Received: 8 April 2024 / Accepted: 4 September 2025 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2025 In Memory of Jean-Pierre Demailly (1957–2022). Runze Zhang runze.zhang@unice.fr Sheng Rao likeanyone@whu.edu.cn 1 School of Mathematics and statistics, Wuhan University, Wuhan 430072, China 2 CNRS, Laboratoire J.-A. Dieudonné, Université Côte d’Azur, Parc Valrose, 06108 Nice Cedex 2, France On extension of closed complex (basic) differential forms: (basic) Hodge numbers and (transversely) p -Kähler structures Sheng Rao 1 · Runze Zhang 1,2 Annali di Matematica Pura ed Applicata (1923 -) https://doi.org/10.1007/s10231-025-01610-y Abstract Inspired by a recent work of Wei–Zhu on the extension of closed complex differential forms and Voisin’s usage of the ∂ ∂ -lemma, we obtain several new theorems of deforma- tion invariance of Hodge numbers and reprove the local stabilities of p -Kähler structures with the ∂∂ -property. Our approach is more concerned with the d -closed extension by means of the exponential operator e ι φ . Furthermore, we prove the local stabilities of trans- versely p -Kähler structures with mild ∂ ∂ -property by adapting the power series method to the foliated case, which strengthens the works of El Kacimi Alaoui–Gmira and Raźny on that of the transversely Kähler foliations with homologically orientability. We observe that a transversely Kähler foliation, even without homologically orientability, also satis- fies the ∂ ∂ -property. So even when p = 1 (transversely Kähler), our results are new as we can drop the assumption in question on the initial foliation. Several theorems on the deformation invariance of basic Hodge/Bott–Chern numbers with mild ∂∂ -properties are also presented. Keywords Deformations of complex structures · Deformations and infinitesimal methods · Formal methods · Deformations · Hermitian and Kählerian manifolds · Foliations Mathematics Subject Classification Primary 32G05 · 53C12 · Secondary 13D10 · 14D15 · 53C55 1 3 S. Rao, R. Zhang 1 Introduction The classical deformation theory of compact complex manifolds, developed by Kodaira– Nirenberg–Spencer and Kuranishi [53–55, 58], intensively studies the complex structures ‘close to’ a given one. Subsequently, the theory had been extended to the case of complex pseudogroup structures such as [52, 56]. In particular, many excellent works concerning the deformations of transversely holomorphic foliations and holomorphic foliations appear, see [16, 24, 25, 28, 30, 31, 40–42, 44, 83], etc. Most of these works essentially dealt with the extension of closed complex (basic) differential forms. As an important and direct application, one considers the deformation invariance of Hodge numbers. It becomes a useful tool in the study of the deformation limit problems. For instance, let π : X → ∆ be a holomorphic family over an open disk in C . Popovici [74] (resp. Barlet [13 ]) proved that if all fibers X t are projective (resp. Moishezon) for 0 ̸ = t ∈ ∆ and X 0 satisfies the deformation invariance for Hodge number of type (0, 1) or admits a strongly Gauduchon metric, then X 0 is Moishezon. The first author–Tsai [ 77] prove that if there exist uncountably many Moishezon fibers in the family, then any fiber with either of the above two conditions is still Moishezon. It is well known that each Hodge number takes a constant value along the small dif- ferentiable deformation X t of X 0 when the central fiber X 0 satisfies the (standard) ∂∂ -property or more generally, the Frölicher spectral sequence of X 0 degenerates at the E 1 -level, cf. [47, Section 5.1] or [92, Proposition 9.20]. Recall that the (standard) ∂ ̄ ∂ - property refers to: for every pure-type d -closed form on a compact complex manifold, the properties of d -exactness, ∂ -exactness, ̄ ∂ -exactness and ∂ ̄ ∂ -exactness are equivalent. So it is natural to study this topic with more general conditions on X 0 , such as some ‘weak’ ∂∂ -properties see Sect. 2.1 . Such results were first given in [ 81, 100 ]. Significantly, the weak version of ∂∂ -property was initially introduced in [37] while investigating deformations of balanced manifolds. Recently, Xia studied this further in terms of canonical deformations [97, Theo- rem 1.3], see also a much recent work [93] of Wan–Xia. Drawing on Xia’s work and a recent work of Wei–Zhu [94], we obtain several new theo- rems on the deformation invariance of Hodge numbers: Theorem A Let π : X → B be a differentiable family of compact complex n-dimensional manifolds over a sufficiently small domain in R d as in Definition 3.1 with the central fiber X 0 := π − 1 ( 0 ) and the general fibers X t := π − 1 ( t ) . Consider the function B t −→ h p,q ∂ t (X t ) := dim C H p,q ∂ t (X t ) , for any non-negative integers p, q ≤ n. (1.1) If the injectivity of the mapping ι p , q + 1 BC ,∂ , the surjectivity of the mapping ι p , q BC , ∂ on the central fiber X 0 and the deformation invariance of the ( p , q − 1 ) -Hodge number h p , q − 1 ∂ t ( X t ) hold, then h p , q ∂ t ( X t ) are independent of t. See the notations ι p , q + 1 BC ,∂ , ι p , q BC ,∂ in Sect. 2.1. Examples 4.1, 4.2, and 4.3 show that the deformation invariance of the ( p , q )-Hodge number fails when one of the three conditions in Theorem A is not satisfied, while the other two hold, thanks to the Kuranishi family of the Iwasawa manifold (see [6, Appendix] and [86, Section 1.c]). 