2.1 Hodge theory for Bott–Chern and Aeppli cohomology

We first recall some basic facts about Hodge theory which will be used later (see [Reference SchweitzerSch07, Reference PopoviciPop15] for more detailed discussions). Let X be a compact complex manifold equipped with a fixed Hermitian metric, then we have the Hodge star operator

$$\begin{align*}*:A^{p,q}(X)\to A^{n-q,n-p}(X). \end{align*}$$

The Bott–Chern/Aeppli cohomology of X is defined by

(2.1) $$ \begin{align} H^{p,q}_{BC}(X) := \frac{\ker d \cap A^{p,q}(X)}{{ \mathrm{im}\,} { \partial }{ \bar{\partial} } \cap A^{p,q}(X)},\quad H^{p,q}_{A}(X) := \frac{\ker { \partial }{ \bar{\partial} } \cap A^{p,q}(X)}{({ \mathrm{im}\,} { \partial }+ { \mathrm{im}\,} { \bar{\partial} }) \cap A^{p,q}(X)}. \end{align} $$

2.1.1 Bott–Chern/Aeppli Laplacians and Hodge decompositions

The Bott–Chern/Aeppli Laplacian operator of X is defined respectively as

(2.2) $$ \begin{align} \begin{aligned} \square_{BC}&:=({ \partial }{ \bar{\partial} })({ \partial }{ \bar{\partial} })^{*}+({ \partial }{ \bar{\partial} })^{*}({ \partial }{ \bar{\partial} })+ ({ \bar{\partial} }^{*}{ \partial })({ \bar{\partial} }^{*}{ \partial })^{*}+({ \bar{\partial} }^{*}{ \partial })^{*}({ \bar{\partial} }^{*}{ \partial })+ { \bar{\partial} }^{*}{ \bar{\partial} } + { \partial }^{*}{ \partial },\\ \square_{A}&:=({ \partial }{ \bar{\partial} })^{*}({ \partial }{ \bar{\partial} })+({ \partial }{ \bar{\partial} })({ \partial }{ \bar{\partial} })^{*}+ ({ \bar{\partial} }{ \partial }^{*})^{*}({ \bar{\partial} }{ \partial }^{*})+({ \bar{\partial} }{ \partial }^{*})({ \bar{\partial} }{ \partial }^{*})^{*}+ { \bar{\partial} }{ \bar{\partial} }^{*} + { \partial }{ \partial }^{*}, \end{aligned} \end{align} $$

where ${ \partial }^{*}=-*{ \bar {\partial } }*$ , ${ \bar {\partial } }^{*}=-*{ \partial }*$ and $({ \partial }{ \bar {\partial } })^{*}=(-1)^{p+q+1}*{ \partial }{ \bar {\partial } }*$ on $A^{p,q}(X)$ . Set

(2.3) $$ \begin{align} \begin{aligned} \mathcal{H}^{p,q}_{BC}(X):&=\ker\square_{BC}\cap A^{p,q}(X)=\ker d\cap\ker ({ \partial }{ \bar{\partial} })^{*}\cap A^{p,q}(X),\\ \mathcal{H}^{p,q}_{A}(X):&=\ker\square_{A}\cap A^{p,q}(X)=\ker{ \partial }{ \bar{\partial} }\cap\ker d^{*}\cap A^{p,q}(X). \end{aligned} \end{align} $$

Then the Hodge star operator induces an isomorphism

$$\begin{align*}*:H^{p,q}_{BC}(X)\cong\mathcal{H}^{p,q}_{BC}(X)\longrightarrow\mathcal{H}^{n-q,n-p}_{A}(X)\cong H^{n-q,n-p}_{A}(X). \end{align*}$$

Since $\square _{BC}, \square _{A}$ are elliptic, the following orthogonal direct sum decompositions hold:

(2.4) $$ \begin{align} \begin{aligned} A^{\bullet,\bullet}(X)&=\ker\square_{BC}\oplus { \mathrm{im}\,}{ \partial }{ \bar{\partial} }\oplus ({ \mathrm{im}\,}{ \partial }^{*}+{ \mathrm{im}\,}{ \bar{\partial} }^{*})\\ &=\ker\square_{A}\oplus { \mathrm{im}\,}({ \partial }{ \bar{\partial} })^{*}\oplus ({ \mathrm{im}\,}{ \partial }+{ \mathrm{im}\,}{ \bar{\partial} }), \end{aligned} \end{align} $$

which is equivalent to the existence of the (Bott–Chern/Aeppli) Green operator $G_{BC}, G_{A}$ , such that

(2.5) $$ \begin{align} 1=\mathcal{H}_{BC}+\square_{BC}G_{BC}=\mathcal{H}_{A}+\square_{A}G_{A}, \end{align} $$

where $\mathcal {H}_{BC}, \mathcal {H}_{A}$ are linear projections onto corresponding harmonic spaces.

2.1.2 Deformed Bott–Chern/Aeppli Laplacians and Hodge decompositions

Now, let $\pi : (\mathcal {X}, X)\to (B,0)$ be a complex analytic family with Beltrami differentials $\phi (t)\in A^{0,1}(X,T^{1,0})$ . We have the deformed version of Bott–Chern/Aeppli Laplacian operators

(2.6) $$ \begin{align} \begin{aligned} \square_{BC\phi(t)}&:=[{ \partial }{ \bar{\partial} }_{\phi(t)},({ \partial }{ \bar{\partial} }_{\phi(t)})^{*}]+ [{ \bar{\partial} }_{\phi(t)}^{*}{ \partial },({ \bar{\partial} }_{\phi(t)}^{*}{ \partial })^{*}]+ { \bar{\partial} }_{\phi(t)}^{*}{ \bar{\partial} }_{\phi(t)} + { \partial }^{*}{ \partial },\\ \square_{A\phi(t)}&:=[{ \partial }{ \bar{\partial} }_{\phi(t)},({ \partial }{ \bar{\partial} }_{\phi(t)})^{*}]+ [({ \bar{\partial} }_{\phi(t)}{ \partial }^{*})^{*},{ \bar{\partial} }_{\phi(t)}{ \partial }^{*}]+ { \bar{\partial} }_{\phi(t)}{ \bar{\partial} }_{\phi(t)}^{*} + { \partial }{ \partial }^{*}. \end{aligned} \end{align} $$

Since $\square _{BC\phi (t)}, \square _{A\phi (t)}$ are still elliptic, deformed versions of (2.4) and (2.5) also holds:

(2.7) $$ \begin{align} \begin{aligned} A^{\bullet,\bullet}(X)&=\ker\square_{BC\phi(t)}\oplus { \mathrm{im}\,}{ \partial }{ \bar{\partial} }_{\phi(t)}\oplus ({ \mathrm{im}\,}{ \partial }^{*}+{ \mathrm{im}\,}{ \bar{\partial} }_{\phi(t)}^{*})\\ &=\ker\square_{A\phi(t)}\oplus { \mathrm{im}\,}({ \partial }{ \bar{\partial} }_{\phi(t)})^{*}\oplus ({ \mathrm{im}\,}{ \partial }+{ \mathrm{im}\,}{ \bar{\partial} }_{\phi(t)}). \end{aligned} \end{align} $$

Set

(2.8) $$ \begin{align} \begin{aligned} \mathcal{H}^{p,q}_{BC\phi(t)}(X):&=\ker\square_{BC\phi(t)}\cap A^{p,q}(X)=\ker d_{\phi(t)}\cap\ker ({ \partial }{ \bar{\partial} }_{\phi(t)})^{*}\cap A^{p,q}(X),\\ \mathcal{H}^{p,q}_{A\phi(t)}(X):&=\ker\square_{A\phi(t)}\cap A^{p,q}(X)=\ker{ \partial }{ \bar{\partial} }_{\phi(t)}\cap\ker d^{*}_{\phi(t)}\cap A^{p,q}(X). \end{aligned} \end{align} $$

2.1.3 Formal adjoint of ${ \partial }{ \bar {\partial } }_{\phi }$

Proposition 2.1 For any $\phi \in A^{0,1}(X, T^{1,0})$ , let $i_{\phi }^{*}, (\mathcal {L}_{\phi }^{1,0})^{*}, { \bar {\partial } }_{\phi }^{*}, ({ \partial }{ \bar {\partial } }_{\phi })^{*}$ be the formal adjoint of $i_{\phi }, \mathcal {L}_{\phi }^{1,0}, { \bar {\partial } }_{\phi }={ \bar {\partial } }-\mathcal {L}_{\phi }^{1,0}, { \partial }{ \bar {\partial } }_{\phi }$ , respectively. Then the following:

