2.1 Obstruction equations

We will discuss the obstruction equation to extend the d-closed pure-type complex differential forms on a complex manifold to the ones on its small differentiable deformation in this subsection. The argument in this and next subsections is applicable to a general differentiable family of complex manifolds.

For a general $\alpha \in A^{p,q}(X_0)$ , by Proposition 1.3 and the integrability condition (11), one has

(12) $$ \begin{align} d(e^{\iota_{\varphi}|\iota_{\bar{\varphi}}}(\alpha))&=d\circ e^{\iota_{\varphi}}\circ e^{-\iota_{\varphi}}\circ e^{\iota_{\varphi}|\iota_{\bar{\varphi}}}(\alpha)\nonumber\\ &=e^{\iota_{\varphi}}\circ\left([\partial,\iota_{\varphi}]+\bar{\partial}+\partial\right)\circ e^{-\iota_{\varphi}}\circ e^{\iota_{\varphi}|\iota_{\bar{\varphi}}}(\alpha)\nonumber\\ &=e^{\iota_{\varphi}|\iota_{\bar{\varphi}}}\circ\left(e^{-\iota_{\varphi}|-\iota_{\bar{\varphi}}}\circ e^{\iota_{\varphi}}\circ\left([\partial,\iota_{\varphi}]+\bar{\partial}+\partial\right) \circ e^{-\iota_{\varphi}}\circ e^{\iota_{\varphi}|\iota_{\bar{\varphi}}}(\alpha)\right). \end{align} $$

Here,

$$ \begin{align*}e^{-\iota_{\varphi(t)}|-\iota_{\overline{\varphi(t)}}}: A^{p,q}(X_t)\rightarrow A^{p,q}(X_0)\end{align*} $$

is the inverse map of $e^{\iota _{\varphi (t)}|\iota _{\overline {\varphi (t)}}}$ , defined by

(13) $$ \begin{align} \begin{aligned} & e^{-\iota_{\varphi(t)}|-\iota_{\overline{\varphi(t)}}}(s)\\ =&s_{i_1\cdots i_p \bar{j}_1 \cdots \bar{j}_q}(\zeta) \bigg(e^{-\iota_{\varphi(t)}}\Big( \big(dz^{i_1}+\varphi(t)\lrcorner dz^{i_1}\big) \wedge \cdots \wedge \big(dz^{i_p}+\varphi(t) \lrcorner dz^{i_p}\big) \Big) \wedge\\ &\qquad\qquad\qquad e^{-\iota_{\overline{\varphi(t)}}} \Big( \big(d\overline{z}^{j_1}+\overline{\varphi(t)}\lrcorner d\overline{z}^{j_1}\big) \wedge \cdots \wedge \big( d\overline{z}^{j_q}+\overline{\varphi(t)}\lrcorner d\overline{z}^{j_q} \big)\Big)\bigg), \end{aligned} \end{align} $$

where $s\in A^{p,q}(X_t)$ is locally written as

$$ \begin{align*} s=s_{i_1\cdots i_p \bar{j}_1 \cdots \bar{j}_q}(\zeta) &(dz^{i_1}+\varphi(t)\lrcorner dz^{i_1}) \wedge \cdots\wedge (dz^{i_p}+\varphi(t)\lrcorner dz^{i_p})\\ \wedge &(d\overline{z}^{j_1}+\overline{\varphi(t)}\lrcorner d\overline{z}^{j_1})\wedge\cdots\wedge (d\overline{z}^{j_q}+\overline{\varphi(t)}\lrcorner d\overline{z}^{j_q}), \end{align*} $$

and the operators $e^{-\iota _{\varphi (t)}}$ , $e^{-\iota _{\overline {\varphi (t)}}}$ also follow the convention (1) as in the proof of [Reference Rao and Zhao43, Lemma $2.8$ ]. We introduce one more new notation $\Finv $ to denote the simultaneous contraction on each component of a complex differential form. For example, $(\mathbb {1}-\bar {\varphi }\varphi +\bar {\varphi })\Finv \alpha $ means that the operator $(\mathbb {1}-\bar {\varphi }\varphi +\bar {\varphi })$ acts on $\alpha $ simultaneously as:

(14)

if $\alpha $ is locally expressed by:

$$ \begin{align*}\alpha=f_{i_1\cdots i_p\overline{j_1}\cdots\overline{j_q}}dz^{i_1}\wedge \cdots\wedge dz^{i_p}\wedge d\bar{z}^{j_1}\wedge \cdots\wedge d\bar{z}^{j_q}.\end{align*} $$

This new simultaneous contraction is well-defined since $\varphi (t)$ is a global $(1,0)$ -vector valued $(0, 1)$ -form on $X_0$ (See [Reference Morrow and Kodaira39, pp. 150–151]) as reasoned in [Reference Rao and Zhao43, Proof of Lemma 2.8]. Notice that $(\mathbb {1}-\bar {\varphi }\varphi +\bar {\varphi })\Finv \alpha \neq \mathbb {1}\Finv \alpha -\bar {\varphi }\varphi \Finv \alpha +\bar {\varphi }\Finv \alpha $ in general. Using this notation, one can rewrite the extension map $e^{\iota _{\varphi }|\iota _{\bar {\varphi }}}$ in Definition 1.2:

Then one has:

Lemma 2.2. For any $\alpha \in A^{p,q}(X_0)$ ,

(15)

Proof. Following the above notations and the definition of $e^{\iota _{\varphi }|\iota _{\bar {\varphi }}}$ , we have

(16)

On the other hand,

(17)

where the last equality holds by

and

Therefore, (15) is proved by (16) and (17).

Similarly:

Lemma 2.3. For any $\alpha \in A^{p,q}(X_0)$ ,

(18)

where $\left ((\mathbb {1}-\bar {\varphi }\varphi )^{-1}-(\mathbb {1}-\bar {\varphi }\varphi )^{-1}\bar {\varphi }\right )$ acts on $\alpha $ just as (14).

Notice that the more intrinsic proofs of (15) and (18) can be found in the proof of [Reference Rao and Zhao43, Proposition 2.12]. Substituting (15) and (18) into (12), one has

Proposition 2.4. For any $\alpha \in A^{p,q}(X_0)$ ,

(19)

From (18), we know that

$$ \begin{align*}e^{-\iota_{\varphi}|-\iota_{\bar{\varphi}}}\circ e^{\iota_{\varphi}}: A^{p,q}(X_0)\to \bigoplus_{i=0}^{\min\{q,n-p\}}A^{p+i,q-i}(X_0). \end{align*} $$

Thus, by carefully comparing the form types in both sides of (19), we have

(20)

The d-closed condition $d(e^{\iota _{\varphi }|\iota _{\bar {\varphi }}}(\alpha ))=0$ amounts to

$$ \begin{align*}\begin{cases} \bar{\partial}_t(e^{\iota_{\varphi}|\iota_{\bar{\varphi}}}(\alpha))=0,\\ \partial_t(e^{\iota_{\varphi}|\iota_{\bar{\varphi}}}(\alpha))=\overline{\bar{\partial}_t\circ e^{\iota_{\varphi}|\iota_{\bar{\varphi}}}}({\alpha})=0, \end{cases} \end{align*} $$

which, together with (20), implies that

(21)

by the invertibility of the operators $e^{\iota _{\varphi }|\iota _{\bar {\varphi }}}$ , $(\mathbb {1}-\bar {\varphi }\varphi )^{-1}\Finv $ and their conjugations. Remark that the operator $\Finv $ in (21) is just the ordinary contraction operator $\lrcorner $ when acting on $A^{1,1}(X)$ of a complex manifold X. Actually, one can obtain an equivalent expression of (19)

From the original proof of the stability theorem sketched at the beginning of this section, one knows that the system (21) of obstruction equations indeed has a real solution $\alpha (t)\in A^{1,1}(X_t)$ with $\alpha (0)=\omega (0)$ . To get this solution by a power series method, we will formulate an effective obstruction system (23) of equations in Proposition 2.7 for (21).

We will use the commutator formula repeatedly, which is originated from [Reference Tian50, Reference Todorov51] and whose various versions appeared in [Reference Barannikov and Kontsevich11, Reference Clemens13, Reference Friedman21, Reference Li34, Reference Liu, Sun and Yau37] and also [Reference Liu and Rao35, Reference Liu, Rao and Yang36] for vector bundle valued forms.

Lemma 2.5. For $\phi , \psi \in A^{0,1}(X,T^{1,0}_X)$ and $\alpha \in A^{*,*}(X)$ on a complex manifold X,

(22) $$ \begin{align} [\phi,\psi]\lrcorner\alpha=-\partial(\psi\lrcorner(\phi \lrcorner\alpha))-\psi\lrcorner(\phi \lrcorner\partial\alpha) +\phi\lrcorner\partial(\psi\lrcorner\alpha)+\psi \lrcorner\partial(\phi\lrcorner\alpha). \end{align} $$

There are several formulae to be established, whose proofs are given in Appendix 4.2.

We first explain the homogenous notation for a power series to be used here and henceforth. Assuming that $\alpha (t)$ is a power series of (bundle-valued) $(p,q)$ -forms, expanded as

$$ \begin{align*}\alpha(t)=\sum_{k=0}^{\infty}\sum_{i+j=k}\alpha_{i,j}t^i\bar{t}^j,\end{align*} $$

one uses the notation

$$ \begin{align*} \begin{cases} \alpha(t) = \sum^{\infty}_{k=0} \alpha_k, \\[4pt] \alpha_k = \sum_{i+j=k} \alpha_{i,j}t^i \overline{t}^j, \\ \end{cases} \end{align*} $$

where $\alpha _k$ is the k-degree homogeneous part in the expansion of $\alpha (t)$ and all $\alpha _{i,j}$ are smooth (bundle-valued) $(p,q)$ -forms on $X_0$ with $\alpha (0)=\alpha _{0,0}$ . Similarly, according to the expansion (8), one will also adopt this notation to other terms related with $\varphi $ , such as

where $\big ( \left ( \mathbb {1} - \overline {\varphi }\varphi \right )^{\!-1} \lrcorner \varphi \big )_k$ stands for the k-degree homogeneous part of the power series $\left ( \mathbb {1} - \overline {\varphi }\varphi \right )^{\!-1} \lrcorner \varphi $ in $t,\bar t$ . Then we come to the crucial reduction:

Proposition 2.7. If the power series

$$ \begin{align*}\omega(t)=\sum_{k=0}^{\infty}\sum_{i+j=k}\omega_{i,j}t^i\bar{t}^j\end{align*} $$

of $(1,1)$ -forms on $X_0$ is a real formal solution of the system of equations

(23) $$ \begin{align} \begin{cases} \bar{\partial}\omega=\bar{\partial}(\bar{\varphi}\varphi\lrcorner\omega-\varphi\lrcorner\bar{\varphi}\lrcorner\omega)-\partial(\varphi\lrcorner\omega),\\ \partial\omega=\partial(\varphi\bar{\varphi}\lrcorner\omega-\bar{\varphi}\lrcorner\varphi\lrcorner\omega)-\bar{\partial}(\bar{\varphi}\lrcorner\omega), \end{cases} \end{align} $$

up to the degree N, then $\overline {\partial }(\varphi (t)\lrcorner \omega (t))_k=0$ for each $0\leq k\leq N+1$ and thus $\omega (t)$ is also one real formal solution of the system (21) up to the degree N.

We will realize the importance of the $\overline {\partial }$ -closedness $\overline {\partial }(\varphi (t)\lrcorner \omega (t))_{N+1}=0$ , which fulfills (26) in Observation 2.11 under the Kähler condition and guarantees the existence of a real solution of $({23})_{N+1}$ , the equation (23) at the $(N+1)$ th degree.

Proof. Comparing the power series expansion of the equations (21) and (23), we use induction on the degrees to complete the proof.

Denote by $\omega _k$ the homogenous k-part of the power series $\omega (t)$ , that is, $\omega _k=\sum _{i+j=k}\omega _{i,j}t^i\bar {t}^j$ . Without danger of confusion, we will use $\varphi $ and $\omega $ to denote $\varphi (t)$ and $\omega (t)$ , respectively.

