2.1 Deformation theory

For the holomorphic family of compact complex manifolds, we adopt the definition [Reference KodairaKod86, Definition 2.8]; while for the differentiable one, we adopt the following definition.

Beltrami differentials play an important role in deformation theory. A Beltrami differential on $X$ , generally denoted by $\unicode[STIX]{x1D719}$ , is an element in $A^{0,1}(X,T_{X}^{1,0})$ , where $T_{X}^{1,0}$ is the holomorphic tangent bundle of $X$ . Then $\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D719}}$ or $\unicode[STIX]{x1D719}\lrcorner$ denotes the contraction operator with respect to $\unicode[STIX]{x1D719}\in A^{0,1}(X,T_{X}^{1,0})$ or other analogous vector-valued complex differential forms alternatively if there is no confusion. We also use the convention

(4) $$\begin{eqnarray}e^{\spadesuit }=\mathop{\sum }_{k=0}^{\infty }\frac{1}{k!}\spadesuit ^{k},\end{eqnarray}$$

where $\spadesuit ^{k}$ denotes $k$ -time action of the operator $\spadesuit$ . As the dimension of $X$ is finite, the summation in the above formulation is always finite.

We will always consider the differentiable family $\unicode[STIX]{x1D70B}:{\mathcal{X}}\rightarrow B$ of compact complex $n$ -dimensional manifolds over a sufficiently small domain in $\mathbb{R}^{k}$ with the reference fiber $X_{0}:=\unicode[STIX]{x1D70B}^{-1}(0)$ and the general fibers $X_{t}:=\unicode[STIX]{x1D70B}^{-1}(t)$ . For simplicity we set $k=1$ . Denote by $\unicode[STIX]{x1D701}:=(\unicode[STIX]{x1D701}_{j}^{\unicode[STIX]{x1D6FC}}(z,t))$ the holomorphic coordinates of $X_{t}$ induced by the family with the holomorphic coordinates $z:=(z^{i})$ of $X_{0}$ , under a coordinate covering $\{{\mathcal{U}}_{j}\}$ of ${\mathcal{X}}$ , when $t$ is assumed to be fixed, as the standard notions in deformation theory described at the beginning of [Reference Morrow and KodairaMK71, ch. 4]. This family induces a canonical differentiable family of integrable Beltrami differentials on $X_{0}$ , denoted by $\unicode[STIX]{x1D711}(z,t)$ , $\unicode[STIX]{x1D711}(t)$ and $\unicode[STIX]{x1D711}$ interchangeably.

In [Reference Zhao and RaoZR15, Reference Zhao and RaoZR13], the first and third authors introduced an extension map

$$\begin{eqnarray}e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}(t)}|\unicode[STIX]{x1D704}_{\overline{\unicode[STIX]{x1D711}(t)}}}:A^{p,q}(X_{0})\rightarrow A^{p,q}(X_{t}),\end{eqnarray}$$

to play an important role in this paper.

Definition 2.2. For $s\in A^{p,q}(X_{0})$ , we define

$$\begin{eqnarray}e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}(t)}|\unicode[STIX]{x1D704}_{\overline{\unicode[STIX]{x1D711}(t)}}}(s)=s_{i_{1}\cdots i_{p}j_{1}\cdots j_{q}}(z(\unicode[STIX]{x1D701}))(e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}(t)}}(dz^{i_{1}}\wedge \cdots \wedge dz^{i_{p}}))\wedge (e^{\unicode[STIX]{x1D704}_{\overline{\unicode[STIX]{x1D711}(t)}}}(d\overline{z}^{j_{1}}\wedge \cdots \wedge d\overline{z}^{j_{q}})),\end{eqnarray}$$

where $s$ is locally written as

$$\begin{eqnarray}s=s_{i_{1}\cdots i_{p}j_{1}\cdots j_{q}}(z)\,dz^{i_{1}}\wedge \cdots \wedge dz^{i_{p}}\wedge d\overline{z}^{j_{1}}\wedge \cdots \wedge d\overline{z}^{j_{q}}\end{eqnarray}$$

and the operators $e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}(t)}}$ , $e^{\unicode[STIX]{x1D704}_{\overline{\unicode[STIX]{x1D711}(t)}}}$ follow convention (4). It is easy to check that this map is a real linear isomorphism as in [Reference Rao and ZhaoRZ18, Lemma 2.8].

The following proposition is crucial in this paper.

Proposition 2.3 [Reference Liu, Rao and YangLRY15, Theorem 3.4], [Reference Rao and ZhaoRZ18, Proposition 2.2].

Let $\unicode[STIX]{x1D719}\in A^{0,1}(X,T_{X}^{1,0})$ on a complex manifold $X$ . Then on the space $A^{\ast ,\ast }(X)$ ,

(5) $$\begin{eqnarray}d\circ e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D719}}}=e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D719}}}(d+\unicode[STIX]{x2202}\circ \unicode[STIX]{x1D704}_{\unicode[STIX]{x1D719}}-\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D719}}\circ \unicode[STIX]{x2202}-\unicode[STIX]{x1D704}_{\overline{\unicode[STIX]{x2202}}\unicode[STIX]{x1D719}-(1/2)[\unicode[STIX]{x1D719},\unicode[STIX]{x1D719}]}).\end{eqnarray}$$

From the proof of Proposition 2.3, we see that (5) is a natural generalization of the Tian–Todorov lemma [Reference TianTia87, Reference TodorovTod89], whose variants appeared in [Reference FriedmanFri91, Reference Barannikov and KontsevichBK98, Reference LiLi05, Reference Liu, Sun and YauLSY09, Reference ClemensCle05] and also [Reference Liu and RaoLR12, Reference Liu, Rao and YangLRY15] for vector bundle valued forms.

Lemma 2.4. For $\unicode[STIX]{x1D719},\unicode[STIX]{x1D713}\in A^{0,1}(X,T_{X}^{1,0})$ and $\unicode[STIX]{x1D6FC}\in A^{\ast ,\ast }(X)$ on an $n$ -dimensional complex manifold  $X$ ,

$$\begin{eqnarray}[\unicode[STIX]{x1D719},\unicode[STIX]{x1D713}]\lrcorner \unicode[STIX]{x1D6FC}=-\unicode[STIX]{x2202}(\unicode[STIX]{x1D713}\lrcorner (\unicode[STIX]{x1D719}\lrcorner \unicode[STIX]{x1D6FC}))-\unicode[STIX]{x1D713}\lrcorner (\unicode[STIX]{x1D719}\lrcorner \unicode[STIX]{x2202}\unicode[STIX]{x1D6FC})+\unicode[STIX]{x1D719}\lrcorner \unicode[STIX]{x2202}(\unicode[STIX]{x1D713}\lrcorner \unicode[STIX]{x1D6FC})+\unicode[STIX]{x1D713}\lrcorner \unicode[STIX]{x2202}(\unicode[STIX]{x1D719}\lrcorner \unicode[STIX]{x1D6FC}),\end{eqnarray}$$

where

$$\begin{eqnarray}[\unicode[STIX]{x1D719},\unicode[STIX]{x1D713}]:=\mathop{\sum }_{i,j=1}^{n}(\unicode[STIX]{x1D719}^{i}\wedge \unicode[STIX]{x2202}_{i}\unicode[STIX]{x1D713}^{j}+\unicode[STIX]{x1D713}^{i}\wedge \unicode[STIX]{x2202}_{i}\unicode[STIX]{x1D719}^{j})\otimes \unicode[STIX]{x2202}_{j}\end{eqnarray}$$

for $\unicode[STIX]{x1D711}=\sum _{i}\unicode[STIX]{x1D711}^{i}\otimes \unicode[STIX]{x2202}_{i}$ and $\unicode[STIX]{x1D713}=\sum _{i}\unicode[STIX]{x1D713}^{i}\otimes \unicode[STIX]{x2202}_{i}$ .

2.2 The $p$ -Kähler structures

Let $V$ be a complex $n$ -dimensional vector space with its dual space $V^{\ast }$ , i.e., the space of complex linear functionals over $V$ . Denote the complexified space of the exterior $m$ -vectors of $V^{\ast }$ by $\bigwedge _{\mathbb{C}}^{m}V^{\ast }$ , which admits a natural direct sum decomposition

$$\begin{eqnarray}\bigwedge _{\mathbb{C}}^{m}V^{\ast }=\mathop{\sum }_{r+s=m}\bigwedge ^{r,s}V^{\ast },\end{eqnarray}$$

where $\bigwedge ^{r,s}V^{\ast }$ denotes the complex vector space of $(r,s)$ -forms on $V^{\ast }$ . The case $m=1$ exactly reads

$$\begin{eqnarray}\bigwedge _{\mathbb{C}}^{1}V^{\ast }=V^{\ast }\oplus \overline{V^{\ast }},\end{eqnarray}$$

where the natural isomorphism $V^{\ast }\cong \bigwedge ^{1,0}V^{\ast }$ is used. Let $q\in \{1,\ldots ,n\}$ and $p=n-q$ . Clearly, the complex dimension $N$ of $\bigwedge ^{q,0}V^{\ast }$ is equal to the combination number $C_{n}^{q}$ . After a basis $\{\unicode[STIX]{x1D6FD}_{i}\}_{i=1}^{N}$ of the complex vector space $\bigwedge ^{q,0}V^{\ast }$ is fixed, the canonical Plücker embedding as in [Reference Griffiths and HarrisGH78, p. 209] is given by

$$\begin{eqnarray}\begin{array}{@{}cccc@{}}\unicode[STIX]{x1D70C}: & G(q,n) & {\hookrightarrow} & \displaystyle \mathbb{P}\biggl(\bigwedge ^{q,0}V^{\ast }\biggr)\\ & \unicode[STIX]{x1D6EC} & \mapsto & [\ldots ,\unicode[STIX]{x1D6EC}_{i},\ldots ].\end{array}\end{eqnarray}$$

Here $G(q,n)$ denotes the Grassmannian of $q$ -planes in the vector space $V^{\ast }$ and $\mathbb{P}(\bigwedge ^{q,0}V^{\ast })$ is the projectivization of $\bigwedge ^{q,0}V^{\ast }$ . A $q$ -plane in $V^{\ast }$ can be represented by a decomposable $(q,0)$ -form $\unicode[STIX]{x1D6EC}\in \bigwedge ^{q,0}V^{\ast }$ up to a nonzero complex number, and $\{\unicode[STIX]{x1D6EC}_{i}\}_{i=1}^{N}$ are exactly the coordinates of $\unicode[STIX]{x1D6EC}$ under the fixed basis $\{\unicode[STIX]{x1D6FD}_{i}\}_{i=1}^{N}$ . Decomposable $(q,0)$ -forms are those forms in $\bigwedge ^{q,0}V^{\ast }$ that can be expressed as $\unicode[STIX]{x1D6FE}_{1}\bigwedge \cdots \bigwedge \unicode[STIX]{x1D6FE}_{q}$ with $\unicode[STIX]{x1D6FE}_{i}\in V^{\ast }\cong \bigwedge ^{1,0}V^{\ast }$ for $1\leqslant i\leqslant q$ . Set

$$\begin{eqnarray}k=(N-1)-pq\end{eqnarray}$$

to be the codimension of $\unicode[STIX]{x1D70C}(G(q,n))$ in $\mathbb{P}(\bigwedge ^{q,0}V^{\ast })$ , whose locus characterizes the decomposable $(q,0)$ -forms in $\mathbb{P}(\bigwedge ^{q,0}V^{\ast })$ .