• • $i_{\phi }^{*}=(-1)^{p+q+1}*(i_{\bar {\phi }})*$ ;

• • $(\mathcal {L}_{\phi }^{1,0})^{*}=-*(\mathcal {L}_{\bar {\phi }}^{0,1})*$ ;

• • ${ \bar {\partial } }_{\phi }^{*}=-*({ \partial }_{\bar {\phi }})*$ ;

• • $({ \partial }{ \bar {\partial } }_{\phi })^{*}=(-1)^{p+q+1}*({ \partial }_{\bar {\phi }}{ \bar {\partial } })*$ ,

holds on $A^{p,q}(X)$ where

$$\begin{align*}{ \partial }_{\bar{\phi}}:={ \partial }-\mathcal{L}_{\bar{\phi}}^{0,1}={ \partial }-(i_{\bar{\phi}}{ \bar{\partial} }-{ \bar{\partial} } i_{\bar{\phi}}). \end{align*}$$

Proof Let $\xi \in A^{p+1,q-1}(X)$ and $\eta \in A^{p,q}(X)$ , then $0=\xi {\,\wedge \,}\overline {*\eta }$ . As a result,

$$\begin{align*}0=i_{\phi}(\xi{\,\wedge\,}\overline{*\eta})=i_{\phi}\xi{\,\wedge\,}\overline{*\eta}+ \xi{\,\wedge\,} i_{\phi}\overline{*\eta}\Longrightarrow i_{\phi}\xi{\,\wedge\,}\overline{*\eta}=-\xi{\,\wedge\,} i_{\phi}\overline{*\eta}=-\xi{\,\wedge\,} \overline{**^{-1}i_{\bar{\phi}}*\eta}, \end{align*}$$

which implies

$$\begin{align*}(i_{\phi}\xi,\eta)_{L^2}=\int_X i_{\phi}\xi{\,\wedge\,}\overline{*\eta}=-\int_X \xi{\,\wedge\,} i_{\phi}\overline{*\eta}=-\int_X \xi{\,\wedge\,} \overline{**^{-1}i_{\bar{\phi}}*\eta}=-(\xi,*^{-1}i_{\bar{\phi}}*\eta)_{L^2}. \end{align*}$$

Hence,

$$\begin{align*}i_{\phi}^{*}\eta=-*^{-1}i_{\bar{\phi}}*\eta=(-1)^{p+q+1}*i_{\bar{\phi}}*\eta. \end{align*}$$

For $\mathcal {L}_{\phi }^{1,0}=i_{\phi }{ \partial }-{ \partial } i_{\phi }$ , we have

$$ \begin{align*} (\mathcal{L}_{\phi}^{1,0})^{*}&=(i_{\phi}{ \partial }-{ \partial } i_{\phi})^{*}\\ &={ \partial }^{*} i_{\phi}^{*}-i_{\phi}^{*}{ \partial }^{*}\\ &=-*{ \bar{\partial} }*(-1)^{p+q+1}*~i_{\bar{\phi}}~*-(-1)^{p+q}*~i_{\bar{\phi}}~*(-1)*{ \bar{\partial} }*\\ &=*{ \bar{\partial} } i_{\bar{\phi}}*-*i_{\bar{\phi}}{ \bar{\partial} }*=-*\mathcal{L}_{\bar{\phi}}^{0,1}*. \end{align*} $$

It follows that

$$ \begin{align*} { \bar{\partial} }_{\phi}^{*}&=({ \bar{\partial} }-\mathcal{L}_{\phi}^{1,0})^{*}\\ &={ \bar{\partial} }^{*}-(\mathcal{L}_{\phi}^{1,0})^{*}\\ &=-*{ \partial }*~+~*\mathcal{L}_{\bar{\phi}}^{0,1}*=-*({ \partial }_{\bar{\phi}})*, \end{align*} $$

and

$$\begin{align*}({ \partial }{ \bar{\partial} }_{\phi})^{*}={ \bar{\partial} }_{\phi}^{*}{ \partial }^{*}=*{ \partial }_{\bar{\phi}}**{ \bar{\partial} }*=(-1)^{p+q+1}*({ \partial }_{\bar{\phi}}{ \bar{\partial} })*.\\[-34pt] \end{align*}$$

As a direct consequence, we have the following corollary.

Corollary 2.2 The following homomorphism

$$\begin{align*}\mathcal{H}^{p,q}_{BC\phi(t)}(X)\to\mathcal{H}^{n-p,n-q}_{A\phi(t)}(X):\varphi\longmapsto \overline{*\varphi}, \end{align*}$$

is an isomorphism.

Proof This follows since $*(\ker d_{\phi (t)})=\ker d_{\overline {\phi (t)}}^{*}$ and $*(\ker ({ \partial }{ \bar {\partial } }_{\phi (t)})^{*})=\ker { \bar {\partial } }{ \partial }_{\overline {\phi (t)}}$ .

Hence, the duality between deformed Bott–Chern and Aeppli cohomology becomes

(2.9) $$ \begin{align} H^{p,q}_{BC\phi(t)}(X)\cong H^{n-p,n-q}_{A\phi(t)}(X). \end{align} $$

2.2 $d^{*}$ -closed representatives of the deformed Aeppli cohomology

Proof $(1)$ follows from the Hodge decomposition:

$$\begin{align*}\sigma = \mathcal{H}_{A}\sigma + G_{A}\square_{A} \sigma = \mathcal{H}_{A}\sigma + G_{A}({ \partial }\bar{{ \partial }})^{*}{ \partial }\bar{{ \partial }} \sigma = \mathcal{H}_{A}\sigma + G_{A}({ \partial }\bar{{ \partial }})^{*}{ \partial } i_{\phi(t)}{ \partial } \sigma~, \end{align*}$$

where we have used the fact that $\square _{A} \sigma =({ \partial }\bar {{ \partial }})^{*}{ \partial }\bar {{ \partial }} \sigma $ if ${ \partial }^{*}\sigma ={ \bar {\partial } }^{*}\sigma =0$ . $(2)$ Substitute $\sigma =\sigma (t)=\sum _{k}\sigma _k$ in (2.10), we have

(2.11) $$ \begin{align} \left\{ \begin{array}{ll} \sigma_1 &= G_{A}({ \partial }\bar{{ \partial }})^{*} { \partial } i_{\phi_1}{ \partial } \sigma_0, \\ \sigma_2 &= G_{A}({ \partial }\bar{{ \partial }})^{*} ({ \partial } i_{\phi_2}{ \partial } \sigma_0 + { \partial } i_{\phi_1}{ \partial } \sigma_1 ),\\ &\cdots, \\ \sigma_k &= G_{A}({ \partial }\bar{{ \partial }})^{*}\sum_{i+j=k} { \partial } i_{\phi_j} { \partial }\sigma_i ,~~\forall k> 0. \\ \end{array} \right. \end{align} $$

For the convergence of $\sigma (t)$ , we note that

(2.12) $$ \begin{align} \|\sigma_{j}\|_{k+\alpha}=\| G_{A}({ \partial }\bar{{ \partial }})^{*}\sum_{a+b=j} { \partial } i_{\phi_a}{ \partial } \sigma_b \|_{k+\alpha} \leq C \sum_{a+b=j} \|\phi_{a}\|_{k+\alpha} \|\sigma_{b}\|_{k+\alpha}, \end{align} $$

for some constant C depends only on k and $\alpha $ . Now it is left to show the uniqueness. Let $\sigma $ and $\sigma '$ be two solutions to $\sigma =\sigma _0+G_{A}({ \partial }\bar {{ \partial }})^{*} { \partial } i_{\phi }{ \partial } \sigma $ and set $\tau =\sigma -\sigma '$ . Then $\tau =G_{A}({ \partial }\bar {{ \partial }})^{*} { \partial } i_{\phi }{ \partial } \tau $ , we have

(2.13) $$ \begin{align} \|\tau\|_{k+\alpha}\leq c \|\phi (t)\|_{k+\alpha} \|\tau\|_{k+\alpha}, \end{align} $$

for some constant $c>0$ . When $|t|$ is sufficiently small, $\|\phi (t)\|_{k+\alpha }$ is also small. Hence we must have $\tau =0$ . For smoothness of the solution, note that we have

$$\begin{align*}\square_{A}\sigma=({ \partial }\bar{{ \partial }})^{*}{ \partial }\bar{{ \partial }} \sigma=({ \partial }\bar{{ \partial }})^{*}{ \partial } i_{\phi(t)}{ \partial } \sigma, \end{align*}$$

which implies

(2.14) $$ \begin{align} \square_{A}\sigma - ({ \partial }\bar{{ \partial }})^{*}{ \partial } i_{\phi(t)}{ \partial }\sigma= 0, \end{align} $$

which is a standard elliptic equation for small t.