The case $N=0$ is trivial. By induction, assuming that the proposition holds for the degrees $\leq N-1$ , we need to show the proposition for the degree N. That is, if $(23)_k$ has a real solution and

$$ \begin{align*}\overline{\partial}(\varphi\lrcorner \omega)_{k}=0\end{align*} $$

for the degrees $k\leq N$ , then we will show that this solution is also one for the system $(21)_N$ and satisfies $\overline {\partial }(\varphi \lrcorner \omega )_{N+1}=0$ . Without loss of generality, we always assume that $N\geq 4$ since the lower-degree cases are also obtained by the same formulation as follows.

Here is an important observation to be proved in Appendix 4.2.

Observation 2.8.

$$ \begin{align*}\overline{\partial}(\varphi\lrcorner \omega)_{k}=0,\qquad k\leq N+1.\end{align*} $$

So by Lemma 2.5, Proposition 2.6.(1) and the induction assumption, we have

Here, the subscript N is omitted. Using Proposition 2.6.(4) and Observation 2.8, one has

$$ \begin{align*}\left(\partial(\varphi\lrcorner\bar{\varphi}\varphi\lrcorner\omega)+\varphi\lrcorner\partial(\omega-\bar{\varphi}\varphi\lrcorner\omega)+\bar{\partial}(\varphi\lrcorner\bar{\varphi}\lrcorner\omega)\right)_N=0, \end{align*} $$

whose left-hand side is exactly the difference of the first equations in $(23)_N$ and $(21)_N$ . Therefore, this real solution of the first equation in $(23)_N$ is also that in $(21)_N$ , and similarly for the second equations in $(21)_N$ and $(23)_N$ .

2.2 Construction of power series

For the resolution of the system (23), we need several more lemmas. As usual, the $\partial \bar {\partial }$ -lemma refers to: for every pure-type d-closed form on $X_0$ , the properties of d-exactness, $\partial $ -exactness, $\bar {\partial }$ -exactness and $\partial \bar {\partial }$ -exactness are equivalent.

Lemma 2.9. Let X be a complex manifold satisfying the $\partial \overline {\partial }$ -lemma. Consider the system of equations:

(24) $$ \begin{align}\left\{ \begin{aligned} \partial x &= \beta, \\ \overline{\partial} x &= \overline{\gamma}, \\ \end{aligned} \right. \end{align} $$

where $\beta ,\gamma $ are $(p+1,p)$ -forms on X. The system of equations (24) has a solution if and only if the following three statements hold:

1. (1) $\overline {\partial } \beta + \partial \overline {\gamma } = 0$ .

2. (2) The $\partial $ -equation $\partial x = \beta $ has a solution.

3. (3) The $\overline {\partial }$ -equation $\overline {\partial } x = \overline {\gamma }$ has a solution.

Proof. If the system (24) has a solution $\eta $ , then both the $\partial $ - and $\overline {\partial }$ -equations have solutions. And it is easy to see that

$$ \begin{align*} \overline{\partial} \beta + \partial \overline{\gamma} = \overline{\partial} \partial \eta + \partial \overline{\partial} \eta = 0.\end{align*} $$

Conversely, let $\eta _1$ be a solution of the $\partial $ -equation, $\eta _2$ a solution of the $\overline {\partial }$ -equation with $\beta ,\gamma $ satisfying $\overline {\partial } \beta + \partial \overline {\gamma } = 0$ , which yields that $\partial \eta _1 = \beta $ .

We claim that there exists some $\tau \in A^{p,p-1}(X)$ such that $\eta _2 + \overline {\partial } \tau $ satisfies the system (24). In fact, it is obvious that $\eta _2 + \overline {\partial } \tau $ satisfies the second equation of (24). As to the first one, we only need to show that $\beta - \partial \eta _2$ is $\partial \overline {\partial }$ -exact. It is easy to check that

$$ \begin{align*} \overline{\partial}( \beta - \partial \eta_2) =\overline{\partial} \beta + \partial \overline{\gamma} = 0.\end{align*} $$

And note that $\beta - \partial \eta _2$ is $\partial $ -exact by the equality

$$ \begin{align*} \beta - \partial \eta_2= \partial ( \eta_1 - \eta_2).\end{align*} $$

From the $\partial \overline {\partial }$ -Lemma, there exists some $\tau \in A^{p,p-1}(X)$ such that $\beta -\partial \eta _2 = \partial \overline {\partial } \tau $ , equivalently saying that $\eta _2 + \overline {\partial } \tau $ satisfies the first equation of (24). Therefore, the claim is proved and $\eta _2 + \overline {\partial } \tau $ is the solution of the system (24).

Corollary 2.10. Let X be a complex manifold satisfying the $\partial \overline {\partial }$ -lemma. The system of equations:

(25) $$ \begin{align}\left\{ \begin{aligned} \partial x &= \beta,\\ \overline{\partial} x &= \overline{\beta}, \\ \end{aligned} \right. \end{align} $$

where $\beta $ is a $(p+1,p)$ -form on X, has a real solution if and only if the following two statements hold:

1. (1) $\overline {\partial } \beta + \partial \overline {\beta } = 0$ .

2. (2) The $\overline {\partial }$ -equation $\overline {\partial } x = \overline {\beta }$ has a solution.

Since $\beta $ and $\overline {\gamma }$ involved in this paper are mostly $\overline {\partial }$ -exact and $\partial $ -exact, respectively, such as (23), we have:

Observation 2.11. Let $\beta =\overline {\partial } \zeta $ and $\overline {\gamma } = \partial \xi $ for some suitable-type complex differential forms $\zeta $ and $\xi $ , respectively, which automatically fulfill the condition (1) in Lemma 2.9. The conditions (2) and (3) rely on the equalities

(26) $$ \begin{align} \left\{ \begin{aligned} \partial \beta = \partial (\overline{\partial} \zeta) =0, \\ \overline{\partial} \overline{\gamma} = \overline{\partial} (\partial \xi)=0. \\ \end{aligned} \right. \end{align} $$

Then the $\partial \overline {\partial }$ -lemma will produce $\mu $ and $\nu $ , satisfying the equations

(27) $$ \begin{align} \left\{ \begin{aligned} \partial \overline{\partial} \mu = \overline{\partial} \zeta =\beta, \\ \overline{\partial} \partial \nu = \partial \xi = \overline{\gamma}. \\ \end{aligned} \right. \end{align} $$

The combined expression

(28) $$ \begin{align} \overline{\partial} \mu + \partial \nu \end{align} $$

is our choice for the solution of the system (24), which will be slightly modified to

$$ \begin{align*}\overline{\partial} \mu + \partial \bar \mu\end{align*} $$

as the real solution of the system (25), when $\beta $ happens to equal to $\gamma $ .

Recall a useful fact that $\overline {\partial }^*\mathbb {G}_{\overline {\partial }}y$ is the unique solution, minimizing the $L^2$ -norms of all the solutions, of the equation

$$ \begin{align*} \overline{\partial} x=y \end{align*} $$

on a compact complex manifold if the equation admits one, where $x,y$ are complex differential forms of pure types and the operator $\mathbb {G}_{\overline {\partial }}$ denotes the corresponding Green’s operator of the $\overline {\partial }$ -Laplacian $\square $ . In the Kähler case, we choose

$$ \begin{align*} \overline{\partial} \mu = \partial^* \mathbb{G}_{\partial} (\overline{\partial} \zeta)=\partial^* \mathbb{G}_{\partial} \beta \quad\text{and}\quad \partial \nu = \overline{\partial}^* \mathbb{G}_{\overline{\partial}} (\partial \xi) = \overline{\partial}^* \mathbb{G}_{\overline{\partial}} \overline{\gamma},\end{align*} $$

where $\mathbb {G}_{\partial }$ and $\mathbb {G}_{\overline {\partial }}$ coincide, with a uniform symbol $\mathbb {G}$ used afterwards. Then an explicit solution of the system (24) can be taken as

$$ \begin{align*} x = \partial^* \mathbb{G} \overline{\partial} \zeta + \overline{\partial}^* \mathbb{G} \partial \xi =\partial^* \mathbb{G} \beta + \overline{\partial}^* \mathbb{G} \overline{\gamma}. \end{align*} $$

When $\beta $ happens to equal to $\gamma $ , one takes the real solution of the system (25) as

(29) $$ \begin{align} x = \partial^* \mathbb{G} \overline{\partial} \zeta + \overline{\partial}^* \mathbb{G} \partial \overline{\zeta} =\partial^* \mathbb{G} \beta + \overline{\partial}^* \mathbb{G} \overline{\beta} \end{align} $$

and accordingly, notices that the operator $\mathbb {G}$ is real in this case.

By these, one is able to obtain the main result of this section:

Theorem 2.12. The system of equations

(30) $$ \begin{align} \begin{cases} d(e^{\iota_{\varphi}|\iota_{\bar{\varphi}}}(\omega(t)))=0,\\ \omega(t)=\overline{\omega(t)},\\ \omega(0)=\omega_0, \end{cases} \end{align} $$

admits a smooth solution $\omega (t)\in A^{1,1}(X_0)$ , where $\omega _0$ is a Kähler metric on the complex manifold $X_0$ . Therefore, we can construct a smooth Kähler metric $e^{\iota _{\varphi }|\iota _{\bar {\varphi }}}(\omega (t))$ on ${X}_t$ .

Proof. We are going to present such an explicit expression for the solution of the obstruction equation (23), whose existence is assured by Proposition 2.7 and the remarks after it, with the initial metric $\omega (0)=\omega _0$ . The first-order system of equations

$$ \begin{align*} \begin{cases} \bar{\partial}\omega_1=-\partial(\varphi_1\lrcorner\omega_0),\\ \partial\omega_1=-\bar{\partial}(\bar{\varphi}_1\lrcorner\omega_0),\\ \end{cases} \end{align*} $$

admits an explicit real solution, as given by (29),

$$ \begin{align*} \omega_1 = \partial\overline{\partial}^*\mathbb{G}(\varphi_1\lrcorner\omega_0)+\overline{\partial}\partial^*\mathbb{G}(\bar{\varphi}_1\lrcorner\omega_0). \end{align*} $$

By induction, we may assume that (23) has an explicit real solution $\omega _k$ for $k\leq N-1$ . Based the construction (29) above, one gets a real solution of the Nth order equation $(23)_N$

$$ \begin{align*} \omega_N=(\bar{\varphi}\varphi\lrcorner\omega-\varphi\lrcorner\bar{\varphi}\lrcorner\omega)_N+\left(\partial\overline{\partial}^*\mathbb{G}(\varphi\lrcorner\omega)+\overline{\partial}\partial^*\mathbb{G}(\bar{\varphi}\lrcorner\omega)\right)_{N}, \end{align*} $$

using Proposition 2.6.(1).

Hence, we complete the induction and get a formal solution $\omega (t)$ of (30) with explicit expressions. By the Hölder $C^{k,\alpha }$ -convergence and regularity argument in Sections 2.3 and 2.4, the formal power series $\omega (t)$ constructed above is smooth and solves the system (30) of equations.

2.3 Hölder convergence

Consider an important power series in deformation theory of complex structures

(31) $$ \begin{align} A(t)=\frac{\beta}{16\gamma}\sum_{m=1}^\infty\frac{(\gamma t)^m}{m^2}:=\sum_{m=1}^\infty A_m t^m, \end{align} $$

where $\beta , \gamma $ are positive constants to be determined. The power series (31) converges for $|t|<\frac {1}{\gamma }$ and has a nice property:

(32) $$ \begin{align} A^i(t)\ll {\left(\frac{\beta}{\gamma}\right)}^{i-1}A(t). \end{align} $$

See [Reference Morrow and Kodaira39, Lemma 3.6 and its Corollary in Chapter 2] for these basic facts. Here, we use the following notation: For the series with real positive coefficients

$$ \begin{align*}a(t)=\sum_{m=1}^\infty a_m t^m, \quad b(t)=\sum_{m=1}^\infty b_m t^m,\end{align*} $$

say that $a(t)$ dominates $b(t)$ , written as $b(t)\ll a(t)$ , if $ b_m\leq a_m$ . But for a power series of (bundle-valued) complex differential forms

$$ \begin{align*}\eta(t)=\sum_{i+j\geq 0}^\infty \eta_{i,j} t^i\bar{t}^j,\end{align*} $$

the notation

$$ \begin{align*}\|\eta(t)\|_{k, \alpha}\ll A(t)\end{align*} $$

means

$$ \begin{align*}\sum_{i+j=m}\|\eta_{i,j}\|_{k, \alpha} \leq A_m\end{align*} $$

with the $C^{k, \alpha }$ -norm $\|\cdot \|_{k, \alpha }$ as defined on [Reference Morrow and Kodaira39, p. 159]. In this manner, for a power series of Beltrami differential $\psi (t)=\sum _{i+j=1}^\infty \psi _{i,j} t^i\bar t^j$ , the notation

$$ \begin{align*}\|\psi(t)\|_{k, \alpha}\ll A(t)\end{align*} $$

indicates $\sum _{i+j=m}\|\psi _{i,j}\|_{k, \alpha } \leq A_m$ , and similarly for $\bar {\psi }(t)$ . This notation is also used to compare two or more power series of (bundle-valued) complex differential forms degree by degree, such as $\|\psi (t)\|_{k, \alpha }\ll \|\eta (t)\|_{k, \alpha }\cdot \|\rho (t)\|_{k, \alpha }$ for three such power series.

