# Deformation limits of projective manifolds: Hodge numbers and strongly Gauduchon metrics

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DOI: 10.1007/s00222-013-0449-0

- Cite this article as:
- Popovici, D. Invent. math. (2013) 194: 515. doi:10.1007/s00222-013-0449-0

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## Abstract

This paper is intended to start a series of works aimed at proving that if in a (smooth) complex analytic family of compact complex manifolds all the fibres, except one, are supposed to be projective, then the remaining (limit) fibre must be Moishezon. A new method of attack, whose starting point originates in Demailly’s work, is introduced. While we hope to be able to address the general case in the near future, two important special cases are established here: the one where the Hodge numbers *h*^{0,1} of the fibres are supposed to be locally constant and the one where the limit fibre is assumed to be a *strongly Gauduchon manifold*. The latter is a rather weak metric assumption giving rise to a new, rather general, class of compact complex manifolds that we hereby introduce and whose relevance to this type of problems we underscore.

## 1 Introduction

A complex analytic family of compact complex manifolds is a proper holomorphic submersion \(\pi: \mathcal{X} \rightarrow \varDelta \) between complex manifolds \(\mathcal{X}\) and *Δ* [11]. Thus all the fibres are (smooth) compact complex manifolds of equal dimensions. The base manifold *Δ* will be assumed to be an open ball containing the origin in some complex space ℂ^{m}. This paper introduces a method for proving the following long-conjectured statement and establishes two major special cases (cf. Theorems 1.2 and 1.4). The general case is intended for later work by reducing it to the situation of Theorem 1.4.

### Conjecture 1.1

*Let*\(\pi: \mathcal{X} \longrightarrow \varDelta \)*be a complex analytic family of compact complex manifolds such that the fibre**X*_{t}:=*π*^{−1}(*t*) *is projective for every**t*∈*Δ*^{⋆}:=*Δ*∖{0}. *Then**X*_{0}:=*π*^{−1}(0) *is Moishezon*.

Recall that a compact complex manifold *X* is said to be Moishezon if there exists a proper holomorphic bimeromorphic map (i.e. a holomorphic modification) \(\mu:\tilde{X}\rightarrow X\) such that \(\tilde{X}\) is a projective manifold. This condition is equivalent to the existence of *n* algebraically independent meromorphic functions on *X* where \(n=\operatorname{dim}_{\mathbb{C}}X\) [14]. A Moishezon manifold becomes projective after finitely many blow-ups with smooth centres [14]. Thus Conjecture 1.1 predicts that projective manifolds can degenerate only mildly (i.e. to Moishezon manifolds) in the deformation limit. Note that the statement is optimal since, by Hironaka’s example [9], the limit fibre *X*_{0} need not be Kähler, let alone projective. (A posteriori, when *X*_{0} has been shown to be Moishezon, *X*_{0} cannot be Kähler unless it is projective—see [14].)

We shall first prove Conjecture 1.1 under the extra assumption that the Hodge number \(h^{0, 1}(t):=\operatorname{dim} H^{0, 1}(X_{t}, \mathbb{C})\) is independent of *t*∈*Δ*.

### Theorem 1.2

*Let*\(\pi: \mathcal{X} \rightarrow \varDelta \)*be a complex analytic family of compact complex manifolds such that the fibre**X*_{t}:=*π*^{−1}(*t*) *is projective for every**t*∈*Δ*^{⋆}. *Suppose that**h*^{0, 1}(0)=*h*^{0, 1}(*t*) *for**t**close to* 0. *Then**X*_{0}*is Moishezon*.

A Moishezon manifold is well known to admit a Hodge decomposition and to have its Hodge–Frölicher spectral sequence degenerate at \(E_{1}^{\bullet}\). This implies that, once Conjecture 1.1 has been proved, all the Hodge numbers \(h^{p, \, q}(t):=\operatorname{dim} H^{p, \, q}(X_{t}, \, \mathbb{C})\), *p*,*q*=0,…,*n*, will be locally constant as *t* varies in *Δ*. In particular, the situation considered in Theorem 1.2 is a posteriori seen to always occur. Theorem 1.2 implicitly shows that, if only *h*^{0, 1}(*t*) is assumed not to depend on *t*, all the *h*^{p, q}(*t*) are independent of *t*.

It is worth noticing that Theorem 1.2 also proves the special case of Conjecture 1.1 where all the fibres *X*_{t} are assumed to be compact complex surfaces in the following strengthened form that fails for higher dimensional fibres.

### Corollary 1.3

*Let*\(\pi: \mathcal{X} \longrightarrow \varDelta \)*be a complex analytic family of compact complex* surfaces *such that the fibre**X*_{t}:=*π*^{−1}(*t*) *is projective for every**t*∈*Δ*^{⋆}:=*Δ*∖{0}. *Then**X*_{0}:=*π*^{−1}(0) *is projective*.

Indeed, the Hodge–Frölicher spectral sequence of any compact complex surface is known to degenerate at \(E_{1}^{\bullet}\). Consequently, all the Hodge numbers *h*^{p,q}(*t*) are locally constant in a family of surfaces. In particular, the situation considered in Theorem 1.2 occurs, hence *X*_{0} is Moishezon. On the other hand, since the Betti numbers *b*_{k} of the fibres are always constant, the first Betti number *b*_{1} of *X*_{0} must be even. Now, by Kodaira’s classification of surfaces and Siu’s result [16] (see also [3, 12]), every compact complex surface with *b*_{1} even is Kähler. Being both Moishezon and Kähler, the limit surface *X*_{0} must be projective [14].

Our method for the resolution of Conjecture 1.1 will evolve from a strategy devised in broad outline and propounded over the years by J.-P. Demailly aiming at producing a Kähler current on the limit fibre *X*_{0}. Recall that a *d*-closed (1,1)-current *T* is said to be a *Kähler current* (a term coined in [10]) if *T*≥*εω* for some *ε*>0 and some positive-definite *C*^{∞} Hermitian (1,1)-form *ω*>0 on the ambient manifold. This is a strong notion of strict positivity for currents. Within the class of compact complex manifolds, the existence of a Kähler current characterises *Fujiki class*\(\mathcal{C}\)*manifolds* (i.e. those admitting a holomorphic modification to a compact Kähler manifold, much as Moishezon manifolds modify to projective ones) by a result of [6], while the existence of a Kähler current with integral De Rham cohomology class characterises Moishezon manifolds ([10], see also [5]). Thus, the pair Fujiki class \(\mathcal{C}\)/Moishezon bears a striking similarity to the pair Kähler/projective: by Kodaira’s Embedding Theorem, projective manifolds are precisely those compact complex manifolds carrying a Kähler metric with integral De Rham cohomology class. The former pair can be seen as the *current* version of the latter, while the latter term in each pair is the *integral class* version of the former.

The thrust of Demailly’s Morse inequalities ([4] and further developments) is to produce a Kähler current in a given cohomology class when the class satisfies comparatively weak positivity assumptions. This idea had motivated our previous work [15] which is to be made a crucial use of in the present paper. It will give a Kähler current on *X*_{0} lying in a certain integral cohomology 2-class in which a (1,1)-current has been produced beforehand by a limiting process.

It is in order to enable this limiting process that the notion of a *strongly Gauduchon metric* is introduced in Sect. 4 (cf. Definition 4.1). Two characterisations of compact complex manifolds carrying such a metric (henceforth called *strongly Gauduchon manifolds*) are obtained in Propositions 4.2 and 4.3 showing this new notion to be quite flexible and general. The second special case of Conjecture 1.1 that this paper establishes is proved in Sect. 4 under this rather weak assumption on the limit fibre.