Now we list notation for several types of positivity and refer the readers to [Reference Harvey and KnappHK74, Reference HarveyHar75, Reference DemaillyDem12] for more details. A $(q,q)$ -form $\unicode[STIX]{x1D6E9}$ in $\bigwedge ^{q,q}V^{\ast }$ is defined to be strictly positive (respectively, positive) if

$$\begin{eqnarray}\unicode[STIX]{x1D6E9}=\unicode[STIX]{x1D70E}_{q}\mathop{\sum }_{i,j=1}^{N}\unicode[STIX]{x1D6E9}_{i\bar{j}}\unicode[STIX]{x1D6FD}_{i}\wedge \bar{\unicode[STIX]{x1D6FD}}_{j},\end{eqnarray}$$

where $\unicode[STIX]{x1D6E9}_{ij}$ is a positive (respectively, semi-positive) hermitian matrix of size $N\times N$ with $N=C_{n}^{q}$ under the basis $\{\unicode[STIX]{x1D6FD}_{i}\}_{i=1}^{N}$ of the complex vector space $\bigwedge ^{q,0}V^{\ast }$ and $\unicode[STIX]{x1D70E}_{q}$ is defined to be the constant $2^{-q}(\sqrt{-1})^{q^{2}}$ . According to this definition, the fundamental form of a hermitian metric on a complex manifold is actually a strictly positive $(1,1)$ -form everywhere. A $(p,p)$ -form $\unicode[STIX]{x1D6E4}\in \bigwedge ^{p,p}V^{\ast }$ is called weakly positive if the volume form

$$\begin{eqnarray}\unicode[STIX]{x1D6E4}\wedge \unicode[STIX]{x1D70E}_{q}\unicode[STIX]{x1D70F}\wedge \bar{\unicode[STIX]{x1D70F}}\end{eqnarray}$$

is positive for every nonzero decomposable $(q,0)$ -form $\unicode[STIX]{x1D70F}$ of $V^{\ast }$ , while a $(q,q)$ -form $\unicode[STIX]{x1D6F6}\in \bigwedge ^{q,q}V^{\ast }$ is said to be strongly positive if $\unicode[STIX]{x1D6F6}$ is a convex combination

$$\begin{eqnarray}\unicode[STIX]{x1D6F6}=\mathop{\sum }_{s}\unicode[STIX]{x1D6FE}_{s}\sqrt{-1}\unicode[STIX]{x1D6FC}_{s,1}\wedge \bar{\unicode[STIX]{x1D6FC}}_{s,1}\wedge \cdots \wedge \sqrt{-1}\unicode[STIX]{x1D6FC}_{s,q}\wedge \bar{\unicode[STIX]{x1D6FC}}_{s,q},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}_{s,i}\in V^{\ast }$ and $\unicode[STIX]{x1D6FE}_{s}\geqslant 0$ . As shown in [Reference DemaillyDem12, ch. III.§ 1.A], the sets of weakly positive and strongly positive forms are closed convex cones, and by definition, the weakly positive cone is dual to the strongly positive cone via the pairing

$$\begin{eqnarray}\bigwedge ^{p,p}V^{\ast }\times \bigwedge ^{q,q}V^{\ast }\longrightarrow \mathbb{C}.\end{eqnarray}$$

Then all weakly positive forms are real. An element $\unicode[STIX]{x1D6EF}$ in $\bigwedge ^{p,p}V^{\ast }$ is called transverse, if the volume form

$$\begin{eqnarray}\unicode[STIX]{x1D6EF}\wedge \unicode[STIX]{x1D70E}_{q}\unicode[STIX]{x1D70F}\wedge \bar{\unicode[STIX]{x1D70F}}\end{eqnarray}$$

is strictly positive for every nonzero decomposable $(q,0)$ -form $\unicode[STIX]{x1D70F}$ of $V^{\ast }$ . There exist many names for this terminology and we refer to [Reference Alessandrini and BassanelliAB91, Appendix] for a list.

This positivity notation on complex vector spaces can be extended pointwise to complex differential forms on a complex manifold. Let $M$ be an $n$ -dimensional complex manifold. Then we have the following definition.

The readers are referred to [Reference SullivanSul76] for more related concepts (such as differential form transversal to the cone structure on a real differentiable manifold) to $p$ -Kähler structures.

3.1 The $(p,q)$ th mild $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$ -lemma and its relevance

This subsection is to study various $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$ -lemmata related to local stabilities of complex structures, their properties, differences and roles there in some special case. More details can be found in [Reference Rao, Wan and ZhaoRWZ16, §3.1] and the references therein.

Now we introduce a new notion.

So we can state our main theorem.

According to Lemma 1.2, Theorem 3.2 unifies the local stabilities of Kähler structures [Reference Kodaira and SpencerKS60, Theorem 15] and balanced structures under the $(n-1,n)$ th mild $\unicode[STIX]{x2202}\bar{\unicode[STIX]{x2202}}$ -lemma [Reference Rao, Wan and ZhaoRWZ16, Theorem 1.5], which is an obvious generalization of Wu’s result [Reference WuWu06, Theorem 5.13] that the balanced structure is stable under small deformation when the $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$ -lemma holds, to $p$ -Kähler mild $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$ -structures for $1\leqslant p\leqslant n-1$ .

Let $X$ be a compact complex manifold of complex dimension $n$ with the following commutative diagram.

(6)

Recall that Dolbeault cohomology groups $H_{\overline{\unicode[STIX]{x2202}}}^{\bullet ,\bullet }(X)$ of $X$ are defined by

$$\begin{eqnarray}H_{\overline{\unicode[STIX]{x2202}}}^{\bullet ,\bullet }(X):=\frac{\ker \overline{\unicode[STIX]{x2202}}}{\operatorname{im}\overline{\unicode[STIX]{x2202}}},\end{eqnarray}$$

with $H_{\unicode[STIX]{x2202}}^{\bullet ,\bullet }(X)$ similarly defined, while Bott–Chern and Aeppli cohomology groups are defined as

$$\begin{eqnarray}H_{\text{BC}}^{\bullet ,\bullet }(X):=\frac{\ker \unicode[STIX]{x2202}\cap \ker \overline{\unicode[STIX]{x2202}}}{\operatorname{im}\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}}\quad \text{and}\quad H_{\text{A}}^{\bullet ,\bullet }(X):=\frac{\ker \unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}}{\operatorname{im}\unicode[STIX]{x2202}+\operatorname{im}\overline{\unicode[STIX]{x2202}}},\end{eqnarray}$$

respectively. The dimensions of $H_{dR}^{p+q}(X)$ , $H_{\overline{\unicode[STIX]{x2202}}}^{p,q}(X)$ , $H_{\text{BC}}^{p,q}(X)$ , $H_{\text{A}}^{p,q}(X)$ and $H_{\unicode[STIX]{x2202}}^{p,q}(X)$ over $\mathbb{C}$ are denoted by $b_{p+q}(X)$ , $h_{\overline{\unicode[STIX]{x2202}}}^{p,q}(X)$ , $h_{\text{BC}}^{p,q}(X)$ , $h_{\text{A}}^{p,q}(X)$ and $h_{\unicode[STIX]{x2202}}^{p,q}(X)$ , respectively, and the first four of them are usually called $(p\,+\,q)$ th Betti numbers, $(p,q)$ -Hodge numbers, Bott–Chern numbers and Aeppli numbers, respectively. So the (standard) $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$ -lemma is equivalent to the injectivities of the mappings

$$\begin{eqnarray}\unicode[STIX]{x1D704}_{\text{BC},dR}^{p,q}:H_{\text{BC}}^{p,q}(X)\rightarrow H_{dR}^{p+q}(X)\end{eqnarray}$$

for all $p,q$ , or to the isomorphisms of all the maps in diagram (6) by [Reference Deligne, Griffiths, Morgan and SullivanDGMS75, Remark 5.16].

Notice that the $(1,2)$ th mild $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$ -lemma is different from the $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$ -lemma on a complex manifold. It is easy to see that the $(1,2)$ th mild $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$ -lemma amounts to the injectivity of the mapping

$$\begin{eqnarray}\unicode[STIX]{x1D704}_{\text{BC},\unicode[STIX]{x2202}}^{1,2}:H_{\text{BC}}^{1,2}(X)\rightarrow H_{\unicode[STIX]{x2202}}^{1,2}(X).\end{eqnarray}$$

Then by [Reference Angella and KasuyaAK17, Tables 5 and 6 in Appendix A] and [Reference KasuyaKas13, the case $B$ in Example 1], we have the following example.

There are another three similar conditions relating to the local stabilities of complex structures. The $(p,p+1)th$ weak $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$ -lemma on a compact complex manifold $X$ , first introduced by Fu and Yau [Reference Fu and YauFY11] for $(p,p+1)=(n-1,n)$ , says that for any real $(p,p)$ -form $\unicode[STIX]{x1D713}$ such that $\overline{\unicode[STIX]{x2202}}\unicode[STIX]{x1D713}$ is $\unicode[STIX]{x2202}$ -exact, there is a $(p-1,p)$ -form $\unicode[STIX]{x1D703}$ , satisfying

$$\begin{eqnarray}\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}\unicode[STIX]{x1D703}=\overline{\unicode[STIX]{x2202}}\unicode[STIX]{x1D713}.\end{eqnarray}$$

And the $(p,q)th$ strong $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$ -lemma on $X$ , first proposed by Angella–Ugarte [Reference Angella and UgarteAU17] in the case $(p,q)=(n-1,n)$ , states that the induced mapping $\unicode[STIX]{x1D704}_{\text{BC},\text{A}}^{p,q}:H_{\text{BC}}^{p,q}(X)\rightarrow H_{\text{A}}^{p,q}(X)$ by the identity map is injective, which is equivalent to the statement that for any $d$ -closed $(p,q)$ -form $\unicode[STIX]{x1D6E4}$ of the type $\unicode[STIX]{x1D6E4}=\unicode[STIX]{x2202}\unicode[STIX]{x1D709}\,+\,\overline{\unicode[STIX]{x2202}}\unicode[STIX]{x1D713}$ , there exists a $(p-1,q-1)$ -form $\unicode[STIX]{x1D703}$ such that

$$\begin{eqnarray}\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}\unicode[STIX]{x1D703}=\unicode[STIX]{x1D6E4}.\end{eqnarray}$$

Angella–Ugarte [Reference Angella and UgarteAU17, Proposition 4.8] showed the deformation openness of the $(n-1,n)$ th strong $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$ -lemma. Besides, the condition that the induced mapping $\unicode[STIX]{x1D704}_{\text{BC},\overline{\unicode[STIX]{x2202}}}^{p,q}:H_{\text{BC}}^{p,q}(X)\rightarrow H_{\overline{\unicode[STIX]{x2202}}}^{p,q}(X)$ by the identity map is injective, is first presented by Angella–Ugarte [Reference Angella and UgarteAU16] in the case $(p,q)=(n-1,n)$ to study local conformal balanced structures and global ones, which we may call the $(p,q)th$ dual mild $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$ -lemma.

After a simple check, we have the following observation.

All these four ‘ $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$ -lemmata’ hold if the compact complex manifold $X$ satisfies the standard $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$ -lemma. And either the $(p,p+1)$ th mild or dual mild $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$ -lemma implies the weak one, while [Reference Rao, Wan and ZhaoRWZ16, Corollary 3.9] implies that the $(n-1,n)$ th mild $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$ -lemma and the dual mild one are unrelated.

By [Reference Alessandrini and BassanelliAB90], a small deformation of the Iwasawa manifold, which satisfies the $(2,3)$ th weak $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$ -lemma but does not satisfy the mild one from Example 3.5, may not be balanced. Thus, the condition ‘ $(n-1,n)$ th mild $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$ -lemma’ in Corollary 1.3(ii) cannot be replaced by the weak one.

The next example shows that neither the $(n-1,n)$ th weak $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$ -lemma nor the mild one is deformation open. And it shows that the condition in [Reference Fu and YauFY11, Theorem 6] is not a necessary one for the deformation openness of balanced structures as mentioned in [Reference Ugarte and VillacampaUV15, the discussion ahead of Example 3.7]. Recall that [Reference Fu and YauFY11, Theorem 6] says that the balanced structure is deformation open, if the $(n-1,n)$ th weak $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$ -lemma holds on the general fibers $X_{t}$ for $t\neq 0$ . Fortunately, Corollary 1.3(ii) can be applied to this example. See also [Reference Angella and UgarteAU17, Remark 4.7], where Corollary 1.3(ii) can also be applied.

Meanwhile, a $2n$ -dimensional nilmanifold endowed with a left-invariant abelian complex structure satisfies the $(n-1,n)$ th mild $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$ -lemma but never satisfies the $(n-1,n)$ th dual mild $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$ -lemma. It shows that the deformation openness of balanced structures with the reference fiber a nilmanifold endowed with a left-invariant abelian balanced Hermitian structure is easily obtained by Corollary 1.3(ii), but not from [Reference Angella and UgarteAU17, Theorem 4.9], which says that if $X_{0}$ admits a locally conformal balanced metric and satisfies the $(n-1,n)$ th strong $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$ -lemma, then $X_{t}$ is balanced for small $t$ .

Moreover, the deformation invariance of the $(n-1,n-1)$ th Bott–Chern numbers $h_{\text{BC}}^{n\,-\,1,n\,-\,1}(X_{t})$ can assure the deformation openness of balanced structures as shown in [Reference Angella and UgarteAU17, Proposition 4.1]. Inspired by Wu’s result [Reference WuWu06, Theorem 5.13], we have the following generalization.

Nevertheless, Corollary 1.3(ii) may be applied to some cases with deformation variance of the $(n-1,n-1)$ th Bott–Chern numbers. The newly constructed example that follows is one case among nilmanifolds, while the manifold in [Reference Angella and UgarteAU17, Example 4.10], satisfying the $(2,3)$ th strong $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$ -lemma, is a solvable manifold but not a nilmanifold by [Reference Rao, Wan and ZhaoRWZ16, Corollary 3.9].