$(3)$ Assume $\sigma \in \ker { \partial }{ \bar {\partial } }_{\phi (t)}\cap { \mathrm {im}\,}({ \partial }\bar {{ \partial }})^{*}$ . Then we have $\mathcal {H}_{A}\sigma =0$ and $\sigma =\mathcal {H}_{A}\sigma + G_{A}({ \partial }\bar {{ \partial }})^{*}{ \partial } i_{\phi (t)}{ \partial } \sigma $ by $(1)$ . It follows from $(2)$ that $\sigma =0$ .

Similarly, assume $\sigma \in \ker d_{\phi (t)}\cap { \mathrm {im}\,} d^{*}$ . Then

$$\begin{align*}\sigma=\sigma_0 - G_{BC}({ \bar{\partial} }^{*}{ \partial }{ \partial }^{*}+{ \bar{\partial} }^{*}){ \partial } i_{\phi(t)} \sigma \end{align*}$$

with $\sigma _0:=\mathcal {H}_{BC}\sigma $ . Applying the Bott–Chern harmonic projection $\mathcal {H}_{BC}$ to the both sides of this equation, we immediately get $\sigma _0=0$ . Hence we must have $\sigma =0$ .

Assume $\tilde {\sigma }\in { \mathrm {im}\,} d\cap { \mathrm {im}\,}({ \partial }\bar {{ \partial }}_{\phi (t)})^{*}\cap A^{p,q}(X)$ . Then it follows from Proposition 2.1 that

$$\begin{align*}*\tilde{\sigma}\in{ \mathrm{im}\,} d^{*}\cap{ \mathrm{im}\,}({ \partial }_{\overline{\phi(t)}}{ \bar{\partial} })\cap A^{n-q,n-p}(X) \end{align*}$$

which means

$$\begin{align*}\overline{*\tilde{\sigma}}\in{ \mathrm{im}\,} d^{*}\cap{ \mathrm{im}\,}{ \partial }{ \bar{\partial} }_{\phi(t)}\cap A^{n-p,n-q}(X). \end{align*}$$

So in order to prove ${ \mathrm {im}\,} d\cap { \mathrm {im}\,}({ \partial }\bar {{ \partial }}_{\phi (t)})^{*}=0$ we only need to show ${ \mathrm {im}\,} d^{*}\cap { \mathrm {im}\,}{ \partial }{ \bar {\partial } }_{\phi (t)}=0$ . This is clear since

$$\begin{align*}{ \mathrm{im}\,} d^{*}\cap{ \mathrm{im}\,}{ \partial }{ \bar{\partial} }_{\phi(t)}\subseteq { \mathrm{im}\,} d^{*}\cap\ker d_{\phi(t)}=0.\\[-34pt] \end{align*}$$

Proposition 2.4 Let $\sigma =\sigma (t)$ be a solution of the equation (2.10) in Proposition 2.3. Then for any small $t\in B$ , we have

(2.15) $$ \begin{align} { \partial }{ \bar{\partial} }_{\phi(t)}\sigma(t) = 0 \Leftrightarrow \mathcal{H}_{BC}\Big({ \partial } i_{\phi(t)}{ \partial }\sigma(t)\Big)=0. \end{align} $$

Moreover, we have ${ \partial } i_{\phi (t)}{ \partial }\sigma (t)\in \ker d$ for any small $t\in B$ .

Proof If ${ \partial }{ \bar {\partial } }_{\phi (t)}\sigma (t) = 0$ , then ${ \partial }{ \bar {\partial } }\sigma (t)={ \partial } i_{\phi (t)}{ \partial }\sigma (t)$ which implies $\mathcal {H}_{BC}\Big ({ \partial } i_{\phi (t)}{ \partial }\sigma (t)\Big )=0$ .

Conversely, suppose $\mathcal {H}_{BC}\Big ({ \partial } i_{\phi (t)}{ \partial }\sigma (t)\Big )=0$ . Then

$$ \begin{align*} { \partial } i_{\phi(t)}{ \partial }\sigma(t) &=G_{BC}\square_{BC}{ \partial } i_{\phi(t)}{ \partial }\sigma(t)\\ &=G_{BC}\Big( ({ \partial }{ \bar{\partial} })({ \partial }{ \bar{\partial} })^{*}+ ({ \bar{\partial} }^{*}{ \partial })({ \bar{\partial} }^{*}{ \partial })^{*}+{ \bar{\partial} }^{*}{ \bar{\partial} }\Big) { \partial } i_{\phi(t)}{ \partial }\sigma(t). \end{align*} $$

Set

$$\begin{align*}\psi(t):={ \partial }{ \bar{\partial} }_{\phi(t)}\sigma(t)={ \partial }{ \bar{\partial} }\sigma(t)-{ \partial } i_{\phi(t)}{ \partial }\sigma(t). \end{align*}$$

Recall that since the Maurer–Cartan equation ${ \bar {\partial } }\phi (t)=\frac {1}{2}[\phi (t),\phi (t)]$ holds, we have

(2.16) $$ \begin{align} \mathcal{L}^{1,0}_{{ \bar{\partial} }\phi(t)}=\frac{1}{2}\mathcal{L}^{1,0}_{[\phi(t),\phi(t)]}=\frac{1}{2}[\mathcal{L}^{1,0}_{\phi(t)},\mathcal{L}^{1,0}_{\phi(t)}]=\mathcal{L}^{1,0}_{\phi(t)}\mathcal{L}^{1,0}_{\phi(t)}, \end{align} $$

By substituting (2.10) and using (2.16), we get

$$ \begin{align*} \psi(t)&={ \partial }{ \bar{\partial} }\Big(G_{A}({ \partial }\bar{{ \partial }})^{*}{ \partial } i_{\phi(t)}{ \partial } \sigma(t)\Big)-{ \partial } i_{\phi(t)}{ \partial }\sigma(t)\\ &=G_{BC}{ \partial }{ \bar{\partial} }({ \partial }\bar{{ \partial }})^{*}{ \partial } i_{\phi(t)}{ \partial } \sigma(t)-{ \partial } i_{\phi(t)}{ \partial }\sigma(t)\\ &=-G_{BC}\Big( { \bar{\partial} }^{*}{ \partial }{ \partial }^{*}+{ \bar{\partial} }^{*} \Big){ \bar{\partial} }{ \partial } i_{\phi(t)}{ \partial }\sigma(t)\\ &=G_{BC}\Big( { \bar{\partial} }^{*}{ \partial }{ \partial }^{*}+{ \bar{\partial} }^{*} \Big){ \partial }{ \bar{\partial} }\mathcal{L}^{1,0}_{\phi(t)}\sigma(t)\\ &=G_{BC}\Big( { \bar{\partial} }^{*}{ \partial }{ \partial }^{*}+{ \bar{\partial} }^{*} \Big){ \partial }(-\mathcal{L}^{1,0}_{\phi(t)}{ \bar{\partial} }+\mathcal{L}^{1,0}_{{ \bar{\partial} }\phi(t)} )\sigma(t)\\ &=G_{BC}\Big( { \bar{\partial} }^{*}{ \partial }{ \partial }^{*}+{ \bar{\partial} }^{*} \Big){ \partial }(-i_{\phi(t)}{ \partial }{ \bar{\partial} }+\mathcal{L}^{1,0}_{\phi(t)}\mathcal{L}^{1,0}_{\phi(t)} )\sigma(t)\\ &=G_{BC}\Big( { \bar{\partial} }^{*}{ \partial }{ \partial }^{*}+{ \bar{\partial} }^{*} \Big){ \partial } i_{\phi(t)}(-{ \partial }{ \bar{\partial} }+{ \partial } i_{\phi(t)}{ \partial } )\sigma(t)\\ &=-G_{BC}\Big( { \bar{\partial} }^{*}{ \partial }{ \partial }^{*}+{ \bar{\partial} }^{*} \Big){ \partial } i_{\phi(t)}\psi(t) \end{align*} $$

which implies

$$\begin{align*}\|\psi(t)\|_{k+\alpha}=\| G_{BC}\Big( { \bar{\partial} }^{*}{ \partial }{ \partial }^{*}+{ \bar{\partial} }^{*} \Big){ \partial } i_{\phi(t)}\psi(t) \|_{k+\alpha} \leq C \|\phi(t)\|_{k+\alpha}\|\psi(t)\|_{k+\alpha}, \end{align*}$$

where C depends only on k and $\alpha $ . Now choose small $|t|$ so that $C \|\phi (t)\|_{k+\alpha }<1$ , then we get $\|\psi (t)\|_{k+\alpha }< \|\psi (t)\|_{k+\alpha }$ which is a contradiction if $\psi (t)\neq 0$ . So, we must have $\psi (t)=0$ whenever $|t|$ is small enough. This proves (2.15).