### Theorem 1.4

*Let*\(\mathcal{X}\rightarrow \varDelta \)*be a complex analytic family of compact complex manifolds such that the fibre**X*_{t}:=*π*^{−1}(*t*) *is projective for every**t*∈*Δ*^{⋆}. *Suppose that**X*_{0}*is a strongly Gauduchon manifold*. *Then**X*_{0}*is Moishezon*.

The *strongly Gauduchon* assumption on *X*_{0} is different in nature to the non-jumping assumption made on *h*^{0,1}(*t*) in Theorem 1.2. As the currents whose existence is ruled out by the *strongly Gauduchon* assumption (see Proposition 4.3) are rather exceptional, this does not appear to be too strong a hypothesis when the complex dimension of the fibres is ≥3. In the case of families of complex surfaces, the *strongly Gauduchon* assumption on *X*_{0} amounts to the Kähler assumption which, clearly, we have no interest in making. However, as noticed in Corollary 1.3, the surface-fibre case of Conjecture 1.1 follows from well-known facts in the theory of compact complex surfaces and from Theorem 1.2 (whose proof is much easier in the surface case). The limit surface *X*_{0} is even projective.

The method of this paper naturally throws up new ideas to mount an attack on the following long-conjectured question.

### Question 1.5

*Let*\(\pi: \mathcal{X} \longrightarrow \varDelta \)*be a complex analytic family of compact complex manifolds such that the fibre**X*_{t}:=*π*^{−1}(*t*) *is Kähler for every**t*∈*Δ*^{⋆}:=*Δ*∖{0}. *Then*, *is**X*_{0}:=*π*^{−1}(0) *a Fujiki class*\(\mathcal{C}\)*manifold*?

Conjecture 1.1 predicts an affirmative answer to what can be seen as the *integral class* version of this question. It clearly suffices to prove Conjecture 1.1 for a 1-dimensional base *Δ*⊂ℂ (i.e. an open disc in ℂ) that we can shrink at will about the origin. This choice of *Δ* will be implicit throughout the paper.

Regarding the method of this work, a word of explanation may be in order. On the face of it, it would seem that embedding all the projective fibres *X*_{t} with *t*≠0 into the same projective space (which is possible thanks, for example, to Siu’s effective Matsusaka Big Theorem [17]) might lead to a quick proof of Conjecture 1.1. However, one would then run up against the difficulty of having to extend across the origin objects that are holomorphically defined on the punctured disc *Δ*^{⋆}. It is hard to see how this can be done without controlling the volumes of the projective submanifolds involved (which might a priori explode) near the origin. Such a uniform volume control would be equivalent to the uniform mass control obtained in Proposition 2.3, so one would be faced with the same difficulty as ours. Furthermore, the present method has the advantage of lending itself to generalisation when *X*_{t} is only assumed to be Kähler for *t*≠0 (cf. situation in Question 1.5).

### Notation and terminology

Given a smooth family of Hermitian metrics (*γ*_{t})_{t∈Δ} on the fibres (*X*_{t})_{t∈Δ}, the formal adjoints \(d_{t}^{\star}, \partial_{t}^{\star}, \bar{\partial}_{t}^{\star}\) associated with *d*, *∂*_{t} and respectively \(\bar{\partial}_{t}\) will be calculated with respect to the metric *γ*_{t}. They give rise to Laplace–Beltrami operators \(\varDelta _{t}=d\,d_{t}^{\star} + d_{t}^{\star}\, d\), \(\varDelta '_{t}=\partial_{t}\partial_{t}^{\star} + \partial_{t}^{\star} \partial_{t}\), \(\varDelta ''_{t}=\bar{\partial}_{t}\,\bar{\partial}_{t}^{\star} + \bar{\partial}_{t}^{\star} \bar{\partial}_{t}\) acting on *C*^{∞} forms of *X* of any degree *k*=1,…,*n* or any *J*_{t}-bidegree (*p*,*q*), *p*,*q*=1,…,*n*. The respective spaces of these forms will be denoted by \(C^{\infty}_{k}(X, \mathbb{C})\) and \(C^{\infty}_{p, q}(X_{t}, \mathbb{C})\). Given a form *u*, its component of type (*p*,*q*) with respect to the complex structure *J*_{t} will be denoted by \(u_{t}^{p, q}\). The *λ*-eigenspace of \(\varDelta ''_{t} : C^{\infty}_{p, q}(X_{t}, \mathbb{C})\rightarrow C^{\infty}_{p, q}(X_{t}, \mathbb{C})\) will be denoted by \(E^{p, q}_{\varDelta ''_{t}}(\lambda)\). Similarly for \(\varDelta _{t}' : C^{\infty}_{p, q}(X_{t}, \mathbb {C})\rightarrow C^{\infty}_{p, \, q}(X_{t}, \mathbb{C})\). Dolbeault cohomology groups of *J*_{t}-(*p*,*q*)-classes will be denoted by *H*^{p, q}(*X*_{t},ℂ), while *H*^{k}(*X*,ℂ) will stand for De Rham cohomology. The respective dimensions of these ℂ-vector spaces are the usual Hodge numbers *h*^{p,q}(*t*) and Betti numbers *b*_{k}. By the Kähler assumption on *X*_{t} for every *t*≠0, every *h*^{p,q}(*t*) is constant on *Δ*^{⋆} after possibly shrinking *Δ* about 0. But it may a priori happen that *h*^{p,q}(0)>*h*^{p,q}(*t*) for *t*≠0, although this case is *a posteriori* ruled out by Conjecture 1.1.

## 2 Preliminaries

As is well known in deformation theory [11], all the fibres *X*_{t}:=*π*^{−1}(*t*) are *C*^{∞}-diffeomorphic to a fixed compact differentiable manifold *X*. In other words, the family of complex manifolds (*X*_{t})_{t∈Δ} can be seen as one differentiable manifold *X* equipped with a family of complex structures (*J*_{t})_{t∈Δ} varying in a holomorphic way with *t*. In particular, for every *k*, the De Rham cohomology groups *H*^{k}(*X*_{t},ℂ) of all the fibres can be identified with a fixed *H*^{k}(*X*,ℂ), while the Dolbeault cohomology groups *H*^{p,q}(*X*_{t},ℂ) depend on *t*∈*Δ*.

As the fibres *X*_{t} are assumed to be projective for *t*≠0, the following fact is classical (although we have been unable to find a reference).

### Remark 2.1

There exists a non-zero integral De Rham cohomology 2-class *α*∈*H*^{2}(*X*,ℤ) such that, for every *t*∈*Δ*^{⋆}, *α* can be represented by a 2-form which is of *J*_{t}-type (1,1).

Moreover, *α* can be chosen in such a way that, for every *t*∈*Δ*∖*Σ*, *α* is the first Chern class of an ample line bundle *L*_{t}→*X*_{t}, where *Σ*={0}∪*Σ*′⊂*Δ* and *Σ*′=⋃*Σ*_{ν} is a countable union of proper analytic subsets *Σ*_{ν}⊆̷*Δ*^{⋆}.

