D(X, X \ E).

We may write c = TZ(a) + ρ, ρ ∈ ker ¯r∗. Thus by the second condition in(11), ψ([a]) = [c]. For the injectivity, take [a] such that TZ(a) = τ∗(c) forsome c ∈ Hp,q

D(X,X \ Z). Then a = ¯r∗ TZ(a) = ¯r∗ τ∗(c) and by the first

condition in (11), a ∈ im χ.

As in the case of de Rham, we see that H•,•∂

(E) is a free H•,•∂

(Z)-modulewith basis 1, α, . . . , αk−2, ak−1(Q), α = a1(T ). Thus we have

Hp−1,q−1

∂(E)/χHp−k,q−k

∂(Z) ≃

k−2⊕

i=0

αi · τ∗EHp−i−1,q−i−1

∂(Z) ⊂ Hp−1,q−1

∂(E).

(20)Under the assumption (11), the restriction of the ∂-Gysin morphism

(iE)∗ = j∗ TE : Hp−1.q−1

∂(E)→ Hp,q

∂(X) gives a splitting η:

0 // Hp,q

∂(X)

τ∗// Hp,q

∂(X)

π// Hp,q

∂(X)/τ∗Hp,q

∂(X) // 0

⊕k−2i=0 α

i · τ∗EHp−i−1,q−i−1

∂(Z).

≀OOη

jj

and we have:

Proposition 19. Under the assumption (11), there is an isomorphism

Hp,q

∂(X) ⊕

k−2⊕

i=0

Hp−i−1,q−i−1

∂(Z)

∼−→ Hp,q

∂(X),

which is given by (x, (zi)k−2i=0 ) 7→ τ∗x+

∑k−2i=0 (iE)∗(α

i · τ∗Ezi) for x ∈ Hp,q

∂(X)

and zi ∈ Hp−i−1,q−i−1

∂(Z).

The theorem follows from Propositions 16 and 19 (cf. Definition 4. 1).

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, whetheror not X has the same property. The problem is that the cohomology of Econtributes to the cohomology of X 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 ofLemma 15, the injectivity of τ∗ on the relative cohomology is equivalentto that of χ. From the definition of χ, we see that this is also equivalentto the injectivity of τ∗

E. The injectivity of χ can also be proven as follows,

independently of the injectivity of τ∗.Recalling that τE : E = P(N) → Z is a P

k−1-bundle, we have the integra-tion along the fibres (τE)∗ : H

h−2dR (E) → Hh−2k

dR (Z). First we claim that, for(τE)∗ : H

2k−2dR (E)→ H0

dR(Z), we have

(τE)∗ck−1(Q) = 1. (21)

This can be seen from (17), the projection formula and the facts that(τE)∗γi = 0, for i = 0, . . . , k − 2, by dimension reason, and (τE)∗γk−1 =(−1)k−1, as γ restricted to each fibre is the first Chern class of the tautolog-ical bundle on P

k−1. Then by the projection formula and (21),

(τE)∗ χ(a) = (τE)∗(ck−1(Q) ` τ∗

E(a)) = (τE)∗c

k−1(Q) ` a = a.

Thus the composition (τE)∗ χ is the identity morphism of Hh−2kdR (Z) so

that χ is injective. Thus τ∗E

is also injective. If E is Kähler, this follows from[Wel74, Theorem 4.1].(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 coho-mology with Z-coefficients. Presumably, the Kähler condition is necessarythere to show that χ or τ∗

Eis injective using the above-mentioned theorem

[Wel74, Theorem 4.1]. The novelty here is the elimination of this restrictionby a result of [Tar19] or Lemma 15, which also gives a precise relation be-tween the Thom classes of Z and E and this in turn gives a precise relationbetween 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 of χ similarly as forχ (cf. (1) above). However this does not directly imply the injectivity ofτ∗ on the relative cohomology. The injectivity of χ is equivalent to that ofτ∗E: Hp−k,q−k

∂(Z) → Hp−k,q−k

∂(E). If E is Kähler, the latter again follows

from [Wel74, Theorem 4.1].(5) The condition regarding the holomorphically-contractible neighbourhoodof Z in Theorem 13 holds, for example, if Z is a point (see Example 21), orif 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 dia-gram (10), which may be verified for the top degree cohomology using theprojection formula.

