Inventiones mathematicae

, Volume 194, Issue 3, pp 515–534

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

Authors

    • Institut de Mathématiques de ToulouseUniversité Paul Sabatier
Article

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

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 h0,1 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 ℂm. 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 fibreXt:=π−1(t) is projective for everytΔ:=Δ∖{0}. ThenX0:=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 X0 need not be Kähler, let alone projective. (A posteriori, when X0 has been shown to be Moishezon, X0 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 tΔ.

Theorem 1.2

Let\(\pi: \mathcal{X} \rightarrow \varDelta \)be a complex analytic family of compact complex manifolds such that the fibreXt:=π−1(t) is projective for everytΔ. Suppose thath0, 1(0)=h0, 1(t) fortclose to 0. ThenX0is 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 h0, 1(t) is assumed not to depend on t, all the hp, 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 Xt 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 fibreXt:=π−1(t) is projective for everytΔ:=Δ∖{0}. ThenX0:=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 hp,q(t) are locally constant in a family of surfaces. In particular, the situation considered in Theorem 1.2 occurs, hence X0 is Moishezon. On the other hand, since the Betti numbers bk of the fibres are always constant, the first Betti number b1 of X0 must be even. Now, by Kodaira’s classification of surfaces and Siu’s result [16] (see also [3, 12]), every compact complex surface with b1 even is Kähler. Being both Moishezon and Kähler, the limit surface X0 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 X0. Recall that a d-closed (1,1)-current T is said to be a Kähler current (a term coined in [10]) if Tεω 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 X0 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 fibreXt:=π−1(t) is projective for everytΔ. Suppose thatX0is a strongly Gauduchon manifold. ThenX0is Moishezon.

The strongly Gauduchon assumption on X0 is different in nature to the non-jumping assumption made on h0,1(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 X0 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 X0 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 fibreXt:=π−1(t) is Kähler for everytΔ:=Δ∖{0}. Then, isX0:=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 Xt 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 Xt 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 (γt)tΔ on the fibres (Xt)tΔ, 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 γt. 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 Jt-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 Jt 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 Jt-(p,q)-classes will be denoted by Hp, q(Xt,ℂ), while Hk(X,ℂ) will stand for De Rham cohomology. The respective dimensions of these ℂ-vector spaces are the usual Hodge numbers hp,q(t) and Betti numbers bk. By the Kähler assumption on Xt for every t≠0, every hp,q(t) is constant on Δ after possibly shrinking Δ about 0. But it may a priori happen that hp,q(0)>hp,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 Xt:=π−1(t) are C-diffeomorphic to a fixed compact differentiable manifold X. In other words, the family of complex manifolds (Xt)tΔ can be seen as one differentiable manifold X equipped with a family of complex structures (Jt)tΔ varying in a holomorphic way with t. In particular, for every k, the De Rham cohomology groups Hk(Xt,ℂ) of all the fibres can be identified with a fixed Hk(X,ℂ), while the Dolbeault cohomology groups Hp,q(Xt,ℂ) depend on tΔ.

As the fibres Xt 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 αH2(X,ℤ) such that, for every tΔ, α can be represented by a 2-form which is of Jt-type (1,1).

Moreover, α can be chosen in such a way that, for every tΔΣ, α is the first Chern class of an ample line bundle LtXt, where Σ={0}∪Σ′⊂Δ and Σ′=⋃Σν is a countable union of proper analytic subsets Σν⊆̷Δ.