For any complex differential form $\phi $ , we have two a priori elliptic estimates

(33) $$ \begin{align} \|\overline{\partial}^*\phi\|_{k-1, \alpha}\leq C_1\|\phi\|_{k, \alpha} \end{align} $$

and

(34) $$ \begin{align} \|\mathbb{G}\phi\|_{k, \alpha}\leq C_{k,\alpha}\|\phi\|_{k-2, \alpha}, \end{align} $$

where $\mathbb {G}$ is the associated Green’s operator to the operator $\overline {\partial }$ , $k>1$ , $C_1$ and $C_{k,\alpha }$ depend on only on k and $\alpha $ , not on $\phi $ . (See [Reference Morrow and Kodaira39, Proposition $2.3$ in Chapter $4$ ].)

According to the proof of Theorem 2.12, for $r\geq 1$ , one real solution of the obstruction equation (23) is

(35) $$ \begin{align} \omega_r=(\bar{\varphi}\varphi\lrcorner\omega-\varphi\lrcorner\bar{\varphi}\lrcorner\omega)_r+\left(\partial\overline{\partial}^*\mathbb{G}(\varphi\lrcorner\omega)+\overline{\partial}\partial^*\mathbb{G}(\bar{\varphi}\lrcorner\omega)\right)_{r}, \end{align} $$

and obviously

(36) $$ \begin{align} \omega^{(r)}=(\bar{\varphi}\varphi\lrcorner\omega-\varphi\lrcorner\bar{\varphi}\lrcorner\omega)^{(r)}+\left(\partial\overline{\partial}^*\mathbb{G}(\varphi\lrcorner\omega)+\overline{\partial}\partial^*\mathbb{G}(\bar{\varphi}\lrcorner\omega)\right)^{(r)} \end{align} $$

are real complex differential forms. Here for a power series of (bundle-valued) complex differential forms

$$ \begin{align*}\eta(t)=\sum_{i+j= 0}^\infty \eta_{i,j} t^i\bar{t}^j,\end{align*} $$

one denotes by $\eta ^{(r)}$ the summation

$$ \begin{align*}\sum_{i+j=1}^{r} \eta_{i,j} t^i\bar{t}^j:=\eta_1+\cdots+\eta_r.\end{align*} $$

Recall the holomorphic family

(37) $$ \begin{align} \varphi(t) = \sum_{i=1}^{\infty}\varphi_{i}t^i:=\sum_{j=1}^{\infty}\varphi_j(t), \end{align} $$

for $|t|< \rho $ a small positive constant, of Beltrami differentials representing Kuranishi family develop as in (9) and (10):

$$ \begin{align*}\varphi_1(t)=t\eta, \end{align*} $$

where $\eta $ is a base of for $\mathbb {H}^{0,1}(X_0,T^{1,0}_{X_0})$ , and for $i\geq 2$ ,

$$ \begin{align*}\varphi_i=\frac{1}{2}\overline{\partial}^*\mathbb{G}\sum_{j+l=i}[\varphi_j,\varphi_{l}]. \end{align*} $$

Then it satisfies a nice convergence property:

$$ \begin{align*}\|\varphi(t)\|_{k, \alpha}\ll A(t)\end{align*} $$

as given in the proof of [Reference Morrow and Kodaira39, Proposition 2.4 in Chapter $4$ ]. In this proof, $\beta $ and $\frac {\beta }{\gamma }$ should satisfy that

(38) $$ \begin{align} \beta\geq b,\ \frac{\beta}{\gamma}\leq b_k, \end{align} $$

where the constants $b,b_k>0$ and $b_k$ depends on k. Here, we follow the idea of proving the convergence of $\varphi (t)$ there to obtain that

$$ \begin{align*}\|\omega^{(r)}\|_{k, \alpha}\ll A(t),\ \text{for any}\ r\geq 1,\end{align*} $$

which implies the desired convergence $\|\omega ^{(+\infty )}\|_{k, \alpha }\ll A(t)$ immediately. Assume that they are chosen so that

$$ \begin{align*}\|\omega^{(r-2)}\|_{k, \alpha},\|\omega^{(r-1)}\|_{k, \alpha}\ll A(t).\end{align*} $$

By the expression (36) and the two a priori elliptic estimates (33) and (34), one has

$$ \begin{align*} \|\omega^{(r)}\|_{k, \alpha} &=\|(\bar{\varphi}\varphi\lrcorner\omega^{(r-2)}-\varphi\lrcorner\bar{\varphi}\lrcorner\omega^{(r-2)})^{(r)}+\left(\partial\overline{\partial}^*\mathbb{G}(\varphi\lrcorner\omega^{(r-1)})+\overline{\partial}\partial^*\mathbb{G}(\bar{\varphi}\lrcorner\omega^{(r-1)})\right)^{(r)}\\ &\quad +(\bar{\varphi}\varphi\lrcorner\omega_0-\varphi\lrcorner\bar{\varphi}\lrcorner\omega_0)^{(r)}+\left(\partial\overline{\partial}^*\mathbb{G}(\varphi\lrcorner\omega_0)+\overline{\partial}\partial^*\mathbb{G}(\bar{\varphi}\lrcorner\omega_0)\right)^{(r)}\|_{k, \alpha}\\ &\ll 2\|\bar{\varphi}^{(r-2)}\|_{k, \alpha}\|\varphi^{(r-2)}\|_{k, \alpha}\|\omega^{(r-2)}\|_{k, \alpha} +2 C_1C_{k,\alpha}\|\varphi^{(r-1)}\|_{k, \alpha}\|\omega^{(r-1)}\|_{k, \alpha}\\ &\quad + 2\|\bar{\varphi}^{(r-1)}\|_{k, \alpha}\|\varphi^{(r-1)}\|_{k, \alpha}\|\omega_0\|_{k, \alpha} +2 C_1C_{k,\alpha}\|\varphi^{(r)}\|_{k, \alpha}\|\omega_0\|_{k, \alpha}. \end{align*} $$

Then, we use induction and (32) to get:

$$ \begin{align*} \|\omega^{(r)}\|_{k, \alpha} &\ll 2(A(t)+C_1C_{k,\alpha})A^2(t)+2\|\omega_0\|_{k,\alpha}(A(t)+C_1C_{k,\alpha})A(t)\\ &\ll 2(A(t)+C_1C_{k,\alpha})\left(\frac{\beta}{\gamma}+\|\omega_0\|_{k,\alpha}\right)A(t). \end{align*} $$

Hence, we may choose $\beta ,\gamma ,\|\omega _0\|_{k, \alpha }$ so that the following inequalities hold:

$$ \begin{align*}2(A(t)+C_1C_{k,\alpha})\left(\frac{\beta}{\gamma}+\|\omega_0\|_{k,\alpha}\right)<1,\end{align*} $$

Equation (38) and $\omega ^{(1)},\omega ^{(2)}\ll A(t)$ according to the above formulation, which are obviously possible as long as t is small and we notice that the Hölder norm depends on the choice of the local coordinate charts when defined a differential manifold as pointed out on [Reference Kodaira28, p. $275$ ]. Therefore, for small t, $\omega (t)$ converges in the $C^{k, \alpha }$ -norm and thus its positivity follows.

2.4 Regularity argument

In this subsection, we proceed to the regularity argument for the power series constructed as above since there is possibly no uniform lower bound for the convergence radius obtained in the last subsection in the $C^{k,\alpha }$ -norm as k converges to $+\infty $ . We resort to the elliptic operator, the $\overline {\partial }$ -Laplacian

(39) $$ \begin{align} \square=\overline{\partial}^*\overline{\partial}+\overline{\partial}\overline{\partial}^*. \end{align} $$

Here, the dual operators are defined with respect to the fixed original Kähler metric $\omega _0$ . By the classical Kähler identity, Hodge decomposition and the induced commutativity of the associated operators, one has

$$ \begin{align*} \square\omega &=\square(\bar{\varphi}\varphi\lrcorner\omega-\varphi\lrcorner\bar{\varphi}\lrcorner\omega) +\square\partial\overline{\partial}^*\mathbb{G}(\varphi\lrcorner\omega)+\square\overline{\partial\overline{\partial}^*\mathbb{G}(\varphi\lrcorner\omega)}\\[5pt] &=\square(\bar{\varphi}\varphi\lrcorner\omega-\varphi\lrcorner\bar{\varphi}\lrcorner\omega) +\partial\overline{\partial}^*(\varphi\lrcorner\omega)+\overline{\partial\overline{\partial}^*(\varphi\lrcorner\omega)} \end{align*} $$

according to the solution (35).

Consequently, $\omega $ is a solution of the two-order partial differential equation

(40) $$ \begin{align} \square\omega-\square(\bar{\varphi}\varphi\lrcorner\omega-\varphi\lrcorner\bar{\varphi}\lrcorner\omega) -\partial\overline{\partial}^*(\varphi\lrcorner\omega)-\overline{\partial}\partial^*(\bar\varphi\lrcorner\omega)=0. \end{align} $$

Similarly to the argument on [Reference Kodaira28, p. 281], writing out the last three terms in the left-hand side of (40) locally, we can find that the expressions of their principal parts (i.e., the highest-order terms) contain factors from $\bar {\varphi }\varphi $ , $\varphi $ , or $\bar {\varphi }$ . Since $\varphi (t)\rightarrow 0$ as $t\rightarrow 0$ , taking sufficiently small $\epsilon $ -disk $\Delta _{\epsilon }\subseteq \mathbb {C}$ , we can assume that the equation (40) is a linear elliptic partial differential equation of $\omega $ on $X_0$ when noticing the ellipticity of an operator only concerns about its principal part. Thus, the interior estimates [Reference Douglis and Nirenberg20] for elliptic systems of partial differential equations give rise to the desired regularity of $\omega (t)$ , that is, $\omega (t)$ is smooth on $X_0$ for each t smaller than a uniform upper bound to be obtained similarly to [Reference Kodaira28, Appendix 8]. Then $\omega (t)$ can be regarded as a real analytic family of $(1,1)$ -forms in t and it is smooth on t by [Reference Krantz and Parks30, Proposition 2.2.3].

3.1 The $(n-1,n)$ th mild $\partial \bar {\partial }$ -lemma and examples

We are going to study a new kind of “ $\partial \overline {\partial }$ -lemma,” its relations with various analogous conditions in the literature and its examples involved.

Let X be a compact complex manifold of (complex) dimension n with the following commutative diagram

Bott–Chern and Aeppli cohomology groups of X are defined as

$$ \begin{align*}H^{\bullet,\bullet}_{\mathrm{BC}}(X)=:\frac{\ker \partial\cap \ker\overline{\partial}}{\mathrm{im} \partial\overline{\partial}}\quad \text{and}\quad H^{\bullet,\bullet}_{\mathrm{A}}(X)=:\frac{\ker \partial\overline{\partial}}{\mathrm{im} \partial+\mathrm{im}\overline{\partial}},\end{align*} $$

respectively, while $H^{\bullet ,\bullet }_{\partial }(X)$ is defined similarly. Note that the inequalities

$$ \begin{align*}\dim_{\mathbb{C}}H^{n-1,n}_{A}(X) \leq \dim_{\mathbb{C}}H^{n-1,n}_{\partial}(X) \leq \dim_{\mathbb{C}}H^{n-1,n}_{BC}(X)\end{align*} $$

hold on any compact complex manifold [Reference Angella and Ugarte10, Corollary 3.3].