### Proof

*α*∈

*H*

^{2}(

*X*,ℝ), let

*S*

_{α}⊂

*Δ*

^{⋆}denote the set of points

*t*∈

*Δ*

^{⋆}such that

*α*can be represented by a

*J*

_{t}-type (1,1)-form. For every

*t*∈

*Δ*

^{⋆},

*X*

_{t}is compact Kähler (even projective), so there exists a Hodge decomposition

*H*

^{2}(

*X*,ℂ)=

*H*

^{2,0}(

*X*

_{t},ℂ)⊕

*H*

^{1,1}(

*X*

_{t},ℂ)⊕

*H*

^{0,2}(

*X*

_{t},ℂ) with \(H^{2, 0}(X_{t}, \mathbb{C})=\overline{H^{0, 2}(X_{t}, \mathbb {C})}\). Thus, a given

*α*∈

*H*

^{2}(

*X*,ℝ) contains a

*J*

_{t}-type (1,1)-form if and only if its projection onto

*H*

^{0,2}(

*X*

_{t},ℂ) vanishes. This means that

*S*

_{α}is the set of zeroes of the section \(\sigma _{\alpha}\in\varGamma(\varDelta ^{\star}, \, R^{2}\pi_{\star}\mathcal {O}_{\mathcal{X}})\) induced by

*α*. By the Kähler assumption on every

*X*

_{t}with

*t*≠0, the map \(\varDelta ^{\star}\ni t\mapsto\operatorname{dim} H^{0, 2}(X_{t}, \mathbb{C})\) is locally constant and therefore the restriction of the higher direct image sheaf \(R^{2}\pi_{\star}\mathcal{O}_{\mathcal{X}}\) to

*Δ*

^{⋆}is locally free. As

*J*

_{t}varies holomorphically with

*t*,

*σ*

_{α}is a holomorphic section of the associated holomorphic vector bundle over

*Δ*

^{⋆}. This clearly implies that

*S*

_{α}is an analytic subset of

*Δ*

^{⋆}for every

*α*∈

*H*

^{2}(

*X*,ℝ). On the other hand, the projectiveness assumption on every

*X*

_{t}with

*t*≠0 entails the equality

*α*∈

*H*

^{2}(

*X*,ℤ) such that

*α*is an ample class on some fibre \(X_{t_{0}}\),

*t*

_{0}≠0 (depending on

*α*). Now, a proper analytic subset is Lebesgue negligible. If

*S*

_{α}were a proper subset of

*Δ*

^{⋆}for every such

*α*∈

*H*

^{2}(

*X*,ℤ), the left-hand side in (1) would be a countable union of Lebesgue negligible subsets, hence a Lebesgue negligible subset of

*Δ*

^{⋆}, contradicting the equality to

*Δ*

^{⋆}. Therefore, there must exist

*α*∈

*H*

^{2}(

*X*,ℤ) which can be represented by a

*J*

_{t}-type (1,1)-form for every

*t*≠0 (i.e.

*S*

_{α}=

*Δ*

^{⋆}) and which is an ample class on at least one fibre \(X_{t_{0}}\),

*t*

_{0}=

*t*

_{0}(

*α*)≠0.

Now, it is a standard fact that the ampleness property is open with respect to the countable analytic Zariski topology of the punctured base *Δ*^{⋆} (over which the fibres are projective). This follows from the Nakai–Moishezon criterion (according to which ampleness can be tested as numerical strict positivity on all classes of analytic cycles of the given projective manifold *X*_{t}, *t*≠0) and Barlet’s theory of cycle spaces [1] which implies that the cohomology classes {[*Z*]} of analytic cycles *Z*⊂*X*_{t} with *t*≠0 are the same on all fibres *X*_{t}, *t*≠0, except possibly on a countable union of analytic subsets of exceptional fibres which may have more classes of cycles than the generic fibre (see e.g. [6, §5] where the argument is extended to Kähler fibres using the transcendental version of the Nakai–Moishezon criterion obtained as the main result of that work).

Hence, as *α* is an ample class on some fibre \(X_{t_{0}}\) with *t*_{0}≠0, *α* must be an ample class on every fibre *X*_{t} with *t*∈*Δ*∖*Σ*, where *Σ*={0}∪*Σ*′ for a countable union *Σ*′=⋃*Σ*_{ν} of proper analytic subsets *Σ*_{ν}⊆̷*Δ*^{⋆}. □

### Construction of Kähler metrics (*ω*_{t})_{t∈Δ∖Σ} in the class *α*

*t*∈

*Δ*. Fix a class

*α*∈

*H*

^{2}(

*X*, ℤ) as above and set

*α*. Moreover,

*v*>0 since

*α*is the first Chern class of an ample line bundle

*L*

_{t}on

*X*

_{t}and \(v=L_{t}^{n}>0\) is the volume of

*L*

_{t}for every

*t*∈

*Δ*∖

*Σ*. Finally, the differential operator

*d*of

*X*admits a separate splitting

*J*

_{t}of

*X*.

*C*

^{∞}family (

*dV*

_{t})

_{t∈Δ}of

*C*

^{∞}volume forms

*dV*

_{t}>0 on

*X*

_{t}normalised such that \(\int_{X_{t}}dV_{t}=1\). We can apply Yau’s theorem [19] on the Calabi conjecture to the class

*α*viewed as a Kähler class on

*X*

_{t}for every

*t*∈

*Δ*∖

*Σ*. Thus, for

*t*∈

*Δ*∖

*Σ*, we get a

*C*

^{∞}2-form

*ω*

_{t}∈

*α*=

*c*

_{1}(

*L*

_{t}) on

*X*which is a Kähler form with respect to the complex structure

*J*

_{t}(i.e.

*dω*

_{t}=0,

*ω*

_{t}is of type (1, 1) and positive definite with respect to

*J*

_{t}) such that

The first step in this attack on Conjecture 1.1 will be to show that the family of Kähler forms (*ω*_{t})_{t∈Δ∖Σ} is bounded in mass (in a suitable sense that will be made precise below) as *t* approaches 0. By weak compactness, this family will contain a subsequence that is weakly convergent to a current *T*. The limit current *T* must be of *J*_{0}-type (1,1) and must lie in the given class *α*∈*H*^{2}(*X*,ℤ). This current, possibly with wild singularities, will only satisfy mild positivity properties on *X*_{0}. However, the so-called *singular Morse inequalities* for integral classes that we obtained in [15] will imply the existence of a Kähler current on *X*_{0} lying in the same cohomology class *α* as *T*. The class being integral, this is equivalent to *α* being the first Chern class of a big line bundle over *X*_{0} whose existence, in turn, amounts to *X*_{0} being Moishezon.

*X*with \(\operatorname{dim}_{\mathbb{C}}X=n\), recall that the volume of a holomorphic line bundle

*L*→

*X*, a birational invariant measuring the asymptotic growth of spaces of global holomorphic sections of high tensor powers of

*L*, is standardly defined as

*L*is ample, the volume is known to be given by

*v*(

*L*)=∫

_{X}

*c*

_{1}(

*L*)

^{n}:=

*L*

^{n}, motivating notation (2). Theorem 1.3. in [15] gives the following metric characterisation of the volume:

*T*≥0 in the first Chern class of

*L*and

*T*

_{ac}stands for the absolutely continuous part of

*T*in the Lebesgue decomposition of its measure coefficients. The interesting inequality in (5) is “≥” (

*singular Morse inequalities*). Now, the following three facts are well known:

*v*(

*L*)>0 if and only if the line bundle

*L*is big, by definition of bigness. A line bundle

*L*→

*X*is big if and only if it can be equipped with a (possibly singular) Hermitian metric

*h*whose curvature current

*T*:=

*iΘ*

_{h}(

*L*) is >0 on

*X*(i.e. a Kähler current), by [5] for a projective

*X*and [10] for a general

*X*. A compact complex manifold carries a big line bundle if and only if it is Moishezon, by [14]. As any

*d*-closed current of type (1,1) whose De Rham cohomology class is integral is always the curvature current of some holomorphic line bundle equipped with a (possibly singular) Hermitian metric, an equivalent way of formulating the “≥” part of (5) above is the following.