Example 3.8. Let $G$ be the simply connected nilpotent Lie group determined by a ten-dimensional $3$ -step nilpotent Lie algebra $\mathfrak{g}$ endowed with a left-invariant abelian complex structure $J$ , satisfying the structure equation

$$\begin{eqnarray}\left\{\begin{array}{@{}l@{}}d\unicode[STIX]{x1D6FE}^{1}=d\unicode[STIX]{x1D6FE}^{2}=d\unicode[STIX]{x1D6FE}^{3}=0,\quad \\ d\unicode[STIX]{x1D6FE}^{4}=\unicode[STIX]{x1D6FE}^{1\bar{3}},\quad \\ d\unicode[STIX]{x1D6FE}^{5}=\unicode[STIX]{x1D6FE}^{3\bar{4}},\quad \end{array}\right.\end{eqnarray}$$

where the natural decomposition with respect to $J$ yields

$$\begin{eqnarray}\mathfrak{g}_{\mathbb{C}}=\mathfrak{g}\otimes _{\mathbb{R}}\mathbb{C}=\mathfrak{g}_{J}^{1,0}\oplus \mathfrak{g}_{J}^{0,1};\quad \mathfrak{g}_{\mathbb{ C}}^{\ast }=\mathfrak{g}^{\ast }\otimes _{\mathbb{ R}}\mathbb{C}=\mathfrak{g}_{J}^{\ast (1,0)}\oplus \mathfrak{g}_{J}^{\ast (0,1)},\end{eqnarray}$$

$\{\unicode[STIX]{x1D6FE}^{i}\}_{i=1}^{5}$ is the basis of $\mathfrak{g}^{\ast (1,0)}$ and the convention $\unicode[STIX]{x1D6FE}^{1\bar{3}}=\unicode[STIX]{x1D6FE}^{1}\wedge \overline{\unicode[STIX]{x1D6FE}^{3}}$ is used here and afterwards. Define a lattice $\unicode[STIX]{x1D6E4}$ in $G$ , determined by the rational span of $\{\unicode[STIX]{x1D6FE}^{i},\bar{\unicode[STIX]{x1D6FE}}^{i}\}_{i=1}^{5}$ . Then $M:=\unicode[STIX]{x1D6E4}\backslash G$ is a compact nilmanifold with the abelian complex structure $J$ given above. It is easy to check that

$$\begin{eqnarray}\unicode[STIX]{x1D6FA}=\unicode[STIX]{x1D6FE}^{1234\overline{1234}}+\unicode[STIX]{x1D6FE}^{1235\overline{1235}}+\unicode[STIX]{x1D6FE}^{1245\overline{1245}}+\unicode[STIX]{x1D6FE}^{1345\overline{1345}}+\unicode[STIX]{x1D6FE}^{2345\overline{2345}}\end{eqnarray}$$

is a left-invariant balanced metric on $(\mathfrak{g},J)$ , descending to $M$ . Denote the basis of $\mathfrak{g}^{1,0}$ dual to $\{\unicode[STIX]{x1D6FE}^{i}\}_{i=1}^{5}$ by $\{\unicode[STIX]{x1D703}_{i}\}_{i=1}^{5}$ . The equation $d\unicode[STIX]{x1D714}(\unicode[STIX]{x1D703},\unicode[STIX]{x1D703}^{\prime })=-\unicode[STIX]{x1D714}([\unicode[STIX]{x1D703},\unicode[STIX]{x1D703}^{\prime }])$ for $\unicode[STIX]{x1D714}\in \mathfrak{g}_{\mathbb{C}}^{\ast }$ and $\unicode[STIX]{x1D703},\unicode[STIX]{x1D703}^{\prime }\in \mathfrak{g}_{\mathbb{C}}$ , establishes the equalities

$$\begin{eqnarray}[\bar{\unicode[STIX]{x1D703}}_{3},\unicode[STIX]{x1D703}_{1}]=\unicode[STIX]{x1D703}_{4},\quad [\bar{\unicode[STIX]{x1D703}}_{4},\unicode[STIX]{x1D703}_{3}]=\unicode[STIX]{x1D703}_{5}.\end{eqnarray}$$

According to [Reference Console, Fino and PoonCFP06, Theorem 3.6], the linear operator $\overline{\unicode[STIX]{x2202}}$ on $\mathfrak{g}^{1,0}$ , defined in [Reference Console, Fino and PoonCFP06, §3.2], amounts to

$$\begin{eqnarray}\overline{\unicode[STIX]{x2202}}:\mathfrak{g}^{1,0}\rightarrow \mathfrak{g}^{\ast (0,1)}\otimes \mathfrak{g}^{1,0}:\overline{\unicode[STIX]{x2202}}V=\bar{\unicode[STIX]{x1D6FE}}^{i}\otimes [\bar{\unicode[STIX]{x1D703}}_{i},V]^{1,0}\quad \text{for}~V\in \mathfrak{g}^{1,0},\end{eqnarray}$$

which induces an isomorphism $H^{1}(M,T_{M}^{1,0})\cong H_{\overline{\unicode[STIX]{x2202}}}^{1}(\mathfrak{g}^{1,0})$ . Therefore, from Kodaira–Spencer’s deformation theory, an analytic deformation $M_{t}$ of $M$ can be constructed by use of the integrable left-invariant Beltrami differential

$$\begin{eqnarray}\unicode[STIX]{x1D711}(t)=(t_{1}\bar{\unicode[STIX]{x1D6FE}}^{4}+t_{2}\bar{\unicode[STIX]{x1D6FE}}^{5})\otimes \unicode[STIX]{x1D703}_{2}+(t_{3}\bar{\unicode[STIX]{x1D6FE}}^{4}+t_{4}\bar{\unicode[STIX]{x1D6FE}}^{5})\otimes \unicode[STIX]{x1D703}_{5}\end{eqnarray}$$

for $t=(t_{1},t_{2},t_{3},t_{4})$ and $|t_{4}|<1$ , which satisfies $\overline{\unicode[STIX]{x2202}}\unicode[STIX]{x1D711}(t)=\frac{1}{2}[\unicode[STIX]{x1D711}(t),\unicode[STIX]{x1D711}(t)]$ and the so-called Schouten-Nijenhuis bracket $[\cdot ,\cdot ]$ (cf. [Reference Console, Fino and PoonCFP06, Formula (4.1)]) works as

$$\begin{eqnarray}[\bar{\unicode[STIX]{x1D6FE}}\otimes \unicode[STIX]{x1D703},\bar{\unicode[STIX]{x1D6FE}}^{\prime }\otimes \unicode[STIX]{x1D703}^{\prime }]=\bar{\unicode[STIX]{x1D6FE}}^{\prime }\wedge \unicode[STIX]{x1D704}_{\unicode[STIX]{x1D703}^{\prime }}d\bar{\unicode[STIX]{x1D6FE}}\otimes \unicode[STIX]{x1D703}+\bar{\unicode[STIX]{x1D6FE}}\wedge \unicode[STIX]{x1D704}_{\unicode[STIX]{x1D703}}d\bar{\unicode[STIX]{x1D6FE}}^{\prime }\otimes \unicode[STIX]{x1D703}^{\prime }\quad \text{for}~\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D6FE}^{\prime }\in \mathfrak{g}^{\ast (1,0)},\unicode[STIX]{x1D703},\unicode[STIX]{x1D703}^{\prime }\in \mathfrak{g}^{1,0}.\end{eqnarray}$$

Then the general fibers $M_{t}$ are still nilmanifolds, determined by the Lie algebra $\mathfrak{g}$ with respect to the decompositions

$$\begin{eqnarray}\mathfrak{g}_{\mathbb{C}}=\mathfrak{g}\otimes _{\mathbb{R}}\mathbb{C}=\mathfrak{g}_{\unicode[STIX]{x1D711}(t)}^{1,0}\oplus \mathfrak{g}_{\unicode[STIX]{x1D711}(t)}^{0,1};\quad \mathfrak{g}_{\mathbb{ C}}^{\ast }=\mathfrak{g}^{\ast }\otimes _{\mathbb{ R}}\mathbb{C}=\mathfrak{g}_{\unicode[STIX]{x1D711}(t)}^{\ast (1,0)}\oplus \mathfrak{g}_{\unicode[STIX]{x1D711}(t)}^{\ast (0,1)},\end{eqnarray}$$

where the basis of $\mathfrak{g}_{\unicode[STIX]{x1D711}(t)}^{\ast (1,0)}$ is given by $\unicode[STIX]{x1D6FE}^{i}(t)=e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}(t)}}(\unicode[STIX]{x1D6FE}^{i})=(\unicode[STIX]{x1D7D9}+\unicode[STIX]{x1D711}(t))\lrcorner \unicode[STIX]{x1D6FE}^{i}$ for $1\leqslant i\leqslant 5$ . Hence, the structure equation of $\{\unicode[STIX]{x1D6FE}^{i}(t)\}_{i=1}^{5}$ is

$$\begin{eqnarray}\left\{\begin{array}{@{}l@{}}d\unicode[STIX]{x1D6FE}^{1}(t)=d\unicode[STIX]{x1D6FE}^{3}(t)=0,\quad \\ d\unicode[STIX]{x1D6FE}^{2}(t)=-t_{1}\unicode[STIX]{x1D6FE}^{3\bar{1}}(t)-t_{2}\unicode[STIX]{x1D6FE}^{4\bar{3}}(t),\quad \\ d\unicode[STIX]{x1D6FE}^{4}(t)=\unicode[STIX]{x1D6FE}^{1\bar{3}}(t),\quad \\ d\unicode[STIX]{x1D6FE}^{5}(t)=\unicode[STIX]{x1D6FE}^{3\bar{4}}(t)-t_{3}\unicode[STIX]{x1D6FE}^{3\bar{1}}(t)-t_{4}\unicode[STIX]{x1D6FE}^{4\bar{3}}(t),\quad \end{array}\right.\end{eqnarray}$$

where $\unicode[STIX]{x1D6FE}^{3\bar{1}}(t)$ denotes $\unicode[STIX]{x1D6FE}^{3}(t)\wedge \overline{\unicode[STIX]{x1D6FE}^{1}(t)}$ , similarly for others. It is well known from [Reference Console and FinoCF01, Reference RollenskeRol09, Reference AngellaAng13] that the Bott–Chern cohomologies of nilmanifolds with abelian complex structures and their small deformation can be calculated via left-invariant differential forms. Remark 1.7 tells us that the dimension of the space of the $d$ -closed left-invariant $(4,4)$ -forms is invariant along the deformation $M_{t}$ , which is equal to $21$ . And one can calculate the $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$ -exact terms directly by use of the structure equation

$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x2202}_{t}\overline{\unicode[STIX]{x2202}}_{t}(\mathfrak{g}_{\unicode[STIX]{x1D711}(t)}^{\ast (3,3)})\nonumber\\ \displaystyle & & \displaystyle \quad =\langle \!-(1+|t_{4}|^{2})\unicode[STIX]{x1D6FE}^{1234\overline{1234}}(t)-|t_{2}|^{2}\unicode[STIX]{x1D6FE}^{1345\overline{1345}}+t_{2}\overline{t}_{4}\unicode[STIX]{x1D6FE}^{1345\overline{1234}}(t)+t_{4}\overline{t}_{2}\unicode[STIX]{x1D6FE}^{1234\overline{1345}}(t),\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\unicode[STIX]{x1D6FE}^{1235\overline{1235}}(t)-|t_{1}|^{2}\unicode[STIX]{x1D6FE}^{1345\overline{1345}}(t)-|t_{3}|^{2}\unicode[STIX]{x1D6FE}^{1234\overline{1234}}(t)+t_{1}\overline{t}_{3}\unicode[STIX]{x1D6FE}^{1345\overline{1234}}(t)+t_{3}\overline{t}_{1}\unicode[STIX]{x1D6FE}^{1234\overline{1345}}(t),\nonumber\\ \displaystyle & & \displaystyle \qquad -\,t_{1}\unicode[STIX]{x1D6FE}^{1345\overline{1234}}(t)+t_{2}\unicode[STIX]{x1D6FE}^{1345\overline{1235}}(t)+t_{3}\unicode[STIX]{x1D6FE}^{1234\overline{1234}}(t)-t_{4}\unicode[STIX]{x1D6FE}^{1234\overline{1235}}(t),\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\overline{t}_{1}\unicode[STIX]{x1D6FE}^{1234\overline{1345}}(t)+\overline{t}_{2}\unicode[STIX]{x1D6FE}^{1235\overline{1345}}(t)+\overline{t}_{3}\unicode[STIX]{x1D6FE}^{1234\overline{1234}}(t)-\overline{t}_{4}\unicode[STIX]{x1D6FE}^{1235\overline{1234}}(t)\!\rangle .\nonumber\end{eqnarray}$$

It is clear that $\dim \unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}(\mathfrak{g}_{J}^{\ast (3,3)})=2$ and $\dim \unicode[STIX]{x2202}_{t}\overline{\unicode[STIX]{x2202}}_{t}(\mathfrak{g}_{\unicode[STIX]{x1D711}(t)}^{\ast (3,3)})=4$ for general $t$ . Therefore, the Bott–Chern number $h_{\text{BC}}^{4,4}(M_{t})$ varies from $19$ to $17$ along the deformation $M_{t}$ .

It may not be difficult to find an example of a non-Kähler $p$ -Kähler manifold in the Fujiki class for $1<p<n-1$ in the literature, and thus it satisfies the $(p,p\,+\,1)$ th mild $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$ -lemma, for example [Reference Alessandrini and BassanelliAB92, §4]. However, motivated by Example 3.8, we ask the following question.

Finally, from the perspective of Corollary 1.3(ii), we may have a clear picture of Angella–Ugarte’s result [Reference Angella and UgarteAU17, Theorem 4.9]. Actually, Observation 3.4 tells us that the $(n\,-\,1,n)$ th strong $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$ -lemma decomposes into the mild one and the dual mild one. A locally conformal balanced metric can be transformed into a balanced one by the $(n-1,n)$ th dual mild $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$ -lemma, from [Reference Angella and UgarteAU16, Theorem 2.5]. Then the $(n\,-\,1,n)$ th mild $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$ -lemma assures the deformation openness of balanced structures originally from the transformed balanced metric on the reference fiber, thanks to Corollary 1.3(ii).