Proof

To see this well-known fact, for any given class αH2(X,ℝ), let SαΔ denote the set of points tΔ such that α can be represented by a Jt-type (1,1)-form. For every tΔ, Xt is compact Kähler (even projective), so there exists a Hodge decomposition H2(X,ℂ)=H2,0(Xt,ℂ)⊕H1,1(Xt,ℂ)⊕H0,2(Xt,ℂ) with \(H^{2, 0}(X_{t}, \mathbb{C})=\overline{H^{0, 2}(X_{t}, \mathbb {C})}\). Thus, a given αH2(X,ℝ) contains a Jt-type (1,1)-form if and only if its projection onto H0,2(Xt,ℂ) vanishes. This means that Sα 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 Xt 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 Jt varies holomorphically with t, σα is a holomorphic section of the associated holomorphic vector bundle over Δ. This clearly implies that Sα is an analytic subset of Δ for every αH2(X,ℝ). On the other hand, the projectiveness assumption on every Xt with t≠0 entails the equality
$$ \bigcup_{\alpha} S_{\alpha} = \varDelta ^{\star}, $$
(1)
where the union is taken over all the integral classes αH2(X,ℤ) such that α is an ample class on some fibre \(X_{t_{0}}\), t0≠0 (depending on α). Now, a proper analytic subset is Lebesgue negligible. If Sα were a proper subset of Δ for every such αH2(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 αH2(X,ℤ) which can be represented by a Jt-type (1,1)-form for every t≠0 (i.e. Sα=Δ) and which is an ample class on at least one fibre \(X_{t_{0}}\), t0=t0(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 Xt, t≠0) and Barlet’s theory of cycle spaces [1] which implies that the cohomology classes {[Z]} of analytic cycles ZXt with t≠0 are the same on all fibres Xt, 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 t0≠0, α must be an ample class on every fibre Xt with tΔΣ, where Σ={0}∪Σ′ for a countable union Σ′=⋃Σν of proper analytic subsets Σν⊆̷Δ. □

Construction of Kähler metrics (ωt)tΔΣ in the class α

Let \(n:=\operatorname{dim}_{\mathbb{C}}\,X_{t}\), tΔ. Fix a class αH2(X, ℤ) as above and set
$$ v:=\int_X \alpha^n>0. $$
(2)
By Stokes’ theorem, this integral is clearly independent of the choice of representative of α. Moreover, v>0 since α is the first Chern class of an ample line bundle Lt on Xt and \(v=L_{t}^{n}>0\) is the volume of Lt for every tΔΣ. Finally, the differential operator d of X admits a separate splitting
$$d=\partial_t + \bar{\partial}_t, \quad t\in \varDelta , $$
for each complex structure Jt of X.
Here is an outline of our approach. Consider a C family (dVt)tΔ of C volume forms dVt>0 on Xt 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 Xt for every tΔΣ. Thus, for tΔΣ, we get a C 2-form ωtα=c1(Lt) on X which is a Kähler form with respect to the complex structure Jt (i.e. t=0, ωt is of type (1, 1) and positive definite with respect to Jt) such that
$$ \omega^n_t(x)=v\, dV_t(x), \quad x\in X_t. $$
(3)

The first step in this attack on Conjecture 1.1 will be to show that the family of Kähler forms (ωt)tΔΣ 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 J0-type (1,1) and must lie in the given class αH2(X,ℤ). This current, possibly with wild singularities, will only satisfy mild positivity properties on X0. However, the so-called singular Morse inequalities for integral classes that we obtained in [15] will imply the existence of a Kähler current on X0 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 X0 whose existence, in turn, amounts to X0 being Moishezon.

The second step in this attack on Conjecture 1.1 will thus consist in a crucial application of the following theorem which was the main result in [15]. Given an arbitrary compact complex manifold X with \(\operatorname{dim}_{\mathbb{C}}X=n\), recall that the volume of a holomorphic line bundle LX, a birational invariant measuring the asymptotic growth of spaces of global holomorphic sections of high tensor powers of L, is standardly defined as
$$ v(L):=\limsup_{k\rightarrow+\infty }\frac{n!}{k^n}\, h^0 \bigl(X, L^k\bigr). $$
(4)
If L is ample, the volume is known to be given by v(L)=∫Xc1(L)n:=Ln, motivating notation (2). Theorem 1.3. in [15] gives the following metric characterisation of the volume:
$$ v(L)=\sup\int_XT_{ac}^n, $$
(5)
where the supremum is taken over all positive currents T≥0 in the first Chern class of L and Tac 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 LX is big if and only if it can be equipped with a (possibly singular) Hermitian metric h whose curvature current T:=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])