Finally, since by (2.10) we have ${ \partial }{ \bar {\partial } }\sigma (t)={ \partial }{ \bar {\partial } } G_{A}({ \partial }\bar {{ \partial }})^{*}{ \partial } i_{\phi (t)}{ \partial } \sigma $ , it follows that

$$ \begin{align*} &{ \partial } i_{\phi(t)}{ \partial }\sigma(t)\\ &=\mathcal{H}_{BC}\Big({ \partial } i_{\phi(t)}{ \partial }\sigma(t)\Big)+G_{BC}\Big( ({ \partial }{ \bar{\partial} })({ \partial }{ \bar{\partial} })^{*}+ ({ \bar{\partial} }^{*}{ \partial })({ \bar{\partial} }^{*}{ \partial })^{*}+{ \bar{\partial} }^{*}{ \bar{\partial} }\Big) { \partial } i_{\phi(t)}{ \partial }\sigma(t)\\ &=\mathcal{H}_{BC}\Big({ \partial } i_{\phi(t)}{ \partial }\sigma(t)\Big)+\Big(({ \partial }{ \bar{\partial} })G_{A}({ \partial }{ \bar{\partial} })^{*}+G_{BC}({ \bar{\partial} }^{*}{ \partial })({ \bar{\partial} }^{*}{ \partial })^{*}+G_{BC}{ \bar{\partial} }^{*}{ \bar{\partial} }\Big) { \partial } i_{\phi(t)}{ \partial }\sigma(t)\\ &=\mathcal{H}_{BC}\Big({ \partial } i_{\phi(t)}{ \partial }\sigma(t)\Big)+{ \partial }{ \bar{\partial} }\sigma(t)+G_{BC}\Big(({ \bar{\partial} }^{*}{ \partial })({ \bar{\partial} }^{*}{ \partial })^{*}+{ \bar{\partial} }^{*}{ \bar{\partial} }\Big) { \partial } i_{\phi(t)}{ \partial }\sigma(t), \end{align*} $$

which combined with (2.15) will imply

$$\begin{align*}G_{BC}\Big(({ \bar{\partial} }^{*}{ \partial })({ \bar{\partial} }^{*}{ \partial })^{*}+{ \bar{\partial} }^{*}{ \bar{\partial} }\Big) { \partial } i_{\phi(t)}{ \partial }\sigma(t)=0. \end{align*}$$

As a result, we get

$$\begin{align*}&{\partial } i_{\phi(t)}{ \partial }\sigma(t)\\ &=\mathcal{H}_{BC}\Big({ \partial } i_{\phi(t)}{ \partial }\sigma(t)\Big)+G_{BC}({ \partial }{ \bar{\partial} })({ \partial }{ \bar{\partial} })^{*}{ \partial } i_{\phi(t)}{ \partial }\sigma(t)\in\ker d.\\[-34pt] \end{align*}$$

Remark 2.5 In the Bott–Chern case, similar proofs will show that if $\sigma =\sigma (t)$ is a solution to $\sigma =\sigma _0 - G_{BC}({ \bar {\partial } }^{*}{ \partial }{ \partial }^{*}+{ \bar {\partial } }^{*}) { \partial } i_{\phi (t)} \sigma $ for some given $\sigma _0\in \mathcal {H}^{p,q}_{BC}(X)$ , then

$$\begin{align*}d_{\phi (t)} \sigma =0 \Leftrightarrow \mathcal {H}_{A} \mathcal{L}^{1,0}_{\phi(t)} \sigma =0.\end{align*}$$

For a vector space A and its subspaces $V,W,U$ , if we have a direct sum decomposition

$$\begin{align*}A=V\oplus W, \end{align*}$$

then the following homomorphism

(2.17) $$ \begin{align} \frac{U}{U\cap W}\longrightarrow V\cap(U+W):[u]\longmapsto v, \end{align} $$

where $u=v+w\in V\oplus W$ , is an isomorphism. In case that $U\cap W=\{0\}$ , then $U\cong V\cap (U+W)$ . This simple observation will be used frequently.

Proposition 2.6 The natural map

(2.18) $$ \begin{align} \frac{\ker d^{*}\cap\ker { \partial }{ \bar{\partial} }_{\phi(t)} \cap A^{p,q}(X)}{\ker d^{*}\cap ({ \mathrm{im}\,} { \partial }+ { \mathrm{im}\,} { \bar{\partial} }_{\phi(t)}) \cap A^{p,q}(X)}\longrightarrow H_{A\phi(t)}^{p,q}(X) \end{align} $$

is an isomorphism.

Proof By (2.4) and (2.7), we have the following orthogonal direct sum decompositionFootnote ²

$$\begin{align*}A^{p,q}(X)=\left(\ker d^{*}\cap\ker{ \partial }{ \bar{\partial} }_{\phi(t)}\right)\oplus ({ \mathrm{im}\,}{ \partial }+{ \mathrm{im}\,}{ \bar{\partial} }+{ \mathrm{im}\,}({ \partial }{ \bar{\partial} }_{\phi(t)})^{*}) \end{align*}$$

which implies

$$\begin{align*}\ker{ \partial }{ \bar{\partial} }_{\phi(t)}=\left(\ker d^{*}\cap\ker{ \partial }{ \bar{\partial} }_{\phi(t)}\right)\oplus \left(\ker{ \partial }{ \bar{\partial} }_{\phi(t)}\cap({ \mathrm{im}\,}{ \partial }+{ \mathrm{im}\,}{ \bar{\partial} }+{ \mathrm{im}\,}({ \partial }{ \bar{\partial} }_{\phi(t)})^{*})\right), \end{align*}$$

and

$$ \begin{align*} { \mathrm{im}\,}{ \partial }+{ \mathrm{im}\,}{ \bar{\partial} }_{\phi(t)}&=\left(\ker d^{*}\cap({ \mathrm{im}\,}{ \partial }+{ \mathrm{im}\,}{ \bar{\partial} }_{\phi(t)})\right)\oplus\\ &\left(({ \mathrm{im}\,}{ \partial }+{ \mathrm{im}\,}{ \bar{\partial} }_{\phi(t)})\cap({ \mathrm{im}\,}{ \partial }+{ \mathrm{im}\,}{ \bar{\partial} }+\ker{ \partial }^{*}\cap\ker{ \bar{\partial} }_{\phi(t)}^{*})\right). \end{align*} $$

Moreover, according to (2.17), we have

$$\begin{align*}\ker{ \partial }{ \bar{\partial} }_{\phi(t)}\cap\left({ \mathrm{im}\,}{ \partial }+{ \mathrm{im}\,}{ \bar{\partial} }+{ \mathrm{im}\,}({ \partial }{ \bar{\partial} }_{\phi(t)})^{*}\right)\cong\frac{{ \mathrm{im}\,}{ \partial }+{ \mathrm{im}\,}{ \bar{\partial} }}{({ \mathrm{im}\,}{ \partial }+{ \mathrm{im}\,}{ \bar{\partial} })\cap{ \mathrm{im}\,}({ \partial }{ \bar{\partial} }_{\phi(t)})^{*}} \end{align*}$$

and

$$\begin{align*}({ \mathrm{im}\,}{ \partial }+{ \mathrm{im}\,}{ \bar{\partial} }_{\phi(t)})\cap({ \mathrm{im}\,}{ \partial }+{ \mathrm{im}\,}{ \bar{\partial} }+\ker{ \partial }^{*}\cap\ker{ \bar{\partial} }_{\phi(t)}^{*}) \cong\frac{{ \mathrm{im}\,}{ \partial }+{ \mathrm{im}\,}{ \bar{\partial} }}{({ \mathrm{im}\,}{ \partial }+{ \mathrm{im}\,}{ \bar{\partial} })\cap\ker{ \partial }^{*}\cap\ker{ \bar{\partial} }_{\phi(t)}^{*}}. \end{align*}$$