### Theorem 2.2

(Rewording of Theorem 1.3. in [15])

*Let*

*X*

*be a compact complex manifold*, \(\operatorname{dim}_{\mathbb {C}}X=n\).

*If there exists a*

*d*-

*closed*(1,1)-

*current*

*T*

*on*

*X*

*whose De Rham cohomology class is integral and which satisfies*

*then the cohomology class of*

*T*

*contains a Kähler current S*.

*Implicitly*,

*X*

*is Moishezon*.

This is the form in which we will use the result of [15] on *X*_{0}. The limit current *T* obtained on the limit fibre *X*_{0} at the end of the first step will be shown to satisfy the mild positivity conditions (i) and (ii) of Theorem 2.2 on *X*_{0}. By that theorem, the integral class *α* of *T* must contain a Kähler current, proving that *X*_{0} is Moishezon.

The assumption made on *h*^{0,1}(*t*):=dim *H*^{0,1}(*X*_{t},ℂ) in Theorem 1.2 enables one to uniformly bound the masses of the Kähler forms (*ω*_{t})_{t∈Δ∖Σ} with respect to a family of Gauduchon metrics on the fibres *X*_{t} varying in a *C*^{∞} way with the parameter *t*∈*Δ*. This is because the invariance of *h*^{0,1}(*t*) amounts to the existence of a uniform positive lower bound for the smallest positive eigenvalue of the anti-holomorphic Laplacian \(\varDelta ''_{t}\) as *t* varies in a neighbourhood of 0 in *Δ*. Hence, the inverses of these small positive eigenvalues are uniformly bounded above and so are the masses of the Kähler forms (*ω*_{t})_{t∈Δ∖Σ}. This will occupy Sect. 3.

Rather than proving the invariance of *h*^{0,1}(*t*) on a priori grounds (a tall order that falls largely beyond the scope of these papers), we deal with Conjecture 1.1 by working directly on strongly Gauduchon metrics and the spectra of the associated Laplace operators. The method yields the desired uniform mass boundedness of the family of Kähler forms (*ω*_{t})_{t∈Δ∖Σ} even in the mythical case where *h*^{0,1}(*t*) would jump at *t*=0. Explicitly, we prove the following fact that can be regarded as the main technical result of this work.

### Proposition 2.3

*Under the hypotheses of either Theorem*1.2

*or Theorem*1.4

*and after possibly shrinking*

*Δ*

*about*0,

*there exists a family*(

*γ*

_{t})

_{t∈Δ}

*of Gauduchon metrics varying in a*

*C*

^{∞}

*way with*

*t*

*on the fibres*(

*X*

_{t})

_{t∈Δ}

*and satisfying the following uniform mass boundedness property*.

*For every*

*t*∈

*Δ*∖

*Σ*,

*choose any*

*J*

_{t}-

*Kähler form*

*ω*

_{t}

*belonging to the class*

*α*∈

*H*

^{2}(

*X*, ℤ)

*given by Remark*2.1.

*Then there exists a constant*

*C*>0

*independent of*

*t*∈

*Δ*∖

*Σ*

*such that*

If *h*^{0,1}(*t*) is independent of *t*∈*Δ*, any choice of smooth family of Gauduchon metrics will do (cf. Proposition 3.1). In general, a family of special (i.e. *strongly Gauduchon*) metrics has to be constructed (cf. Proposition 4.5).

Notice that the real class analogue of Remark 2.1 no longer holds in the more general context of Question 1.5 as there are examples of families with Kähler fibres for which no non-zero real De Rham cohomology 2-class which is of type (1,1) for all the complex structures involved exists. Thus, the constant class *α* has to be replaced with a *C*^{∞} family of real classes *α*_{t}∈*H*^{2}(*X*,ℝ), *t*≠0, whose volumes *v*_{t} remain uniformly bounded below by a positive constant near the origin in *Δ*^{⋆}. This can be arranged by standard arguments. Now, the significant fact is that our Proposition 2.3 still holds in this more general context if a suitable family of Kähler classes *α*_{t}, *t*≠0, replaces the constant class *α*. This is because the projective assumption on *X*_{t} with *t*≠0 is not made full use of in the proof of Proposition 2.3, but only the Kähler assumption and the resulting \(\partial\bar{\partial}\)-lemma are used. This means that, when Conjecture 1.1 has been proved, the only extra hurdle that will yet have to be cleared before an (affirmative?) answer to Question 1.5 can be given is Demailly’s conjecture on *transcendental Morse inequalities*: the singular Morse inequalities that we obtained in [15] and listed above as Theorem 2.2 are expected to hold without the *integral class* assumption on *T* (hence *X* would be Fujiki class \(\mathcal{C}\)). We hope to be able to address this conjecture in future work.

## 3 The case of constant *h*^{0,1}(*t*), *t*∈*Δ*

In this section we prove Theorem 1.2. The proof falls into two steps.

*Step 1: Produce a weak-limit current**T*≥0 *on**X*_{0}*from* (*ω*_{t})_{t∈Δ∖Σ}.

*X*

_{t}=(

*X*,

*J*

_{t}). In other words, there exists a family of 2-forms (

*γ*

_{t})

_{t∈Δ}on

*X*, varying in a

*C*

^{∞}way with

*t*∈

*Δ*, such that each

*γ*

_{t}is a positive-definite, type (1,1)-form with respect to

*J*

_{t}and satisfies the Gauduchon condition on

*X*

_{t}: \(\partial_{t}\bar{\partial}_{t}\gamma_{t}^{n-1}=0\). To see this, let us briefly scan the argument of [7] in our family context. Let \((\omega'_{t})_{t\in \varDelta }\) be any family of Hermitian metrics varying in a

*C*

^{∞}way with

*t*on (

*X*

_{t})

_{t∈Δ}. Consider the Laplace-type operator acting on smooth functions:

*f*>0 on

*X*or

*f*<0 on

*X*or

*f*=0 on

*X*. The existence of a

*C*

^{∞}function

*f*

_{t}:

*X*→(0,+∞) such that \(P^{\star}_{\omega_{t}'}(f_{t})=0\) is equivalent to the Hermitian metric \(f_{t}^{\frac{1}{n-1}} \omega_{t}'\) being Gauduchon. Now, \((P^{\star}_{\omega_{t}'})_{t\in \varDelta }\) is a

*C*

^{∞}family of elliptic operators on the fibres

*X*

_{t}with kernels of constant dimensions (= 1). By Kodaira and Spencer (see e.g. [11, Theorem 7.4, p. 326]), the kernels define a

*C*

^{∞}vector bundle \(\varDelta \ni t\mapsto\ker(P^{\star}_{\omega_{t}'})\). Then it suffices to pick \(f_{0}\in\ker(P^{\star}_{\omega_{0}'|C^{\infty}(X, \mathbb{R})})\) such that

*f*

_{0}>0 and to extend it to a

*C*

^{∞}local section

*Δ*∋

*t*↦

*f*

_{t}of the

*C*

^{∞}real bundle \(\varDelta \ni t\mapsto\ker(P^{\star}_{\omega_{t}'|C^{\infty}(X, \mathbb{R})})\) which is a trivial bundle if

*Δ*has been shrunk sufficiently about 0. By continuity,

*f*

_{t}>0 for all

*t*sufficiently close to 0∈

*Δ*, defining a family \(\gamma_{t}:=f_{t}^{\frac{1}{n-1}} \omega_{t}'\),

*t*∈

*Δ*, of Gauduchon metrics varying in a

*C*

^{∞}way with

*t*on the fibres

*X*

_{t}.