Definition 3.1. The compact complex manifold X satisfies the $(n-1,n)$ th mild  $\partial \bar {\partial }$ -lemma, if the mapping $\iota _{BC,\partial }^{n-1,n}:H^{n-1,n}_{BC}(X) \rightarrow H^{n-1,n}_{\partial }(X)$ , induced by the identity map, is injective. Equivalently, for any $(n-2,n)$ -complex differential form $\xi $ , there exists an $(n-2,n-1)$ -form $\theta $ such that

$$ \begin{align*} \partial \overline{\partial} \theta = \partial \xi. \end{align*} $$

Since $\iota _{BC,\partial }^{n-1,n}:H^{n-1,n}_{BC}(X) \rightarrow H^{n-1,n}_{\partial }(X)$ is always surjective, the following conditions are equivalent:

$$ \begin{align*} \begin{aligned} \text{the}\ (n-1,n)\text{th mild}\ \partial\bar{\partial}\text{-lemma} &\Longleftrightarrow \iota_{BC,\partial}^{n-1,n}:H^{n-1,n}_{BC}(X) \rightarrow H^{n-1,n}_{\partial}(X)\ \text{an isomorphism} \\ &\Longleftrightarrow \dim_{\mathbb{C}} H^{n-1,n}_{BC}(X) = \dim_{\mathbb{C}} H^{n-1,n}_{\partial}(X). \end{aligned}\end{align*} $$

Let us give an example of a non- $\partial \overline {\partial }$ -manifold which satisfies the (1,2)-mild $\partial \overline {\partial }$ -lemma.

Proof. It is shown in [Reference Angella and Kasuya7, Table 5 in Appendix A] and [Reference Kasuya27, the case B in Example 1] that the bases of $H^{2,1}_{BC}(X)$ and $H^{2,1}_{\overline {\partial }}(X)$ can be illustrated as follows:

$$ \begin{align*} \begin{aligned} H^{2,1}_{BC}(X) &= \langle [dz_{12\bar{3}}]_{BC}, [e^{-2z_1}dz_{12\bar{2}}]_{BC}, [e^{2z_1}dz_{13\bar{3}}]_{BC}, [dz_{12\bar{3}}]_{BC},[dz_{13\bar{2}}]_{BC} \rangle,\\ H^{2,1}_{\overline{\partial}}(X) &= \langle [dz_{12\bar{3}}]_{\overline{\partial}}, [e^{-2z_1}dz_{12\bar{2}}]_{\overline{\partial}}, [e^{2z_1}dz_{13\bar{3}}]_{\overline{\partial}}, [dz_{12\bar{3}}]_{\overline{\partial}},[dz_{13\bar{2}}]_{\overline{\partial}} \rangle, \end{aligned}\end{align*} $$

which indicates that $\iota _{BC,\overline {\partial }}^{2,1}$ is actually an isomorphism. It is obvious from [Reference Angella and Kasuya7, Table 6 in Appendix A] that $\dim _{\mathbb {C}}H^{1,1}_{BC}(X)=3$ and $\dim _{\mathbb {C}}H^{1,1}_{\overline {\partial }}(X)=5$ , which implies that the $\partial \overline {\partial }$ -lemma doesn’t hold on X.

There are three more similar conditions in relevance with the local stability of balanced structures. The $(n-1,n)$ th weak  $\partial \overline {\partial }$ -lemma on the compact complex manifold X, introduced by Fu–Yau [Reference Fu and Yau23], says that if for any real $(n-1,n-1)$ -form $\psi $ such that $\overline {\partial } \psi $ is $\partial $ -exact, there exists an $(n-2,n-1)$ -form $\theta $ , satisfying

$$ \begin{align*} \partial \overline{\partial} \theta = \overline{\partial} \psi. \end{align*} $$

And the  $(n-1,n)$ th strong  $\partial \overline {\partial }$ -lemma, proposed by Angella–Ugarte [Reference Angella and Ugarte10], states that the mapping $\iota ^{n-1,n}_{BC,A}: H^{n-1,n}_{BC}(X) \rightarrow H^{n-1,n}_{A}(X)$ , induced by the identity map, is injective, which is equivalent to that for any $\partial $ -closed $(n-1,n)$ -form $\Gamma $ of the type $\Gamma =\partial \xi +\overline {\partial } \psi $ , there exists an $(n-2,n-1)$ -form $\theta $ such that

$$ \begin{align*} \partial \overline{\partial} \theta = \Gamma. \end{align*} $$

Angella–Ugarte [Reference Angella and Ugarte10, Theorem 3.1] show that the $(n-1,n)$ th strong $\partial \overline {\partial }$ -lemma amounts to the sGG condition, carefully studied in [Reference Popovici and Ugarte42], and the vanishing of the first $\partial \overline {\partial }$ -degree $\Delta ^1(X)$ , introduced in [Reference Angella and Tomassini8], with the deformation openness of the $(n-1,n)$ th strong $\partial \overline {\partial }$ -lemma proved in [Reference Angella and Ugarte10, Proposition 4.8]. They also show in [Reference Angella and Ugarte10, Corollary 3.3] the equivalence:

$$ \begin{align*} \begin{aligned} \text{the}\ (n-1,n)\text{th strong}\ \partial\overline{\partial}\text{-lemma} &\Longleftrightarrow \dim_{\mathbb{C}}H^{n-1,n}_{BC}(X) = \dim_{\mathbb{C}}H^{n-1,n}_{A}(X) \\ &\Longleftrightarrow \dim_{\mathbb{C}}H^{0,1}_{BC}(X) = \dim_{\mathbb{C}}H^{0,1}_{A}(X) \\ &\Longleftrightarrow b_1=2\dim_{\mathbb{C}}H^{0,1}_{A}(X). \end{aligned}\end{align*} $$

Besides, the condition that the induced mapping $\iota ^{n-1,n}_{BC,\overline {\partial }}: H^{n-1,n}_{BC}(X) \rightarrow H^{n-1,n}_{\overline {\partial }}(X)$ by the identity map is injective, is presented by Angella–Ugarte [Reference Angella and Ugarte9] to study local conformal balanced structures and global ones, which we may call the  $(n-1,n)$ th dual mild  $\partial \overline {\partial }$ -lemma.

After a simple check, we have the following observation:

And the mild one and the dual mild one both imply the weak one. All the four “ $\partial \overline {\partial }$ -lemmata” hold if the compact complex manifold X satisfies the standard $\partial \overline {\partial }$ -lemma.

We refer the readers to [Reference Angella and Kasuya7, Reference Latorre, Ugarte and Villacampa32, Reference Rollenske45, Reference Ugarte and Villacampa53] as the background materials on the theory of nilmanifolds and solvmanifolds to be focused on in the rest of this subsection.

Recall that a nilmanifold M with left-invariant complex structure is a compact quotient of a simply-connected nilpotent Lie group G of real even dimension by a lattice $\Gamma $ of maximal rank, whose Lie algebra $\mathfrak {g}$ admits an integrable complex structure J. It is clear that the invariant complex structure J on G descends to M in a natural way and it is given by an endomorphism $J:\mathfrak {g}\rightarrow \mathfrak {g}$ of the Lie algebra $\mathfrak {g}$ such that $J^2=-\mathbb {1}$ , satisfying the “Nijenhuis condition”

$$ \begin{align*}[JX,JY]=J[JX,Y]+J[X,JY]+[X,Y],\end{align*} $$

for any $X,Y\in \mathfrak {g}$ .

Let $\mathfrak {g}_{\mathbb {C}}$ be the complexification of $\mathfrak {g}$ and $\mathfrak {g}_{\mathbb {C}}^*$ its dual. We denote by $\mathfrak {g}^{1,0}$ and $\mathfrak {g}^{0,1}$ the eigenspaces corresponding to the eigenvalues $\pm \sqrt {-1}$ of J as an endomorphism of $\mathfrak {g}_{\mathbb {C}}^*$ , respectively. The decomposition

$$ \begin{align*}\mathfrak{g}_{\mathbb{C}}^*=\mathfrak{g}^{1,0}\bigoplus\mathfrak{g}^{0,1}\end{align*} $$

gives rise to a natural bigraduation on the complexified exterior algebra

$$ \begin{align*}\bigwedge^*\mathfrak{g}_{\mathbb{C}}^*=\bigoplus_{p,q}\bigwedge^{p,q} \mathfrak{g}^* =\bigoplus_{p,q}\bigwedge^p \mathfrak{g}^{1,0}\bigoplus\bigwedge^q\mathfrak{g}^{0,1}.\end{align*} $$

We will still use the Chevalley–Eilenberg differential d of the Lie algebra to denote its extension to the complexified exterior algebra, that is, $d:\bigwedge ^*\mathfrak {g}_{\mathbb {C}}^*\rightarrow \bigwedge ^{*+1}\mathfrak {g}_{\mathbb {C}}^*.$ It is well known that the endomorphism J is a complex structure if and only if

$$ \begin{align*}d\mathfrak{g}^{1,0}\subset\bigwedge^{2,0} \mathfrak{g}^*\bigoplus\bigwedge^{1,1} \mathfrak{g}^*.\end{align*} $$

As for nilpotent Lie algebras $\mathfrak {g}$ , Salamon [Reference Salamon46] proves the equivalence: J is a complex structure on $\mathfrak {g}$ if and only if $\mathfrak {g}^{1,0}$ has a basis $\{\varpi ^i\}_{i=1}^n$ such that $d\varpi ^1=0$ and

$$ \begin{align*}d\varpi^i\in \mathcal{I}(\varpi^1,\dots,\varpi^{i-1}),\ \text{for}\ i=2,\dots,n,\end{align*} $$

where $\mathcal {I}(\varpi ^1,\dots ,\varpi ^{i-1})$ is the ideal in $\wedge ^*\mathfrak {g}_{\mathbb {C}}^*$ generated by $\{\varpi ^1,\dots ,\varpi ^{i-1}\}$ . The work [Reference Cordero, Fernández, Gray and Ugarte16] shows that a complex structure J isnilpotent if and only if $\mathfrak {g}^{1,0}$ admits a basis $\{\varpi ^i\}_{i=1}^n$ with $d\varpi ^1=0$ and

$$ \begin{align*}d\varpi^i\in \wedge^2\langle\varpi^1,\dots,\varpi^{i-1},\bar{\varpi}^1,\cdots,\bar{\varpi}^{i-1})\rangle,\ \text{for}\ i=2,\dots,n;\end{align*} $$

otherwise J is non-nilpotent. Abelian complex structures satisfy additionally that $d\mathfrak {g}^{1,0}\subset \bigwedge ^{1,1} \mathfrak {g}^*$ . A nilpotent complex structure is complex parallelizable if and only if $d\mathfrak {g}^{1,0}\subset \bigwedge ^{2,0} \mathfrak {g}^*$ .

Inspired by Ugarte–Villacampa [Reference Ugarte and Villacampa53, Proposition 3.4 and Corollary 3.5], one has:

Proof. Let $\xi $ be an $(n-2,n)$ -complex differential form on M. From the isomorphism $H^{n-2,n}_{\overline {\partial }}(M,J) \cong H^{n-2,n}_{\overline {\partial }}(\mathfrak {g},J)$ and [Reference Ugarte and Villacampa53, Remark 3.3], there exists an $(n-2,n-1)$ -form $\theta $ on M such that

$$ \begin{align*} \xi = \xi_{\nu} + \overline{\partial} \theta, \end{align*} $$

where $\xi _{\nu }$ denotes the image of $\xi $ under the symmetrization process $A^{p,q}(M) \rightarrow \bigwedge ^{p,q}(\mathfrak {g^*})$ . The $(n-1,n)$ th mild $\partial \overline {\partial }$ -lemma on $(\mathfrak {g},J)$ implies that

$$ \begin{align*} \partial \overline{\partial} \tilde{\theta} = \partial \xi_{\nu} \end{align*} $$

for some $\tilde {\theta } \in \bigwedge ^{n-2,n-1}(\mathfrak {g^*})$ . It follows that

$$ \begin{align*} \partial \xi = \partial \overline{\partial} (\tilde{\theta} + \theta). \\[-37pt]\end{align*} $$

Similarly with [Reference Ugarte and Villacampa53, Proposition 3.2], we have

Proof. Suppose that the $(n-1,n)$ th mild $\partial \overline {\partial }$ -lemma holds on $(M,J)$ , that is, for any $(n-2,n)$ -form $\xi $ on M, there exists an $(n-2,n-1)$ -form $\theta $ such that

$$ \begin{align*} \partial \overline{\partial} \theta = \partial \xi. \end{align*} $$

Using the symmetrization process (cf. the proof of [Reference Ugarte and Villacampa53, Proposition 3.2]) on the both sides, we have

$$ \begin{align*} \partial \overline{\partial} \theta_{\nu} = \partial \xi_{\nu}, \end{align*} $$

which contradicts the assumption on the Lie algebra level $(\mathfrak {g},J)$ . Here, the equalities

$$ \begin{align*} (\partial \alpha)_{\nu} = \partial \alpha_{\nu}\quad \text{and} \quad (\overline{\partial} \alpha)_{\nu} = \overline{\partial} \alpha_{\nu}\end{align*} $$

for any $\alpha \in A^{p,q}(M)$ are used as in the proof of [Reference Ugarte and Villacampa53, Proposition 3.2].