Hence,

$$\begin{align*}H_{A\phi(t)}^{p,q}(X)\cong \frac{\ker d^{*}\cap\ker{ \partial }{ \bar{\partial} }_{\phi(t)} }{\ker d^{*}\cap({ \mathrm{im}\,}{ \partial }+{ \mathrm{im}\,}{ \bar{\partial} }_{\phi(t)}) } \oplus \frac{({ \mathrm{im}\,}{ \partial }+{ \mathrm{im}\,}{ \bar{\partial} })\cap\ker{ \partial }^{*}\cap\ker{ \bar{\partial} }_{\phi(t)}^{*} }{({ \mathrm{im}\,}{ \partial }+{ \mathrm{im}\,}{ \bar{\partial} })\cap{ \mathrm{im}\,}({ \partial }{ \bar{\partial} }_{\phi(t)})^{*} }. \end{align*}$$

The conclusion now follows because $({ \mathrm {im}\,}{ \partial }+{ \mathrm {im}\,}{ \bar {\partial } })\cap \ker { \partial }^{*}\cap \ker { \bar {\partial } }_{\phi (t)}^{*}=0$ by $(3)$ of Proposition 2.3. Indeed, note that

$$\begin{align*}({ \mathrm{im}\,}{ \partial }+{ \mathrm{im}\,}{ \bar{\partial} })\cap\ker{ \partial }^{*}\cap\ker{ \bar{\partial} }_{\phi(t)}^{*}=\overline{*(\ker d_{\phi(t)}\cap{ \mathrm{im}\,} d^{*})}.\\[-34pt] \end{align*}$$

3.1 Dolbeault/Bott–Chern/Aeppli deformations of $(p,q)$ -forms

Let $\pi : (\mathcal {X}, X)\to (B,0)$ be a complex analytic family such that for each $t\in B$ the complex structure on $X_t$ is represented by Beltrami differential $\phi (t)$ . In order to motivate the definition of Aeppli deformations of $(p,q)$ -forms, we will also recall the definitions of Dolbeault/Bott–Chern deformations of $(p,q)$ -forms studied in [Reference XiaXia22, Reference XiaXia21].

Definition 3.1 Given a $(p,q)$ -form

$$\begin{align*}y\in \ker{ \bar{\partial} }/\ker d/\ker{ \partial }{ \bar{\partial} } \end{align*}$$

and $T\subseteq B$ , which is an analytic subset of B containing $0$ ,

$$\begin{align*}\text{a } \textit{Dolbeault/Bott--Chern/Aeppli deformation} \text{ of } y \text{ w.r.t. } \pi: (\mathcal{X}, X)\to (B,0) \text{ on } T \end{align*}$$

is a family of $(p,q)$ -forms $\sigma (t)$ such that

1. 1. $\sigma (t)$ is holomorphic in $t\in T$ and $\sigma (0) = y$ ;

2. 2. $\forall t\in T$ , the following holds respectively:

   • – ${ \bar {\partial } }_{\phi (t)}\sigma (t) = 0$ (Dolbeault);

   • – $d_{\phi (t)}\sigma (t) = 0$ (Bott–Chern);

   • – ${ \partial }{ \bar {\partial } }_{\phi (t)}\sigma (t) = 0$ (Aeppli).

A Dolbeault/Bott–Chern/Aeppli deformation of a class

$$\begin{align*}\alpha\in H_{{ \bar{\partial} }}^{p,q}(X)/H_{BC}^{p,q}(X)/H_{A}^{p,q}(X) \end{align*}$$

(w.r.t. $\pi $ ) on T is a triple $(y,\sigma (t),T)$ which consisting of a representative $y\in \alpha $ and a Dolbeault/Bott–Chern/Aeppli deformation $\sigma (t)$ of y (w.r.t. $\pi $ ) on T. Two deformations $(y,\sigma (t),T)$ and $(y',\sigma ' (t),T)$ of $[y]$ of $\alpha $ on T are equivalent if

$$\begin{align*}[\sigma (t) - \sigma' (t)] = 0 \in H_{{ \bar{\partial} }_{\phi(t)}}^{p,q}(X)/H_{BC\phi(t)}^{p,q}(X)/H_{A\phi(t)}^{p,q}(X),~\forall t\in T. \end{align*}$$

3.1.1 Canonical deformations

Definition 3.2 Assume X is equipped with a fixed Hermitian metric h. Correspondingly, we have the following notions for canonical deformations:

• Dolbeault A Dolbeault deformation $\sigma (t)$ of $y\in \ker { \bar {\partial } }$ on T is called canonical if $\sigma (t)$ is of the form

  $$\begin{align*}\sigma(t)=\sigma^h(t)+{ \bar{\partial} }_{\phi(t)}u, \end{align*}$$
  such that

  • • $\sigma ^h(t) = \mathcal {H}_{{ \bar {\partial } }}y + { \bar {\partial } }^{*}G \mathcal {L}_{\phi (t)}^{1,0}\sigma ^h(t),~\forall t\in T$ ;

  • • ${ \bar {\partial } } u=y-\mathcal {H}_{{ \bar {\partial } }}y$ .

• Bott–Chern A Bott–Chern deformation $\sigma (t)$ of $y\in \ker d$ on T is called canonical if $\sigma (t)$ is of the form

  $$\begin{align*}\sigma(t)=\sigma^h(t)+{ \partial }{ \bar{\partial} }_{\phi(t)}u, \end{align*}$$
  such that

  • • $\sigma ^h(t) = \mathcal {H}_{BC}y - G_{BC}({ \bar {\partial } }^{*}{ \partial }{ \partial }^{*}+{ \bar {\partial } }^{*}) { \partial } i_{\phi (t)} \sigma ^h(t),~\forall t\in T$ ;

  • • ${ \partial }{ \bar {\partial } } u=y-\mathcal {H}_{BC}y$ .

• Aeppli An Aeppli deformation $\sigma (t)$ of $y\in \ker { \partial }{ \bar {\partial } }$ on T is called canonical if $\sigma (t)$ is of the form

  $$\begin{align*}\sigma(t)=\sigma^h(t)+{ \partial } v+{ \bar{\partial} }_{\phi(t)}u, \end{align*}$$
  such that

  • • $\sigma ^h(t) = \mathcal {H}_{A}y + G_{A}({ \partial }\bar {{ \partial }})^{*}{ \partial } i_{\phi (t)}{ \partial } \sigma ^h(t),~\forall t\in T$ ;

  • • ${ \partial } v+{ \bar {\partial } } u=y-\mathcal {H}_{A}y$ .

Under this definition, Dolbeault/Bott–Chern/Aeppli-exact forms can always deform canonically into a family of deformed Dolbeault/Bott–Chern/Aeppli-exact forms. For example, in the Aeppli case, if $y={ \partial } v+{ \bar {\partial } } u$ , then $\sigma (t)={ \partial } v+{ \bar {\partial } }_{\phi (t)}u$ is clearly a canonical (Aeppli) deformation of y on B. In fact, we have the following proposition.

Proposition 3.1 Let $\alpha \in H_{{ \bar {\partial } }}^{p,q}(X)/H_{BC}^{p,q}(X)/H_{A}^{p,q}(X)$ . Suppose a Hermitian metric on X is fixed. Then any two canonical Dolbeault/Bott–Chern/Aeppli deformation of $\alpha $ (on some common domain T) are equivalent.

Proof Let us take the case of canonical Aeppli deformation as an example, the other two cases are similar. Indeed, let $y, y'\in \alpha $ and $\alpha \in H_{A}^{p,q}(X)$ . If $\sigma (t)=\sigma ^h(t)+{ \partial } v+{ \bar {\partial } }_{\phi (t)}u$ and $\sigma '(t)=\sigma ^{'h}(t)+{ \partial } v'+{ \bar {\partial } }_{\phi (t)}u'$ are canonical Aeppli deformation of $y, y'$ (on T), respectively. Then by Proposition 2.3, we have $\sigma ^h(t)=\sigma ^{'h}(t)$ since $\mathcal {H}_{A}y=\mathcal {H}_{A}y'$ . Now,

$$\begin{align*}\sigma(t)-\sigma'(t)={ \partial }(v-v')+{ \bar{\partial} }_{\phi(t)}(u-u')\in{ \mathrm{im}\,}{ \partial }+{ \mathrm{im}\,}{ \bar{\partial} }_{\phi(t)}, \end{align*}$$

from which we conclude that $\sigma (t)$ is equivalent to $\sigma '(t)$ .