Fix any such family (*γ*_{t})_{t∈Δ}. It is against these Gauduchon metrics that the masses of forms will be measured. We can now prove Proposition 2.3 in the special case of this section.

### Proposition 3.1

*Let*

*α*∈

*H*

^{2}(

*X*,ℤ)

*a class given by Remark*2.1.

*For every*

*t*∈

*Δ*∖

*Σ*,

*let*

*ω*

_{t}

*be an arbitrary*

*J*

_{t}-

*Kähler form belonging to the class*

*α*.

*If*

*h*

^{0,1}(

*t*)

*is independent of*

*t*∈

*Δ*,

*there exists a constant*

*C*>0

*independent of*

*t*∈

*Δ*∖

*Σ*

*such that the masses of the*

*ω*

_{t}

*’s with respect to the*\(\gamma_{t}^{n-1}\)

*’s satisfy*:

*after possibly shrinking*

*Δ*

*about*0.

Notice that the choice (3) of Kähler forms *ω*_{t} in the given class *α* by means of Yau’s theorem is not needed here. It will come in later on.

### Proof

*ω*

_{t}>0 and

*γ*

_{t}>0. Let \(\tilde{\omega}\) be any

*d*-closed real 2-form in the De Rham class

*α*. As

*ω*

_{t}and \(\tilde{\omega}\) are De Rham cohomologous real 2-forms, there exists a smooth real 1-form

*β*

_{t}on

*X*

_{t}such that:

*t*∈

*Δ*∖

*Σ*, the mass of

*ω*

_{t}splits as:

*γ*

_{t})

_{t∈Δ}vary in a

*C*

^{∞}way with

*t*∈

*Δ*, the first term in the right-hand side of (9) is bounded as

*t*varies in a neighbourhood of 0. We are thus reduced to showing the boundedness of the second term as

*t*∈

*Δ*∖

*Σ*approaches 0. The difficulty stems from the fact that the family (

*ω*

_{t})

_{t∈Δ∖Σ}of Kähler forms (and implicitly (

*β*

_{t})

_{t∈Δ∖Σ}) need not extend to the limit fibre

*X*

_{0}as

*X*

_{0}is not assumed to be Kähler. Using Stokes’ theorem we get: where \(\beta_{t}=\beta_{t}^{1, 0} + \beta_{t}^{0, 1}\) is the decomposition of

*β*

_{t}into components of types (1,0) and (0,1). As \(\beta_{t}=\overline{\beta_{t}}\) (i.e.

*β*

_{t}is a real form), \(\beta_{t}^{1, 0}=\overline{\beta_{t}^{0, 1}}\) and the two terms in the right-hand side above are conjugate to each other. It thus suffices to show the boundedness of the integral containing \(\beta_{t}^{0, 1}\) as

*t*∈

*Δ*∖

*Σ*approaches 0.

*β*

_{t}of (8) is not unique. We will make a particular choice of

*β*

_{t}. The Kähler form

*ω*

_{t}being of

*J*

_{t}-type (1,1), equating the components of

*J*

_{t}-type (0,2) in (8), we see that \(\beta_{t}^{0, 1}\) must solve the equation:

*t*∈

*Δ*

^{⋆}, choose \(\beta_{t}^{0, 1}\) to be the solution of equation (11) of minimal

*L*

^{2}norm with respect to the metric

*γ*

_{t}of

*X*

_{t}. (Notice that (11) is solvable in \(\beta_{t}^{0,\, 1}\) for every

*t*≠0 because

*α*contains a

*J*

_{t}-type (1,1)-form for every

*t*≠0. However, it need not be solvable for

*t*=0 as

*α*is not known to contain a

*J*

_{0}-type (1,1)-form.) Set \(\beta_{t}^{1, 0}:=\overline{\beta_{t}^{0, 1}}\) and \(\beta_{t}:=\beta_{t}^{1, \, 0} + \beta_{t}^{0, 1}\). Clearly,

*β*

_{t}is a real 1-form on

*X*but it need not solve (8). However, the \(\partial\bar{\partial}\)-lemma (which holds on every

*X*

_{t}with

*t*≠0 by the Kähler assumption) shows that

*β*

_{t}satisfies (8) on

*X*

_{t}up to a \(\partial_{t}\bar{\partial}_{t}\)-exact (1,1)-form for each

*t*∈

*Δ*∖

*Σ*. Indeed, \(\tilde{\omega} + d\, \beta_{t}\) is of

*J*

_{t}-type (1,1) since its (0,2)-component is \(\tilde{\omega}_{t}^{0, 2} + \bar{\partial}_{t}\beta_{t}^{0, 1}=0\) by (11) and its (2,0)-component also vanishes by conjugation. It follows that \(\tilde{\omega} + d \beta_{t} - \omega_{t}\) is of

*J*

_{t}-type (1,1) and

*d*-exact, hence also \(\partial_{t}\bar{\partial}_{t}\)-exact by the \(\partial\bar{\partial}\)-lemma. Thus

*β*

_{t}solves the equation:

*φ*

_{t}on

*X*. Now, since

*γ*

_{t}has been chosen such that \(\partial_{t}\bar{\partial}_{t}\gamma _{t}^{n-1}=0\) (the Gauduchon condition), Stokes’ theorem gives:

In other words, \(\partial\bar{\partial}\)-exact (1,1)-forms have no mass against the relevant power of a Gauduchon metric. Thus relation (9) holds thanks to (12). As explained above, the proof reduces to showing the boundedness of the integral containing \(\beta_{t}^{0, \, 1}\) in the right-hand side of (10) as *t*∈*Δ*^{⋆} approaches 0.

*t*∈

*Δ*, let \(\bar{\partial_{t}}^{\star}\) denote the formal adjoint of \(\bar{\partial_{t}}\) with respect to the global

*L*

^{2}scalar product defined by the Gauduchon metric

*γ*

_{t}of

*X*

_{t}. We get a

*C*

^{∞}family \((\varDelta ''_{t})_{t\in \varDelta }\) of associated anti-holomorphic Laplace–Beltrami operators defined as

*L*

^{2}norm of (11) is known to be given by the formula:

*G*

_{t}denotes the Green operator of \(\varDelta ''_{t}\). Clearly, the family of operators \((\bar{\partial_{t}}^{\star})_{t\in \varDelta }\) varies in a

*C*

^{∞}way with

*t*. By the Hodge Fundamental Theorem (which does not require the Kähler property), the Hodge isomorphism holds:

*J*

_{t}-(0,1)-forms. By a well-known result of Kodaira and Spencer (see [11, Theorem 7.6, p. 344]), the family of Green operators (

*G*

_{t})

_{t∈Δ}of a

*C*

^{∞}family of strongly elliptic operators \((\varDelta ''_{t})_{t\in \varDelta }\) is

*C*

^{∞}with respect to

*t*∈

*Δ*if the dimensions of the kernel spaces \(\mathcal{H}^{0, 1}(X_{t}, \mathbb{C})\) are independent of

*t*∈

*Δ*. This is indeed the case here as, by assumption, \(h^{0, 1}(t)=\operatorname{dim} H^{0, 1}(X_{t}, \mathbb{C})\) is independent of

*t*∈

*Δ*, and \(\operatorname{dim}\mathcal{H}^{0, 1}(X_{t}, \mathbb{C})=h^{0, 1}(t)\) by the Hodge isomorphism.