LetXbe a compact complex manifold, \(\operatorname{dim}_{\mathbb {C}}X=n\). If there exists ad-closed (1,1)-currentTonXwhose De Rham cohomology class is integral and which satisfies
$$\textup{(i)}\quad T\geq0\quad \mbox{\textit{on}} \ X; \qquad \textup{(ii)}\quad \int _XT_{ac}^n>0, $$
then the cohomology class ofTcontains a Kähler current S. Implicitly, Xis Moishezon.

This is the form in which we will use the result of [15] on X0. The limit current T obtained on the limit fibre X0 at the end of the first step will be shown to satisfy the mild positivity conditions (i) and (ii) of Theorem 2.2 on X0. By that theorem, the integral class α of T must contain a Kähler current, proving that X0 is Moishezon.

The assumption made on h0,1(t):=dim H0,1(Xt,ℂ) in Theorem 1.2 enables one to uniformly bound the masses of the Kähler forms (ωt)tΔΣ with respect to a family of Gauduchon metrics on the fibres Xt varying in a C way with the parameter tΔ. This is because the invariance of h0,1(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 (ωt)tΔΣ. This will occupy Sect. 3.

Rather than proving the invariance of h0,1(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 (ωt)tΔΣ even in the mythical case where h0,1(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 shrinkingΔabout 0, there exists a family (γt)tΔof Gauduchon metrics varying in aCway withton the fibres (Xt)tΔand satisfying the following uniform mass boundedness property. For everytΔΣ, choose anyJt-Kähler formωtbelonging to the classαH2(X, ℤ) given by Remark 2.1. Then there exists a constantC>0 independent oftΔΣsuch that
$$ 0<\int_{X_t}\omega_t\wedge \gamma_t^{n-1}\leq C<+\infty, \quad \mbox{\textit{for all}} \ t\in \varDelta \setminus\varSigma. $$
(6)

If h0,1(t) is independent of tΔ, 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 αtH2(X,ℝ), t≠0, whose volumes vt 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 αt, t≠0, replaces the constant class α. This is because the projective assumption on Xt 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 h0,1(t), tΔ

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

Step 1: Produce a weak-limit currentT≥0 onX0from (ωt)tΔΣ.