Example 21 (Blow-up in a point; see also [YY17, Proposition 3.6]). Thevery particular case when Z is a point is easier, and follows by the descriptionof the Dolbeault cohomology in [GH78]. For completeness we outline theproof in this situation. Let X be a compact complex manifold and considerτ : X → X the blow-up of X on a point p. If X admits a Hodge structure,then also X does.


We denote by E = Pn−1 = τ−1(p) the exceptional divisor of the blow-

up. We recall that the de Rham and Dolbeault cohomologies of X and Xare related as follows (see [GH78, pages 473-474]): for k /∈ 0 , 2n, for(p, q) /∈ (0, 0) , (n, n),

H•(X,C) = H•(X,C)⊕H•(E,C)

andH•,•

∂(X) = H•,•

∂(X)⊕H•,•

∂(E).

In particular, hp,p(X) = hp,p(X)+1 and hp,q(X) = hp,q(X) for p 6= q. Since,by hypothesis, X satisfies the ∂∂-lemma and E clearly does, we have that

Hk(X,C) = Hk(X,C)⊕Hk(E,C)

=⊕

p+q=k

Hp,q

∂(X)⊕

⊕

r+s=k

Hr,s

∂(E)

=⊕

t+v=k

(

Ht,v

∂(X)⊕Ht,v

∂(E)

)

=⊕

t+v=k

Ht,v

∂(X)

and

Hp,q

∂(X) = Hp,q

∂(X)⊕Hp,q

∂(E)

= Hq,p

∂(X)⊕Hq,p

∂(E) = Hq,p

∂(X).

Question 22. We ask whether a submanifold of a manifold satisfying the∂∂-Lemma, still satisfies the ∂∂-Lemma. Note that, in general, existence ofHodge structures is not preserved by blow-ups: Claire Voisin suggested to usan example that appears in [Vul12] by Victor Vuletescu: take the blow-upof a Hopf surface inside S

3 × S3 × P

1. (Compare also [YY17, ConcludingRemarks].)

Question 23. We ask whether if X and Z satisfy the ∂∂-Lemma, then we canperform constructions like the deformation to the normal cone for (X,Z) thatstill satisfies the ∂∂-Lemma. We recall that the deformation to the normalcone by MacPherson [Ful84, Chapter 5] allows to modify the pair (X,Z)to the pair (NZ|X , Z) as deformation, where clearly Z has the propertyof admitting a holomorphically contractible neighbourhood in its normalbundle NZ|X . We briefly recall the construction, see also [Suw09, Section 8]:consider a 1-dimensional disc D; define X∗ := BlZ×0(X×D)\BlZ×0(X×0) that provides a deformation path through X∗

t = X to X∗0 = NZ|X . We

notice that NZ|X is clearly non-compact. We also recall that satisfying the∂∂-Lemma is an open property under deformations [Voi02, Proposition 9.21],but in general it is not closed [AK17b].

Remark 24. If Questions 23 and 22 have positive answers, if we can avoid thetechnical assumption (11), and if we can prove the naturality of the inducedHodge structures on the blow-up, then our argument would give that the ∂∂-Lemma property is defined inside the localization of the category of holo-morphic maps with respect to bimeromorphisms, equivalently, modifications.More precisely: Let f : M → N be a bimeromorphic map between compactcomplex manifolds of the same dimension. Then M satisfies the ∂∂-Lemmaif and only if N does. (The same would be true assuming M Kähler without


Question 22.) Indeed, this will follow by the Weak Factorization Theoremfor bimeromorphic maps between compact complex manifolds [AKMW02,Theorem 0.3.1], [Wlo03]. It states that f can be functorially factored as asequence of blow-ups and blow-downs with non-singular centres. For a blow-up ϕ : V ′ → V ′′, we have that: if V ′ satisfies the ∂∂-Lemma, then V ′′ doesby [DGMS75, Theorem 5.22]; if V ′′ satisfies the ∂∂-Lemma, then V ′ does byTheorem 13. Compare [RYY17, Question 1.2] and [Ste18c, Corollary 28].