1 3 On extension of closed complex (basic) differential forms: (basic) Hodge... Following [81, Notation 3.5], we say that a compact complex manifold X satisfies S p,q (resp. B p,q ), if for any ∂ -closed ∂g ∈ A p,q ( X ) , the equation ∂x = ∂g (1.2) has a solution (resp. a ∂ -exact solution) of pure-type complex differential form. Similarly, a compact complex manifold X is said to satisfy S p,q (resp. B p,q ), if for any ∂ -closed g ∈ A p − 1 ,q (X ) , the Eq. (1.2) has a solution (resp. a ∂ -exact solution) of pure-type complex differential form. The special cases of the ( p , 0) and (0, q ) types may allow for a weakening of the condi- tions in Theorem A: Theorem B (Theorems 4.5 and 4.6 ) With the setting of Theorem A, (1) if X 0 satisfies B p, 1 (i.e., the mapping ι p, 1 BC ,∂ is injective) and S p +1 , 0 , then h p, 0 ∂ t ( X t ) are deformation invariant; (2) if X 0 satisfies B 1 ,q (i.e., the mapping ι 0 ,q BC ,∂ is surjective) and h 0 ,q − 1 ∂ t ( X t ) satisfies the deformation invariance, then h 0 ,q ∂ t ( X t ) are independent of t. Note that Theorem B (2) first appeared as [ 81, Theorem 3.7]. So we can also obtain [81, Remark 3.8, Corollary 3.9] as a consequence. Furthermore, an example applicable to Theo- rem B(1) in Remark 4.7 shows that the function of Theorem B(1) for ( p , 0)-Hodge numbers goes beyond Kodaira–Spencer’s squeeze [55, Theorem 13] sometimes. Before setting out the strategy to prove Theorems A and B , we first briefly state some knowledge of analytic deformation theory of complex structures to be introduced in detail in Sect. 3.1. Let π : X → B be a differentiable family as aforementioned inducing a canonical differentiable family of integrable Beltrami differentials on X 0 , denoted by φ ( z, t ) , φ ( t ) , and φ interchangeably. Now consider the exponential operator of contraction operators e ι φ := ∞ ∑ k =0 ι k φ ( t ) k ! , and one can show that e ι φ : A p, 0 (X 0 ) −→ A p, 0 (X t ) However, e ι φ can’t preserve ( p , q )-forms to X t when 1 ≤ q ≤ n . A useful fact due to [35, Theorem 5.1] or [94 , Section 3.1] shows that it actually maps from a filtration on X 0 to one on X t , that is e ι φ : F p A k ( X 0 ) = ⊕ p ≤s ≤k A s,k − s ( X 0 ) −→ F p A k ( X t ) According to [94, 97 ], one can define the projection operator 1 3 S. Rao, R. Zhang P φ : A 0 , 1 (X 0 ) −→ A 0 , 1 (X t ) We then use the exponential operator e ι φ and the projection operator P φ to define the exten- sion operator ρ φ : A p,q ( X 0 ) −→ A p,q ( X t ) (1.3) as in (3.4 ). To prove the deformation invariance of Hodge numbers, the first author– Zhao [81, 100] introduced an extension map e ι φ | ι φ : A p,q ( X 0 ) −→ A p,q ( X t ) , (1.4) which can preserve all ( p , q )-forms and played an important role in their subsequent papers [78, 79]. By comparing the explicit local expressions, one can deduce the relationship between these two extension maps: ρ φ = e ι φ | ι φ ◦ ( 1 − φφ ) − 1 Ⅎ , where the notation Ⅎ denotes the simultaneous contraction on each component of a complex differential form. An application of Kuranishi’s completeness theorem [58] can reduce our Theorems A and B to the Kuranishi family, which contains all sufficiently small differentiable defor - mations of X 0 in some sense (see Sect. 3.2). So to prove them, we just need to consider a Kuranishi family of deformations of X 0 over ∆ ε := { t ∈ C | | t | ≤ ε } with small ε , and there exists a family of integrable Beltrami differentials { φ (t ) } | t |≤ ε depending on t holomorphi- cally and describing the variations of complex structures on X 0 Recently, Wei–Zhu [94] applied the ∂∂ -Hodge theory to extend a d -closed ( p , q )-form on X 0 to a d -closed filtrated ( p + q ) -form on X t , whose ( p , q )-part on X t is ∂ t -closed via the extension operator ρ φ in (1.3). This is surely an interesting and important result. We will reprove their theorem in Sect. 3.3, from the perspective of Bott–Chern theory. Theorem C [94 , Theorem 1.1] Let X 0 be a compact complex manifold that satisfies B p , q + 1 . Given a d-closed μ 0 ∈ A p , q ( X 0 ) , one can construct μ ( z , t ) ∈ A p , q ( X 0 ) satisfying d ( e ι φ ( μ ( z , t ))) = 0 on X 0 (or X t ) and μ ( z , 0 ) = μ 0 ( z ) , which is holomorphic in small t. Furthermore, the extension ρ φ ( μ ( z , t )) ∈ A p , q ( X t ) is ̄ ∂ t -closed. Now we are ready to describe our strategy to consider the deformation invariance of Hodge numbers briefly. The Kodaira–Spencer’s upper semi-continuity theorem [ 55, The- orem 4] tells us that the function (1.1) is always upper