The proof of Gauduchon’s theorem [7] implies the existence of a smooth family of Gauduchon metrics on the fibres Xt=(X,Jt). In other words, there exists a family of 2-forms (γt)tΔ on X, varying in a C way with tΔ, such that each γt is a positive-definite, type (1,1)-form with respect to Jt and satisfies the Gauduchon condition on Xt: \(\partial_{t}\bar{\partial}_{t}\gamma_{t}^{n-1}=0\). 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 (Xt)tΔ. Consider the Laplace-type operator acting on smooth functions:
$$P_{\omega_t'}:=i \varLambda_{\omega'_t}\bar{\partial}_t \partial_t : C^{\infty}(X, \mathbb{C})\rightarrow C^{\infty}(X, \mathbb{C}), $$
where \(\varLambda_{\omega'_{t}}\) is the \(\omega'_{t}\)-adjoint of the multiplication by \(\omega'_{t}\). The adjoint of \(P_{\omega_{t}'}\) is
$$P^{\star}_{\omega_t'} : C^{\infty}(X, \mathbb{C}) \rightarrow C^{\infty}(X, \mathbb{C}), \quad P^{\star }_{\omega_t'}(f)=i \, \star\bar{\partial}_t\partial_t \biggl(f \frac {\omega_t^{'n-1}}{(n-1)!} \biggr), $$
where \(\star=\star_{\omega'_{t}} : C^{\infty}_{n, n}(X_{t}, \mathbb{C})\rightarrow C^{\infty}(X_{t}, \mathbb{C})\) is the Hodge-star operator (an isometry) associated with \(\omega'_{t}\). The operators \(P_{\omega_{t}'}\) and \(P^{\star}_{\omega_{t}'}\) are elliptic, ≥0, and of vanishing index (as the principal symbols are self-adjoint). Moreover, \(\ker P_{\omega_{t}'}=\mathbb{C}\) (i.e. the constant functions) by the obvious inclusion \(\mathbb{C}\subset\ker P_{\omega_{t}'}\) and the maximum principle. Hence, by ellipticity and vanishing index, \(\operatorname{dim} \ker P^{\star}_{\omega_{t}'}=1\). Furthermore, the proof of [7] shows that any function \(f\in \ker (P^{\star}_{\omega_{t}'|C^{\infty}(X, \mathbb{R})})\) must satisfy: f>0 on X or f<0 on X or f=0 on X. The existence of a C function ft: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 Xt 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 f0>0 and to extend it to a C local section Δtft 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, ft>0 for all t sufficiently close to 0∈Δ, defining a family \(\gamma_{t}:=f_{t}^{\frac{1}{n-1}} \omega_{t}'\), tΔ, of Gauduchon metrics varying in a C way with t on the fibres Xt.

Fix any such family (γt)tΔ. 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

LetαH2(X,ℤ) a class given by Remark 2.1. For everytΔΣ, letωtbe an arbitraryJt-Kähler form belonging to the classα. Ifh0,1(t) is independent oftΔ, there exists a constantC>0 independent oftΔΣsuch that the masses of theωt’s with respect to the\(\gamma_{t}^{n-1}\)’s satisfy:
$$ 0<\int_{X_t}\omega_t\wedge \gamma_t^{n-1}\leq C < +\infty, \quad\mbox{\textit{for all}}\ t\in \varDelta \setminus\varSigma $$
(7)
after possibly shrinkingΔabout 0.