Now the *J*_{t}-type (0,2)-components \((\tilde{\omega}_{t}^{0, 2})_{t\in \varDelta }\) of the fixed 2-form \(\tilde{\omega}\) vary in a *C*^{∞} way with *t* (up to *t*=0) since the complex structures (*J*_{t})_{t∈Δ} do. As the composed operators \(G_{t} \bar {\partial_{t}}^{\star}\) have the same property, the forms \(\beta_{t}^{0, 1} = - G_{t} \bar{\partial}_{t}^{\star} \tilde{\omega}_{t}^{0, 2}\) (cf. (13)) extend smoothly across *t*=0 to a family \((\beta_{t}^{0, 1})_{t\in \varDelta }\) of forms which vary in a *C*^{∞} way with *t*∈*Δ*. This clearly implies the boundedness in a neighbourhood of *t*=0 of the second term in the right-hand side of (10). Taking conjugates, the same is true of the first term in the right-hand side of (10). This completes the proof of Proposition 3.1. □

As the family of positive forms (*ω*_{t})_{t∈Δ∖Σ} is bounded in mass, it is weakly compact. Thus it contains a weakly convergent subsequence \(\omega_{t_{k}}\rightarrow T\), with *Δ*∖*Σ*∋*t*_{k}→0 as *k*→+∞. The limit current *T*≥0 is closed, positive and of type (1,1) for the limit complex structure *J*_{0} of *X*_{0}. By the weak continuity of De Rham classes { }, \(\{\omega_{t_{k}}\} \rightarrow\{T\}\). As \(\{\omega_{t_{k}}\}=\alpha\) for all *k*, we see that *T*∈*α*.

We have thus produced a closed positive (1,1)-current *T*≥0 on *X*_{0} in the given integral De Rham cohomology class *α*.

*Step 2: Prove that the integral class* {*T*} *contains a Kähler current*.

*ω*

_{t},

*t*∈

*Δ*∖

*Σ*, are chosen in the given Kähler class

*α*by means of Yau’s theorem [19] as explained in (3), the identity \(\omega_{t_{k}}^{n}=v\, dV_{t_{k}}\) and (14) give:

*T*satisfies condition (ii) of Theorem 2.2. It is for this sole purpose that Yau’s theorem [19] has been used.

Summing up, the limit current *T* is of type (1,1) (for *J*_{0}), has an integral De Rham cohomology class *α* and satisfies the mild positivity assumptions (i) and (ii) of Theorem 2.2 on singular Morse inequalities. By that theorem applied on *X*_{0}, *α* must contain a Kähler current, hence *X*_{0} must be Moishezon.

The proof of Theorem 1.2 is complete. □

### Remark 3.2

*h*

^{0,1}(

*t*) at

*t*=0, one is faced with the following difficulty in proving the existence of a uniform upper bound (7) for the masses of the Kähler forms (

*ω*

_{t})

_{t∈Δ∖Σ}. For every

*t*∈

*Δ*, the Laplace operator \(\varDelta ''_{t}\) acting on

*J*

_{t}-(0,1)-forms of

*X*is elliptic and therefore has a compact resolvent and a discrete spectrum

*λ*

_{k}(

*t*)→+∞ as

*k*→+∞. By the Hodge isomorphism, the multiplicity of zero as an eigenvalue of \(\varDelta ''_{t}\) equals

*h*

^{0,1}(

*t*). By results of Kodaira and Spencer (see [11, Lemmas 7.5–7.7 and Proof of Theorem 7.2, p. 338–343]), for every small

*ε*>0, the number

*m*∈ℕ

^{⋆}of eigenvalues (counted with multiplicities) of \(\varDelta ''_{t}\) contained in the interval [0,

*ε*) is independent of

*t*if

*t*∈

*Δ*is sufficiently close to 0 (say

*δ*

_{ε}-close). If

*ε*>0 has been chosen so small that 0 is the only eigenvalue of \(\varDelta ''_{0}\) contained in [0,

*ε*), it follows that

*m*=

*h*

^{0,1}(0)≥

*h*

^{0,1}(

*t*) for

*t*sufficiently close to 0 (the upper-semicontinuity property). Consequently, for

*t*near 0,

*h*

^{0,1}(0)=

*h*

^{0,1}(

*t*) if and only if 0 is the only eigenvalue of \(\varDelta ''_{t}\) lying in [0,

*ε*). In other words, if

*h*

^{0,1}(0)>

*h*

^{0,1}(

*t*) when

*t*(≠0) is near 0, choosing increasingly small

*ε*>0 gives eigenvalues of \(\varDelta ''_{t}\):

*ε*

_{t}→0) when

*t*→0, where

*N*=

*h*

^{0,1}(0)−

*h*

^{0,1}(

*t*). Now, formula (13) for \(\beta_{t}^{0, 1}\) involves the Green operator

*G*

_{t}which is the inverse of the restriction of \(\varDelta ''_{t}\) to the orthogonal complement of its kernel. The inverses \(1/\lambda _{k_{j}}(t)\rightarrow+\infty\) of the small eigenvalues of \(\varDelta ''_{t}\) are eigenvalues for

*G*

_{t}. Thus, if \(\bar{\partial}_{t}^{\star }\tilde{\omega}_{t}^{0, 2}\) has non-trivial projections onto the eigenspaces \(E^{0, \, 1}_{\varDelta ''_{t}}(\lambda_{k_{j}(t)})\), these projections get multiplied by \(1/\lambda_{k_{j}}(t)\) when

*G*

_{t}acts on \(\bar{\partial}_{t}^{\star}\tilde{\omega}_{t}^{0, 2}\). Then \(\beta _{t}^{0, 1}\) need not be bounded as

*t*approaches 0, unless the said projections can be proved to tend to zero sufficiently quickly to offset \(1/\lambda_{k_{j}}(t)\rightarrow+\infty\) when

*t*approaches 0. This may cause the mass of

*ω*

_{t}(cf. (9)) to get arbitrarily large in the limit as

*t*→0.

Thus the remaining difficulty in proving Conjecture 1.1 is to prove the uniform mass boundedness of Proposition 3.1 without the non-jumping assumption on *h*^{0,1}(*t*).

## 4 The strongly Gauduchon case

In this section we transform the topological problem with the possible jump of *h*^{0,1}(*t*) at *t*=0 into a more tractable metric problem. For this purpose, we introduce the notion of *strongly Gauduchon manifold* and prove Theorem 1.4.

### 4.1 Setting the method in motion

*γ*

_{t})

_{t∈Δ}of

*J*

_{t}-Gauduchon metrics varying in a

*C*

^{∞}way with

*t*. As explained after the identity (10),

*Step 1*(hence everything that follows) in the proofs of Theorems 1.2 and 1.4 can be run if we can guarantee the boundedness of the following integral, that will henceforth be termed the

*main quantity*:

*t*approaches 0. The difficulty is that the family \((\beta^{0, 1}_{t})_{t\in \varDelta ^{\star}}\) of

*J*

_{t}−(0,1)-forms constructed as the minimal

*L*

^{2}-norm solutions of (11) (extended to all

*t*≠0) need not be bounded as

*t*approaches 0 if

*h*

^{0,1}(0)>

*h*

^{0,1}(

*t*),

*t*≠0 (cf. Remark 3.2). Thus, \(\partial_{t}\beta_{t}^{0, 1}\) may “explode” near

*t*=0. However, \(\bar{\partial}_{t}\beta_{t}^{0, 1}=\tilde{\omega }_{t}^{0, 2}\) is bounded and extends smoothly to

*t*=0 by construction, since the family \((\tilde{\omega}_{t}^{0, 2})_{t\in \varDelta }\) of

*J*

_{t}−(0,2)-components of the fixed 2-form \(\tilde{\omega}\) varies in a

*C*

^{∞}way with

*t*∈

*Δ*. This means that if we can convert \(\partial_{t}\beta_{t}^{0, 1}\) to \(\bar{\partial}_{t}\beta_{t}^{0, 1}\) in (18), we will obtain the desired boundedness of the

*main quantity*

*I*

_{t}, hence Proposition 2.3 and implicitly a proof of Conjecture 1.1. This very simple observation is the starting point of our method to tackle the general case.