4. The orbifold case

We now consider the orbifold case, applying the Stelzig arguments to theorbifold Dolbeault cohomology studied in [Bai54, Bai56]. Recall that anorbifold, also called V-manifold [Sat56], is a singular complex space whosesingularities are locally isomorphic to quotient singularities C

n/G, whereG ⊂ GL(n,C) is a finite subgroup. Tensors on an orbifold are defined tobe locally G-invariant. In particular, this yields the notions of orbifold deRham cohomology and orbifold Dolbeault cohomology, for which we haveboth a sheaf-theoretic and an analytic interpretation [Sat56, Bai54, Bai56],and Hodge decomposition in cohomology defines the orbifold ∂∂-Lemmaproperty.

The following result generalizes the contents of Theorem 1 to orbifolds ofglobal-quotient type, namely, X/G, where X is a complex manifold and G isa finite group of biholomorphisms of X. We can interpret this case as thesmooth case with the further action of a group G: for example, an orbifoldmorphism Z/H → X/G is just an equivariant map Z → X. The orb-ifold Dolbeault cohomology of X/G is the cohomology of the complex of G-invariant forms,

(

(∧•,•X)G, ∂)

. The notion of ∂∂-Lemma for orbifolds refersto the cohomological decomposition for the double complex

(

(∧•,•X)G, ∂, ∂)

.This result follows directly by the work of Jonas Stelzig and it will let usconstruct new examples of compact complex manifolds satisfying the ∂∂-Lemma, as resolutions of orbifolds obtained starting from compact quotientsof solvable Lie groups. (Here, by asking that jo : Zo = Z/G → Xo = X/Gis a suborbifold, we mean that Z is a G-invariant submanifold of X, and theembedding j : Z → X is G-equivariant.)

Theorem 25 (see [Ste18c]). Let Xo = X/G be a compact complex orbifold ofcomplex dimension n, and jo : Zo = Z/G → Xo be a suborbifold of complexdimension d and codimension k := n − d, and consider τ o : XZo → Xo theblow-up of Xo along the centre Zo. If both Xo and Zo satisfy the ∂∂-Lemma,then also XZo does satisfy the ∂∂-Lemma.

Proof. We first notice that XZo itself is a (possibly smooth) orbifold of global-quotient type. Indeed, by the universal property of blow-up, see e.g. [GH78,page 604], the action G X yields the action G XZ the blow-up of Xalong Z. The proof then follows by considering the E1-quasi-isomorphism∧•,•XZ ≃1 ∧•,•X ⊕

⊕k−1j=1 ∧•−j,•−jZ. This means that there is a mor-

phism of double complexes that induces an isomorphism at the first pageE1 of the Frölicher spectral sequence, that is, the Dolbeault cohomology, see[Ste18c, Definition D]. The fact that there is an E1-quasi-isomorphism asabove is [Ste18c, Theorem 23], see also [Ste18b]. Since the action of G is


compatible with the above morphism, we get also an E1-quasi-isomorphism(∧•,•XZ)

G ≃1 (∧•,•X)G ⊕⊕k−1j=1(∧•−j,•−jZ)G. Recall that the Dolbeault

and the de Rham cohomologies of the orbifold are computed as the coho-mologies of the complex of G-invariant forms, as said above. Therefore theproperties of Hodge decomposition for X and Z reflects on the property ofHodge decomposition for XZ by means of the above quasi-isomorphism.

Example 26 (resolution of an orbifold covered by the Iwasawa manifold). Inthis example, starting from a smooth compact complex manifold which doesnot satisfy the ∂∂-lemma, we construct a simply-connected smooth compactcomplex manifold that does.

Consider the complex Heisenberg group

G :=

1 z1 z30 1 z20 0 1

: z1, z2, z3 ∈ C

.

It is a nilpotent Lie group, and it is endowed with a bi-invariant complexstructure defined by the coframe of (1, 0)-forms

ϕ1 := dz1, ϕ2 := dz2, ϕ3 := dz3 − z1dz2.They have structure equations

dϕ1 = 0, dϕ2 = 0, dϕ3 = −ϕ1 ∧ ϕ2.