Remark 3.2 In our previous work [Reference XiaXia22, Reference XiaXia21], canonical deformations have slightly different meanings. They coincide with Definition 3.2 for (Dolbeault/Bott–Chern) harmonic forms. In view of the principle “exact forms/zero classes can always deforms into a family of deformed exact forms/zero classes,” the definitions given as above seems better suited for our purpose. In particular, it follows immediately from Definition 3.2 that canonical Dolbeault/Bott–Chern/Aeppli deformations of exact forms always exist and are equivalent to the trivial deformation. This is exactly the content of [Reference XiaXia21, Lemma 5.6] (in the Dolbeault case). On the other hand, we point out that under Definition 3.2 canonical deformation of a given closed form is not unique in general. However, by Proposition 3.1 the equivalence classes of canonical deformations of a given closed form/class are unique as long as we fix the Hermitian metric on X.

3.2 The dimension of $\ker ({ \partial }{ \bar {\partial } })^{*}\cap { \mathrm {im}\,}{ \partial }{ \bar {\partial } }_{\phi (t)}$

Recall the following proposition:

Proposition 3.4 In the context of Proposition 3.3, let

$$\begin{align*}\check{g}_t:\mathcal{H}^{p,q}_{A}(X)\longrightarrow \ker({ \partial }{ \bar{\partial} })^{*}\cap{ \mathrm{im}\,}{ \partial }{ \bar{\partial} }_{\phi(t)}\cap A^{p+1,q+1}(X) \end{align*}$$

be the restriction of $g_t$ on $\mathcal {H}^{p,q}_{A}(X)$ . Then $\check {g}_t$ is surjective with

$$\begin{align*}\ker\check{g}_t=\mathcal{H}^{p,q}_{A}(X)\cap\left(\ker{ \partial }{ \bar{\partial} }_{\phi(t)}+{ \mathrm{im}\,}({ \partial }{ \bar{\partial} })^{*}\right). \end{align*}$$

Moreover, we have

$$ \begin{align*} \dim\mathcal{H}^{p,q}_{A}(X)&=\dim\ker{ \partial }^{*}\cap\ker{ \bar{\partial} }^{*}\cap\ker{ \partial }{ \bar{\partial} }_{\phi(t)}\cap A^{p,q}(X)\\&\quad+ \dim \ker({ \partial }{ \bar{\partial} })^{*}\cap{ \mathrm{im}\,}{ \partial }{ \bar{\partial} }_{\phi(t)}\cap A^{p+1,q+1}(X). \end{align*} $$

Proof It follows immediately from $\ker g_t=\ker { \partial }{ \bar {\partial } }\cap \left (\ker { \partial }{ \bar {\partial } }_{\phi (t)}+{ \mathrm {im}\,}({ \partial }{ \bar {\partial } })^{*}\right )$ that

$$\begin{align*}\ker\check{g}_t=\mathcal{H}^{p,q}_{A}(X)\cap\left(\ker{ \partial }{ \bar{\partial} }_{\phi(t)}+{ \mathrm{im}\,}({ \partial }{ \bar{\partial} })^{*}\right). \end{align*}$$

To show $\check {g}_t$ is surjective it is enough to show

$$\begin{align*}\frac{\mathcal{H}^{p,q}_{A}(X)}{\mathcal{H}^{p,q}_{A}(X)\cap\left(\ker{ \partial }{ \bar{\partial} }_{\phi(t)}+{ \mathrm{im}\,}({ \partial }{ \bar{\partial} })^{*}\right)}\cong \frac{\ker{ \partial }{ \bar{\partial} }\cap A^{p,q}(X)}{\ker{ \partial }{ \bar{\partial} }\cap\left(\ker{ \partial }{ \bar{\partial} }_{\phi(t)}+{ \mathrm{im}\,}({ \partial }{ \bar{\partial} })^{*}\right)\cap A^{p,q}(X)}. \end{align*}$$

Indeed, it follows from (see $(3)$ of Proposition 2.3)

$$\begin{align*}{ \mathrm{im}\,} d\cap{ \mathrm{im}\,}({ \partial }\bar{{ \partial }}_{\phi(t)})^{*}=\ker{ \partial }{ \bar{\partial} }_{\phi(t)}\cap{ \mathrm{im}\,}({ \partial }{ \bar{\partial} })^{*}=0 \end{align*}$$

that

$$ \begin{align*} \ker{ \partial }{ \bar{\partial} }&= \mathcal{H}^{p,q}_{A}(X)\oplus({ \mathrm{im}\,}{ \partial }+ { \mathrm{im}\,}{ \bar{\partial} })\\ &\cong \mathcal{H}^{p,q}_{A}(X)\oplus[\ker{ \partial }{ \bar{\partial} }_{\phi(t)}\cap({ \mathrm{im}\,}{ \partial }+{ \mathrm{im}\,}{ \bar{\partial} }+{ \mathrm{im}\,}({ \partial }{ \bar{\partial} }_{\phi(t)})^{*})] \end{align*} $$

and

$$ \begin{align*} &\ker{ \partial }{ \bar{\partial} }\cap\left(\ker{ \partial }{ \bar{\partial} }_{\phi(t)}+{ \mathrm{im}\,}({ \partial }{ \bar{\partial} })^{*}\right)\cong\ker{ \partial }{ \bar{\partial} }_{\phi(t)}\\ &\quad =\left\{\ker{ \partial }{ \bar{\partial} }_{\phi(t)}\cap ({ \mathrm{im}\,}{ \partial }+{ \mathrm{im}\,}{ \bar{\partial} }+{ \mathrm{im}\,}({ \partial }{ \bar{\partial} }_{\phi(t)})^{*})\right\}\oplus\\ &\left\{\ker{ \partial }{ \bar{\partial} }_{\phi(t)}\cap[{ \mathrm{im}\,}({ \partial }{ \bar{\partial} }_{\phi(t)})^{*}+\ker{ \partial }^{*}\cap\ker{ \bar{\partial} }^{*}\cap\ker{ \partial }{ \bar{\partial} }_{\phi(t)}]\right\}\\ &\quad \cong \left\{\ker{ \partial }{ \bar{\partial} }_{\phi(t)}\cap ({ \mathrm{im}\,}{ \partial }+{ \mathrm{im}\,}{ \bar{\partial} }+{ \mathrm{im}\,}({ \partial }{ \bar{\partial} }_{\phi(t)})^{*})\right\}\oplus \left(\ker{ \partial }^{*}\cap\ker{ \bar{\partial} }^{*}\cap\ker{ \partial }{ \bar{\partial} }_{\phi(t)}\right). \end{align*} $$

On the other hand, we note that

$$ \begin{align*} \mathcal{H}^{p,q}_{A}(X)\cap\left(\ker{ \partial }{ \bar{\partial} }_{\phi(t)}+{ \mathrm{im}\,}({ \partial }{ \bar{\partial} })^{*}\right)&=\mathcal{H}^{p,q}_{A}(X)\cap\left(\ker d^{*}\cap\ker{ \partial }{ \bar{\partial} }_{\phi(t)}+{ \mathrm{im}\,}({ \partial }{ \bar{\partial} })^{*}\right)\\ (\text{since}~\ker d^{*}=\mathcal{H}^{p,q}_{A}(X)\oplus{ \mathrm{im}\,}({ \partial }{ \bar{\partial} })^{*})&\cong\ker d^{*}\cap\ker{ \partial }{ \bar{\partial} }_{\phi(t)}\\ &=\ker{ \partial }^{*}\cap\ker{ \bar{\partial} }^{*}\cap\ker{ \partial }{ \bar{\partial} }_{\phi(t)}.\\[-34pt] \end{align*} $$

4.1 Jumping of cohomology dimensions and canonical deformations

For any $t\in B$ and a vector subspace $V\subseteq \mathcal {H}_{BC}^{p,q}(X)$ , the following was introduced in [Reference XiaXia21, Definition 4.5]:

(4.1) $$ \begin{align} \begin{aligned} V_{BC,t}^{p,q}:= &\Big\{ \sigma_0\in V \mid \sigma(t)\in\ker d_{\phi(t)},~\text{where}~\sigma (t)=\sum_{k\geq0} \sigma_k~\text{with}\\ &\sigma_{k}=-G_{BC}({ \bar{\partial} }^{*}{ \partial }{ \partial }^{*}+{ \bar{\partial} }^{*})\sum_{i+j=k} { \partial } i_{\phi_j} \sigma_i,~\forall k> 0 \Big\}. \end{aligned} \end{align} $$

When $V=\mathcal {H}_{BC}^{p,q}(X)$ , we have (c.f. [Reference XiaXia21, Proposition 4.7])

(4.2) $$ \begin{align} V_{BC,t}^{p,q}\cong \ker({ \partial }{ \bar{\partial} })^{*}\cap\ker d_{\phi(t)} \cap A^{p,q}(X), \end{align} $$

which implies

(4.3) $$ \begin{align} \begin{aligned} v^{p,q}_t&:=\dim H_{BC}^{p,q}(X)-\dim \ker d_{\phi(t)}\cap\ker({ \partial }\bar{{ \partial }})^{*}\cap A^{p,q}(X)\\ &=\dim H_{BC}^{p,q}(X)-\dim V_{BC,t}^{p,q}. \end{aligned} \end{align} $$

These constructions for Bott–Chern deformations have Aeppli analogs as follows.

Definition 4.1 For any $t\in B$ and a vector subspace $W\subseteq \mathcal {H}_{A}^{p,q}(X)$ , we set

$$ \begin{align*} W_{t}^{p,q}:= &\Big\{ \sigma_0\in W \mid \sigma(t)\in\ker { \partial }{ \bar{\partial} }_{\phi(t)},\\ &\text{where}~\sigma (t)=\sum_{k} \sigma_k~\text{with}~\sigma_k=G_{A}({ \partial }\bar{{ \partial }})^{*}\sum_{i+j=k} { \partial } i_{\phi_j} { \partial }\sigma_i,~\forall k>0 \Big\}, \end{align*} $$

and

$$ \begin{align*} f_t^{p,q}: &W_{t}^{p,q} \longrightarrow \frac{\ker d^{*}\cap\ker { \partial }{ \bar{\partial} }_{\phi(t)} \cap A^{p,q}(X)}{\ker d^{*}\cap ({ \mathrm{im}\,} { \partial }+ { \mathrm{im}\,} { \bar{\partial} }_{\phi(t)}) \cap A^{p,q}(X)}\cong H_{A\phi(t)}^{p,q}(X),\\ &\sigma_0\longmapsto \sigma(t)=\sum_{k} \sigma_k,~\text{where}~\sigma_{k}=G_{A}({ \partial }\bar{{ \partial }})^{*}\sum_{i+j=k} { \partial } i_{\phi_j} { \partial }\sigma_i,~\forall k>0. \end{align*} $$

Remark 4.1 In view of Proposition 3.1, the requirement of W being a subspace of the harmonic space $\mathcal {H}_{A}^{p,q}(X)$ may be removed. More precisely, for any subspace

$$\begin{align*}W_0={ \mathbb{C} }\{[\sigma_0^1],\ldots,[\sigma_0^l]\}\subset H_{A}^{p,q}(X), \end{align*}$$

of dimension l. Set $\tilde {W}:={ \mathbb {C} }\{\sigma _0^1,\ldots ,\sigma _0^l\}$ and

$$\begin{align*}\tilde{W}_{t}^{p,q}:= \Big\{ \sigma_0\in \tilde{W} \mid \sigma(t)\in\ker { \partial }{ \bar{\partial} }_{\phi(t)}, \text{where } \sigma (t) \text{ is a canonical deformation of } \sigma_0\Big\}, \end{align*}$$

Then it follows from Proposition 3.1 that we have an isomorphism

$$ \begin{align*} \mathcal{H}_{A}: &\tilde{W}_{t}^{p,q} \longrightarrow W_{t}^{p,q},\\ &\sigma_0\longmapsto \mathcal{H}_{A}\sigma_0, \end{align*} $$

where $W_{t}^{p,q}$ is as in Definition 4.1 for $W={ \mathbb {C} }\{\mathcal {H}_{A}\sigma _0^1,\ldots ,\mathcal {H}_{A}\sigma _0^l\}$ .

By definition, the space $W_{t}^{p,q}$ .According to Proposition 2.3, the following linear mapping

$$ \begin{align*} \tilde{f}_t^{p,q}: &W_{t}^{p,q} \longrightarrow \ker d^{*}\cap\ker { \partial }{ \bar{\partial} }_{\phi(t)} \cap A^{p,q}(X),\\ &\sigma_0\longmapsto \sigma(t)=\sum_{k} \sigma_k,~\text{where}~\sigma_{k}=G_{A}({ \partial }\bar{{ \partial }})^{*}\sum_{i+j=k} { \partial } i_{\phi_j} { \partial }\sigma_i,~\forall k\neq 0, \end{align*} $$

is an isomorphism for any small $t\in B$ . In particular, when $W=\mathcal {H}_{A}^{p,q}(X)$ ,

(4.4) $$ \begin{align} \begin{aligned} w_{t}^{p,q}&:=\dim H_{A}^{p,q}(X)-\dim\ker{ \partial }{ \bar{\partial} }_{\phi(t)}\cap\ker d^{*}\cap A^{p,q}(X)\\ &=\dim H_{A}^{p,q}(X) - \dim W_{t}^{p,q}, \end{aligned} \end{align} $$

and $f_t^{p,q}$ is surjective with $\ker f_t^{p,q}\cong \ker d^{*}\cap ({ \mathrm {im}\,} { \partial }+ { \mathrm {im}\,} { \bar {\partial } }_{\phi (t)})\cap A^{p,q}(X)$ . On the other hand, we have the following section.

4.2 The jumping formulas

Theorem 4.2 Let $\pi : (\mathcal {X}, X)\to (B,0)$ be a complex analytic family such that for each $t\in B$ the complex structure on $X_t$ is represented by Beltrami differential $\phi (t)$ . Suppose X is equipped with a fixed Hermitian metric. Then for each $(p,q)\in \mathbb {N}\times \mathbb {N}$ , we have the jumping formula for the deformed Bott–Chern cohomology:

(4.5) $$ \begin{align} \dim H_{BC}^{p,q}(X)=\dim H_{BC\phi(t)}^{p,q}(X)+v^{p,q}_t+w^{p-1,q-1}_t, \end{align} $$

and the jumping formula for the deformed Aeppli cohomology:

(4.6) $$ \begin{align} \dim H_{A}^{p,q}(X)=\dim H_{A\phi(t)}^{p,q}(X)+v^{n-p,n-q}_t+w^{n-p-1,n-q-1}_t. \end{align} $$

Proof First, we know that

$$\begin{align*}H_{BC\phi(t)}^{p,q}(X)\cong\frac{\ker({ \partial }{ \bar{\partial} })^{*}\cap\ker d_{\phi(t)} \cap A^{p,q}(X)}{\ker({ \partial }{ \bar{\partial} })^{*}\cap{ \mathrm{im}\,}{ \partial }{ \bar{\partial} }_{\phi(t)} \cap A^{p,q}(X)}. \end{align*}$$

It follows from this, (4.3) and Proposition 3.4 that

$$ \begin{align*} \dim H_{BC\phi(t)}^{p,q}(X)&=\dim H_{BC}^{p,q}(X)-v^{p,q}_t-\dim\ker({ \partial }{ \bar{\partial} })^{*}\cap{ \mathrm{im}\,}{ \partial }{ \bar{\partial} }_{\phi(t)} \cap A^{p,q}(X)\\ &=\dim H_{BC}^{p,q}(X)-v^{p,q}_t-w^{p,q}_t. \end{align*} $$

This is (4.5). For (4.6), since

$$\begin{align*}H_{BC}^{p,q}(X)\cong H_{A}^{n-q,n-p}(X),\quad H_{A}^{p,q}(X)\cong H_{A}^{q,p}(X),\quad \forall (p,q)\in \mathbb{N}\times \mathbb{N} \end{align*}$$

and $H_{BC\phi (t)}^{p,q}(X)\cong H_{A\phi (t)}^{n-p,n-q}(X)$ by Corollary 2.2, it follows from (4.5) that for each $(p,q)\in \mathbb {N}\times \mathbb {N}$ ,

$$ \begin{align*} \dim H_{A}^{n-p,n-q}(X)&=\dim H_{A}^{n-q,n-p}(X)=\dim H_{BC}^{p,q}(X)\\ &=\dim H_{BC\phi(t)}^{p,q}(X)+v^{p,q}_t+w^{p-1,q-1}_t\\ &=\dim H_{A\phi(t)}^{n-p,n-q}(X)+v^{p,q}_t+w^{p-1,q-1}_t. \end{align*} $$

Equation (4.6) follows from this by replacing $p, q$ with $n-p, n-q$ , respectively.