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

Proof

The lower bound is obvious as ωt>0 and γt>0. Let \(\tilde{\omega}\) be any d-closed real 2-form in the De Rham class α. As ωt and \(\tilde{\omega}\) are De Rham cohomologous real 2-forms, there exists a smooth real 1-form βt on Xt such that:
$$ \omega_t=\tilde{\omega} + d\beta_t \quad \mbox{on} \ X_t, \ \mbox{for every} \ t\in \varDelta \setminus\varSigma. $$
(8)
Thus, for each tΔΣ, the mass of ωt splits as:
$$ \int_{X_t}\omega_t\wedge \gamma_t^{n-1} = \int_{X_t}\tilde{\omega} \wedge\gamma_t^{n-1} + \int_{X_t}d \beta_t\wedge\gamma_t^{n-1}. $$
(9)
As the forms (γt)tΔ vary in a C way with tΔ, 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 tΔΣ approaches 0. The difficulty stems from the fact that the family (ωt)tΔΣ of Kähler forms (and implicitly (βt)tΔΣ) need not extend to the limit fibre X0 as X0 is not assumed to be Kähler. Using Stokes’ theorem we get:
https://static-content.springer.com/image/art%3A10.1007%2Fs00222-013-0449-0/MediaObjects/222_2013_449_Equ10_HTML.gif
(10)
where \(\beta_{t}=\beta_{t}^{1, 0} + \beta_{t}^{0, 1}\) is the decomposition of βt into components of types (1,0) and (0,1). As \(\beta_{t}=\overline{\beta_{t}}\) (i.e. βt 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 tΔΣ approaches 0.
Now the solution βt of (8) is not unique. We will make a particular choice of βt. The Kähler form ωt being of Jt-type (1,1), equating the components of Jt-type (0,2) in (8), we see that \(\beta_{t}^{0, 1}\) must solve the equation:
$$ \bar{\partial_t}\beta_t^{0, 1} = - \tilde{\omega}_t^{0, 2} \quad \mbox{on}\ X_t, \ \mbox{for}\ t\in \varDelta \setminus\varSigma. $$
(11)
Conversely, for every tΔ, choose \(\beta_{t}^{0, 1}\) to be the solution of equation (11) of minimal L2 norm with respect to the metric γt of Xt. (Notice that (11) is solvable in \(\beta_{t}^{0,\, 1}\) for every t≠0 because α contains a Jt-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 J0-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, βt is a real 1-form on X but it need not solve (8). However, the \(\partial\bar{\partial}\)-lemma (which holds on every Xt with t≠0 by the Kähler assumption) shows that βt satisfies (8) on Xt up to a \(\partial_{t}\bar{\partial}_{t}\)-exact (1,1)-form for each tΔΣ. Indeed, \(\tilde{\omega} + d\, \beta_{t}\) is of Jt-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 Jt-type (1,1) and d-exact, hence also \(\partial_{t}\bar{\partial}_{t}\)-exact by the \(\partial\bar{\partial}\)-lemma. Thus βt solves the equation:
$$ \omega_t=\tilde{\omega} + d\beta_t + i\partial_t\bar{\partial}_t\varphi_t \quad \mbox{on} \ X_t, \ \mbox{for} \ t\in \varDelta \setminus\varSigma, $$
(12)
for some smooth function φt on X. Now, since γt has been chosen such that \(\partial_{t}\bar{\partial}_{t}\gamma _{t}^{n-1}=0\) (the Gauduchon condition), Stokes’ theorem gives:
$$\int_{X_t}i\partial_t\bar{\partial}_t \varphi_t\wedge\gamma_t^{n-1} =\int _{X_t}\varphi_t\wedge i\partial_t \bar{\partial}_t\gamma_t^{n-1}=0. $$

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 tΔ approaches 0.

For every tΔ, let \(\bar{\partial_{t}}^{\star}\) denote the formal adjoint of \(\bar{\partial_{t}}\) with respect to the global L2 scalar product defined by the Gauduchon metric γt of Xt. We get a C family \((\varDelta ''_{t})_{t\in \varDelta }\) of associated anti-holomorphic Laplace–Beltrami operators defined as
$$\varDelta ''_t = \bar{\partial_t} \bar{\partial_t}^{\star} + \bar{\partial_t}^{\star} \bar{\partial_t} \quad \mbox{on} \ X_t=(X, \, J_t), \ t\in \varDelta . $$
The (unique) solution of minimal L2 norm of (11) is known to be given by the formula:
$$ \beta_t^{0, 1} = - G_t \bar {\partial_t}^{\star} \tilde{\omega_t}^{0, 2}, \quad t\in \varDelta \setminus\varSigma, $$
(13)
where Gt 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:
$$H^{0, 1}(X_t, \mathbb{C}) \simeq\mathcal{H}^{0, 1}(X_t, \mathbb{C}), \quad t\in \varDelta , $$
where \(\mathcal{H}^{0, 1}(X_{t}, \mathbb{C}):=\ker \varDelta ''_{t}\) is the space of harmonic Jt-(0,1)-forms. By a well-known result of Kodaira and Spencer (see [11, Theorem 7.6, p. 344]), the family of Green operators (Gt)tΔ of a C family of strongly elliptic operators \((\varDelta ''_{t})_{t\in \varDelta }\) is C with respect to tΔ if the dimensions of the kernel spaces \(\mathcal{H}^{0, 1}(X_{t}, \mathbb{C})\) are independent of tΔ. 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 tΔ, and \(\operatorname{dim}\mathcal{H}^{0, 1}(X_{t}, \mathbb{C})=h^{0, 1}(t)\) by the Hodge isomorphism.