semi-continuous for t ∈ ∆ ε and thus, to approach the deformation invariance of h p,q ∂ t ( X t ) , we only need to obtain the lower semi-continuity. And, our main strategy is to look for an injective extension map from H p,q ∂ ( X 0 ) to H p,q ∂ t ( X t ) with the help of Theorem C . More precisely, we first need to find a nice uniquely- chosen d -closed representative μ 0 of the given initial Dolbeault cohomology class in H p,q ∂ ( X 0 ) (see Lemma 4.9), and then apply power series method and Bott–Chern theory to construct μ ( z, t ) ∈ A p,q ( X 0 ) , such that it is smooth in ( z , t ) and holomorphic in small t 1 3 On extension of closed complex (basic) differential forms: (basic) Hodge... and ρ φ ( μ ( z, t )) is ̄ ∂ t -closed in A p,q ( X t ) by Theorem C. Finally, we try to verify that the extension map H p,q ∂ ( X 0 ) −→ H p,q ∂ t (X t ) : [ μ 0 ] ∂ −→ [ ρ φ ( μ (z, t ))] ∂ t is injective. Our Theorem A and the first assertion of Theorem B on the deformation invariance of Hodge numbers are different from the results in [ 81], cf. Remarks 4.4, 4.12 and 4.13. Our approach focuses more on the d -closed extension, while they concentrated on the specific ∂ -extension from A p,q ( X 0 ) to A p,q ( X t ) (see [81, Proposition 1.2]) by use of their extension map (1.4). As another application of the extension of closed complex differential forms in Theorem C, we study the local stabilities of several special complex structures. Inspired by the proof of [92, Theorem 9.23], we can take advantage of Theorem C and also the deformation open- ness of the ∂∂ -property to prove the local stabilities of p - Kähler structures (for this concept, one can refer to Appendix A for more details) with the ∂ ∂ -property. Theorem D [79 , Theorem 4.9] For any positive integer p ≤ n − 1 , any small differentiable deformation X t of a p-Kähler manifold X 0 satisfying the ∂∂ -property is still p-Kählerian. Notice that [79] presented a power series proof for Kodaira–Spencer’s local stabilities theorem of Kähler structures via their extension map (1.4), which is a problem at latest dated back to [70 , Remark 1 on p. 180]: ‘A good problem would be to find an elementary proof (for example, using power series methods). Our proof uses nontrivial results from partial differential equations’. In our proof, our primary focus is on the d -closed extension, by means of the exponential operator e ι φ , which is more natural and succinct in some sense. However, we still need to use [55, Theorem 7] to guarantee the positivity of the constructed explicit p -Kähler form, see Sect. 5.2 for more details. A challenge problem proposed by [94] is how to prove a direct corollary of Theorem D, that is Corollary 5.2(i), without using [55, Theorem 7]. It is worth mentioning that in [78 ], the first author–Wan–Zhao looked deeper into the local stabilities of p -Kähler structure when the central fiber X 0 satisfies some ‘weak’ ∂∂ -properties. More concretely, for any positive integer p ≤ n − 1 , any small differentiable deformation X t of an n -dimensional p -Kähler manifold X 0 satisfying the ( p, p + 1) -th mild ∂∂ -property is still p -Kählerian [78, Theorem 1.1]. However, in our approach, it seems that the standard ∂∂ -property condition on X 0 in Theorem D can’t be weakened, see Remark 5.4. As is well known, the Bott–Chern numbers are always upper semi-continuous with respect to the ordinary topology in a small differentiable family. Since the ordinary topol - ogy is much finer than the analytic Zariski topology, it’s natural to ask: Question A (Question 5.7) For a holomorphic family { X t } , is the function B t −→ h p,q BC (X t ) := dim C H p,q BC (X t ) upper semi-continuous with respect to the analytic Zariski topology? 1 3 S. Rao, R. Zhang To solve this question, we wish to use Grauert’s upper semi-continuity theorem 5.6. In other words, one needs to find some holomorphic vector bundle V on X , such that H p,q BC ( X t ) ≃ H q (X t , V | X t ) This seems hard to achieve due to the results in [5, 14], see Sect. 5.3 for more details. Recently, Xia [96 , Theorem 1.1 and Remark 3.6] confirmed Ques - tion 1.1 when the type is ( p , 0) or (0, q ). One motivation to affirm Question 1.1 is to obtain Theorem 5.9, as long as one notices that the ∂∂ -property in Theorem D actually can be replaced by the deformation invariance of ( p , p )-Bott–Chern numbers as shown in [79, Remark 4.13]. As is widely recognized, the transversely Kähler structures hold a central position within the field of foliation theory and are closely linked to a wealth of geometric structures. For instance, Vaisman manifolds, LVM manifolds (a generalized version of Calabi–Eckmann manifolds), and Sasakian manifolds all possess transversely Kähler structures despite not being Kähler themselves, see [15, 38, 64, 66, 71, 73], etc. Recently, the transversely balanced structures (or, more generally, the transversely Gaud- uchon structures, see Remark 6.11), are also actively