*∂*

_{t}-exact (

*n*,

*n*−1)-form \(\partial_{t}\gamma_{t}^{n-1}\) is

*d*-closed. Indeed, it is

*∂*

_{t}-closed in a trivial way and is \(\bar{\partial}_{t}\)-closed by the Gauduchon assumption on

*γ*

_{t}. For

*t*≠0, the \(\partial\bar{\partial}\)-lemma holds on

*X*

_{t}(thanks to the Kähler assumption) and yields the \(\bar{\partial}_{t}\)-exactness of \(\partial_{t}\gamma _{t}^{n-1}\). Thus,

*C*

^{∞}

*J*

_{t}-(

*n*,

*n*−2)-form

*ζ*

_{t}can be chosen as the minimal

*L*

^{2}-norm solution of the above equation (with respect to

*γ*

_{t}), generating a

*C*

^{∞}family \((\zeta_{t})_{t\in \varDelta ^{\star}}\) defined off

*t*=0. By Stokes’ theorem, the

*main quantity*now reads:

The situation is now the reverse of that in (18): we have rendered the factor depending on \(\beta_{t}^{0, 1}\) bounded, as \(\bar{\partial}_{t}\beta_{t}^{0, 1}=\tilde{\omega}_{t}^{0, 2}\) varies in a *C*^{∞} way with *t* up to *t*=0, but we are now faced with the task of ensuring the boundedness of the new factor *ζ*_{t} near *t*=0. The difficulty stems from the fact that, at this point, the \(\partial\bar{\partial}\)-lemma is not known to hold on *X*_{0}. However, if \(\partial_{0}\gamma_{0}^{n-1}\) were known to be \(\bar{\partial}_{0}\)-exact, the proof of Conjecture 1.1 could be completed.

We will now highlight a hypothesis on *X*_{0}, different to the one considered in the previous section, that guarantees the \(\bar{\partial}_{0}\)-exactness of \(\partial_{0}\gamma_{0}^{n-1}\). We digress briefly to introduce a new type of Gauduchon metrics satisfying an extra property.

### 4.2 Strongly Gauduchon metrics

### Definition 4.1

*X*be a compact complex manifold, \(\operatorname{dim}_{\mathbb{C}} X=n\).

- (i)
A

*C*^{∞}positive-definite (1,1)-form*γ*on*X*will be said to be a*strongly Gauduchon metric*if the (*n*,*n*−1)-form*∂γ*^{n−1}is \(\bar{\partial}\)-exact on*X*. - (ii)
If

*X*carries such a metric,*X*will be said to be a*strongly Gauduchon manifold*.

Notice that the Gauduchon condition only requires *∂γ*^{n−1} to be \(\bar{\partial}\)-closed on *X*. Hence, every *strongly Gauduchon* metric is a Gauduchon metric. Now, if the \(\partial\bar{\partial}\)-lemma holds on *X* (as is the case if, for example, *X* is Kähler), the converse statement holds as well (see argument above), and therefore the two notions coincide in that case. However, we will now show that the *strongly Gauduchon* condition is strictly stronger than the Gauduchon condition in general. Furthermore, unlike Gauduchon metrics which exist on any compact complex manifold, *strongly Gauduchon* metrics need not exist in general. We will give a necessary and sufficient condition on the manifold *X* ensuring the existence of a *strongly Gauduchon* metric. The method, proceeding by duality and an application of the Hahn–Banach separation theorem in locally convex spaces, is the classical one introduced by Sullivan in [18] and used in several instances in [3, 8, 12, 13].

We begin with the following simple observation.

### Proposition 4.2

*A compact complex manifold**X**of complex dimension**n**carries a strongly Gauduchon metric**γ**if and only if there exists a real**d*-*closed**C*^{∞}*form**Ω**of degree* 2*n*−2 *on**X**such that its component of type* (*n*−1,*n*−1) *satisfies**Ω*^{n−1,n−1}>0 *on X*.

### Proof

*n*−2)-form

*Ω*, the (2

*n*−1)-form

*dΩ*is also real, hence its components of type (

*n*,

*n*−1) and respectively (

*n*−1,

*n*) are conjugate to each other. Thus, the condition

*dΩ*=0 amounts to

If a *strongly Gauduchon* metric *γ* exists on *X*, we set *Ω*^{n−1,n−1}:=*γ*^{n−1}. This is a smooth form of type (*n*−1,*n*−1) and *Ω*^{n−1,n−1}>0. By the *strongly Gauduchon* condition on *γ*, *∂Ω*^{n−1,n−1} is \(\bar{\partial}\)-exact on *X*. Hence, one can find a smooth form *Ω*^{n,n−2} of type (*n*,*n*−2) on *X* such that \(\partial \varOmega^{n-1, n-1}=-\bar{\partial}\varOmega^{n, n-2}\). By setting \(\varOmega^{n-2, n}:=\overline{\varOmega^{n, n-2}}\) and *Ω*=*Ω*^{n,n−2}+*Ω*^{n−1,n−1}+*Ω*^{n−2,n}, we get the desired form of degree 2*n*−2.

Conversely, if there exists a (2*n*−2)-form *Ω* on *X* as in the statement, the assumption *Ω*^{n−1,n−1}>0 allows one to extract the root of order *n*−1 in the following sense. A very useful remark of Michelsohn [13, pp. 279–280] in linear algebra asserts that there is a unique positive-definite smooth form *γ* of type (1,1) on *X* such that *Ω*^{n−1,n−1}=*γ*^{n−1}. By the assumption *dΩ*=0 and its equivalent formulation (21), we see that *γ* satisfies the *strongly Gauduchon* condition. □