3.2 Modification stabilities

We consider the modification on a compact complex manifold defined as follows and refer the reader to [Reference UenoUen75, §2] for its general definition on complex spaces.

Definition 3.10. A modification of an $n$ -dimensional compact complex manifold $M$ is a holomorphic map

$$\begin{eqnarray}\unicode[STIX]{x1D707}:\tilde{M}\rightarrow M\end{eqnarray}$$

so that:

1. (i) $\tilde{M}$ is also an $n$ -dimensional compact complex manifold;

2. (ii) there exists an analytic subset $S\subseteq M$ of codimension greater than or equal to $1$ such that $\unicode[STIX]{x1D707}\mid _{\tilde{M}\setminus \unicode[STIX]{x1D707}^{-1}(S)}:\tilde{M}\setminus \unicode[STIX]{x1D707}^{-1}(S)\rightarrow M\setminus S$ is a biholomorphism.

It is a classical result, [Reference ParshinPar66] or [Reference Deligne, Griffiths, Morgan and SullivanDGMS75, Theorem 5.22], that if the modification of a complex manifold is a $\unicode[STIX]{x2202}\bar{\unicode[STIX]{x2202}}$ -manifold, then so is this manifold. Therefore, each compact complex manifold in the Fujiki class ${\mathcal{C}}$ (i.e. admitting a Kähler modification) is a $\unicode[STIX]{x2202}\bar{\unicode[STIX]{x2202}}$ -manifold. The converse is an open question as in [Reference AlessandriniAle17, Introduction]: Is the modification of a $\unicode[STIX]{x2202}\bar{\unicode[STIX]{x2202}}$ -manifold still a $\unicode[STIX]{x2202}\bar{\unicode[STIX]{x2202}}$ -manifold? A recent result [Reference Yang and YangYY17, Theorem 1.3] of S. Yang and X. Yang, by means of a blow-up formula for Bott–Chern cohomologies and the characterizations by Angella and Tomassini [Reference Angella and TomassiniAT13] and Angella and Tardini [Reference Angella and TardiniAT17] of $\unicode[STIX]{x2202}\bar{\unicode[STIX]{x2202}}$ -manifolds, and also [Reference Rao, Yang and YangRYY17, Main Theorem 1.1] by S. Yang, X. Yang and the first author confirm this question in dimension three, that is, the modification of a $\unicode[STIX]{x2202}\bar{\unicode[STIX]{x2202}}$ -threefold is still a $\unicode[STIX]{x2202}\bar{\unicode[STIX]{x2202}}$ -threefold. See also, more recently, [Reference Angella, Suwa, Tardini and TomassiniASTT17, Theorem 2.1]. These results provide us with more classes of complex manifolds satisfying mild $\unicode[STIX]{x2202}\bar{\unicode[STIX]{x2202}}$ -lemmata. Moreover, it is natural to ask the following analogous question.

Now we present a modification stability of $(p,q)$ th mild $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$ -lemma on a compact complex manifold. Let $M$ be a complex manifold. One has the $\mathbb{K}$ -valued de Rham complex $(A^{\bullet }(M)_{\mathbb{K}},d)$ for $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$ . Fixing a $d$ -closed $1$ -form $\unicode[STIX]{x1D719}\in A^{1}(M)_{\mathbb{K}}$ , we consider another complex. Namely, define

$$\begin{eqnarray}d_{\unicode[STIX]{x1D719}}=d+L_{\unicode[STIX]{x1D719}},\end{eqnarray}$$

where $L_{\unicode[STIX]{x1D719}}:=\unicode[STIX]{x1D719}\wedge \bullet$ . The cochain complex

$$\begin{eqnarray}(A^{\bullet }(M)_{\mathbb{K}},d_{\unicode[STIX]{x1D719}})\end{eqnarray}$$

can be regarded as the de Rham complex with values in the topologically trivial flat bundle $M\times \mathbb{K}$ with the connection form $\unicode[STIX]{x1D719}$ . We study cohomologies and Hodge theory for general complex manifolds with twisted differentials. More precisely, for $\unicode[STIX]{x1D703}_{1},\unicode[STIX]{x1D703}_{2}\in H_{\text{BC}}^{1,0}(M)$ , consider the bi-differential $\mathbb{Z}$ -graded complex

$$\begin{eqnarray}(A^{\bullet }(M)_{\mathbb{C}},\unicode[STIX]{x2202}_{(\unicode[STIX]{x1D703}_{1},\unicode[STIX]{x1D703}_{2})},\overline{\unicode[STIX]{x2202}}_{(\unicode[STIX]{x1D703}_{1},\unicode[STIX]{x1D703}_{2})}),\end{eqnarray}$$

where

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{(\unicode[STIX]{x1D703}_{1},\unicode[STIX]{x1D703}_{2})} & := & \displaystyle \unicode[STIX]{x2202}+L_{\unicode[STIX]{x1D703}_{2}}+L_{\overline{\unicode[STIX]{x1D703}_{1}}},\nonumber\\ \displaystyle \overline{\unicode[STIX]{x2202}}_{(\unicode[STIX]{x1D703}_{1},\unicode[STIX]{x1D703}_{2})} & := & \displaystyle \overline{\unicode[STIX]{x2202}}-L_{\overline{\unicode[STIX]{x1D703}_{2}}}+L_{\unicode[STIX]{x1D703}_{1}}.\nonumber\end{eqnarray}$$

It is easy to check that

$$\begin{eqnarray}\unicode[STIX]{x2202}_{(\unicode[STIX]{x1D703}_{1},\unicode[STIX]{x1D703}_{2})}\unicode[STIX]{x2202}_{(\unicode[STIX]{x1D703}_{1},\unicode[STIX]{x1D703}_{2})}=\overline{\unicode[STIX]{x2202}}_{(\unicode[STIX]{x1D703}_{1},\unicode[STIX]{x1D703}_{2})}\overline{\unicode[STIX]{x2202}}_{(\unicode[STIX]{x1D703}_{1},\unicode[STIX]{x1D703}_{2})}=\unicode[STIX]{x2202}_{(\unicode[STIX]{x1D703}_{1},\unicode[STIX]{x1D703}_{2})}\overline{\unicode[STIX]{x2202}}_{(\unicode[STIX]{x1D703}_{1},\unicode[STIX]{x1D703}_{2})}+\overline{\unicode[STIX]{x2202}}_{(\unicode[STIX]{x1D703}_{1},\unicode[STIX]{x1D703}_{2})}\unicode[STIX]{x2202}_{(\unicode[STIX]{x1D703}_{1},\unicode[STIX]{x1D703}_{2})}=0.\end{eqnarray}$$

Angella and Kasuya [Reference Angella and KasuyaAK14] investigated cohomological properties of this bi-differential complex and considered (more than) two cohomologies:

$$\begin{eqnarray}H^{\bullet }(A^{\bullet }(M)_{\mathbb{C}};\unicode[STIX]{x2202}_{(\unicode[STIX]{x1D703}_{1},\unicode[STIX]{x1D703}_{2})},\overline{\unicode[STIX]{x2202}}_{(\unicode[STIX]{x1D703}_{1},\unicode[STIX]{x1D703}_{2})};\unicode[STIX]{x2202}_{(\unicode[STIX]{x1D703}_{1},\unicode[STIX]{x1D703}_{2})}\overline{\unicode[STIX]{x2202}}_{(\unicode[STIX]{x1D703}_{1},\unicode[STIX]{x1D703}_{2})}):=\frac{\ker \unicode[STIX]{x2202}_{(\unicode[STIX]{x1D703}_{1},\unicode[STIX]{x1D703}_{2})}\cap \ker \overline{\unicode[STIX]{x2202}}_{(\unicode[STIX]{x1D703}_{1},\unicode[STIX]{x1D703}_{2})}}{\operatorname{im}\unicode[STIX]{x2202}_{(\unicode[STIX]{x1D703}_{1},\unicode[STIX]{x1D703}_{2})}\overline{\unicode[STIX]{x2202}}_{(\unicode[STIX]{x1D703}_{1},\unicode[STIX]{x1D703}_{2})}}\end{eqnarray}$$

and

$$\begin{eqnarray}H^{\bullet }(A^{\bullet }(M)_{\mathbb{C}};\unicode[STIX]{x2202}_{(\unicode[STIX]{x1D703}_{1},\unicode[STIX]{x1D703}_{2})}):=\frac{\ker \unicode[STIX]{x2202}_{(\unicode[STIX]{x1D703}_{1},\unicode[STIX]{x1D703}_{2})}}{\operatorname{im}\unicode[STIX]{x2202}_{(\unicode[STIX]{x1D703}_{1},\unicode[STIX]{x1D703}_{2})}},\end{eqnarray}$$

which are simply denoted by $H_{\text{BC}}^{\bullet }(M,\unicode[STIX]{x1D703}_{1},\unicode[STIX]{x1D703}_{2})$ and $H_{\unicode[STIX]{x2202}}^{\bullet }(M,\unicode[STIX]{x1D703}_{1},\unicode[STIX]{x1D703}_{2})$ , respectively. If one sets $\unicode[STIX]{x1D703}_{1}=\unicode[STIX]{x1D703}_{2}=0$ , $H_{\text{BC}}^{\bullet }(M,\unicode[STIX]{x1D703}_{1},\unicode[STIX]{x1D703}_{2})$ and $H_{\unicode[STIX]{x2202}}^{\bullet }(M,\unicode[STIX]{x1D703}_{1},\unicode[STIX]{x1D703}_{2})$ are just the ordinary Bott–Chern cohomology $H_{\text{BC}}^{\bullet ,\bullet }(M)$ and $H_{\unicode[STIX]{x2202}}^{\bullet ,\bullet }(M)$ of $M$ , respectively.

Following Wells in [Reference WellsWel74, Theorem 3.1], Angella and Kasuya proved the following proposition.

Proposition 3.12 [Reference Angella and KasuyaAK14, Theorem 2.4].

Let $\unicode[STIX]{x1D707}:\tilde{M}\rightarrow M$ be a modification of a compact complex manifold $M$ . Then the induced maps

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}_{\text{BC}}^{\ast }:H_{\text{BC}}^{\bullet }(M,\unicode[STIX]{x1D703}_{1},\unicode[STIX]{x1D703}_{2}) & \rightarrow & \displaystyle H_{\text{BC}}^{\bullet }(\tilde{M},\unicode[STIX]{x1D707}^{\ast }\unicode[STIX]{x1D703}_{1},\unicode[STIX]{x1D707}^{\ast }\unicode[STIX]{x1D703}_{2}),\nonumber\\ \displaystyle \unicode[STIX]{x1D707}_{\unicode[STIX]{x2202}}^{\ast }:H_{\unicode[STIX]{x2202}}^{\bullet }(M,\unicode[STIX]{x1D703}_{1},\unicode[STIX]{x1D703}_{2}) & \rightarrow & \displaystyle H_{\unicode[STIX]{x2202}}^{\bullet }(\tilde{M},\unicode[STIX]{x1D707}^{\ast }\unicode[STIX]{x1D703}_{1},\unicode[STIX]{x1D707}^{\ast }\unicode[STIX]{x1D703}_{2})\nonumber\end{eqnarray}$$

are injective.

We reformulate Definition 3.1 for the $(p,q)$ th mild $\unicode[STIX]{x2202}\bar{\unicode[STIX]{x2202}}$ -lemma as follows.

Proof. Taking $\unicode[STIX]{x1D703}_{1}=\unicode[STIX]{x1D703}_{2}=0$ , we have the following commutative diagram.

By the $(p,q)$ th mild $\unicode[STIX]{x2202}\bar{\unicode[STIX]{x2202}}$ -lemma assumption on $\tilde{M}$ , the map $\unicode[STIX]{x1D704}_{\text{BC},\unicode[STIX]{x2202}}^{p,q}$ for $\tilde{M}$ is injective and so are $\unicode[STIX]{x1D707}_{\text{BC}}^{\ast },\unicode[STIX]{x1D707}_{\unicode[STIX]{x2202}}^{\ast }$ by Proposition 3.12. So the map $\unicode[STIX]{x1D704}_{\text{BC},\unicode[STIX]{x2202}}^{p,q}$ for $M$ is injective, i.e., $M$ satisfies the $(p,q)$ th mild $\unicode[STIX]{x2202}\bar{\unicode[STIX]{x2202}}$ -lemma.◻

4.1 Kuranishi family and Beltrami differentials

We introduce some basics on the Kuranishi family of complex structures in this subsection, which is extracted from [Reference Rao and ZhaoRZ18, Reference Rao, Wan and ZhaoRWZ16] and obviously originally from [Reference KuranishiKur64].