studied, as evidenced by [11, 34], etc. In light of this, it naturally becomes a logical progression to extend the concept of compact p -Kähler manifolds, as originally introduced by Alessandrini–Andreatta [1 , Defi - nition 1.11], to the transverse context. This extension leads us to the introduction of trans - versely p - Kähler foliations (Definition 6.9 ). Remarkably, this overarching notion unifies the two aforementioned structures, specifically when p takes values of 1 and r − 1 (in the con- text of a transversely holomorphic foliation of codimension r ), respectively (Remark 6.11). When delving into the intermediate cases ( 1 < p < r − 1 ), a thought-provoking ques- tion naturally arises: Does there exist a non-trivial (specifically, with 1 < p < r − 1 , and non-transversely Kähler) transversely p -Kähler foliation? The answer to this question is affirmative, as demonstrated in Example 6.12. So, it seems that the transversely p -Kähler structures represent an interesting subject in foliation theory, especially within the realm of non-transversely Kähler geometry. In the second part of this paper, we are mainly concerned with the local stabilities of transversely p -Kähler structures. El Kacimi Alaoui proved that small deformations of a compact Kähler orbifold as an orbifold are still Kähler [28], which is a generalization of Kodaira–Spencer’s result of sta- bility for compact Kähler manifolds. Later, El Kacimi Alaoui–Gmira [30] proved a much more general result as follows: let F be a homologically orientable Hermitian foliation on a compact manifold and F t a deformation of F by a transversely holomorphic foliation with fixed differentiable type, parametrized by an open neighborhood of 0 in R d . Suppose that a transversely Hermitian metric σ of F = F 0 is Kählerian. Then F t has a transversely Kähler metric σ t for every t sufficiently close to 0, depending differentiably on t with σ 0 = σ . See also [83, Theorem 5.2]. One can refer to Sects. 6.1–6.5 for relevant definitions. Notice that if the deformation F t is arbitrary, then the above assertion doesn’t hold, due to the example built in [32], namely an analytic family {F t } t ∈ C of holomorphic foliations on a compact complex nilmanifold such that F 0 is transversely Kähler, but for any t ̸ = 0 , F t has no transversely Kähler structure. See also the discussion in [83, Section 6] based on the examples from [43, 71]. Inspired by [78], we extend the ‘weak’ ∂ ∂ -properties to the foliated version (Defini - tion 6.21), and then prove the following local stabilities theorem by adapting the power series method to our setting. The main ingredients in this proof are the transversely ellip- 1 3 On extension of closed complex (basic) differential forms: (basic) Hodge... tic operator theory initiated from [29] and the Kuranishi family constructed by [31], see Sect. 6.6 for more details. This is our main result in this paper. Theorem E (Main Result) Let {F t } t ∈ U be a smooth family of transversely Hermitian struc- tures on a compact foliated manifold ( M , F ) with fixed differentiable type ( F is of complex codimension r), parametrized by an open neighborhood U of 0 in R d . If F = F 0 is trans - versely p-Kähler and satisfies the ( p , p + 1 ) -th mild ∂ ̄ ∂ -property for 1 ≤ p ≤ r − 1 , then F t is also transversely p-Kähler for every t sufficiently close to 0. This result strengthens the works of [30, 83] on that of the transversely Kähler folia- tions with homologically orientability, and can be viewed as a generalization of [78, Theo- rem 1.1] to the foliated case. In a certain sense, the approach we have adopted has resulted in greater efficiency compared to the methods used in [ 30, 83], cf. Remark 6.33(a). Notice that even when p = 1 (transversely Kähler) Theorem E is new. Corollary F With the setting of Theorem E , if F = F 0 is transversely Kähler, then F t is also transversely Kähler for every t sufficiently close to 0. Compared with the previous works [30, 83], Corollary F can drop the homologically orientability assumption. To this end, We observe that a transversely Kähler foliation, even without homologically orientability, also satisfies the ∂∂ -property and, therefore, satisfies (1, 2)-th mild ∂ ̄ ∂ -property, cf. Sect. 6.4 and Remark 6.33(b). Finally, several theorems concerning the deformation invariance of basic Hodge/Bott– Chern numbers with ‘weak’ ∂ ∂ -properties are displayed, see Theorem 6.35. In these theo- rems, the homologically orientability assumption is necessary, as indicated in Sect. 6.7. We also require the foliation to remain unchanged in this context. For further insights, one can refer to Question 6.36 posed by Raźny (and also the preceding discussions) regarding the invariance of basic Hodge numbers under arbitrary deformations of transversely Kähler foliations. Convention All compact complex (or smooth) manifolds in this paper are assumed to be connected unless mentioned otherwise and π : X → B will always