Now the Jt-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 (Jt)tΔ 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 tΔ. 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 (ωt)tΔΣ is bounded in mass, it is weakly compact. Thus it contains a weakly convergent subsequence \(\omega_{t_{k}}\rightarrow T\), with ΔΣtk→0 as k→+∞. The limit current T≥0 is closed, positive and of type (1,1) for the limit complex structure J0 of X0. By the weak continuity of De Rham classes { }, \(\{\omega_{t_{k}}\} \rightarrow\{T\}\). As \(\{\omega_{t_{k}}\}=\alpha\) for all k, we see that Tα.

We have thus produced a closed positive (1,1)-current T≥0 on X0 in the given integral De Rham cohomology class α.

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

This is where the singular Morse inequalities for integral classes (Theorem 2.2) come into play. The semicontinuity property of the top power of the absolutely continuous part of (1,1)-currents (see e.g. [2]) reads:
$$ T_{ac}(x)^n \geq\limsup_{k\rightarrow+\infty} \omega_{t_k}(x)^n, \quad \mbox{for almost every}\ x\in X_0. $$
(14)
Now, if the Kähler forms ωt, tΔΣ, 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:
$$T_{ac}(x)^n \geq v\, \limsup_{k\rightarrow+\infty} dV_{t_k}(x) = v\, dV_0(x),\quad\mbox{for almost every}\ x\in X_0. $$
Hence
$$ \int_{X_0}T_{ac}^n \geq v >0. $$
(15)
In particular, 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 J0), 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 X0, α must contain a Kähler current, hence X0 must be Moishezon.

The proof of Theorem 1.2 is complete. □

Remark 3.2

When trying to dispense with the non-jumping hypothesis that was made in Theorem 1.2 on h0,1(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 (ωt)tΔΣ. For every tΔ, the Laplace operator \(\varDelta ''_{t}\) acting on Jt-(0,1)-forms of X is elliptic and therefore has a compact resolvent and a discrete spectrum
$$ 0=\lambda_0(t)\leq\lambda_1(t) \leq \dots\leq\lambda_k(t) \leq\cdots $$
(16)
with λk(t)→+∞ as k→+∞. By the Hodge isomorphism, the multiplicity of zero as an eigenvalue of \(\varDelta ''_{t}\) equals h0,1(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 tΔ is sufficiently close to 0 (say δε-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=h0,1(0)≥h0,1(t) for t sufficiently close to 0 (the upper-semicontinuity property). Consequently, for t near 0, h0,1(0)=h0,1(t) if and only if 0 is the only eigenvalue of \(\varDelta ''_{t}\) lying in [0,ε). In other words, if h0,1(0)>h0,1(t) when t(≠0) is near 0, choosing increasingly small ε>0 gives eigenvalues of \(\varDelta ''_{t}\):
$$ 0<\lambda_{k_1}(t)\leq \lambda_{k_2(t)} \leq\cdots\leq\lambda_{k_N}(t):= \varepsilon_t < \varepsilon, \quad t\in \varDelta ^{\star}, $$
(17)
that converge to zero (i.e. εt→0) when t→0, where N=h0,1(0)−h0,1(t). Now, formula (13) for \(\beta_{t}^{0, 1}\) involves the Green operator Gt 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 Gt. 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 Gt 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 ωt (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 h0,1(t).

4 The strongly Gauduchon case

In this section we transform the topological problem with the possible jump of h0,1(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