By (the proof of) Kuranishi’s completeness theorem [Reference KuranishiKur64], for any compact complex manifold $X_{0}$ , there exists a complete holomorphic family $\unicode[STIX]{x1D71B}:{\mathcal{K}}\rightarrow T$ of complex manifolds at the reference point $0\in T$ in the sense that for any differentiable family $\unicode[STIX]{x1D70B}:{\mathcal{X}}\rightarrow B$ with $\unicode[STIX]{x1D70B}^{-1}(s_{0})=\unicode[STIX]{x1D71B}^{-1}(0)=X_{0}$ , there exist a sufficiently small neighborhood $E\subseteq B$ of $s_{0}$ , and smooth maps $\unicode[STIX]{x1D6F7}:{\mathcal{X}}_{E}\rightarrow {\mathcal{K}}$ , $\unicode[STIX]{x1D70F}:E\rightarrow T$ with $\unicode[STIX]{x1D70F}(s_{0})=0$ such that the diagram

commutes, $\unicode[STIX]{x1D6F7}$ maps $\unicode[STIX]{x1D70B}^{-1}(s)$ biholomorphically onto $\unicode[STIX]{x1D71B}^{-1}(\unicode[STIX]{x1D70F}(s))$ for each $s\in E$ , and

$$\begin{eqnarray}\unicode[STIX]{x1D6F7}:\unicode[STIX]{x1D70B}^{-1}(s_{0})=X_{0}\rightarrow \unicode[STIX]{x1D71B}^{-1}(0)=X_{0}\end{eqnarray}$$

is the identity map. This family is called Kuranishi family and is constructed as follows. Let $\{\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D708}}\}_{\unicode[STIX]{x1D708}=1}^{m}$ be a base for $\mathbb{H}^{0,1}(X_{0},T_{X_{0}}^{1,0})$ , where some suitable hermitian metric is fixed on $X_{0}$ and $m\geqslant 1$ . Otherwise the complex manifold $X_{0}$ would be rigid, i.e., for any differentiable family $\unicode[STIX]{x1D705}:{\mathcal{M}}\rightarrow P$ with $s_{0}\in P$ and $\unicode[STIX]{x1D705}^{-1}(s_{0})=X_{0}$ , there is a neighborhood $V\subseteq P$ of $s_{0}$ such that $\unicode[STIX]{x1D705}:\unicode[STIX]{x1D705}^{-1}(V)\rightarrow V$ is trivial. Then one can construct a holomorphic family

$$\begin{eqnarray}\unicode[STIX]{x1D711}(t)=\mathop{\sum }_{|I|=1}^{\infty }\unicode[STIX]{x1D711}_{I}t^{I}:=\mathop{\sum }_{j=1}^{\infty }\unicode[STIX]{x1D711}_{j}(t),\quad I=(i_{1},\ldots ,i_{m}),t=(t_{1},\ldots ,t_{m})\in \mathbb{C}^{m},\end{eqnarray}$$

for $|t|<\unicode[STIX]{x1D70C}$ a small positive constant, of Beltrami differentials as follows:

$$\begin{eqnarray}\unicode[STIX]{x1D711}_{1}(t)=\mathop{\sum }_{\unicode[STIX]{x1D708}=1}^{m}t_{\unicode[STIX]{x1D708}}\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D708}}\end{eqnarray}$$

and for $|I|\geqslant 2$ ,

$$\begin{eqnarray}\unicode[STIX]{x1D711}_{I}=\frac{1}{2}\overline{\unicode[STIX]{x2202}}^{\ast }\mathbb{G}\mathop{\sum }_{J+L=I}[\unicode[STIX]{x1D711}_{J},\unicode[STIX]{x1D711}_{L}].\end{eqnarray}$$

It is clear that $\unicode[STIX]{x1D711}(t)$ satisfies the equation

$$\begin{eqnarray}\unicode[STIX]{x1D711}(t)=\unicode[STIX]{x1D711}_{1}+{\textstyle \frac{1}{2}}\overline{\unicode[STIX]{x2202}}^{\ast }\mathbb{G}[\unicode[STIX]{x1D711}(t),\unicode[STIX]{x1D711}(t)].\end{eqnarray}$$

Let

$$\begin{eqnarray}T=\{t\mid \mathbb{H}[\unicode[STIX]{x1D711}(t),\unicode[STIX]{x1D711}(t)]=0\}.\end{eqnarray}$$

So for each $t\in T$ , $\unicode[STIX]{x1D711}(t)$ satisfies

$$\begin{eqnarray}\bar{\unicode[STIX]{x2202}}\unicode[STIX]{x1D711}(t)={\textstyle \frac{1}{2}}[\unicode[STIX]{x1D711}(t),\unicode[STIX]{x1D711}(t)],\end{eqnarray}$$

and determines a complex structure $X_{t}$ on the underlying differentiable manifold of $X_{0}$ . More importantly, $\unicode[STIX]{x1D711}(t)$ represents the complete holomorphic family $\unicode[STIX]{x1D71B}:{\mathcal{K}}\rightarrow T$ of complex manifolds. Roughly speaking, the Kuranishi family $\unicode[STIX]{x1D71B}:{\mathcal{K}}\rightarrow T$ contains all sufficiently small differentiable deformations of $X_{0}$ .

By means of these, one can reduce the local stability Theorem 4.1 to the Kuranishi family by shrinking $E$ if necessary, that is, it suffices to construct a $p$ -Kähler metric on each $X_{t}$ . From now on, we use $\unicode[STIX]{x1D711}(t)$ and $\unicode[STIX]{x1D711}$ interchangeably to denote this holomorphic family of integrable Beltrami differentials, and assume $m=1$ for simplicity.

4.2 Obstruction equation and construction of power series

Now we begin to prove the $d$ -closed smooth extension of $(p,q)$ -forms as in Theorem 4.1 by using the power series method.

As both $e^{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}}$ and $e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}}$ are invertible operators when $t$ is sufficiently small, it follows that for any $\unicode[STIX]{x1D6FA}\in A^{p,q}(X_{0})$ ,

(8) $$\begin{eqnarray}e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FA})=e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}}\circ e^{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}}\circ e^{-\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}}\circ e^{-\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}}\circ e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FA}).\end{eqnarray}$$

Set

(9) $$\begin{eqnarray}\tilde{\unicode[STIX]{x1D6FA}}=e^{-\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}}\circ e^{-\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}}\circ e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FA}),\end{eqnarray}$$

where $\unicode[STIX]{x1D6FA}$ and $\tilde{\unicode[STIX]{x1D6FA}}$ apparently have a one-to-one correspondence. Here we follow the notation: $\overline{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711}=\unicode[STIX]{x1D711}\lrcorner \overline{\unicode[STIX]{x1D711}}$ , $\unicode[STIX]{x1D7D9}$ is the identity operator defined as

$$\begin{eqnarray}\unicode[STIX]{x1D7D9}=\frac{1}{p+q}\biggl(\mathop{\sum }_{i}^{n}dz^{i}\otimes \frac{\unicode[STIX]{x2202}~}{\unicode[STIX]{x2202}z^{i}}+\mathop{\sum }_{i}^{n}d\bar{z}^{i}\otimes \frac{\unicode[STIX]{x2202}~}{\unicode[STIX]{x2202}\bar{z}^{i}}\biggr)\end{eqnarray}$$

when it acts on $(p,q)$ -forms of a complex manifold, and similarly for others. And it is easy to check that the operator

$$\begin{eqnarray}e^{-\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}}\circ e^{-\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}}\circ e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}\end{eqnarray}$$

preserves the form types and thus $\tilde{\unicode[STIX]{x1D6FA}}$ is still a $(p,q)$ -form. In fact, for any $(p,q)$ -form $\unicode[STIX]{x1D6FC}$ on $X_{0}$ , we will find

$$\begin{eqnarray}\displaystyle & & \displaystyle e^{-\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}}\circ e^{-\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}}\circ e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FC})\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D6FC}_{i_{1}\cdots i_{p}\overline{j_{1}}\cdots \overline{j_{q}}}\,dz^{i_{1}}\wedge \cdots \wedge dz^{i_{p}}\wedge (\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})\lrcorner \,d\bar{z}^{j_{1}}\wedge \cdots \wedge (\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})\lrcorner \,d\bar{z}^{j_{q}}\in A^{p,q}(X_{0}),\nonumber\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FC}_{i_{1}\cdots i_{p}\overline{j_{1}}\cdots \overline{j_{q}}}\,dz^{i_{1}}\wedge \cdots \wedge \,dz^{i_{p}}\wedge \,d\bar{z}^{j_{1}}\wedge \cdots \wedge \,d\bar{z}^{j_{q}}$ . Together with (8) and (9), Proposition 2.3 implies that

(10) $$\begin{eqnarray}\displaystyle d(e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FA})) & = & \displaystyle d\circ e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}}\circ e^{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}}(\tilde{\unicode[STIX]{x1D6FA}})\nonumber\\ \displaystyle & = & \displaystyle e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}}\circ (\bar{\unicode[STIX]{x2202}}+[\unicode[STIX]{x2202},\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}]+\unicode[STIX]{x2202})\circ e^{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}}(\tilde{\unicode[STIX]{x1D6FA}})\nonumber\\ \displaystyle & = & \displaystyle e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}}(\bar{\unicode[STIX]{x2202}}_{\unicode[STIX]{x1D711}}+\unicode[STIX]{x2202})\mathop{\sum }_{k=0}^{+\infty }A_{k}\nonumber\\ \displaystyle & = & \displaystyle e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}}\biggl(\bar{\unicode[STIX]{x2202}}_{\unicode[STIX]{x1D711}}A_{0}+\mathop{\sum }_{k=0}^{+\infty }(\unicode[STIX]{x2202}A_{k}+\bar{\unicode[STIX]{x2202}}_{\unicode[STIX]{x1D711}}A_{k+1})\biggr),\end{eqnarray}$$

where

$$\begin{eqnarray}A_{k}:=\frac{(\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}})^{k}}{k!}(\tilde{\unicode[STIX]{x1D6FA}})\end{eqnarray}$$

is a $(p+k,q-k)$ -form and

$$\begin{eqnarray}\bar{\unicode[STIX]{x2202}}_{\unicode[STIX]{x1D711}}:=\bar{\unicode[STIX]{x2202}}+[\unicode[STIX]{x2202},\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}].\end{eqnarray}$$

Thus, $d(e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FA}))=0$ amounts to

$$\begin{eqnarray}\bar{\unicode[STIX]{x2202}}_{\unicode[STIX]{x1D711}}A_{0}=0,\quad \unicode[STIX]{x2202}A_{k}+\bar{\unicode[STIX]{x2202}}_{\unicode[STIX]{x1D711}}A_{k+1}=0,\quad k=0,1,2,\ldots .\end{eqnarray}$$

Proposition 4.2. For any given $(p,q)$ -form $\unicode[STIX]{x1D6FA}$ ,

$$\begin{eqnarray}d(e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FA}))=0\end{eqnarray}$$

is equivalent to

(11) $$\begin{eqnarray}\mathop{\sum }_{k=0}^{\infty }\biggl(\bar{\unicode[STIX]{x2202}}\circ \frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{k-1}}{(k-1)!}+\unicode[STIX]{x2202}\circ \frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{k}}{k!}\biggr)A_{n-p-(l-k)}=0,\end{eqnarray}$$

where $\max \{1,n-p-q\}\leqslant l\leqslant \min \{2n-p-q,n+1\}$ , $\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{k}=0$ for $k<0$ and $0!=1$ .

Proof. Recall that $\tilde{\unicode[STIX]{x1D6FA}}=e^{-\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}}\circ e^{-\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}}\circ e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FA})$ and then

$$\begin{eqnarray}\displaystyle d(e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FA})) & = & \displaystyle d\circ e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}}\circ e^{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}}(\tilde{\unicode[STIX]{x1D6FA}})\nonumber\\ \displaystyle & = & \displaystyle (\bar{\unicode[STIX]{x2202}}\circ e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}}\circ e^{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}}+\unicode[STIX]{x2202}\circ e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}}\circ e^{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}})(\tilde{\unicode[STIX]{x1D6FA}})\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{k_{1},k_{2}=0}^{+\infty }\biggl(\bar{\unicode[STIX]{x2202}}\circ \frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{k_{1}}}{k_{1}!}+\unicode[STIX]{x2202}\circ \frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{k_{1}}}{k_{1}!}\biggr)\circ \frac{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}^{k_{2}}}{k_{2}!}(\tilde{\unicode[STIX]{x1D6FA}}).\nonumber\end{eqnarray}$$

Note that the part of degree $(+(n-p-l+1),-(n-p-l))$ in the operator $d\circ e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}}\circ e^{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}}$ is

(12) $$\begin{eqnarray}\mathop{\sum }_{k=0}^{\infty }\biggl(\bar{\unicode[STIX]{x2202}}\circ \frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{k-1}}{(k-1)!}+\unicode[STIX]{x2202}\circ \frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{k}}{k!}\biggr)\circ \frac{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}^{n-p-l+k}}{(n-p-l+k)!}(\tilde{\unicode[STIX]{x1D6FA}})\end{eqnarray}$$

since

$$\begin{eqnarray}\displaystyle (+(n-p-l+1),-(n-p-l)) & = & \displaystyle (n-p-l+k)(1,-1)+(k-1)(-1,1)+(0,1)\nonumber\\ \displaystyle & = & \displaystyle (n-p-l+k)(1,-1)+k(-1,1)+(1,0).\nonumber\end{eqnarray}$$

This is exactly the left-hand side of (11). So $d(e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FA}))=0$ is equivalent to the vanishing of (12) for each $l$ such that

$$\begin{eqnarray}(p,q)+(+(n-p-l+1),-(n-p-l))\in [0,n]\times [0,n],\end{eqnarray}$$

i.e., $\max \{1,n-p-q\}\leqslant l\leqslant \min \{2n-p-q,n+1\}$ .◻

Unfortunately, the system (11) of obstruction equations consists of too many equations, and is difficult to solve. We try to reduce it to one with only two equations as in Proposition 4.5.