Let ξ 6= 1 be a cubic root of the unity, and Λ be the lattice generated by1 and ξ. Consider the subgroup Γ in G consisting of matrices with entriesin Λ. The compact quotient M := G/Γ is a holomorphically-parallelizablenilmanifold [Nak75]. By [Nom54, Sak76], the de Rham and Dolbeault coho-mologies of M are the same as the cohomologies of the Iwasawa manifold,which are computed for instance in [Sch07].

We consider the following action of the finite group Z3 on G:

σ : (z1, z2, z3) 7→ (ξz1, ξz2, ξ2z3).

It is easy to check that the action is linear, and since ξ2 = −1− ξ then Γ isσ-invariant. Therefore we get an action on the quotient M , and a complexspace Mo := M/ 〈σ〉 with orbifold singularities. The action on the globalco-frame of (1, 0)-forms becomes

σ∗(ϕ1) = ξϕ1, σ∗(ϕ2) = ξϕ2, σ∗(ϕ3) = ξ2ϕ3.

We compute the orbifold de Rham and Dolbeault cohomologies by takingthe σ-invariant forms. We have

∧•Mo = (∧•M)〈σ〉

= ∧〈1, ϕ13, ϕ23, ϕ11, ϕ12, ϕ21, ϕ22, ϕ33, ϕ13, ϕ23, ϕ123, ϕ312〉as an algebra, with the only non-trivial differentials

dϕ33 = ϕ123 − ϕ312, dϕ123 = dϕ312 = ϕ1212.

It is straightforward to check that this complex satisfies the ∂∂-Lemma, thatis, the orbifold Mo satisfies the ∂∂-lemma.

Now we resolve the singularities of Mo in order to obtain a simply-connected smooth compact complex manifold satisfying the ∂∂-lemma. The


procedure is similar to the one described in [FM08]. The singular locus ofMo consists of 33 isolated singular points. By blowing-up each point, we getan exceptional divisor CP

2/Z3, where the action is given by

σ : [z1 : z2 : z3] 7→ [z1 : z2 : ξz3].

The singular locus consists now of the isolated point q := [0 : 0 : 1] andof the complex projective line L := [z1 : z2 : 0] ⊂ CP

2, which both admita holomorphically contractible neighbourhood. Finally, by blowing-up theq’s and the L’s, we get a smooth model M . Thanks to Theorem 25, theperformed operations mantain the ∂∂-Lemma property.

In fact, the same argument as [FM08, Proposition 2.3] adapted to ourmanifold M , which is a principal 2-torus bundle over a 4-torus, yields thatM is simply-connected. Moreover, the metric

ω :=

√−12

3∑

j=1

ϕj ∧ ϕj

on M is σ-invariant and so it descends to the orbifold Mo. We can alsoobtain M by blowing-up M and then by quotienting by Z3. Therefore, ωyields a balanced metric on M thanks to [AB96].

Finally, we notice that M 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-Kählermanifold M such that: it is non-Kähler, in fact it does not belong to classC of Fujiki; it satisfies the ∂∂-Lemma; and it is endowed with a balancedmetric.

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(Daniele Angella) Dipartimento di Matematica e Informatica “Ulisse Dini”,Università degli Studi di Firenze, viale Morgagni 67/a, 50134 Firenze, Italy

E-mail address: [email protected]


(Tatsuo Suwa) Department of Mathematics, Hokkaido University, Sapporo060-0810, Japan


(Nicoletta Tardini) Dipartimento di Matematica e Informatica “Ulisse Dini”,Università degli Studi di Firenze, viale Morgagni 67/a, 50134 Firenze, Italy


Current address: Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Unitàdi Matematica e Informatica, Università degli Studi di Parma, Parco Area delle Scienze53/A, 43124 Parma, Italy

(Adriano Tomassini) Dipartimento di Scienze Matematiche, Fisiche e Infor-matiche, Unità di Matematica e Informatica, Università degli Studi di Parma,Parco Area delle Scienze 53/A, 43124 Parma, Italy


Abstract. arXiv:1712.08889v3 [math.DG] 18 Aug 2020 · ples are provided by Clemens manifolds...

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Transcript of Abstract. arXiv:1712.08889v3 [math.DG] 18 Aug 2020 · ples are provided by Clemens manifolds...