denote a differentia - ble or holomorphic family of n -dimensional compact complex manifolds, whose central fiber is ( X 0 , z ) with local holomorphic coordinates z := ( z i ) i =1 ,...,n and general fiber is X t := π − 1 (t ) 2 Variations of ∂ ̄ ∂ -property and ∂ ̄ ∂ -equation In this section, we collect some basics to be used later. Throughout this section, we will always denote by X a compact complex manifold of complex dimension n 2.1 Cohomology groups and variations of ∂ ∂ -property We will often use the commutative diagram: 1 3 S. Rao, R. Zhang (2.1) Recall that Dolbeault cohomology groups H • , • ∂ ( X ) of X are defined by: H • , • ∂ ( X ) := ker ∂ im ∂ , with H • , • ∂ ( X ) likewise defined, while Bott – Chern and Aeppli cohomology groups are defined as H • , • BC ( X ) := ker ∂ ∩ ker ∂ im ∂∂ and H • , • A ( X ) := ker ∂∂ im ∂ + im ∂ , (2.2) respectively. The dimensions of H p + q dR ( X ) , H p,q ∂ ( X ) , H p,q BC ( X ) , H p,q A ( X ) and H p,q ∂ ( X ) over C are denoted by b p + q ( X ) , h p,q ∂ ( X ) , h p,q BC ( X ) , h p,q A ( X ) and h p,q ∂ ( X ) , respectively, and the first four of them are usually called the ( p + q ) - th Betti numbers , ( p , q )- th Hodge numbers , Bott – Chern numbers and Aeppli numbers of X , respectively. From the very definition of these cohomology groups, the following equalities clearly hold h p,q BC = h q,p BC = h n − q,n − p A = h n − p,n − q A , h n − p,n − q ∂ = h p,q ∂ = h q,p ∂ = h n − q,n − p ∂ So the (standard) ∂∂ - property , which means for every pure-type d -closed form on a com- pact complex manifold, the properties of d -exactness, ∂ -exactness, ̄ ∂ -exactness and ∂ ̄ ∂ -exactness are equivalent, is equivalent to the following mappings ι p,q BC ,dR : H p,q BC (X ) −→ H p + q dR (X ) are injective for all p , q , or to the isomorphisms of all the maps in Diagram (2.1) by [22, Remark 5.16]. Recall that a compact complex manifold X satisfies S p,q (resp. B p,q ) if for any ∂ -closed ∂g ∈ A p,q ( X ) , the Eq. (1.2) has a solution (resp. a ∂ -exact solution) of pure-type complex differential form. Similarly, a compact complex manifold X is said to satisfy S p,q (resp. B p,q ), if for any ∂ -closed g ∈ A p − 1 ,q (X ) , the Eq. (1.2) has a solution (resp. a ∂ -exact solu- tion) of pure-type complex differential form. It is easy to verify the following implications: B p,q ⇒ S p,q ⇓ ⇓ B p,q ⇒ S p,q 1 3 On extension of closed complex (basic) differential forms: (basic) Hodge... And it is apparent that a compact complex manifold X , where the ∂∂ -property holds, satis- fies B p,q for any ( p , q ). It is easy to check that the following statements are equivalent: (1) the injectivity of ι p,q BC ,∂ holds on X ⇔ X satisfies B p,q ; (2) the injectivity of ι p,q ∂, A holds on X ⇔ X satisfies S p,q ; (3) the surjectivity of ι p − 1 ,q BC , ∂ holds on X ⇔ X satisfies B p,q 2.2 Hodge theory on compact complex manifolds: Bott–Chern Let X be a compact complex manifold. The Bott–Chern cohomology group has been intro- duced in (2.2). Also, the Bott – Chern Laplacian is given by □ BC := ∂∂∂ ∗ ∂ ∗ + ∂ ∗ ∂ ∗ ∂∂ + ∂ ∗ ∂∂ ∗ ∂ + ∂ ∗ ∂∂ ∗ ∂ + ∂ ∗ ∂ + ∂ ∗ ∂, (2.3) and G BC is the associated Green’s operator of this fourth order Kodaira–Spencer operator. Then we have the Hodge decomposition of □ BC on X : A p,q ( X ) = ker □ BC ⊕ im (∂∂ ) ⊕ ( im ∂ ∗ + im ∂ ∗ ) , whose three parts are orthogonal to each other with respect to the L 2 -scalar product defined by a Hermitian metric on X , combined with the equality 1 = H BC + □ BC G BC = H BC + G BC □ BC , where H BC is the harmonic projection operator . And it should be noted that ker □ BC = ker ∂ ∩ ker ∂ ∩ ker (∂∂ ) ∗ We get the following two observations: (1) □ BC ∂∂ (∂∂ ) ∗ = ∂∂ (∂∂ ) ∗ □ BC ; (2) G BC ∂∂ (∂∂ ) ∗ = ∂∂ (∂∂ ) ∗ G BC For the resolution of ∂ ∂ -equations, we need a crucial lemma. Lemma 2.1 [75 , Theorem 4.1] Let ( X , ω ) be a compact Hermitian manifold and α a pure- type complex differential form. If the ∂ ∂ -equation ∂ ∂ x = α admits a pure-type solution, then ( ∂∂ ) ∗ G BC α is a solution that uniquely minimizes the L 2 -norm among all solutions with respect to ω Besides, the equalities hold G BC ( ∂∂ ) = (∂∂ ) G A and (∂∂ ) ∗ G BC = G A (∂∂ ) ∗ , 1 3 S. Rao, R. Zhang where G BC and G A are the associated Green’s operators of □ BC and □ A , respectively. Here □ BC is defined in ( 2.3) and □ A is the second Kodaira–Spencer operator (often also called Aeppli Laplacian) □ A = ∂ ∗ ∂ ∗ ∂∂ + ∂∂∂ ∗ ∂ ∗ + ∂∂ ∗ ∂∂ ∗ + ∂∂ ∗ ∂∂ ∗ + ∂∂ ∗ + ∂∂ ∗ 3 Extension of closed forms: Bott–Chern approach Wei–Zhu [94] applied the ∂∂ -Hodge theory to extend a d -closed ( p , q )-form on X 0 to a d -closed filtrated ( p + q ) -form on X t , whose ( p , q )-part on X t is ∂ t -closed by type projec- tion. We will reprove their theorem in this section by applying Bott–Chern theory. 