We will use the homogenous notation for a power series here and henceforth. Assuming that $\unicode[STIX]{x1D6FC}(t)$ is a power series of (bundle-valued) $(p,q)$ -forms, expanded as

$$\begin{eqnarray}\unicode[STIX]{x1D6FC}(t)=\mathop{\sum }_{k=0}^{\infty }\mathop{\sum }_{i+j=k}\unicode[STIX]{x1D6FC}_{i,j}t^{i}\bar{t}^{j},\end{eqnarray}$$

we use the notation

$$\begin{eqnarray}\left\{\begin{array}{@{}l@{}}\displaystyle \unicode[STIX]{x1D6FC}(t)=\mathop{\sum }_{k=0}^{\infty }\unicode[STIX]{x1D6FC}_{k},\quad \\ \displaystyle \unicode[STIX]{x1D6FC}_{k}=\mathop{\sum }_{i+j=k}\unicode[STIX]{x1D6FC}_{i,j}t^{i}\overline{t}^{j},\quad \end{array}\right.\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}_{k}$ is the $k$ -order homogeneous part in the expansion of $\unicode[STIX]{x1D6FC}(t)$ and all $\unicode[STIX]{x1D6FC}_{i,j}$ are smooth (bundle-valued) $(p,q)$ -forms on $X_{0}$ with $\unicode[STIX]{x1D6FC}(0)=\unicode[STIX]{x1D6FC}_{0,0}$ .

Lemma 4.4. If $d(e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FA}))_{N_{1}}=0$ for any $N_{1}\leqslant N$ , then

$$\begin{eqnarray}(\bar{\unicode[STIX]{x2202}}_{\unicode[STIX]{x1D711}}A_{0})_{N_{1}}=0,\quad (\unicode[STIX]{x2202}A_{k}+\bar{\unicode[STIX]{x2202}}_{\unicode[STIX]{x1D711}}A_{k+1})_{N_{1}}=0,\quad k=0,1,2,\ldots\end{eqnarray}$$

for any $N_{1}\leqslant N$ .

Proof. From (10), it follows that

$$\begin{eqnarray}e^{-\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}}d(e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FA}))=\bar{\unicode[STIX]{x2202}}_{\unicode[STIX]{x1D711}}A_{0}+\mathop{\sum }_{k=0}^{+\infty }(\unicode[STIX]{x2202}A_{k}+\bar{\unicode[STIX]{x2202}}_{\unicode[STIX]{x1D711}}A_{k+1}).\end{eqnarray}$$

For any $N_{1}\leqslant N$ ,

$$\begin{eqnarray}0=(e^{-\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}}d(e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FA})))_{N_{1}}=(\bar{\unicode[STIX]{x2202}}_{\unicode[STIX]{x1D711}}A_{0})_{N_{1}}+\mathop{\sum }_{k=0}^{+\infty }(\unicode[STIX]{x2202}A_{k}+\bar{\unicode[STIX]{x2202}}_{\unicode[STIX]{x1D711}}A_{k+1})_{N_{1}}.\end{eqnarray}$$

By comparing degrees, we complete the proof. ◻

As for (10), one can also have

$$\begin{eqnarray}d(e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FC}))=e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}\circ (e^{-\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|-\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}\circ e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}}\circ ([\unicode[STIX]{x2202},\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}]+\bar{\unicode[STIX]{x2202}}+\unicode[STIX]{x2202})\circ e^{-\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}}\circ e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FC})).\end{eqnarray}$$

A long local calculation shows that

(13)

Here we use the notation , first introduced in[Reference Rao, Wan and ZhaoRWZ16, §2.1], to denote the simultaneous contraction on each component of a complex differential form. For example,  means that the operator $(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711}+\bar{\unicode[STIX]{x1D711}})$ acts on $\unicode[STIX]{x1D6FC}$ simultaneously as

if $\unicode[STIX]{x1D6FC}$ is locally expressed by

$$\begin{eqnarray}\unicode[STIX]{x1D6FC}=f_{i_{1}\cdots i_{p}\overline{j_{1}}\cdots \overline{j_{q}}}dz^{i_{1}}\wedge \cdots \wedge dz^{i_{p}}\wedge d\bar{z}^{j_{1}}\wedge \cdots \wedge d\bar{z}^{j_{q}}.\end{eqnarray}$$

This new simultaneous contraction is well defined since $\unicode[STIX]{x1D711}(t)$ is a global $(1,0)$ -vector valued $(0,1)$ -form on $X_{0}$ (see [Reference Morrow and KodairaMK71, pp. 150–151]) as reasoned in [Reference Rao and ZhaoRZ18, Proof of Lemma 2.8]. Moreover, we know that

Thus, by carefully comparing the types of forms in both sides of (13), we have

(14)

See [Reference Rao and ZhaoRZ18, Proposition 2.13] and [Reference Rao, Wan and ZhaoRWZ16, (2.14)] for more details of (14).

Here and henceforth we denote by $(\unicode[STIX]{x1D6FC})^{p,q}$ the $(p,q)$ -type part of a $(p+q)$ -degree complex differential form $\unicode[STIX]{x1D6FC}$ .

Proposition 4.5. The obstruction equation $d(e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FA}))=0$ is also equivalent to

(15) $$\begin{eqnarray}\left\{\begin{array}{@{}l@{}}\displaystyle \mathop{\sum }_{k=0}^{\infty }\biggl(\bar{\unicode[STIX]{x2202}}\circ \frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{k-1}}{(k-1)!}+\unicode[STIX]{x2202}\circ \frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{k}}{k!}\biggr)\circ \frac{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}^{k}}{k!}(\tilde{\unicode[STIX]{x1D6FA}})=0,\quad \\ \displaystyle \mathop{\sum }_{k=0}^{\infty }\biggl(\bar{\unicode[STIX]{x2202}}\circ \frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{k-1}}{(k-1)!}+\unicode[STIX]{x2202}\circ \frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{k}}{k!}\biggr)\circ \frac{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}^{k-1}}{(k-1)!}(\tilde{\unicode[STIX]{x1D6FA}})=0.\quad \end{array}\right.\end{eqnarray}$$

Proof. From the proof of Proposition 4.2, it is easy to see that the left-hand side of the first equation in (15) is $(d(e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FA})))^{p+1,q}$ , while the other one is $(d(e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FA})))^{p,q+1}$ . Thus, (15) holds if $d(e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FA}))=0$ .

Conversely, we assume that (15) holds. By (14) and (10), we compare types of forms to get

(16)

Similarly, we get

where $\overline{\bullet }$ in the last term means that $\unicode[STIX]{x2202},\bar{\unicode[STIX]{x2202}},\unicode[STIX]{x1D711},\bar{\unicode[STIX]{x1D711}}$ are replaced by $\bar{\unicode[STIX]{x2202}},\unicode[STIX]{x2202},\bar{\unicode[STIX]{x1D711}},\unicode[STIX]{x1D711}$ , respectively, while $\unicode[STIX]{x1D6FA}$ takes no conjugation.

If (15) holds, i.e., $(d(e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FA})))^{p+1,q}=0$ and $(d(e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FA})))^{p,q+1}=0$ , we will prove

$$\begin{eqnarray}d(e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FA}))=0\end{eqnarray}$$

by induction on orders. Obviously,

$$\begin{eqnarray}d(e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FA}))_{0}=d\unicode[STIX]{x1D6FA}_{0}=0.\end{eqnarray}$$

Now we assume that for any $N_{1}\leqslant N$ , $d(e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FA}))_{N_{1}}=0$ . By Lemma 4.4, we have

$$\begin{eqnarray}(\bar{\unicode[STIX]{x2202}}_{\unicode[STIX]{x1D711}}A_{0})_{N_{1}}=0,\quad (\unicode[STIX]{x2202}A_{k}+\bar{\unicode[STIX]{x2202}}_{\unicode[STIX]{x1D711}}A_{k+1})_{N_{1}}=0,\quad k=0,1,2,\ldots\end{eqnarray}$$

for any $N_{1}\leqslant N$ . For the $(N+1)$ th order, (16) and the induction assumption for any $N_{1}\leqslant N$ imply

Similarly, we have $(\unicode[STIX]{x2202}_{t}e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FA}))_{N+1}=0$ . So

$$\begin{eqnarray}d(e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FA}))_{N+1}=(\bar{\unicode[STIX]{x2202}}_{t}e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FA}))_{N+1}+(\unicode[STIX]{x2202}_{t}e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FA}))_{N+1}=0.\end{eqnarray}$$

Thus, we complete the proof. ◻

Remark 4.6. For $p=q=1$ , if solving $de^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FA})=0$ for the orders ${\leqslant}N$ , we proceed to the $(N+1)$ th order. It is necessary to prove

$$\begin{eqnarray}\bar{\unicode[STIX]{x2202}}(\unicode[STIX]{x1D711}\lrcorner \unicode[STIX]{x1D6FA})_{N+1}=(\bar{\unicode[STIX]{x2202}}e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FA}))_{N+1}^{0,3}=(de^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FA}))_{N+1}^{0,3}=0.\end{eqnarray}$$

From (10), we have

$$\begin{eqnarray}(de^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FA}))_{N+1}^{0,3}=\mathop{\biggl(\mathop{\sum }_{k=-1}^{+\infty }\frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{k+2}}{(k+2)!}(\unicode[STIX]{x2202}A_{k}+\bar{\unicode[STIX]{x2202}}_{\unicode[STIX]{x1D711}}A_{k+1})\biggr)}\nolimits_{N+1}=\biggl(\frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{3}}{3!}\circ \unicode[STIX]{x2202}(\bar{\unicode[STIX]{x1D711}}\lrcorner \unicode[STIX]{x1D6FA})\biggr)_{N+1}=0,\end{eqnarray}$$

where the last equality follows from Lemma 4.4.

Now we begin to solve (15) with two more lemmas. For the resolution of $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$ -equations, we need a lemma due to [Reference PopoviciPop15, Theorem 4.1] (or [Reference Rao, Wan and ZhaoRWZ16, Lemma 3.14]).

Lemma 4.7. Let $(X,\unicode[STIX]{x1D714})$ be a compact Hermitian complex manifold with the pure-type complex differential forms $x$ and $y$ . Assume that the $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$ -equation

(17) $$\begin{eqnarray}\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}x=y\end{eqnarray}$$

admits a solution. Then an explicit solution of the $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$ -equation (17) can be chosen as

$$\begin{eqnarray}(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})^{\ast }\mathbb{G}_{\text{BC}}y,\end{eqnarray}$$

which uniquely minimizes the $L^{2}$ -norms of all the solutions with respect to $\unicode[STIX]{x1D714}$ . Besides, the equalities

$$\begin{eqnarray}\mathbb{G}_{\text{BC}}(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})=(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})\mathbb{G}_{\text{A}}\quad \text{and}\quad (\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})^{\ast }\mathbb{G}_{\text{BC}}=\mathbb{G}_{\text{A}}(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})^{\ast }\end{eqnarray}$$

hold, where $\mathbb{G}_{\text{BC}}$ and $\mathbb{G}_{\text{A}}$ are the associated Green’s operators of $\Box _{\text{BC}}$ and $\Box _{\text{A}}$ , respectively. Here $\Box _{\text{BC}}$ is defined in (7) and $\Box _{\text{A}}$ is the second Kodaira–Spencer operator (often also called Aeppli Laplacian)

$$\begin{eqnarray}\Box _{\text{A}}=\unicode[STIX]{x2202}^{\ast }\overline{\unicode[STIX]{x2202}}^{\ast }\overline{\unicode[STIX]{x2202}}\unicode[STIX]{x2202}+\overline{\unicode[STIX]{x2202}}\unicode[STIX]{x2202}\unicode[STIX]{x2202}^{\ast }\overline{\unicode[STIX]{x2202}}^{\ast }+\overline{\unicode[STIX]{x2202}}\unicode[STIX]{x2202}^{\ast }\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}^{\ast }+\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}^{\ast }\overline{\unicode[STIX]{x2202}}\unicode[STIX]{x2202}^{\ast }+\overline{\unicode[STIX]{x2202}}\overline{\unicode[STIX]{x2202}}^{\ast }+\unicode[STIX]{x2202}\unicode[STIX]{x2202}^{\ast }.\end{eqnarray}$$

Sketch of Proof for Lemma 4.7.

We shall use the Hodge decomposition of $\Box _{\text{BC}}$ on $X$ ,

$$\begin{eqnarray}A^{p,q}(X)=\ker \Box _{\text{BC}}\oplus \operatorname{Im}(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})\oplus (\operatorname{Im}\unicode[STIX]{x2202}^{\ast }+\operatorname{Im}\overline{\unicode[STIX]{x2202}}^{\ast }),\end{eqnarray}$$

whose three parts are orthogonal to each other with respect to the $L^{2}$ -scalar product defined by $\unicode[STIX]{x1D714}$ , combined with the equality

$$\begin{eqnarray}\unicode[STIX]{x1D7D9}=\mathbb{H}_{\text{BC}}+\Box _{\text{BC}}\mathbb{G}_{\text{BC}},\end{eqnarray}$$

where $\mathbb{H}_{\text{BC}}$ is the harmonic projection operator. Then two observations follow:

1. (1) $\Box _{\text{BC}}\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})^{\ast }=\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})^{\ast }\Box _{\text{BC}}$ ;

2. (2) $\mathbb{G}_{\text{BC}}\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})^{\ast }=\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})^{\ast }\mathbb{G}_{\text{BC}}$ .