The following result of the $(n-1,n)$ th dual mild $\partial \overline {\partial }$ -lemma is almost the same as the mild one in Propositions 3.4 and 3.6, for which we omit the proofs.

Proof. Let J be the complex structure in the category $(i)$ , which satisfies the $(2,3)$ th weak $\partial \overline {\partial }$ -lemma by the proof of [Reference Ugarte and Villacampa53, Proposition 3.6]. It is easy to check that, on the Lie algebra level,

$$ \begin{align*}\partial \big( \bigwedge^{1,3}(\mathfrak{g}^*)\big) = \langle \omega^{12\bar{1}\bar{2}\bar{3}} \rangle,\quad \overline{\partial} \big( \bigwedge^{1,2}(\mathfrak{g}^*) \big)=0\quad \text{and} \quad \overline{\partial} \big( \bigwedge^{2,2}(\mathfrak{g}^*) \big)=0.\end{align*} $$

However,

$$ \begin{align*} \partial \omega^{3\bar{1}\bar{2}\bar{3}} = \omega^{12\bar{1}\bar{2}\bar{3}} \notin \partial \overline{\partial} \big( \bigwedge^{1,2}(\mathfrak{g}^*) \big)=0,\end{align*} $$

and thus $(2,3)$ th mild $\partial \overline {\partial }$ -lemma does not hold on the nilmanifold, from Proposition 3.6. Here, $\omega ^{3\bar {1}\bar {2}\bar {3}}$ denotes $\omega ^3 \wedge \overline {\omega ^1} \wedge \overline {\omega ^2} \wedge \overline {\omega ^3}$ as the notation used in [Reference Ugarte and Villacampa53]. Also, it is clear that $(\mathfrak {g},J)$ satisfies the $(2,3)$ th dual mild $\partial \overline {\partial }$ -lemma, with $H^{p,q}_{BC}(M,J) \cong H^{p,q}_{BC}(\mathfrak {g},J)$ for any $p,q$ assured by [Reference Angella6, Theorem 3.8]. Therefore, the $(2,3)$ -th dual mild $\partial \overline {\partial }$ -lemma holds on the nilmanifold by Proposition 3.7.

An invariant balanced Hermitian structure F on a nilmanifold M is the one, coming from a balanced Hermitian structure on the Lie algebra $\mathfrak {g}^*$ .

Proof. It is known, from [Reference Ugarte and Villacampa53, Proposition 3.6], $(M,J)$ satisfies the $(2,3)$ th weak $\partial \overline {\partial }$ -lemma if and only if J is abelian, complex parallelizable or non-nilpotent. Example 3.8 shows that the complex parallelizable case satisfies the $(2,3)$ th dual mild $\partial \overline {\partial }$ -lemma but does not satisfy the mild one. Corollary 3.5 and [Reference Angella and Ugarte9, Proposition 2.9] say that a $2n$ -dimensional nilmanifold endowed with an invariant abelian complex structure satisfies the $(n-1,n)$ th mild $\partial \overline {\partial }$ -lemma but never satisfies the $(n-1,n)$ th dual mild $\partial \overline {\partial }$ -lemma, especially for the situation of the six-dimension (i.e., complex dimension $3$ ) here. Hence, the only left to check is the non-nilpotent case.

As shown in the proof of [Reference Ugarte and Villacampa53, Proposition 3.6], the equality holds

$$ \begin{align*} \partial \overline{\partial} \big( \bigwedge^{1,2}(\mathfrak{g}^{*}) \big) = \langle \omega^{13\bar{1}\bar{2}\bar{3}}\rangle = \partial \big( \bigwedge^{1,3}(\mathfrak{g}^*) \big),\end{align*} $$

according to the category $(iii)$ of [Reference Ugarte and Villacampa53, Proposition 2.3], and the natural inclusion

$$ \begin{align*}\big(\bigwedge^{*,*}(\mathfrak{g}^*),\overline{\partial}\big)\hookrightarrow \big(A^{*,*}(M),\overline{\partial}\big)\end{align*} $$

induces an isomorphism in the $\overline {\partial }$ -cohomology in this case. Therefore, $(M,J)$ with the non-nilpotent complex structure J satisfies the $(2,3)$ -th mild $\partial \overline {\partial }$ -lemma by Proposition 3.4. Meanwhile, it should be noted that

$$ \begin{align*} \overline{\partial} \omega^{13\bar{2}\bar{3}} = \pm \sqrt{-1} \omega^{12\bar{1}\bar{2}\bar{3}}\quad \text{and} \quad \partial \omega^{12\bar{1}\bar{2}\bar{3}} =0,\end{align*} $$

according to the category $(iii)$ of [Reference Ugarte and Villacampa53, Proposition 2.3]. Hence, we have

$$ \begin{align*} \text{the}\ \partial\text{-closed}\ (2,3)\text{-form}\ \overline{\partial} \omega^{13\bar{2}\bar{3}} = \pm \sqrt{-1} \omega^{12\bar{1}\bar{2}\bar{3}} \notin \partial \overline{\partial} \big( \bigwedge^{1,2}(\mathfrak{g}^{*}) \big) = \langle \omega^{13\bar{1}\bar{2}\bar{3}}\rangle.\end{align*} $$

Then Proposition 3.7 tells us that $(M,J)$ with J non-nilpotent does not satisfy the $(2,3)$ th dual mild $\partial \overline {\partial }$ -lemma.

Explicit examples of different complex structures in Proposition 3.9 have been provided in [Reference Ugarte and Villacampa53]. Directly from Observation 3.3 and Proposition 3.9, one has:

Let $\pi :\mathcal {X}\to B$ be a differentiable family of compact n-dimensional complex manifolds with the reference fiber $\pi ^{-1}(t_0)=X_0$ and the general fibers $X_t:=\pi ^{-1}(t)$ . Here, B denotes a sufficiently small domain in $\mathbb {R}^k$ . Fu–Yau [Reference Fu and Yau23, Theorem 6] show that the balanced structure is deformation open, assuming that the $(n-1,n)$ th weak $\partial \overline {\partial }$ -lemma holds on the general fibers $X_t$ for $t \neq 0$ . Angella–Ugarte [Reference Angella and Ugarte10, Theorem 4.9] prove that if $X_0$ admits a locally conformal balanced metric and satisfies the $(n-1,n)$ th strong $\partial \overline {\partial }$ -lemma, then $X_t$ is balanced for t small. Our main result in this section, whose proof is postponed to the next subsection, is

It is well known from [Reference Alessandrini and Bassanelli3] that small deformation of the Iwasawa manifold, which satisfies the $(2,3)$ th weak $\partial \overline {\partial }$ -lemma but does not satisfy the mild one from Example 3.8, may not be balanced. Thus, the condition “ $(n-1,n)$ th mild $\partial \overline {\partial }$ -lemma” in Theorem 3.12 cannot be replaced by the weak one. It is an obvious generalization of Wu’s result [Reference Wu55, Theorem 5.13] that the balanced condition is preserved under small deformation if the reference fiber satisfies the $\partial \overline {\partial }$ -lemma. Based on Corollary 3.5 and Proposition 3.9, one obtains:

Examples and results such as [Reference Angella and Ugarte10, Proposition 4.4, Remark 4.6, Remark 4.7, and Example 4.10] and [Reference Fu and Yau23, Corollary 8 and Corollary 9] become consequences of Theorem 3.12 and Corollary 3.13. And it is interesting to note that six-dimensional nilmanifolds endowed with an invariant abelian or non-nilpotent balanced Hermitian structure provide solutions of the Strominger system with respect to the Bismut connection or the Chern connection in the anomaly cancellation condition, respectively. See [Reference Ugarte and Villacampa53, Section 5] and [Reference Ugarte and Villacampa52, Section 4] for more details.

The example above proves that neither the $(n-1,n)$ th weak $\partial \overline {\partial }$ -lemma nor the mild one is deformation open. And it shows that the condition in [Reference Fu and Yau23, Theorem 6] is not a necessary one for the deformation openness of balanced structures as mentioned in [Reference Ugarte and Villacampa53, the discussion ahead of Example 3.7]. Fortunately, Corollary 3.13 can be applied to this example. See also [Reference Angella and Ugarte10, Remark 4.7], where Corollary 3.13 can also be applied.

Meanwhile, from Corollary 3.5 and [Reference Angella and Ugarte9, Proposition 2.9], a $2n$ -dimensional nilmanifold endowed with an invariant abelian complex structure satisfies the $(n-1,n)$ th mild $\partial \overline {\partial }$ -lemma but never satisfies the $(n-1,n)$ th dual mild $\partial \overline {\partial }$ -lemma. It implies that the deformation openness of balanced structures with the reference fiber a $2n$ -dimensional nilmanifold endowed with an invariant abelian balanced Hermitian structure can be obtained by Corollary 3.13, but it cannot be derived from [Reference Angella and Ugarte10, Theorem 4.9].

Besides, it is known that the deformation invariance of the dimensions of the $(n-1,n-1)$ th Bott–Chern group $H^{n-1,n-1}_{BC}(X_t)$ can assure the deformation openness of balanced structures as shown in [Reference Angella and Ugarte10, Proposition 4.1]. See also Proposition 4.1, which is a kind of generalization of this result. However, [Reference Angella and Ugarte10, Example 4.10] shows that small deformation of a completely-solvable Nakamura threefold, which is balanced and satisfies the $(2,3)$ th strong $\partial \overline {\partial }$ -lemma, is also balanced. The $(2,2)$ th Bott–Chern number varies along this deformation. Fortunately, Theorem 3.12 is applicable to this case and also possibly to some cases with deformation variance of $(n-1,n-1)$ th Bott–Chern number.

Finally, from the perspective of Theorem 3.12, we may have a clear picture of Angella–Ugarte’s result [Reference Angella and Ugarte10, Theorem 4.9], which states that if $X_0$ admits a locally conformal balanced metric and satisfies the $(n-1,n)$ th strong $\partial \overline {\partial }$ -lemma, then $X_t$ is balanced for t small. Actually, the $(n-1,n)$ th strong $\partial \overline {\partial }$ -lemma decomposes into the mild one and the dual mild one, according to Observation 3.3. A locally conformal balanced metric can be transformed into a balanced one by the $(n-1,n)$ -th dual mild $\partial \overline {\partial }$ -lemma, from [Reference Angella and Ugarte9, Theorem 2.5]. Then the $(n-1,n)$ th mild $\partial \overline {\partial }$ -lemma assures that the deformation openness of balanced structures starts from the transformed balanced metric on the reference fiber, thanks to Theorem 3.12.

3.2 Proof for stability of balanced structures with mild $\partial \overline {\partial }$ -lemma

In this subsection, we will prove the local stability Theorem 3.12 of balanced structures with the balanced reference fiber $X_0$ with the $(n-1,n)$ th mild $\partial \bar {\partial }$ -lemma. Similarly to the Kähler case, we will reduce the proof to Kuranishi family as described in the beginning of Section 2.

Our goal is to construct a power series $\Omega (t)\in A^{n-1,n-1}(X_0)$ such that

(41) $$ \begin{align} \begin{cases} d(e^{\iota_{\varphi}|\iota_{\bar{\varphi}}}(\Omega(t)))=0,\\ \Omega(t)=\overline{\Omega(t)},\\ \Omega(0)=\omega^{n-1}, \end{cases} \end{align} $$

and show the Hölder $C^{k,\alpha }$ -convergence and regularity of the power series $\Omega (t)$ . Then it is clear that $e^{\iota _{\varphi }|\iota _{\bar {\varphi }}}(\Omega (t)))$ will be a positive $(n-1,n-1)$ -form on $X_t$ for t small, due to the positivity of its initial complex differential form $\omega ^{n-1}$ and the convergence argument. By the real property of $e^{\iota _{\varphi }|\iota _{\bar {\varphi }}}$ as in [Reference Rao and Zhao43, Lemma $2.8$ ], it suffices to solve the following system of equations

(42) $$ \begin{align} \begin{cases} d(e^{\iota_{\varphi}|\iota_{\bar{\varphi}}}(\Omega(t)))=0,\\ \Omega(0)=\omega^{n-1},\\ \end{cases} \end{align} $$

since $\frac {\Omega (t)+\overline {\Omega (t)}}{2}$ from one solution $\Omega (t)$ of (42) becomes one of the system (41). The resolution of the system (42) below is a bit different from the one for the Kähler case, which relies more on the form type $(n-1,n-1)$ .

As both $e^{\iota _{(\mathbb {1}-\bar {\varphi }\varphi )^{-1}\bar {\varphi }}}$ and $e^{\iota _{\varphi }}$ are invertible operators when t is sufficiently small, it follows that for any $\Omega \in A^{n-1,n-1}(X_0)$ ,

(43)

Set

(44)

where $\Omega $ and $\tilde {\Omega }$ are apparently one-to-one correspondence. And it is easy to check that the operator

preserves the form types and thus $\tilde {\Omega }$ is still an $(n-1,n-1)$ -form. In fact, for any $(p,q)$ -form $\alpha $ on $X_0$ , we will find

where $\alpha =\alpha _{i_1\cdots i_p\overline {j_1}\cdots \overline {j_q}}dz^{i_1}\wedge \cdots \wedge dz^{i_p}\wedge d\bar {z}^{j_1}\wedge \cdots \wedge d\bar {z}^{j_q}$ . Together with (43) and (44), we obtain that

(45)

where Proposition 1.3 is used in the second equality of (45) and the third equality results from the form type of $\tilde {\Omega }$ .

By the invertibility of the operator $e^{\iota _{\varphi }}$ and the form-type comparison, the equation $d(e^{\iota _{\varphi }|\iota _{\bar {\varphi }}}(\Omega ))=0$ amounts to

(46) $$ \begin{align} \begin{cases} (\bar{\partial}+[\partial, \iota_{\varphi}])\tilde{\Omega}=0,\\ \partial\tilde{\Omega}+(\bar{\partial}+\partial\circ \iota_{\varphi})\circ \iota_{(\mathbb{1}-\bar{\varphi}\varphi)^{-1}\bar{\varphi}}(\tilde{\Omega})=0. \end{cases} \end{align} $$

Then the second equation in (46) and Lemma 2.5 imply

(47)

where the form type of $\tilde {\Omega }$ is also used in the fourth equality of (47). Substituting (47) into (46), one obtains that (46) is equivalent to

(48)

For the resolution of $\partial \overline {\partial }$ -equations, we need a lemma due to [Reference Popovici41, Theorem $4.1$ ]:

Lemma 3.15. Let $(X,\omega )$ be a compact Hermitian complex manifold with the pure-type complex differential forms x and y. Assume that the $\partial \overline {\partial }$ -equation

(49) $$ \begin{align} \partial \overline{\partial} x =y \end{align} $$

admits a solution. Then an explicit solution of the $\partial \overline {\partial }$ -equation (49) can be chosen as

$$ \begin{align*}(\partial\overline{\partial})^*\mathbb{G}_{BC}y,\end{align*} $$

which uniquely minimizes the $L^2$ -norms of all the solutions with respect to $\omega $ . Besides, the equalities hold

$$ \begin{align*}\mathbb{G}_{BC}(\partial\overline{\partial}) = (\partial\overline{\partial}) \mathbb{G}_{A}\quad \text{and} \quad (\partial\overline{\partial})^*\mathbb{G}_{BC} = \mathbb{G}_{A}(\partial\overline{\partial})^*,\end{align*} $$

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

$$ \begin{align*}\square_{{A}}=\partial^*\overline{\partial}^*\overline{\partial}\partial+\overline{\partial}\partial\partial^*\overline{\partial}^*+\overline{\partial}\partial^*\partial\overline{\partial}^*+\partial\overline{\partial}^*\overline{\partial}\partial^*+\overline{\partial}\overline{\partial}^*+\partial\partial^*.\end{align*} $$

Proof. We shall use the Hodge decomposition of $\square _{BC}$ on X:

(50) $$ \begin{align} A^{p,q}(X)=\ker\square_{BC}\oplus \text{Im}~(\partial\overline{\partial}) \oplus(\text{Im}~\partial^*+\text{Im}~\overline{\partial}^*), \end{align} $$

whose three parts are orthogonal to each other with respect to the $L^2$ -scalar product defined by $\omega $ , combined with the equality

where $\mathbb {H}_{BC}$ is the harmonic projection operator. And it should be noted that

(51) $$ \begin{align} \ker\square_{BC}=\ker\partial\cap\ker\overline{\partial}\cap\ker(\partial\overline{\partial})^*. \end{align} $$

Then two observations follow:

1. (1) $\square _{BC}\partial \overline {\partial }(\partial \overline {\partial })^*=\partial \overline {\partial }(\partial \overline {\partial })^*\square _{BC}$ .

2. (2) $\mathbb {G}_{BC}\partial \overline {\partial }(\partial \overline {\partial })^*=\partial \overline {\partial }(\partial \overline {\partial })^*\mathbb {G}_{BC}.$

It is clear that (1) implies (2). Actually, (1) yields

$$ \begin{align*} \mathbb{G}_{BC}\square_{BC}\partial\overline{\partial}(\partial\overline{\partial})^*\mathbb{G}_{BC}=\mathbb{G}_{BC}\partial\overline{\partial}(\partial\overline{\partial})^*\square_{BC}\mathbb{G}_{BC}. \end{align*} $$

A routine check shows that

and

while, the statement (1) is proved by a direct calculation:

$$ \begin{align*}\square_{BC}\partial\overline{\partial}(\partial\overline{\partial})^*=(\partial\overline{\partial})(\partial\overline{\partial})^*(\partial\overline{\partial})(\partial\overline{\partial})^*=\partial\overline{\partial}(\partial\overline{\partial})^*\square_{BC}.\end{align*} $$

Hence, one has

where $y \in \text {Im}~\partial \overline {\partial }$ due to the solution-existence of the $\partial \overline {\partial }$ -equation.

To see that the solution $(\partial \overline {\partial })^*\mathbb {G}_{BC}y$ is the unique $L^2$ -norm minimum, we resort to the Hodge decomposition of the operator $\square _{A}$ :

(52) $$ \begin{align} A^{p,q}(X)=\ker\square_{A} \oplus (Im~\partial +Im~\overline{\partial} ) \oplus Im~(\partial\overline{\partial})^*, \end{align} $$

where $\ker \square _{A} = \ker (\partial \overline {\partial }) \cap \ker \partial ^* \cap \ker \overline {\partial }^*$ . Let z be an arbitrary solution of the $\partial \overline {\partial }$ -equation (49), which decomposes into three components $z_1 + z_2 +z_3$ with respect to the Hodge decomposition (52) of $\square _{A}$ . By the Hodge theory of $\square _{A}$ , the equality holds

$$ \begin{align*} \ker (\partial\overline{\partial}) = \ker \square_{A} \oplus ( \text{Im}\partial + \text{Im}\overline{\partial} ) ,\end{align*} $$

which implies that $\partial \overline {\partial } (z_1 + z_2)=0$ . Hence, it follows that

$$ \begin{align*} \partial \overline{\partial} z = \partial \overline{\partial} z_3 =y. \end{align*} $$

After noticing that $\overline {\partial }^* z_3 = \partial ^* z_3 =0$ , we get

$$ \begin{align*} (\partial\overline{\partial})^* y = (\partial\overline{\partial})^* \partial \overline{\partial} z_3 = \square_{A} z_3, \end{align*} $$

which implies that

Then it is obvious that

(53) $$ \begin{align} \square_{BC} (\partial\overline{\partial})=\partial \overline{\partial} (\partial\overline{\partial})^* \partial \overline{\partial}=(\partial\overline{\partial})\square_{A} \end{align} $$

with $\mathbb {G}_{BC}(\partial \overline {\partial })=(\partial \overline {\partial })\mathbb {G}_{A}$ established as well. Taking adjoint operators of both sides in (53), we will find that

$$ \begin{align*} (\partial\overline{\partial})^* \square_{BC} = \square_{A} (\partial\overline{\partial})^*,\end{align*} $$

implying the equality

$$ \begin{align*}(\partial\overline{\partial})^*\mathbb{G}_{BC}=\mathbb{G}_{A}(\partial\overline{\partial})^*.\end{align*} $$

And thus, we obtain that

$$ \begin{align*} z_3 = \mathbb{G}_{A} (\partial\overline{\partial})^* y = (\partial\overline{\partial})^*\mathbb{G}_{BC} y. \end{align*} $$

Therefore,

$$ \begin{align*} \|z\|^2 = \|z_1\|^2 +\|z_2\|^2+\|z_3\|^2\geq \|z_3\|^2 = \|(\partial\overline{\partial})^*\mathbb{G}_{BC}y \|^2, \end{align*} $$

and the equality holds if and only if $z_1 =z_2 =0$ , that is, $z=z_3=(\partial \overline {\partial })^*\mathbb {G}_{BC} y$ .

Now we arrive at:

Proof. It suffices to resolve (48) with initial value $\tilde {\Omega }(0)=\omega ^{n-1}$ . Actually, the solution $\tilde {\Omega }(t)$ of the system (48), satisfying that $\tilde {\Omega }(0)=\omega ^{n-1}$ , corresponds to the one of (42), by the relation below as in (44)

Therefore, we focus on

and solve this system of equations also by the iteration method used in the Kähler case.

Assume that $\tilde {\Omega }(t)$ can develop into a power series as follows:

$$ \begin{align*} \begin{cases} \tilde{\Omega}(t)=\sum_{k=0}^{\infty}\tilde{\Omega}_k,\\ \tilde{\Omega}_k=\sum_{i+j=k}\tilde{\Omega}_{i,j}t^i\bar{t}^j, \end{cases} \end{align*} $$

where $\tilde {\Omega }_k$ is the k-degree homogeneous part in the expansion of $\tilde {\Omega }(t)$ and $\tilde {\Omega }_{i,j}$ are all smooth $(n-1,n-1)$ -forms on $X_0$ . Here, we follow the notation described before Proposition 2.7. And in (37) the holomorphic family of integrable Beltrami differentials on $X_0$ develops a power series of Beltrami differentials in t as

$$ \begin{align*}\varphi(t) = \sum_{i=1}^{\infty}\varphi_{i}t^i. \end{align*} $$

Hence, we need to solve

(54)

for any $k\geq 0$ .

The case $k=0$ of the equations (54) holds since $\omega $ is a balanced metric. By induction, we assume that the system (54) of equations is solved for each $k\leq l$ and the solutions are denoted by $\{\tilde {\Omega }_k\}_{k\leq l}\in A^{n-1,n-1}(X_0)$ . By the form-type consideration, one observes that the $(n-1,n)$ th mild $\partial \overline {\partial }$ -lemma produces $\mu $ and $\nu $ , such that the following equalities

hold as the ones (27) in Observation 2.11. Then Lemma 3.15 enables us to determine the two explicit solutions

(55)

Hence, the $(l+1)$ th order of the system (54) is solved, yielding that

in the same manner as (28), where $\mu $ and $\nu $ are given by (55). Therefore, we complete the proof.

Then we come to the Hölder convergence argument for $\tilde {\Omega }(t)$ . For $l=1,2,\dots $ , one l-degree canonical formal solution of (54) is given by induction:

(56)

and obviously

(57)

In this proof, we follow the notations in Section 2.3.