3.1 Beltrami differentials and extension map By a holomorphic family π : X → B of compact complex manifolds from a complex mani- fold to a connected complex manifold, we mean that π is a proper and surjective holomor- phic submersion, as in [53 , Definition 2.8]; while for differentiable one, we adopt: Definition 3.1 [53 , Definition 4.1] Let X be a differentiable manifold, B a domain of R d and π a smooth map of X onto B . By a differentiable family of n - dimensional compact complex manifolds we mean the triple π : X → B satisfying the following conditions: (a) The rank of the Jacobian matrix of π is equal to d at every point of X ; (b) For each point t ∈ B , π − 1 ( t ) is a compact connected subset of X ; (c) π − 1 ( t ) is the underlying differentiable manifold of the n -dimensional compact complex manifold X t associated to each t ∈ B ; (d) There is a locally finite open covering {U j | j = 1 , 2 , . . . } of X and complex-valued smooth functions ζ 1 j ( p ), . . . , ζ n j (p ) , defined on U j such that for each t , { p −→ (ζ 1 j (p ), . . . , ζ n j (p )) | U j ∩ π − 1 (t ) = ∅} form a system of local holomorphic coordinates of X t Beltrami differential plays an important role in deformation theory. For a compact complex manifold X , we call an element in A 0 , 1 ( X, T 1 , 0 X ) a Beltrami differential , where T 1 , 0 X denotes the holomorphic tangent bundle of X . Then ι φ or φ ⌟ denotes the contraction operator with φ ∈ A 0 , 1 ( X, T 1 , 0 X ) alternatively if there is no confusion. One similarly follows the notation e ♡ = ∞ ∑ k =0 1 k ! ♡ k , where ♡ k denotes k -time action of the operator ♡ . The summation in the above formulation is often finite since the dimension of X is finite. We will always consider a differentiable family π : X → B of compact complex n -dimensional manifolds over a sufficiently small domain in R d with the reference fiber 1 3 On extension of closed complex (basic) differential forms: (basic) Hodge... X 0 := π − 1 (0) and the general fibers X t := π − 1 (t ) For simplicity we set d = 1 . Denote by ζ := ( ζ α j ( z, t )) the holomorphic coordinates of X t induced by the family with the holomor- phic coordinates z := ( z i ) of X 0 , under a coordinate covering {U j } of X , when t is assumed to be fixed, as the standard notions in deformation theory described at the beginning of [ 70, Chapter 4]. This family induces a canonical differentiable family of integrable Beltrami dif - ferentials on X 0 , denoted by φ ( z, t ) , φ ( t ) and φ interchangeably. In the sequel we will state some explicit computations as presented in [70, Chapter 4.1] or [81 , Section 2.1]. A Beltrami differential can be written as φ ( t ) = ( ∂ ∂z ) T ( ∂ζ ∂z ) − 1 ̄ ∂ζ, (3.1) where ∂ ∂z = ∂ ∂z 1 ∂ ∂z n , ∂ζ = ∂ζ 1 ∂ζ n , ∂ζ ∂z stands for the matrix ( ∂ζ α ∂z j ) 1 ≤ α ≤ n 1 ≤ j ≤ n and α, j are the row and column indices. Here ( ∂ ∂z ) T is the transpose of ∂ ∂z and ∂ denotes the Cauchy–Riemann operator with respect to the holomorphic structure on X 0 Locally, φ ( t ) is expressed as φ i j d ̄ z j ⊗ ∂ ∂z i ∈ A 0 , 1 ( X 0 , T 1 , 0 X 0 ) , so it can be considered as a matrix ( φ i j ) 1 ≤ i ≤ n 1 ≤ j ≤ n . By (3.1), this matrix can be explicitly written as φ = ( φ i j ) 1 ≤ i ≤ n 1 ≤ j ≤ n = φ ( t ) ( ∂ ∂ ̄ z j , dz i ) = (( ∂ζ ∂z ) − 1 ( ∂ζ ∂ ̄ z )) i j (3.2) A fundamental fact is that the Beltrami differential φ ( t ) defined as above satisfies the integrability: ∂φ ( t ) = 1 2 [ φ ( t ) , φ ( t )] , and we then call φ ( t ) an integrable Beltrami differential . Now consider the exponential operator of contraction operators e ι φ := ∞ ∑ k =0 ι k φ ( t ) k ! A calculation shows that e ι φ ( dz i 1 ∧ · · · ∧ dz i p ) = ( dz i 1 + φ (t ) ⌟dz i 1 ) ∧ · · · ∧ ( dz i p + φ (t ) ⌟dz i p ) Then one obtains e ι φ : A p, 0 (X 0 ) −→ A p, 0 (X t ) 1 3 S. Rao, R. Zhang by dζ β = ∂ζ β ∂z i dz i + ∂ζ β ∂ ̄ z i d ̄ z i = ∂ζ β ∂z i e ι φ ( dz i ) according to (3.2). However, e ι φ can’t preserve ( p , q )-forms for 1 ≤ q ≤ n. An interesting observation in [35, Theorem 5.1] or [94, Section 3.1] tells that it in fact maps from a filtration on X 0 to one on X t , i.e., e ι φ : F p A k ( X 0 ) = ⊕ p ≤s ≤k A s,k − s ( X 0 ) −→ F p A k ( X t ) Actually, for σ ∈ A s,k − s (X 0 ) ( p ≤ s ≤ k ) with the local expression σ = σ i 1 ··· i s j 1 ··· j k − s dz i 1 ∧ · · · ∧ dz i s ∧ d ̄ z j 1 ∧ · · · ∧ d ̄ z j k − s , one has e ι φ ( σ ) = σ i 1 ··· i s j 1 ··· j k − s e ι φ ( dz i 1 ∧ · · · ∧ dz i s ) ∧ d ̄ z j 1 ∧ · · · ∧ d ̄ z j k − s = σ i 1 ··· i s j 1 ··· j k − s ( dz i 1 + φ ( t ) ⌟ dz i 1 ) ∧ · · · ∧ ( dz i s + φ ( t ) ⌟ dz i s ) ∧ ( ∂ ̄ z j 1 ∂ζ β dζ β + ∂ ̄ z j 1 ∂ ̄ ζ β d ̄ ζ β ) ∧ · · · ∧ ( ∂ ̄ z j k − s ∂ζ β dζ β + ∂ ̄ z j k − s ∂ ̄ ζ β d ̄ ζ β ) , which clearly belongs to F p A k (X t ) Next, we will state the definitions of the projection operator and the extension opera - tor, originating from [94, 97]. For σ ∈ A 0 , 1 (X 0 ) with the local expression σ = σ j d ̄ z j , one defines the projection operator P φ : A 0 , 1 (X 0 ) −→ A 0 , 1 (X t ) by P φ ( σ ) = P φ ( σ j ( ∂ ̄ z j ∂ζ β dζ β + ∂ ̄ z j ∂ ̄ ζ β d ̄ ζ β )) := σ j ( ∂ ̄ z j ∂ ̄ ζ β d ̄ ζ β ) = σ j ( ( 1 − φφ ) − 1 ) j ̄ k ( dz k + φ ( t ) ⌟ dz k ) , (3.3) where φ φ = φ ⌟ φ, φφ = φ ⌟ φ and 1 is the identity matrix. Since φ ( t ) is a well-defined, global (1, 0) vector valued (0, 1)-form on X 0 , P φ is globally well-defined thanks to the last equality in (3.3) from [81, (2.18)]. We then use the exponential operator e ι φ and the projection operator P φ to define the extension operator ρ φ : A p,q ( X 0 ) −→ A p,q ( X t ) 1 3 On extension of closed complex (basic) differential forms: (basic) Hodge... Concretely, for σ ∈ A p,q ( X 0 ) with the local expression σ = σ i 1 ··· i p j 1 ··· j q dz i 1 ∧ · · · ∧ dz i p ∧ d ̄ z j 1 ∧ · · · ∧ d ̄ z j q , one has ρ φ ( σ ) := σ i 1 ··· i p j 1 ··· j q e ι φ ( dz i 1 ∧ · · · ∧ dz i p ) ∧ P φ ( d ̄ z j 1 ) ∧ · · · ∧ P φ ( d ̄ z j q ) = σ i 1 ··· i p j 1 ··· j q ( dz i 1 + φ ( t ) ⌟ dz i 1 ) ∧ · · · ∧ ( dz i p + φ ( t ) ⌟ dz i p ) ∧ ( ( 1 − φφ ) − 1 ) j 1 ̄ k ( dz k + φ ( t ) ⌟ dz k ) ∧ · · · ∧ ( ( 1 − φφ ) − 1 ) j q ̄ k ( dz k + φ ( t ) ⌟ dz k ) (3.4) Likewise, ρ φ is also globally well-defined. In [81, 100 ], Zhao—the first author introduced an extension map e ι φ | ι φ : A p,q ( X 0 ) −→ A p,q ( X t ) , which can preserve all ( p , q )-forms and played an important role in their subsequent papers [78, 79]. More precisely, with the above notations, for σ ∈ A p,q ( X 0 ) , one defines [ 81, Definition 2.8]: e ι φ | ι φ ( σ ) = σ i 1 ··· i p j 1 ··· j q e ι φ ( dz i 1 ∧ · · · ∧ dz i p ) ∧ e ι φ ( d ̄ z j 1 ∧ · · · ∧ d ̄ z j q ) = σ i 1 ··· i p j 1 ··· j q ( dz i 1 + φ ( t ) ⌟ dz i 1 ) ∧ · · · ∧ ( dz i p + φ ( t ) ⌟ dz i p ) ∧ ( d ̄ z j 1 + φ ( t ) ⌟ d ̄ z j 1 ) ∧ · · · ∧ ( d ̄ z j q + φ ( t ) ⌟ d ̄ z j q ) (3.5) In [79 , Section 2.1], the first author–Wan–Zhao introduced a new notation Ⅎ to denote the simultaneous contraction on each component of a complex differential form. 1 By comparing (3.4) and (3.5), we get the relationship between the two extension maps: ρ φ = e ι φ | ι φ ◦ ( 1 − φφ ) − 1 Ⅎ (3.6) the first author–Zhao obtained the ∂ -extension obstruction for ( p , q )-forms of the smooth family via the extension map e ι φ | ι φ (cf. [81, Proposition 2.13]), namely ̄ ∂ t ( e ι φ | ι ̄ φ ( σ )) = e ι φ | ι ̄ φ (( 1 − ̄ φφ ) − 1 Ⅎ ([ ∂, ι φ ] + ̄ ∂ )( 1 − ̄ φφ ) Ⅎ σ ) , (3.7) 1 For example, for σ ∈ A p,q ( X 0 ) with the above notation, ( 1 − ̄ φφ + ̄ φ ) Ⅎ σ means: ( 1 − ̄ φφ + ̄ φ ) Ⅎ ( σ i 1 ··· i p j 1 ··· j q dz i 1 ∧ · · · ∧ dz p p ∧ d ̄ z j 1 ∧ · · · ∧ d ̄ z j q ) = σ i 1 ··· i p j 1 ··· j q ( 1 − ̄ φφ + ̄ φ ) ⌟ dz i 1 ∧ · · · ∧ ( 1 − ̄ φφ + ̄ φ ) ⌟ dz i p ∧ ( 1 − ̄ φφ + ̄ φ ) ⌟ d ̄ z j 1 ∧ · · · ∧ ( 1 − ̄ φφ + ̄ φ ) ⌟ d ̄ z j q 1 3 S. Rao, R. Zhang where σ ∈ A p,q ( X 0 ) . One concludes that (3.7) is equivalent to the following assertion due to (3.6): ρ φ − 1 ̄ ∂ t ρ φ = ∂ − L 1 , 0 φ ( t ) on A p,q (X 0 ) , which is exactly [97, Theorem 2.9, the case of E = Ω p ]. Remark 3.2 In [90], Tu used the pair deformation { ( X t , E t ) } to give a correspondence between E 0 -valued ( p , q )-forms on X 0 and E t -valued ( p , q )-forms on X t by P t : A p,q ( X 0 , E 0 ) −→ A p,q ( X t , E t ) Recall that the pair deformation { ( X t , E t ) } is a holomorphic family of pairs { ( X t , E t ) } where each E t is a holomorphic vector bundle over a compact complex manifold X t . The holomorphic structures of E t and complex structures of X t vary simultaneously. In the case when each E t is the trivial line bundle, P t coincides with (3.6) and, therefore [90, Theo- rem 2] is equivalent to [81, Proposition 2.13]. The following proposition plays a key role in this paper: Proposition 3.3 [ 60, Theorem 3.4] and [ 81 , Proposition 2.2] Let φ ∈ A 0 , 1 ( X , T 1 , 0 X ) on a complex manifold X. Then on A ∗ , ∗ ( X ) , e − ι φ ◦ d ◦ e ι φ = d − L 1 , 0 φ + ι ̄ ∂φ − 1 2 [ φ,φ ] , (3.8) where L 1 , 0 φ := ι φ ∂ − ∂ι φ is the Lie derivative. In particular, if φ is an integrable Beltrami differential (i.e., ̄ ∂φ = 1 2 [ φ, φ ] ), then the last term in ( 3.8 ) vanishes and we get e − ι φ ◦ d ◦ e ι φ = d − L 1 , 0 φ = d + ∂ι φ − ι φ ∂. (3.9) From the proof of Proposition 3.3, we see that (3.8) naturally generalizes the Tian– Todorov Lemma [88, 89], whose variants appear in [12, 18, 36, 61] and also [59, 60] for vector bundle-valued forms. Lemma 3.4 Let φ, ψ ∈ A 0 , 1 ( X , T 1 , 0 X ) and α ∈ A ∗ , ∗ ( X ) on an n-dimensional complex manifold X. Then [ φ, ψ ] ⌟ α = − ∂ ( ψ ⌟ ( φ ⌟ α )) − ψ ⌟ ( φ ⌟ ∂α ) + φ ⌟ ∂ ( ψ ⌟ α ) + ψ ⌟ ∂ ( φ ⌟ α ) , (3.10) where [ φ, ψ ] := n ∑ i,j =1 ( φ i ∧ ∂ i ψ j + ψ i ∧ ∂ i φ j ) ⊗ ∂ j for φ = ∑ i φ i ⊗ ∂ i and ψ = ∑ i ψ i ⊗ ∂ i . Here ∂ i := ∂ ∂ z i and similar for others. 