It is clear that (1) implies (2), while the statement (1) is proved by a direct calculation,

$$\begin{eqnarray}\Box _{\text{BC}}\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})^{\ast }=(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})^{\ast }(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})^{\ast }=\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})^{\ast }\Box _{\text{BC}}.\end{eqnarray}$$

Hence, we have

$$\begin{eqnarray}(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})^{\ast }\mathbb{G}_{\text{BC}}y=\mathbb{G}_{\text{BC}}(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})^{\ast }y=\mathbb{G}_{\text{BC}}\Box _{\text{BC}}y=(\unicode[STIX]{x1D7D9}-\mathbb{H}_{\text{BC}})y=y,\end{eqnarray}$$

where $y\in \operatorname{Im}\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$ due to the solution-existence of the $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$ -equation.

To see that the solution $(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})^{\ast }\mathbb{G}_{\text{BC}}y$ is the unique $L^{2}$ -norm minimum, we resort to the Hodge decomposition of the operator $\Box _{\text{A}}$ ,

(18) $$\begin{eqnarray}A^{p,q}(X)=\ker \Box _{\text{A}}\oplus (\operatorname{Im}\unicode[STIX]{x2202}+\operatorname{Im}\overline{\unicode[STIX]{x2202}})\oplus \operatorname{Im}(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})^{\ast },\end{eqnarray}$$

where $\ker \Box _{\text{A}}=\ker (\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})\cap \ker \unicode[STIX]{x2202}^{\ast }\cap \ker \overline{\unicode[STIX]{x2202}}^{\ast }$ . Let $z$ be an arbitrary solution of the $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$ -equation (17), which decomposes into three components $z_{1}+z_{2}+z_{3}$ with respect to the Hodge decomposition (18) of $\Box _{\text{A}}$ . And we are able to obtain that

$$\begin{eqnarray}z_{3}=\mathbb{G}_{\text{A}}(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})^{\ast }y=(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})^{\ast }\mathbb{G}_{\text{BC}}y.\end{eqnarray}$$

Therefore,

$$\begin{eqnarray}\Vert z\Vert ^{2}=\Vert z_{1}\Vert ^{2}+\Vert z_{2}\Vert ^{2}+\Vert z_{3}\Vert ^{2}\geqslant \Vert z_{3}\Vert ^{2}=\Vert (\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})^{\ast }\mathbb{G}_{\text{BC}}y\Vert ^{2},\end{eqnarray}$$

and the equality holds if and only if $z_{1}=z_{2}=0$ , i.e., $z=z_{3}=(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})^{\ast }\mathbb{G}_{\text{BC}}y$ .◻

Lemma 4.8. Let $X$ be a complex manifold satisfying the $(p,q+1)$ th and $(q,p+1)$ th mild $\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}$ -lemmata. Consider the system of equations

(19) $$\begin{eqnarray}\left\{\begin{array}{@{}l@{}}\unicode[STIX]{x2202}x=\overline{\unicode[STIX]{x2202}}\unicode[STIX]{x1D701},\quad \\ \overline{\unicode[STIX]{x2202}}x=\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D709}},\quad \end{array}\right.\end{eqnarray}$$

where $\unicode[STIX]{x1D701},\unicode[STIX]{x1D709}$ are $(p+1,q-1)$ - and $(q+1,p-1)$ -forms on $X$ , respectively. The system of equations (19) has a solution if and only if

$$\begin{eqnarray}\left\{\begin{array}{@{}l@{}}\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}\unicode[STIX]{x1D701}=0,\quad \\ \overline{\unicode[STIX]{x2202}}\unicode[STIX]{x2202}\overline{\unicode[STIX]{x1D709}}=0.\quad \end{array}\right.\end{eqnarray}$$

Proof. This lemma is inspired by [Reference Rao, Wan and ZhaoRWZ16, Observation 2.11]. The lemmata assumption will produce $\unicode[STIX]{x1D707}\in A^{p,q-1}$ and $\unicode[STIX]{x1D708}\in A^{p-1,q}$ , satisfying the system of equations

$$\begin{eqnarray}\left\{\begin{array}{@{}l@{}}\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}\unicode[STIX]{x1D707}=\overline{\unicode[STIX]{x2202}}\unicode[STIX]{x1D701},\quad \\ \overline{\unicode[STIX]{x2202}}\unicode[STIX]{x2202}\unicode[STIX]{x1D708}=\unicode[STIX]{x2202}\bar{\unicode[STIX]{x1D709}}.\quad \end{array}\right.\end{eqnarray}$$

The combined expression

$$\begin{eqnarray}\overline{\unicode[STIX]{x2202}}\unicode[STIX]{x1D707}+\unicode[STIX]{x2202}\unicode[STIX]{x1D708}\end{eqnarray}$$

is our choice for the solution of the system (19). ◻

By Lemmata 4.7 and 4.8, we have the following proposition.

Proposition 4.9. With the same notation as in Lemmata 4.7 and 4.8, the system of equations (19) has a canonical solution

$$\begin{eqnarray}x=\overline{\unicode[STIX]{x2202}}(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})^{\ast }\mathbb{G}_{\text{BC}}\overline{\unicode[STIX]{x2202}}\unicode[STIX]{x1D701}-\unicode[STIX]{x2202}(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})^{\ast }\mathbb{G}_{\text{BC}}\unicode[STIX]{x2202}\bar{\unicode[STIX]{x1D709}}.\end{eqnarray}$$

Now we assume that $X_{0}$ is a complex manifold that satisfies the $(q,p+1)$ th and $(p,q+1)$ th mild $\unicode[STIX]{x2202}\bar{\unicode[STIX]{x2202}}$ -lemmata. The obstruction (15) can be rewritten as

(20) $$\begin{eqnarray}\left\{\begin{array}{@{}l@{}}\displaystyle \unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D6FA}}+\mathop{\sum }_{k=1}^{\infty }\biggl(\bar{\unicode[STIX]{x2202}}\circ \frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{k-1}}{(k-1)!}+\unicode[STIX]{x2202}\circ \frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{k}}{k!}\biggr)\circ \frac{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}^{k}}{k!}(\tilde{\unicode[STIX]{x1D6FA}})=0,\quad \\ \displaystyle \bar{\unicode[STIX]{x2202}}\tilde{\unicode[STIX]{x1D6FA}}+\unicode[STIX]{x2202}\circ \unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}(\tilde{\unicode[STIX]{x1D6FA}})+\mathop{\sum }_{k=2}^{\infty }\biggl(\bar{\unicode[STIX]{x2202}}\circ \frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{k-1}}{(k-1)!}+\unicode[STIX]{x2202}\circ \frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{k}}{k!}\biggr)\circ \frac{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}^{k-1}}{(k-1)!}(\tilde{\unicode[STIX]{x1D6FA}})=0.\quad \end{array}\right.\end{eqnarray}$$

Set

$$\begin{eqnarray}\tilde{\unicode[STIX]{x1D6FA}}^{\prime }:=\tilde{\unicode[STIX]{x1D6FA}}+\mathop{\sum }_{k=1}^{\infty }\frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{k}}{k!}\circ \frac{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}^{k}}{k!}(\tilde{\unicode[STIX]{x1D6FA}}).\end{eqnarray}$$

Then (20) becomes

$$\begin{eqnarray}\left\{\begin{array}{@{}l@{}}\displaystyle \unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D6FA}}^{\prime }=-\bar{\unicode[STIX]{x2202}}\mathop{\sum }_{k=1}^{\infty }\frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{k-1}}{(k-1)!}\circ \frac{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}^{k}}{k!}(\tilde{\unicode[STIX]{x1D6FA}}),\quad \\ \displaystyle \bar{\unicode[STIX]{x2202}}\tilde{\unicode[STIX]{x1D6FA}}^{\prime }=-\unicode[STIX]{x2202}\mathop{\sum }_{k=0}^{\infty }\frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{k+1}}{(k+1)!}\circ \frac{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}^{k}}{k!}(\tilde{\unicode[STIX]{x1D6FA}}).\quad \end{array}\right.\end{eqnarray}$$

In order to use the $(q,p+1)$ th and $(p,q+1)$ th mild $\unicode[STIX]{x2202}\bar{\unicode[STIX]{x2202}}$ -lemmata, we need to prove

(21) $$\begin{eqnarray}\left\{\begin{array}{@{}l@{}}\displaystyle \unicode[STIX]{x2202}\bar{\unicode[STIX]{x2202}}\mathop{\sum }_{k=1}^{\infty }\frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{k-1}}{(k-1)!}\circ \frac{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}^{k}}{k!}(\tilde{\unicode[STIX]{x1D6FA}})=0,\quad \\ \displaystyle \unicode[STIX]{x2202}\bar{\unicode[STIX]{x2202}}\mathop{\sum }_{k=0}^{\infty }\frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{k+1}}{(k+1)!}\circ \frac{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}^{k}}{k!}(\tilde{\unicode[STIX]{x1D6FA}})=0,\quad \end{array}\right.\end{eqnarray}$$

at the $(N+1)$ th order if it has been solved for the orders ${\leqslant}N$ .

Now we prove (21). Firstly, note that

$$\begin{eqnarray}\mathop{\sum }_{k=0}^{\infty }\frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{k+1}}{(k+1)!}\circ \frac{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}^{k}}{k!}(\tilde{\unicode[STIX]{x1D6FA}})=(e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}}\circ e^{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}}(\tilde{\unicode[STIX]{x1D6FA}}))^{p-1,q+1}=(e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FA}))^{p-1,q+1}.\end{eqnarray}$$

Thus,

$$\begin{eqnarray}\displaystyle \mathop{\biggl(\unicode[STIX]{x2202}\bar{\unicode[STIX]{x2202}}\mathop{\sum }_{k=0}^{\infty }\frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{k+1}}{(k+1)!}\circ \frac{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}^{k}}{k!}(\tilde{\unicode[STIX]{x1D6FA}})\biggr)}\nolimits_{N+1} & = & \displaystyle \unicode[STIX]{x2202}\bar{\unicode[STIX]{x2202}}((e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FA}))^{p-1,q+1})_{N+1}\nonumber\\ \displaystyle & = & \displaystyle (\unicode[STIX]{x2202}\bar{\unicode[STIX]{x2202}}e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FA}))_{N+1}^{p,q+2}\nonumber\\ \displaystyle & = & \displaystyle (\unicode[STIX]{x2202}de^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FA}))_{N+1}^{p,q+2}\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x2202}(de^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FA}))_{N+1}^{p-1,q+2}\nonumber\end{eqnarray}$$

since the obstruction equation is solved for the orders ${\leqslant}N$ , i.e., $d(e^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FA}))_{N_{1}}=0$ for any $N_{1}\leqslant N$ . By Lemma 4.4, we have

$$\begin{eqnarray}(\bar{\unicode[STIX]{x2202}}_{\unicode[STIX]{x1D711}}A_{0})_{N_{1}}=0,\quad (\unicode[STIX]{x2202}A_{k}+\bar{\unicode[STIX]{x2202}}_{\unicode[STIX]{x1D711}}A_{k+1})_{N_{1}}=0,\quad k=0,1,2,\ldots\end{eqnarray}$$

for any $N_{1}\leqslant N$ . It follows from (10) that

$$\begin{eqnarray}(de^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FA}))_{N+1}^{p-1,q+2}=\mathop{\biggl(\mathop{\sum }_{k=-1}^{+\infty }\frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{k+2}}{(k+2)!}(\unicode[STIX]{x2202}A_{k}+\bar{\unicode[STIX]{x2202}}_{\unicode[STIX]{x1D711}}A_{k+1})\biggr)}\nolimits_{N+1}=0.\end{eqnarray}$$

So

$$\begin{eqnarray}\mathop{\biggl(\unicode[STIX]{x2202}\bar{\unicode[STIX]{x2202}}\mathop{\sum }_{k=0}^{\infty }\frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{k+1}}{(k+1)!}\circ \frac{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}^{k}}{k!}(\tilde{\unicode[STIX]{x1D6FA}})\biggr)}\nolimits_{N+1}=\unicode[STIX]{x2202}(de^{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}|\unicode[STIX]{x1D704}_{\bar{\unicode[STIX]{x1D711}}}}(\unicode[STIX]{x1D6FA}))_{N+1}^{p-1,q+2}=0.\end{eqnarray}$$

Hence, we have proved the second equation of (21). Similarly, by the same argument (i.e., replace all $\unicode[STIX]{x1D711}$ (respectively, $\bar{\unicode[STIX]{x1D711}}$ ) by $\bar{\unicode[STIX]{x1D711}}$ (respectively, $\unicode[STIX]{x1D711}$ )), the first equation of (21) also holds.