The notation is carried forward from the previous sections. Fix any family (γt)tΔ of Jt-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:
$$ I_t:=\int_{X_t} \partial_t\beta_t^{0, 1}\wedge \gamma_t^{n-1}= \int_{X_t} \beta_t^{0, 1}\wedge\partial_t \gamma_t^{n-1},\quad t\in \varDelta ^{\star}, $$
(18)
as t approaches 0. The difficulty is that the family \((\beta^{0, 1}_{t})_{t\in \varDelta ^{\star}}\) of Jt−(0,1)-forms constructed as the minimal L2-norm solutions of (11) (extended to all t≠0) need not be bounded as t approaches 0 if h0,1(0)>h0,1(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 Jt−(0,2)-components of the fixed 2-form \(\tilde{\omega}\) varies in a C way with tΔ. 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 quantityIt, 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.
The next observation is that the 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 γt. For t≠0, the \(\partial\bar{\partial}\)-lemma holds on Xt (thanks to the Kähler assumption) and yields the \(\bar{\partial}_{t}\)-exactness of \(\partial_{t}\gamma _{t}^{n-1}\). Thus,
$$ \partial_t\gamma_t^{n-1}= \bar{\partial}_t\zeta_t, \quad t\neq0, $$
(19)
where the CJt-(n,n−2)-form ζt can be chosen as the minimal L2-norm solution of the above equation (with respect to γt), generating a C family \((\zeta_{t})_{t\in \varDelta ^{\star}}\) defined off t=0. By Stokes’ theorem, the main quantity now reads:
$$ I_t=\int_{X_t} \beta_t^{0, 1}\wedge\partial_t \gamma_t^{n-1}=\int_{X_t}\bar{\partial}_t\beta_t^{0, 1}\wedge \zeta_t,\quad t\in \varDelta ^{\star}. $$
(20)

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 ζt near t=0. The difficulty stems from the fact that, at this point, the \(\partial\bar{\partial}\)-lemma is not known to hold on X0. 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 X0, 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

Let X be a compact complex manifold, \(\operatorname{dim}_{\mathbb{C}} X=n\).
  1. (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 ∂γn−1 is \(\bar{\partial}\)-exact on X.

     
  2. (ii)

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

     

Notice that the Gauduchon condition only requires ∂γn−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 manifoldXof complex dimensionncarries a strongly Gauduchon metricγif and only if there exists a reald-closedCformΩof degree 2n−2 onXsuch that its component of type (n−1,n−1) satisfiesΩn−1,n−1>0 on X.

Proof

For a real (2n−2)-form Ω, the (2n−1)-form is also real, hence its components of type (n,n−1) and respectively (n−1,n) are conjugate to each other. Thus, the condition =0 amounts to
$$ \partial\varOmega^{n-1, n-1}=-\bar{\partial}\varOmega^{n, n-2}. $$
(21)

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

Conversely, if there exists a (2n−2)-form Ω on X as in the statement, the assumption Ωn−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 Ωn−1,n−1=γn−1. By the assumption =0 and its equivalent formulation (21), we see that γ satisfies the strongly Gauduchon condition. □

We shall now determine when a (2n−2)-form as in Proposition 4.2 above exists. Let X be any compact complex manifold, dimX=n, and let Ω be any real C form of degree 2n−2 on X. The condition =0 is equivalent, by the duality between d-closed smooth real (2n−2)-forms and real exact 2-currents T=dS on X, to the property
$$ \int_X\varOmega\wedge dS=0, \quad \mbox{for every real}\ 1\mbox{-current}\ S \ \mbox{on} \ X. $$
(22)
On the other hand, the duality between strictly positive, smooth (n−1,n−1)-forms and non-zero positive (1,1)-currents on X shows that the condition Ωn−1,n−1>0 is equivalent to the property
$$ \int_X\varOmega^{n-1, n-1}\wedge T>0, \quad \mbox{for every non-zero}\ (1, 1)\mbox{-current}\ T\geq0 \ \mbox{on} \ X. $$
(23)

Now, if T is of type (1,1), we clearly have ∫XΩn−1,n−1T=∫XΩT. Furthermore, real d-exact 2-currents T=dS 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 ∫XTΘ=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 (2n−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

LetXbe a compact complex manifold, \(\operatorname{dim}_{\mathbb{C}} X=n\). Then, Xcarries a strongly Gauduchon metricγif and only if there is no non-zero currentTof type (1,1) onXsuch thatT≥0 andTisd-exact on X.