1 3 On extension of closed complex (basic) differential forms: (basic) Hodge... Remark 3.5 As shown in (the proof of) [60, Lemma 3.2], Eq. (3.10) means that the Lie derivative L 1 , 0 φ and the contraction ι ψ do not commute; their difference is precisely the contraction with the Lie bracket [ φ, ψ ] , i.e., [ L 1 , 0 φ , ι ψ ] = ι [ φ,ψ ] 3.2 Kuranishi family We introduce some basics on Kuranishi family of complex structures in this subsection originally from [58]. By (the proof of) Kuranishi’s completeness theorem [58], for any compact complex man- ifold X 0 , there exists a complete holomorphic family π : K → T of complex manifolds at the reference point 0 ∈ T in the sense that for any differentiable family π : X → B with π − 1 ( s 0 ) = π − 1 (0) = X 0 , there exist a sufficiently small neighborhood E ⊆ B of s 0 , and smooth maps Φ : X E → K , τ : E → T with τ ( s 0 ) = 0 such that the diagram commutes Φ maps π − 1 ( s ) biholomorphically onto π − 1 ( τ (s )) for each s ∈ E , and Φ : π − 1 (s 0 ) = X 0 −→ π − 1 (0) = X 0 is the identity map. This family is called the Kuranishi family and constructed as follows. Let { η ν } m ν =1 be a base for the harmonic space H 0 , 1 ( X 0 , T 1 , 0 X 0 ) , where some suitable Hermitian metric is fixed on X 0 and m ≥ 1 ; Otherwise the complex manifold X 0 would be rigid , i.e., for any differentiable family φ : M → P with s 0 ∈ P and φ − 1 ( s 0 ) = X 0 , there is a neighborhood V ⊆ P of s 0 such that φ : φ − 1 ( V ) → V is trivial. Then one can construct a holomorphic family φ ( t ) = ∞ ∑ | I | =1 φ I t I := ∞ ∑ j =1 φ j ( t ) , I = ( i 1 , . . . , i m ) , t = ( t 1 , . . . , t m ) ∈ C m , for small t , of Beltrami differentials as follows: φ 1 ( t ) = m ∑ ν =1 t ν η ν and for | I | ≥ 2 , φ I = 1 2 ∂ ∗ G ∑ J +L =I [ φ J , φ L ] Clearly, φ ( t ) satisfies the equation 1 3 S. Rao, R. Zhang φ ( t ) = φ 1 + 1 2 ∂ ∗ G [ φ ( t ) , φ ( t )] Let T = { t | H [ φ ( t ) , φ ( t )] = 0 } where H is the harmonic projection. So for each t ∈ T , φ ( t ) satisfies ̄ ∂φ ( t ) = 1 2 [ φ ( t ) , φ ( t )] , (3.11) and determines a complex structure X t on the underlying differentiable manifold of X 0 More importantly, φ ( t ) represents the complete holomorphic family π : K → T of com- plex manifolds. Roughly speaking, the Kuranishi family π : K → T contains all small dif- ferentiable deformations of X 0 3.3 d-closed extension of (p, q)-form: Bott–Chern approach Assume that a compact complex manifold X 0 satisfies B p,q +1 Now, considering a Kura- nishi family of deformations of X 0 over ∆ ε for small ε , one can find a family of integrable Beltrami differentials { φ ( t ) } | t |≤ ε depending on t holomorphically and describing the varia- tions of complex structure on X 0 In this subsection, we will reprove Theorem C due to [94], from the perspective of Bott– Chern Hodge theory. Proof The proof will be divided into four steps. Step 1 Transfer to the certain differential equations. By Eq. (3.9) in Proposition 3.3, for any μ ∈ A p,q ( X 0 ) , the form e ι φ ( μ ) is d -closed on X 0 (or X t ) if and only if ( d + ∂ι φ − ι φ ∂ ) μ = 0 By comparing the types, one knows that the above equation amounts to { ∂μ = 0, ∂μ = − ∂ ( φ ⌟ μ ) (3.12) Step 2 Study the integral equation. Given any d -closed ( p , q )-form μ 0 ∈ A p,q ( X 0 ) , one studies the integral equation as follows: μ + ∂ (∂∂ ) ∗ G BC ∂ (φ (t ) ⌟μ ) = μ 0 (3.13) 1 3 On extension of closed complex (basic) differential forms: (basic) Hodge... In the sequel we will prove that the Eq. (3.13) possesses a unique solution μ ( z, t ) ∈ A p,q ( X 0 ) which is smooth in ( z , t ) and holomorphic in t ∈ ∆ ε Denote the completion of the norm space ( A p,q ( X 0 ) , ∥ · ∥ k,α ) by E , and consider the linear operator Q φ ( t ) := − ∂ (∂∂ ) ∗ G BC ∂ι φ ( t ) on E . Then (3.13) is equivalent to the following equation ( I − Q φ ( t ) ) μ = μ 0 (3.14) One can easily see that Q φ (t ) satisfies ∥Q φ (t ) ∥ < 1 on E due to the standard estimates for Green’s operator G BC as t ∈ ∆ ε . We then apply [98, Chapter II.1, Theorem 2] to T = I − Q φ (t ) to obtain the unique solution μ ( z, t ) = ( I − Q φ ( t ) ) − 1 μ 0 = μ 0 + ∞ ∑ k =1 Q k φ ( t ) ( μ 0 ) (3.15) of (3.14) for any | t | ≤ ε. The integrable Beltrami differential { φ (t ) } | t |≤ ε can be written as a convergent power series in t since it depends on t holomorphically. Thus, μ ( z, t ) is also a power series in t by (3.15). Moreover, it is easy to verify by the standard elliptic estimates and (3.14) that the ∥ · ∥ k,α -norm of μ ( z, t ) is finite which implies that μ ( z, t ) is convergent for | t | ≤ ε . So μ ( z, t ) is holomorphic in t for | t | ≤ ε. Since we have got the C k -continuity of μ ( z, t ) from above, one can use the standard regularity theory for elliptic differential operator such as in [70, Proposition 2.6 of Chapter 4] to obtain that the unique solution μ ( z, t ) satisfying (3.13) is a smooth ( p , q )-form in ( z , t ) for small t Step 3 Show the solution μ ( z, t ) obtained in Step 2 satisfies ( 3.12). The unique solution μ ( z, t ) obtained in Step 2 clearly satisfies the first equality of ( 3.12) and, therefore we just need to check the second equality. We can locally express φ ( t ) and μ ( z, t ) as ∑ i ≥ 1 φ i t i and ∑ i ≥ 0 μ i t i , respectively, since they are both holomorphic in small