We use an important a priori estimate for the three terms in the left-hand side of the above equality: for any complex differential form $\phi $ ,

(58) $$ \begin{align} \|\mathbb{G}_{BC}\phi\|_{k, \alpha}\leq C_{k,\alpha}\|\phi\|_{k-4, \alpha}, \end{align} $$

where $k>3$ and $C_{k,\alpha }$ depends on only on k and $\alpha $ , not on $\phi $ . See [Reference Kodaira28, Appendix Theorem $7.4$ ] for example. Assume that

$$ \begin{align*}\|\tilde{\Omega}^{(r-1)}\|_{k, \alpha},\|\tilde{\Omega}^{(r-2)}\|_{k, \alpha},\|\tilde{\Omega}^{(r-3)}\|_{k, \alpha}\ll A(t).\end{align*} $$

So, by the expression (57) and the a priori estimate (58),

and similarly,

Hence, we choose $\beta ,\gamma ,\|\tilde {\Omega }_0\|_{k, \alpha }$ so that the following inequalities hold:

$$ \begin{align*}2\left(\frac{\beta}{\gamma}\right)\!\left(\frac{\beta}{\gamma}+\|\tilde{\Omega}_0\|_{k, \alpha}\right), 2C_1^2C_{k,\alpha}\left(\frac{\beta}{\gamma}+\|\tilde{\Omega}_0\|_{k, \alpha}\right), C_1^2C_{k,\alpha}\left(\frac{\beta}{\gamma}+\|\tilde{\Omega}_0\|_{k, \alpha}\right)\!\left(1+\left(\frac{\beta}{\gamma}\right)^2\right)\! \,{<}\,\frac{1}{3},\end{align*} $$

(38) for the integrable Beltrami differential $\varphi (t)$ and

$$ \begin{align*}\|\tilde{\Omega}^{(1)}\|_{k, \alpha},\|\tilde{\Omega}^{(2)}\|_{k, \alpha},\|\tilde{\Omega}^{(3)}\|_{k, \alpha}\ll A(t)\end{align*} $$

according the above formulation, which are obviously possible as t is small. Thus, we obtain that

$$ \begin{align*}\|\tilde{\Omega}^{(r)}\|_{k, \alpha}\ll A(t),\ \text{for any}\ r\geq 1,\end{align*} $$

which implies the desired convergence $\|\tilde {\Omega }^{(+\infty )}\|_{k, \alpha }\ll A(t)$ immediately.

Finally, inspired by the elliptic argument [Reference Kodaira28, Appendix 8], we will apply the interior estimate to obtain the regularity of $\tilde {\Omega }(t)$ , which is a local problem. By the canonical formal solution expression (56) of (54), one just needs to use the following strongly elliptic second-order pseudo-differential equation

(59)

where $\square $ is the $\overline {\partial }$ -Laplacian defined by (39). Recall that an elliptic partial differential operator of order $2m$ is pseudo-differential and so is its inverse, whose order becomes $-2m$ as a pseudo-differential operator.

We cover $X:=X_0$ by coordinates neighborhoods $X_j$ , $j=1,\dots , J$ . Let $z_j=(z_j^1,\dots ,z_j^n)$ be the local holomorphic coordinates on $X_j$ with $z_j^k=x_j^k+\sqrt {-1}x_j^{k+n}$ and put $t=x^{2n+1}+\sqrt {-1}x^{2n+2}\in \Delta _{\epsilon }$ a $\epsilon $ -disk in $\mathbb {C}$ . By these $2n+2$ real coordinates, $X_j\times \Delta _{\epsilon }$ is identified with an open set $U_j$ of a $(2n+2)$ -dimensional torus $\mathbb {T}^{2n+2}$ .

Choose a partition of unity subordinate to $X_j$ , that is, a set $\{\rho _j\}$ of $C^\infty $ functions $\rho _j(x)$ on X so that $\mathop {\mathrm {sup}}\limits \rho _j\subset X_j$ and for any $x\in X$ , $\sum _{j=1}^{J}\rho _j(x)\equiv 1$ . For each $l=1,2,\dots $ , choose a smooth function $\eta ^l(t)$ with values in $[0,1]$ :

(60) $$ \begin{align} \eta^l(t)\equiv \begin{cases} 1,\ \text{for}\ |t|\leq (\frac{1}{2}+\frac{1}{2^{l+1}})r;\\ 0,\ \text{for}\ |t|\geq (\frac{1}{2}+\frac{1}{2^{l}})r, \end{cases} \end{align} $$

where r is a positive constant to be determined. Notice that r is crucially used to give the uniform bound for the convergence radius of $\tilde {\Omega }(t)$ .

Set

(61) $$ \begin{align} \rho_j^l(x,t)=\rho_j(x)\eta^l(t). \end{align} $$

Recall in the proof for $\varphi (t)$ in [Reference Kodaira28, Appendix 8], one should also set

$$ \begin{align*}\chi_j^l(x,t)=\chi_j(x)\eta^l(t),\end{align*} $$

where the smooth function $\chi _j(x)$ with $\mathop {\mathrm {sup}}\limits \chi _j\subset X_j$ is identically equal to $1$ on some neighborhood of the support of $\rho _j$ . But here we will replace its role directly by $\eta ^l(t)$ to avoid the trouble caused by the presence of Green’s operator $\mathbb {G}_{BC}$ .

First we will prove that $\eta ^3\tilde {\Omega }$ is $C^{k+1,\alpha }$ . Consider the equation:

(62) $$ \begin{align} \square(\triangle^h_i\rho_j^3\tilde{\Omega}) =F_1^1+F_2^1+F_3^1, \end{align} $$

where $F_1^1,F_2^1,F_3^1$ denote the three terms with respect to the ones in the right-hand side of (59) after the corresponding operations, respectively, and $i=1,\dots , 2n$ . Here $\triangle ^h_i$ is the difference quotient as [Reference Kodaira28, Appendix (8.14)]. In particular,

In this proof, the “order” refers to the one of a pseudo-differential operator. Since $\square $ is an elliptic linear differential operator whose principal part is of diagonal type, by [Reference Kodaira28, Appendix Theorem 2.3] and (62), one obtains the a priori estimate

(63) $$ \begin{align} \begin{aligned} \|\triangle^h_i\rho_j^3\tilde{\Omega}\|_{k, \alpha} &\leq C_k(\| F_1^1+F_2^1+F_3^1 \|_{k-2, \alpha}+\|\triangle^h_i\rho_j^3\tilde{\Omega}\|_{0})\\ &\leq C_k(\| F_1^1\|_{k-2, \alpha}+\| F_2^1\|_{k-2, \alpha}+\| F_3^1 \|_{k-2, \alpha}+\|\triangle^h_i\rho_j^3\tilde{\Omega}\|_{0}), \end{aligned} \end{align} $$

where $C_k$ is a positive constant, possibly depending on k. Now let us estimate the first three terms in the right-hand side of the above inequality. Here, we just estimate the second one, the most troublesome one, since the other two terms are quite analogous.

We need an equality for the Green’s operator: Let $E:=E(x, D)$ be an elliptic linear partial differential operator on a smooth manifold and $G_E$ , $\mathbb {H}_E$ its associated Green’s operator and orthogonal projection to $\ker E$ defined such as in [Reference Kodaira28, Appendix Definition 7.2]. Then for any smooth function f and differential form $\varpi $ on this manifold,

(64) $$ \begin{align} fG_E\varpi=G_E(f\varpi)-G_E(f\mathbb{H}_E\varpi)+\mathbb{H}_E(fG_E\varpi)+\text{lower-order terms of}\ \varpi. \end{align} $$

Applying this equality to $\rho _j^3\overline {\partial } \overline {\partial }^*\overline {\partial }(\partial \overline {\partial })^* \mathbb {G}_{BC} \overline {\partial } \Big ( \iota _{(\mathbb {1}-\bar {\varphi }\varphi )^{-1}\bar {\varphi }} \big ( \tilde {\Omega }(t) \big ) \Big )$ with $E=\square _{BC}$ , $f=\rho _j^3$ and $\varpi =\overline {\partial } \Big ( \iota _{(\mathbb {1}-\bar {\varphi }\varphi )^{-1}\bar {\varphi }} \big ( \tilde {\Omega }(t) \big ) \Big )$ , one obtains

(65)

where we use the fact that $\overline {\partial } \Big ( \iota _{(\mathbb {1}-\bar {\varphi }\varphi )^{-1}\bar {\varphi }} \big ( \tilde {\Omega }(t) \big ) \Big )$ is $\partial \overline {\partial }$ -exact and the equalities (50) and (51). Thus, by (65) and the useful formula

(66) $$ \begin{align} \eta^{2l-1}\cdot\eta^{2l+1}=\eta^{2l+1},\ l=1,2,\dots, \end{align} $$

one gets the estimate on the first term of $F_2^1$ :

where $L,L'$ are positive numbers. Hence, one obtains:

(67) $$ \begin{align} \| F_2^1\|_{k-2, \alpha}\leq M_2^1A(r)\|\triangle^h_i(\rho_j^3\tilde{\Omega}(t))\|_{k, \alpha} +M_k\|\eta^3\tilde{\Omega}(t)\|_{k, \alpha}\|\eta^1\varphi\|_{k, \alpha}, \end{align} $$

where $M_2^1,M_k$ are positive numbers. Similarly, we are able to get the analogous estimates for $F_1^1$ and $F_3^1$ with the positive constants $M_1^1$ and $M_3^1$ . Thus,

(68) $$ \begin{align} \begin{aligned} &\| \triangle^h_i(\rho_j^3\tilde{\Omega}(t))\|_{k, \alpha}\\ &\quad\leq C_k(M_1^1+M_2^1+M_3^1)A(r)\|\triangle^h_i(\rho_j^3\tilde{\Omega}(t))\|_{k, \alpha}+C_k\|\rho_j^3\tilde{\Omega}(t)\|_{1}+C_kM_k\|\eta^3\tilde{\Omega}(t)\|_{k, \alpha}\|\eta^1\varphi\|_{k, \alpha}. \end{aligned} \end{align} $$

Choose a sufficiently small r such that

(69) $$ \begin{align} C_k(M_1^1+M_2^1+M_3^1)A(r)\leq \frac{1}{2} \end{align} $$

and if $|t|<r$

(70) $$ \begin{align} \tilde{\Omega}(t), \varphi(t) \in C^{k, \alpha}. \end{align} $$

Then

$$ \begin{align*} \eta^3\tilde{\Omega}(t), \eta^1\varphi(t) \in C^{k, \alpha}. \end{align*} $$

Therefore, by (68), one knows that

$$ \begin{align*}\| \triangle^h_i(\rho_j^3\tilde{\Omega}(t))\|_{k, \alpha} \leq 2C_k\|\rho_j^3\tilde{\Omega}(t)\|_{1}+2C_kM_k\|\rho_j^3\tilde{\Omega}(t)\|_{k, \alpha}\|\eta^1\varphi\|_{k, \alpha}, \end{align*} $$

where the right-hand side is bounded and independent of h. Hence, we have proved

$$ \begin{align*}\rho_j^3\tilde{\Omega}(t)\in C^{k+1, \alpha}\end{align*} $$

by [Reference Kodaira28, Appendix Lemma 8.2.(iii)]. Summing with respect j, one knows that $\eta ^3\tilde {\Omega }(t)\in C^{k+1, \alpha }$ by (61).