We now end this digression on a few simple remarks. Given a Hermitian metric (equivalently, a C positive-definite (1,1)-form) γ on a compact complex manifold X, the following four conditions on γ: the Kähler condition (=0), the balanced condition (dγ=0), the strongly Gauduchon condition (∂γn−1 is \(\bar{\partial}\)-exact) and the Gauduchon condition (∂γn−1 is \(\bar{\partial}\)-closed) stand in the following implication hierarchy:
https://static-content.springer.com/image/art%3A10.1007%2Fs00222-013-0449-0/MediaObjects/222_2013_449_Equ24_HTML.gif
(24)

For example, the implication “balanced ⟹ strongly Gauduchon” can be seen as follows. Using the Hodge ⋆ operator that gives isometries ⋆: Λp,qTXΛnq,npTX defined by the Hermitian metric γ on X, we have ⋆γ=γn−1/(n−1)! and d=−⋆ d ⋆. Hence, the balanced condition dγ=0 is equivalent to n−1=0, which in turn is equivalent, thanks to γn−1 being a real form, to ∂γn−1=0 (cf. [13]). This is clearly a stronger condition than the strongly Gauduchon requirement that ∂γn−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 b1 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 b1 odd. □

4.3 Proof of Conjecture 1.1 under the strongly Gauduchon assumption on X0

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 (γt)tΔ, varying in a C way with t, such that γ0 is a strongly Gauduchon metric on X0. Indeed, the above preparations being understood, Proposition 2.3 can be proved under this assumption.

Proposition 4.5

Suppose the limit fibreX0of a family as in Conjecture 1.1 is a strongly Gauduchon manifold. Then, after possibly shrinkingΔabout 0, there exists a family (γt)tΔ, varying in aCway witht, of strongly Gauduchon metrics on the fibres (Xt)tΔ. Implicitly, uniform mass boundedness holds:
$$ 0<\int_{X_t}\omega_t\wedge \gamma_t^{n-1}\leq C < +\infty, \quad \mbox{\textit{for all}}\ t\in \varDelta \setminus\varSigma, $$
(25)
where, for everytΔΣ, ωtis anyJt-Kähler form belonging to the classαH2(X,ℤ) given by Remark 2.1 andC>0 is a constant independent oftΔΣ.

Proof

As the limit fibre X0 is assumed to be strongly Gauduchon, by the above Proposition 4.3 there is no current as described there on X0. Equivalently, there exists a real smooth (2n−2)-form Ω on X such that Ω is d-closed and \(\varOmega_{0}^{n-1, n-1}>0\) on X0 (cf. Proposition 4.2). Now, the Jt-type-(n−1,n−1)-components of Ω vary in a C way with tΔ as the complex structures (Jt)tΔ do. Thus, after possibly shrinking Δ about 0, we still have \(\varOmega_{t}^{n-1, n-1}>0\) on Xt. Furthermore, Michelsohn’s procedure for extracting the root of order n−1 being a purely linear-algebraic argument, the family of corresponding roots (γt)tΔ (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 =0, (21) reads:
$$ \partial_t\gamma_t^{n-1}=- \bar{\partial}_t\varOmega_t^{n, \, n-2}, \quad t\in \varDelta . $$
(26)
Thus (20) shows that the main quantityIt extends to t=0 and reads
$$ I_t=-\int_{X_t}\bar{\partial}_t\beta_t^{0, 1}\wedge \varOmega_t^{n, n-2}=-\int_{X_t}\tilde{ \omega}_t^{0, 2}\wedge\varOmega_t^{n, n-2}, \quad t\in \varDelta . $$
(27)
As the family \((\varOmega_{t}^{n, n-2})_{t\in \varDelta }\) of Jt−(n,n−2)-components of the fixed form Ω varies in a C way with t, so does the family (It)tΔ. 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.

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