*n*−2)-form as in Proposition 4.2 above exists. Let

*X*be any compact complex manifold, dim

_{ℂ}

*X*=

*n*, and let

*Ω*be any real

*C*

^{∞}form of degree 2

*n*−2 on

*X*. The condition

*dΩ*=0 is equivalent, by the duality between

*d*-closed smooth real (2

*n*−2)-forms and real exact 2-currents

*T*=

*d*

*S*on

*X*, to the property

*n*−1,

*n*−1)-forms and non-zero positive (1,1)-currents on

*X*shows that the condition

*Ω*

^{n−1,n−1}>0 is equivalent to the property

Now, if *T* is of type (1,1), we clearly have ∫_{X}*Ω*^{n−1,n−1}∧*T*=∫_{X}*Ω*∧*T*. Furthermore, real *d*-exact 2-currents *T*=*d*
*S* form a closed vector subspace \(\mathcal{A}\) of the locally convex space \(\mathcal{D}'_{\mathbb{R}}(X)\) of real 2-currents on *X*. Meanwhile, if we fix a smooth, strictly positive (*n*−1,*n*−1)-form *Θ* on *X*, positive non-zero (1,1)-currents *T* on *X* can be normalised such that ∫_{X}*T*∧*Θ*=1 and it suffices to guarantee property (23) for normalised currents. Clearly, these normalised positive (1,1)-currents form a compact (in the locally convex topology of weak convergence of currents) convex subset \(\mathcal{B}\) of the locally convex space \(\mathcal{D}'_{\mathbb{R}}(X)\) of real 2-currents on *X*. The Hahn–Banach separation theorem for locally convex spaces (see [8] and the references therein) guarantees the existence of a linear functional vanishing identically on a given closed subset and assuming only positive values on a given compact subset if the two subsets are convex and do not intersect. Hence, in our case, there exists a real smooth (2*n*−2)-form *Ω* on *X* satisfying both conditions (22) and (23) if and only if \(\mathcal {A}\cap\mathcal{B}=\emptyset\). This amounts to there existing no non-trivial exact (1,1)-current *T*=*dS* such that *T*≥0 on *X*. We have thus proved (using Proposition 4.2) the following characterisation of *strongly Gauduchon* manifolds in terms of non-existence of certain currents. This closely parallels similar existence criteria for Kähler metrics [8] and balanced metrics [13].

### Proposition 4.3

*Let**X**be a compact complex manifold*, \(\operatorname{dim}_{\mathbb{C}} X=n\). *Then*, *X**carries a strongly Gauduchon metric**γ**if and only if there is no non*-*zero current**T**of type* (1,1) *on**X**such that**T*≥0 *and**T**is**d*-*exact on X*.

*C*

^{∞}positive-definite (1,1)-form)

*γ*on a compact complex manifold

*X*, the following four conditions on

*γ*: the Kähler condition (

*dγ*=0), the balanced condition (

*d*

^{⋆}

*γ*=0), the

*strongly Gauduchon*condition (

*∂γ*

^{n−1}is \(\bar{\partial}\)-exact) and the Gauduchon condition (

*∂γ*

^{n−1}is \(\bar{\partial}\)-closed) stand in the following implication hierarchy:

For example, the implication “balanced ⟹ strongly Gauduchon” can be seen as follows. Using the Hodge ⋆ operator that gives isometries ⋆: *Λ*^{p,q}*T*^{⋆}*X*⟶*Λ*^{n−q,n−p}*T*^{⋆}*X* defined by the Hermitian metric *γ* on *X*, we have ⋆*γ*=*γ*^{n−1}/(*n*−1)! and *d*^{⋆}=−⋆ *d* ⋆. Hence, the balanced condition *d*^{⋆}*γ*=0 is equivalent to *dγ*^{n−1}=0, which in turn is equivalent, thanks to *γ*^{n−1} being a real form, to *∂γ*^{n−1}=0 (cf. [13]). This is clearly a stronger condition than the *strongly Gauduchon* requirement that *∂γ*^{n−1} be \(\bar{\partial}\)-exact.

Except for Gauduchon metrics, any of the other three kinds of metrics in (24) need not exist in general. Each of the conditions (24) can be given an intrinsic characterisation in terms of non-existence of certain currents (cf. [8, 13], Proposition 4.3 above, for the first three of them respectively). Recall that the fourth condition is known to have a similar characterisation that can be obtained by the same method: a Gauduchon metric *γ* exists on *X* if and only if there is no non-zero (1,1)-current *T* that is both positive and \(\partial \bar{\partial}\)-exact, i.e. \(T=i\partial\bar{\partial}\varphi\geq0\) globally on *X*. The compactness assumption on *X* and the maximum principle for psh functions rule out the existence of such a current, proving that a Gauduchon metric always exists on any compact complex manifold.

On manifolds of complex dimension ≥3, the implications (24) are strict. However, on compact complex surfaces the notions of Kähler and balanced metrics are equivalent [13] and so are the notions of Kähler, balanced and *strongly Gauduchon* surfaces (i.e. surfaces carrying the respective kind of metrics). In particular, we have

### Observation 4.4

*Every strongly Gauduchon compact complex surface is Kähler*.

### Proof

It is well known that a compact complex surface is Kähler if and only if its first Betti number *b*_{1} is even (see [16] and also [3, 12]). Now, it can be easily shown by the same duality method of Sullivan (see, e.g. [12, Théorème 6.1]) that a current as described in Proposition 4.3 always exists on any compact complex surface with *b*_{1} odd. □

### 4.3 Proof of Conjecture 1.1 under the strongly Gauduchon assumption on *X*_{0}

We now pick up where we left off before Definition 4.1. As hinted there, the proof of Conjecture 1.1 would be complete if we were able to choose our family of Gauduchon metrics (*γ*_{t})_{t∈Δ}, varying in a *C*^{∞} way with *t*, such that *γ*_{0} is a *strongly Gauduchon* metric on *X*_{0}. Indeed, the above preparations being understood, Proposition 2.3 can be proved under this assumption.

### Proposition 4.5

*Suppose the limit fibre*

*X*

_{0}

*of a family as in Conjecture*1.1

*is a strongly Gauduchon manifold*.

*Then*,

*after possibly shrinking*

*Δ*

*about*0,

*there exists a family*(

*γ*

_{t})

_{t∈Δ},

*varying in a*

*C*

^{∞}

*way with*

*t*,

*of strongly Gauduchon metrics on the fibres*(

*X*

_{t})

_{t∈Δ}.

*Implicitly*,

*uniform mass boundedness holds*:

*where*,

*for every*

*t*∈

*Δ*∖

*Σ*,

*ω*

_{t}

*is any*

*J*

_{t}-

*Kähler form belonging to the class*

*α*∈

*H*

^{2}(

*X*,ℤ)

*given by Remark*2.1

*and*

*C*>0

*is a constant independent of*

*t*∈

*Δ*∖

*Σ*.

### Proof

*X*

_{0}is assumed to be

*strongly Gauduchon*, by the above Proposition 4.3 there is no current as described there on

*X*

_{0}. Equivalently, there exists a real smooth (2

*n*−2)-form

*Ω*on

*X*such that

*Ω*is

*d*-closed and \(\varOmega_{0}^{n-1, n-1}>0\) on

*X*

_{0}(cf. Proposition 4.2). Now, the

*J*

_{t}-type-(

*n*−1,

*n*−1)-components of

*Ω*vary in a

*C*

^{∞}way with

*t*∈

*Δ*as the complex structures (

*J*

_{t})

_{t∈Δ}do. Thus, after possibly shrinking

*Δ*about 0, we still have \(\varOmega_{t}^{n-1, n-1}>0\) on

*X*

_{t}. Furthermore, Michelsohn’s procedure for extracting the root of order

*n*−1 being a purely linear-algebraic argument, the family of corresponding roots (

*γ*

_{t})

_{t∈Δ}(i.e. \(\gamma_{t}^{n-1}=\varOmega_{t}^{n-1, \, n-1}\)) varies in a

*C*

^{∞}way with

*t*. It is therefore a

*C*

^{∞}family of

*strongly Gauduchon*metrics. Moreover, as

*dΩ*=0, (21) reads:

*main quantity*

*I*

_{t}extends to

*t*=0 and reads

*J*

_{t}−(

*n*,

*n*−2)-components of the fixed form

*Ω*varies in a

*C*

^{∞}way with

*t*, so does the family (

*I*

_{t})

_{t∈Δ}. In view of the explanations given at the beginning of this section, the proof is complete. □

The above arguments add up to a proof of Theorem 1.4.