Next, we shall prove that $\eta ^5\tilde {\Omega }$ is $C^{k+2,\alpha }$ . Consider the equation:

$$ \begin{align*} \square(\triangle^h_i D_\beta(\rho_j^5\tilde{\Omega})) =F_1^2+F_2^2+F_3^2, \end{align*} $$

where $D_\beta =\frac {\partial \ }{\partial x^\beta }$ and $F_1^2,F_2^2,F_3^2$ denote the three terms with respect to the ones in the right-hand side of (59) after the corresponding operations, respectively. In particular,

$$ \begin{align*}F_2^2=D_\beta F_2^1+\text{lower-order terms of}\ \tilde{\Omega}(t).\end{align*} $$

By [Reference Kodaira28, Appendix Theorem 2.3], one obtains the a priori estimate

(71) $$ \begin{align} \begin{aligned} \|\triangle^h_iD_\beta(\rho_j^5\tilde{\Omega})\|_{k, \alpha} &\leq C_k(\| F_1^2+F_2^2+F_3^2 \|_{k-2, \alpha}+\|\triangle^h_iD_\beta(\rho_j^5\tilde{\Omega})\|_{0})\\ &\leq C_k(\| F_1^2\|_{k-2, \alpha}+\| F_2^2\|_{k-2, \alpha}+\| F_3^2 \|_{k-2, \alpha}+\|\triangle^h_iD_\beta(\rho_j^5\tilde{\Omega})\|_{0}), \end{aligned} \end{align} $$

where $C_k$ is the same as in (63). Now let us estimate the first three terms in the right-hand side of the above inequality. Here, we just estimate the second one since the other two terms are quite analogous. We use the equality (64) for the Green’s operator again. Then one gets the estimate on the first term of $F_2^2$ :

where $L,L''$ are positive numbers. Hence, one obtains:

(72) $$ \begin{align} \| F_2^2\|_{k-2, \alpha}\leq M_2^1A(r)\|\triangle^h_iD_\beta(\rho_j^5\tilde{\Omega}(t))\|_{k, \alpha} +M_{k+1}\|\eta^5\tilde{\Omega}(t)\|_{k+1, \alpha}\|\eta^3\varphi\|_{k+1, \alpha}, \end{align} $$

where $M_2^1,M_{k+1}$ are positive numbers. Crucially, $M_2^1$ is exactly the same as in (67), since the range of the function $\eta ^l(t)$ is $[0,1]$ by (60), although $M_{k+1}$ is possibly different from $M_k$ in (67). Similarly, we are able to get the analogous estimates for $F_1^2$ and $F_3^2$ with the positive constants $M_1^1$ and $M_3^1$ , which are the same as in the above argument for $\eta ^3\tilde {\Omega }(t)\in C^{k+1, \alpha }$ . Thus, by (71) and (72),

(73) $$ \begin{align} \begin{aligned} &\| \triangle^h_iD_\beta(\rho_j^5\tilde{\Omega}(t))\|_{k, \alpha}\\ &\quad\leq C_k(M_1^1+M_2^1+M_3^1)A(r)\|\triangle^h_iD_\beta(\rho_j^5\tilde{\Omega}(t))\|_{k, \alpha}+C_k\|\rho_j^5\tilde{\Omega}(t)\|_{2}\\&\qquad+C_kM_{k+1}\|\eta^5\tilde{\Omega}(t)\|_{k+1, \alpha}\|\eta^3\varphi\|_{k+1, \alpha}. \end{aligned} \end{align} $$

Choose the same sufficiently small r as in the above argument for $\eta ^3\tilde {\Omega }(t)\in C^{k+1, \alpha }$ , given by (69) and (70). Therefore, by (73),

$$ \begin{align*}\| \triangle^h_iD_\beta(\rho_j^5\tilde{\Omega}(t))\|_{k, \alpha} \leq 2C_k\|\rho_j^5\tilde{\Omega}(t)\|_{2}+2C_kM_{k+1}\|\eta^5\tilde{\Omega}(t)\|_{k+1, \alpha}\|\eta^3\varphi\|_{k+1, \alpha}, \end{align*} $$

where the right-hand side is bounded when we use the formula (66), $\eta ^3\tilde {\Omega }(t)\in C^{k+1, \alpha }$ that we just proved in the above argument and also the fact that $\eta ^3\varphi (t)\in C^{k+1, \alpha }$ proved in [Reference Kodaira28, Appendix 8]. Hence, we have proved

$$ \begin{align*}D_\beta(\rho_j^5\tilde{\Omega}(t))\in C^{k+1, \alpha}\end{align*} $$

by [Reference Kodaira28, Appendix Lemma 8.2.(iii)] and thus

$$ \begin{align*}\rho_j^5\tilde{\Omega}(t)\in C^{k+2, \alpha}.\end{align*} $$

Summing with respect j, one obtains that $\eta ^5\tilde {\Omega }(t)\in C^{k+2, \alpha }$ by (61). Notice that in this procedure r has not been replaced.

We can also prove that, for any $l=1,2,\dots $ , $\eta ^{2l+1}\tilde {\Omega }$ is $C^{k+l, \alpha }$ , where r can be chosen independent of l. Since $\eta ^{2l+1}(t)$ is identically equal to $1$ on $|t|<\frac {r}{2}$ which is independent of l, $\tilde {\Omega }(t)$ is $C^{\infty }$ on $X_0$ with $|t|<\frac {r}{2}$ . Then $\tilde {\Omega }(t)$ can be considered as a real analytic family of $(n-1,n-1)$ -forms in t and it is smooth on t by [Reference Krantz and Parks30, Proposition 2.2.3] again.

4.1 Obstruction of extension for d-closed and $\partial \overline {\partial }$ -closed forms

Inspired by Wu’s result [Reference Wu55, Theorem 5.13], one has:

Proof. By use of the extension map $e^{\iota _{\varphi }|\iota _{\overline {\varphi }}}$ , we can construct two $(r,s)$ -forms $e^{\iota _{\varphi }|\iota _{\overline {\varphi }}}(\Omega _0)$ and $e^{\iota _{\varphi }|\iota _{\overline {\varphi }}}(\Psi _0)$ on $X_t$ , starting with $\Omega _0$ and $\Psi _0$ , respectively.

Let $F_t$ be the orthogonal projection to $\mathbb {F}_t$ , the kernel of

$$ \begin{align*}\square_{{BC},t}=\partial_t\overline{\partial}_t\overline{\partial}_t^*\partial_t^*+\overline{\partial}_t^*\partial_t^*\partial_t\overline{\partial}_t+\overline{\partial}_t^*\partial_t\partial_t^*\overline{\partial}_t+\partial_t^*\overline{\partial}_t\overline{\partial}_t^*\partial_t+\overline{\partial}_t^*\overline{\partial}_t+\partial_t^*\partial_t\end{align*} $$

and $\mathbb {G}_t$ denote the associated Green’s operator with respect to a smooth family of Hermitian metrics on $X_t$ . Then $F_t$ and $\mathbb {G}_t$ are $C^{\infty }$ differentiable in t since the $\dim \mathbb {F}_t$ is deformation invariant, thanks to [Reference Wu55, Theorem 5.12]. Therefore, the desired d-closed $(r,s)$ -form is

(74) $$ \begin{align} \Omega_t =\big(\partial_t\overline{\partial}_t\overline{\partial}_t^*\partial_t^*\mathbb{G}_t+F_t\big)\big(e^{\iota_{\varphi}|\iota_{\overline{\varphi}}}(\Omega_0)\big) \end{align} $$

as long as one notices that

$$ \begin{align*}\left.\Omega_t\right|{}_{t=0} =\big(\square_{{BC},0}\mathbb{G}_0+F_0\big)\Omega_0=\Omega_0\end{align*} $$

by the Hodge decomposition of the operator $\square _{{BC},0}$ and the d-closedness of $\Omega _0$ .

The construction of $\Psi _t$ is quite similar to the one of $\Omega _t$ in (74). Denote by $\widetilde {F}_t$ the orthogonal projection to $\widetilde {\mathbb {F}}_t$ , the kernel of

$$ \begin{align*} \square_{A,t}= \partial_t\overline{\partial}_t\overline{\partial}_t^*\partial_t^*+\overline{\partial}_t^*\partial_t^*\partial_t\overline{\partial}_t +\partial_t\overline{\partial}_t^*\overline{\partial}_t\partial_t^*+\overline{\partial}_t\partial_t^*\partial_t\overline{\partial}_t^*+\overline{\partial}_t\overline{\partial}_t^*+\partial_t\partial_t^* \end{align*} $$

and by $\widetilde {\mathbb {G}}_t$ the associated Green’s operator with respect to a smooth family of Hermitian metrics on $X_t$ . By the same token, $\widetilde {F}_t$ and $\widetilde {\mathbb {G}}_t$ are also $C^{\infty }$ differentiable in t since the $\dim \widetilde {\mathbb {F}}_t$ is deformation invariant. Then the construction of $\Psi _t$ goes as follows:

$$ \begin{align*} \Psi_t =\Big(\big(\partial_t\overline{\partial}_t\overline{\partial}_t^*\partial_t^*+\partial_t\overline{\partial}_t^*\overline{\partial}_t\partial_t^*+ \overline{\partial}_t\partial_t^*\partial_t\overline{\partial}_t^*+\overline{\partial}_t\overline{\partial}_t^*+\partial_t\partial_t^*\big)\widetilde{\mathbb{G}}_t+\widetilde{F}_t\Big) \Big(e^{\iota_{\varphi}|\iota_{\overline{\varphi}}}(\Psi_0)\Big), \end{align*} $$

where it is easy to see that

$$ \begin{align*} \left.\Psi_t\right|{}_{t=0}= \big( \square_{A,0}\widetilde{\mathbb{G}}_0 + \widetilde{F}_0 \big) \Psi_0 = \Psi_0. \\[-37pt]\end{align*} $$

We can also prove this proposition by another way inspired by the results of [Reference Angella and Ugarte10, Reference Fu and Yau23, Reference Wu55] in the $(n-1,n-1)$ -case, that is, any d-closed $(n-1,n-1)$ -form $\Omega $ on a complex manifold X satisfying the $\partial \overline {\partial }$ -lemma can be extended unobstructed to a d-closed $(n-1,n-1)$ -form on the small differentiable deformation $X_t$ of X.

Let $f_t: X_t\rightarrow X_0$ be a diffeomorphism depending on t with $f_0=$ identity. And then one obtains a d-closed $(2n-2)$ -form $\Omega _t=f_t^*\Omega $ on $X_t$ , which is decomposed as

$$ \begin{align*}\Omega_t=\Omega_t^{n-2,n}+\Omega_t^{n-1,n-1}+\Omega_t^{n,n-2}\end{align*} $$

with respect to the complex structure on $X_t$ . It is easy to check the following properties:

1. (1) $\Omega _t^{n-1,n-1}$ approaches to $\Omega $ as $t\rightarrow 0$ .

2. (2) $\Omega _t^{n-2,n}$ and $\Omega _t^{n,n-2}$ approach to 0 as $t\rightarrow 0$ .

Recall that [Reference Wu55, Theorem 5.12] or [Reference Angella and Tomassini8, Corollary 3.7] says that if $X_0$ satisfies the $\partial \overline {\partial }$ -lemma, so does the general fiber $X_t$ . So we can choose an $(n-2,n-1)$ -form $\Psi _1$ and an $(n-1,n-2)$ -form $\Psi _2$ on $X_t$ such that

$$ \begin{align*}\partial_t\overline{\partial}_t \Psi_1=\overline{\partial}_t \Omega_t^{n-1,n-1}=-\partial_t \Omega_t^{n-2,n},\end{align*} $$

$$ \begin{align*}-\partial_t\overline{\partial}_t \Psi_2=\partial_t \Omega_t^{n-1,n-1}=-\overline{\partial}_t \Omega_t^{n,n-2},\end{align*} $$

where $\Psi _1$ and $\Psi _2$ can be set as

$$ \begin{align*}\Psi_1,\Psi_2\bot_{\omega_t} \ker(\partial_t\overline{\partial}_t).\end{align*} $$

Put

$$ \begin{align*}\widetilde{\Omega}_t=\Omega_t^{n-1,n-1}+ \partial_t\Psi_1+\overline{\partial}_t \Psi_2.\end{align*} $$

Obviously, $\widetilde {\Omega }_t$ is a d-closed $(n-1,n-1)$ -form on $X_t$ . Then Fu–Li–Yau proved the following highly nontrivial estimates in [Reference Fu, Li and Yau22, Sections $4$ and $5$ ]: for some $0<\alpha <1$

$$ \begin{align*}\|\partial_t\Psi_1\|_{C^0(\omega_t)}\leq C\|\partial_t \Omega_t^{n-2,n}\|_{C^{0,\alpha}(\omega_t)}\end{align*} $$

and similarly

$$ \begin{align*}\|\overline{\partial}_t \Psi_2\|_{C^0(\omega_t)}\leq C\|\overline{\partial}_t \Omega_t^{n,n-2}\|_{C^{0,\alpha}(\omega_t)},\end{align*} $$

where C is a uniform constant. By use of these two estimates, one knows that $\widetilde {\Omega }_t$ is indeed d-closed extension of $\Omega $ since $\partial _t \Omega _t^{n-2,n}$ and $\overline {\partial }_t\Omega _t^{n,n-2}$ approaches to zero uniformly as $t\rightarrow 0$ .

It is easy to see that Theorem 1.5 is impossible to obtain by Fu–Yau’s result since the proof would rely on the deformation openness of $(n-1,n)$ th mild $\partial \overline {\partial }$ -lemma, which contradicts with Ugarte–Villacampa’s Example 3.14.

4.2 Un-obstruction of extension for transverse forms

In this subsection, we study some basic properties and local stabilities of p-Kähler structures, which seem more pertinent to the nature of the stability problem of complex structures.

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