By Proposition 4.9, one obtains a formal solution of (20) by induction

(22) $$\begin{eqnarray}\displaystyle \tilde{\unicode[STIX]{x1D6FA}}_{l} & = & \displaystyle -\mathop{\biggl(\mathop{\sum }_{k=1}^{\infty }\frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{k}}{k!}\circ \frac{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}^{k}}{k!}(\tilde{\unicode[STIX]{x1D6FA}})\biggr)}\nolimits_{l}-\overline{\unicode[STIX]{x2202}}(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})^{\ast }\mathbb{G}_{\text{BC}}\overline{\unicode[STIX]{x2202}}\mathop{\biggl(\mathop{\sum }_{k=1}^{\infty }\frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{k-1}}{(k-1)!}\circ \frac{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}^{k}}{k!}(\tilde{\unicode[STIX]{x1D6FA}})\biggr)}\nolimits_{l}\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x2202}(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})^{\ast }\mathbb{G}_{\text{BC}}\unicode[STIX]{x2202}\mathop{\biggl(\mathop{\sum }_{k=0}^{\infty }\frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{k+1}}{(k+1)!}\circ \frac{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}^{k}}{k!}(\tilde{\unicode[STIX]{x1D6FA}})\biggr)}\nolimits_{l}\nonumber\\ \displaystyle & = & \displaystyle -\mathop{\biggl(\mathop{\sum }_{k=1}^{\min \{q,n-p\}}\frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{k}}{k!}\circ \frac{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}^{k}}{k!}(\tilde{\unicode[STIX]{x1D6FA}})\biggr)}\nolimits_{l}-\overline{\unicode[STIX]{x2202}}(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})^{\ast }\mathbb{G}_{\text{BC}}\overline{\unicode[STIX]{x2202}}\mathop{\biggl(\mathop{\sum }_{k=1}^{\min \{q,n-p\}}\frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{k-1}}{(k-1)!}\circ \frac{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}^{k}}{k!}(\tilde{\unicode[STIX]{x1D6FA}})\biggr)}\nolimits_{l}\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x2202}(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})^{\ast }\mathbb{G}_{\text{BC}}\unicode[STIX]{x2202}\mathop{\biggl(\mathop{\sum }_{k=0}^{\min \{q,n-p\}}\frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{k+1}}{(k+1)!}\circ \frac{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}^{k}}{k!}(\tilde{\unicode[STIX]{x1D6FA}})\biggr)}\nolimits_{l}.\end{eqnarray}$$

4.3 Regularity argument

Here we adopt a strategy for a convergence argument [Reference Liu and ZhuLZ18] suggested by Liu, which simplifies our argument involved in [Reference Rao, Wan and ZhaoRWZ16, Reference Rao and ZhaoRZ18].

From the induction expression (22), one obtains the formal expression of $\tilde{\unicode[STIX]{x1D6FA}}$

(23) $$\begin{eqnarray}\displaystyle \tilde{\unicode[STIX]{x1D6FA}} & = & \displaystyle -\mathop{\sum }_{i=1}^{\min \{q,n-p\}}\frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{i}}{i!}\circ \frac{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}^{i}}{i!}(\tilde{\unicode[STIX]{x1D6FA}})-\overline{\unicode[STIX]{x2202}}(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})^{\ast }\mathbb{G}_{\text{BC}}\overline{\unicode[STIX]{x2202}}\mathop{\sum }_{i=1}^{\min \{q,n-p\}}\frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{i-1}}{(i-1)!}\circ \frac{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}^{i}}{i!}(\tilde{\unicode[STIX]{x1D6FA}})\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x2202}(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})^{\ast }\mathbb{G}_{\text{BC}}\unicode[STIX]{x2202}\mathop{\sum }_{i=0}^{\min \{q,n-p\}}\frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{i+1}}{(i+1)!}\circ \frac{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}^{i}}{i!}(\tilde{\unicode[STIX]{x1D6FA}})+\unicode[STIX]{x1D6FA}_{0}.\end{eqnarray}$$

Set

$$\begin{eqnarray}\displaystyle F & = & \displaystyle \unicode[STIX]{x2202}(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})^{\ast }\mathbb{G}_{\text{BC}}\unicode[STIX]{x2202}\mathop{\sum }_{i=0}^{\min \{q,n-p\}}\frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{i+1}}{(i+1)!}\circ \frac{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}^{i}}{i!}\nonumber\\ \displaystyle & & \displaystyle -\,\mathop{\sum }_{i=1}^{\min \{q,n-p\}}\frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{i}}{i!}\circ \frac{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}^{i}}{i!}\nonumber\\ \displaystyle & & \displaystyle -\,\overline{\unicode[STIX]{x2202}}(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})^{\ast }\mathbb{G}_{\text{BC}}\overline{\unicode[STIX]{x2202}}\mathop{\sum }_{i=1}^{\min \{q,n-p\}}\frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{i-1}}{(i-1)!}\circ \frac{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}^{i}}{i!}\nonumber\end{eqnarray}$$

and write

(24) $$\begin{eqnarray}\unicode[STIX]{x1D6FA}_{0}=(\unicode[STIX]{x1D7D9}-F)\tilde{\unicode[STIX]{x1D6FA}}.\end{eqnarray}$$

We claim that $\tilde{\unicode[STIX]{x1D6FA}}(t)$ converges in Hölder norm as $t\rightarrow 0$ by use of the following two a priori elliptic estimates: for any complex differential form $\unicode[STIX]{x1D719}$ ,

$$\begin{eqnarray}\Vert \overline{\unicode[STIX]{x2202}}^{\ast }\unicode[STIX]{x1D719}\Vert _{k-1,\unicode[STIX]{x1D6FC}}\leqslant C_{1}\Vert \unicode[STIX]{x1D719}\Vert _{k,\unicode[STIX]{x1D6FC}}\end{eqnarray}$$

and

$$\begin{eqnarray}\Vert \mathbb{G}_{\text{BC}}\unicode[STIX]{x1D719}\Vert _{k,\unicode[STIX]{x1D6FC}}\leqslant C_{k,\unicode[STIX]{x1D6FC}}\Vert \unicode[STIX]{x1D719}\Vert _{k-4,\unicode[STIX]{x1D6FC}},\end{eqnarray}$$

where $k>3$ and $C_{k,\unicode[STIX]{x1D6FC}}$ depends only on $k$ and $\unicode[STIX]{x1D6FC}$ , not on $\unicode[STIX]{x1D719}$ (cf. [Reference KodairaKod86, Appendix.Theorem 7.4] for example). And we note that $\unicode[STIX]{x1D711}(t)$ converges smoothly to zero as $t\rightarrow 0$ . Thus, by (24), we estimate

$$\begin{eqnarray}\Vert \unicode[STIX]{x1D6FA}_{0}\Vert _{k,\unicode[STIX]{x1D6FC}}\geqslant (1-\unicode[STIX]{x1D716}_{k,\unicode[STIX]{x1D6FC}})\Vert \tilde{\unicode[STIX]{x1D6FA}}\Vert _{k,\unicode[STIX]{x1D6FC}},\end{eqnarray}$$

where $0<\unicode[STIX]{x1D716}_{k,\unicode[STIX]{x1D6FC}}\ll 1$ is some constant depending on $k,\unicode[STIX]{x1D6FC}$ .

Finally, we proceed to the regularity of $\tilde{\unicode[STIX]{x1D6FA}}(t)$ since there is possibly no uniform lower bound for the convergence radius obtained as above in the $C^{k,\unicode[STIX]{x1D6FC}}$ -norm when $k$ converges to $+\infty$ . This argument lies heavily in the elliptic estimates [Reference KodairaKod86, Appendix 8], [Reference Douglis and NirenbergDN55] and also [Reference Rao, Wan and ZhaoRWZ16, §3.2].

Without loss of generality, we just consider the equation

$$\begin{eqnarray}\displaystyle \Box \tilde{\unicode[STIX]{x1D6FA}} & = & \displaystyle -\Box \mathop{\sum }_{k=1}^{\min \{q,n-p\}}\frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{k}}{k!}\circ \frac{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}^{k}}{k!}(\tilde{\unicode[STIX]{x1D6FA}})-\overline{\unicode[STIX]{x2202}}\overline{\unicode[STIX]{x2202}}^{\ast }\overline{\unicode[STIX]{x2202}}(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})^{\ast }\mathbb{G}_{\text{BC}}\overline{\unicode[STIX]{x2202}}\mathop{\sum }_{k=1}^{\min \{q,n-p\}}\frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{k-1}}{(k-1)!}\circ \frac{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}^{k}}{k!}(\tilde{\unicode[STIX]{x1D6FA}})\nonumber\\ \displaystyle & & \displaystyle +\,\Box \unicode[STIX]{x2202}(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})^{\ast }\mathbb{G}_{\text{BC}}\unicode[STIX]{x2202}\mathop{\sum }_{k=0}^{\min \{q,n-p\}}\frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{k+1}}{(k+1)!}\circ \frac{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}^{k}}{k!}(\tilde{\unicode[STIX]{x1D6FA}})\nonumber\end{eqnarray}$$

by applying the $\overline{\unicode[STIX]{x2202}}$ -Laplacian $\Box =\overline{\unicode[STIX]{x2202}}^{\ast }\overline{\unicode[STIX]{x2202}}+\overline{\unicode[STIX]{x2202}}\overline{\unicode[STIX]{x2202}}^{\ast }$ to the expression formula (23) and omitting the lower-order term $\Box \unicode[STIX]{x1D6FA}_{0}$ in this expression. By replacing the roles of

$$\begin{eqnarray}\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}\circ \unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}},\quad \unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}},\quad \unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}+{\textstyle \frac{1}{2}}\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}\circ \unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}\circ \unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}\end{eqnarray}$$

in the analogous strongly elliptic second-order pseudo-differential equation in the regularity argument of [Reference Rao, Wan and ZhaoRWZ16, §3.2]

$$\begin{eqnarray}\displaystyle \Box \tilde{\unicode[STIX]{x1D6FA}}(t) & = & \displaystyle -\Box (\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}\circ \unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}(\tilde{\unicode[STIX]{x1D6FA}}(t)))-\overline{\unicode[STIX]{x2202}}\overline{\unicode[STIX]{x2202}}^{\ast }\overline{\unicode[STIX]{x2202}}(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})^{\ast }\mathbb{G}_{\text{BC}}\overline{\unicode[STIX]{x2202}}(\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}(\tilde{\unicode[STIX]{x1D6FA}}(t)))\nonumber\\ \displaystyle & & \displaystyle +\,\Box \unicode[STIX]{x2202}(\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}})^{\ast }\mathbb{G}_{\text{BC}}\unicode[STIX]{x2202}((\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}+{\textstyle \frac{1}{2}}\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}\circ \unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}\circ \unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}})\tilde{\unicode[STIX]{x1D6FA}}(t)),\nonumber\end{eqnarray}$$

by

$$\begin{eqnarray}\mathop{\sum }_{i=1}^{\min \{q,n-p\}}\frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{i}}{i!}\circ \frac{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}^{i}}{i!},\quad \mathop{\sum }_{i=1}^{\min \{q,n-p\}}\frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{i-1}}{(i-1)!}\circ \frac{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}^{i}}{i!},\quad \mathop{\sum }_{i=0}^{\min \{q,n-p\}}\frac{\unicode[STIX]{x1D704}_{\unicode[STIX]{x1D711}}^{i+1}}{(i+1)!}\circ \frac{\unicode[STIX]{x1D704}_{(\unicode[STIX]{x1D7D9}-\bar{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D711})^{-1}\bar{\unicode[STIX]{x1D711}}}^{i}}{i!},\end{eqnarray}$$

respectively, we prove the following result. For each $l=1,2,\ldots \,$ , choose a smooth function $\unicode[STIX]{x1D702}^{l}(t)$ with values in $[0,1]$

$$\begin{eqnarray}\unicode[STIX]{x1D702}^{l}(t)\equiv \left\{\begin{array}{@{}l@{}}\displaystyle 1\quad \text{for }|t|\leqslant \biggl(\frac{1}{2}+\frac{1}{2^{l+1}}\biggr)r;\quad \\ \displaystyle 0\quad \text{for }|t|\geqslant \biggl(\frac{1}{2}+\frac{1}{2^{l}}\biggr)r,\quad \end{array}\right.\end{eqnarray}$$

where $r$ is a positive constant to be determined. Inductively, by Douglis–Nirenberg’s interior estimates [Reference KodairaKod86, Appendix.Theorem 2.3], [Reference Douglis and NirenbergDN55], for any $l=1,2,\ldots \,$ , $\unicode[STIX]{x1D702}^{2l+1}\tilde{\unicode[STIX]{x1D6FA}}(t)$ is $C^{k+l,\unicode[STIX]{x1D6FC}}$ , where $r$ can be chosen independent of $l$ . Since $\unicode[STIX]{x1D702}^{2l+1}(t)$ is identically equal to $1$ on $|t|<r/2$ , which is independent of $l$ , $\tilde{\unicode[STIX]{x1D6FA}}(t)$ is $C^{\infty }$ on $X_{0}$ with $|t|<r/2$ . Then $\tilde{\unicode[STIX]{x1D6FA}}(t)$ can be considered as a real analytic family of $(p,q)$ -forms in $t$ and thus is smooth on $t$ .