Note that since the Aeppli deformation of $(n,q)$ or $(p,n)$ -forms is always canonically unobstructed, we have $w^{n,q}_t=w^{p,n}_t=0$ for any $t\in B$ .

The jumping formulas provide us a useful criterion to determine when the dimension of these deformed cohomology remains constant under deformation of complex structures. In fact, set

$$ \begin{align*} h_{BC\phi(t)}^{p,q}&=\dim H_{BC\phi(t)}^{p,q}(X),\quad h_{BC}^{p,q}=\dim H_{BC}^{p,q}(X),\\ h_{A\phi(t)}^{p,q}&=\dim H_{A\phi(t)}^{p,q}(X),\quad h_{A}^{p,q}=\dim H_{A}^{p,q}(X), \end{align*} $$

we have the following corollary.

4.3 Semi-continuous property under the analytic Zariski topology

Proof The case for $v^{p,q}_t$ is already known, see [Reference XiaXia21, Remark 4.9]. So, we only need prove this for $w^{p,q}_t$ . Indeed, since $W_{t}^{p,q}$ is isomorphic to $\ker d^{*}\cap \ker { \partial }{ \bar {\partial } }_{\phi (t)} \cap A^{p,q}(X)$ , we have

$$\begin{align*}w^{p,q}_t=\dim H_{A}^{p,q}(X)-\dim W_{t}^{p,q}. \end{align*}$$

On the other hand, assume $\{e_1,\ldots , e_m\}$ is an orthonormal basis of $\mathcal {H}_{A}^{p+1,q+1}(X)$ and $V=\mathcal {H}_{A}^{p,q}(X)$ . Then it follows from Proposition 2.4 that for any $k\in { \mathbb {N} }$

$$ \begin{align*} W_{t}^{p,q}& = \{\sigma_0\in \mathcal{H}_{A}^{p,q}(X) \mid { \partial }{ \bar{\partial} }_{\phi(t)}\sigma(t)=0\}\\ &=\{\sigma_0\in \mathcal{H}_{A}^{p,q}(X) \mid \mathcal{H}_{BC}\Big({ \partial } i_{\phi(t)}{ \partial }\sigma(t)\Big)=0\}\\ &=\{\sigma_0\in \mathcal{H}_{A}^{p,q}(X) \mid \langle { \partial } i_{\phi(t)}{ \partial }\sigma(t), e_i\rangle_{L^2}=0,~1\leq i\leq m\}\\ &=\Big\{\sum_{l=1}^N a_l\sigma_0^l\in \mathcal{H}_{A}^{p,q}(X) \mid \sum_{l=1}^N a_l\langle { \partial } i_{\phi(t)}{ \partial }\sigma^l(t), e_i\rangle_{L^2}=0,~1\leq i\leq m\Big\}, \end{align*} $$

where $\{\sigma _0^l\}_{l=1}^N$ is a basis of $\mathcal {H}_{A}^{p,q}(X)$ and $\sigma (t)=\sum _{k} \sigma _k$ is the canonical deformation of $\sigma _0$ with $\sigma _k=G_{A}({ \partial }\bar {{ \partial }})^{*}\sum _{i+j=k} { \partial } i_{\phi _j} { \partial }\sigma _i,~\forall k>0$ . Similarly, each $\sigma ^l(t)$ is the canonical deformation of $\sigma _0^l$ . As a result, we get

$$\begin{align*}\{t\in B\mid w^{p,q}_t< k\} =\Big\{t\in B\mid\dim H_{A}^{p,q}(X)-\dim W_{t}^{p,q}=\textrm{rank}\Big(u_i^l(t)\Big)_{m\times N}< k\Big\}, \end{align*}$$

where $u_i^l(t)=\langle { \partial } i_{\phi (t)}{ \partial }\sigma ^l(t), e_i\rangle _{L^2}$ . Because each $u_i^l(t)$ is holomorphic in t, we see that for any $k\in { \mathbb {N} }$ , the set $\{t\in B\mid w^{p,q}_t< k\}$ is an analytic subset of B.

Proof We will only give the proof of $(1)$ since $(2)$ can be shown by the exactly the same method. For $(1)$ , by applying the jumping formula (4.5), we get

$$ \begin{align*} \begin{aligned} h_{BC}^{p,q}&=(h_{BC\phi(t)}^{p,q}+v^{p,q}_t-w^{p-1,q}_t)+w^{p-1,q}_t+w^{p-1,q-1}_t,\\ -h_{BC}^{p,q-1}&=-(h_{BC\phi(t)}^{p,q-1}+v^{p,q-1}_t-w^{p-1,q-1}_t)-w^{p-1,q-1}_t-w^{p-1,q-2}_t,\\ &\quad \vdots\\ (-1)^q h_{BC}^{p,0}&=(-1)^q(h_{BC\phi(t)}^{p,0}+v^{p,0}_t-w^{p-1,0}_t)+(-1)^qw^{p-1,0}_t.\\ \end{aligned} \end{align*} $$

Summing all these up, we have

$$\begin{align*}\sum_{i=0}^q(-1)^{q-i} h_{BC}^{p,i}-w^{p-1,q}_t=\sum_{i=0}^q(-1)^{q-i}(h_{BC\phi(t)}^{p,i}+v^{p,i}_t-w^{p-1,i}_t). \end{align*}$$

The conclusion then follows since by Proposition 4.4 we know $w^{p-1,q}_t$ is a lower semi-continuous functions of t under the analytic Zariski topology.

4.4 Remarks on canonical Dolbeault deformations

We end this section with some discussions about canonical Dolbeault deformations.

Let E be a holomorphic tensor bundle on X and the $E_t$ the corresponding tensor bundle on X. It is proved [Reference XiaXia22, Theorem 5.10] by the second author that

(4.7) $$ \begin{align} \dim H^{q}(X,E)=\dim H^{q}(X_t,E_t)+v^{q}_t+v^{q-1}_t, \end{align} $$

where $v^{q}_t:=\dim H^{q}(X,E)-\dim \ker { \bar {\partial } }_{\phi (t)}\cap \ker { \bar {\partial } }^{*}\cap A^{0,q}(X,E) \geq 0$ . By using similar arguments as in the proof of Proposition 4.5, it follows from (4.7) that

(4.8) $$ \begin{align} v^{q}_t=\sum_{i=0}^{q}(-1)^{q-i}\dim H^{i}(X,E)-\sum_{i=0}^{q}(-1)^{q-i}\dim H^{i}(X_t,E_t). \end{align} $$

Since the Dolbeault deformations of E-valued $(0,q)$ -forms are canonically unobstructed iff $v^{q}_t=0$ for any $t\in B$ , we conclude that for Dolbeault deformations of E-valued $(0,q)$ -forms,

$$\begin{align*}\text{canonically unobstructedness}\Longleftrightarrow\sum_{i=0}^{q}(-1)^{q-i}\dim H^{i}(X_t,E_t)~\text{is independent of}~t. \end{align*}$$

In particular, we get the following corollary.

As a result, the notion of canonically unobstructedness (at least for Dolbeault deformations) is independent of the choices of Hermitian metrics.

On the other hand, we see that $\sum _{i=0}^{q}(-1)^{q-i}\dim H^{i}(X_t,E_t)$ is upper semicontinuous (under the analytic Zariski topology) since $v^{q}_t$ is lower semi-continuous. This is in fact a special case of a result due to Flenner [Reference FlennerFle81] where $\{E_t\}_{t\in B}$ may be replaced by a coherent sheaf on the total space $\mathcal {X}$ , see also [Reference Bertin, Demailly, Illusie and PetersBDIP02, Theorem 5.10].

6.1 The Iwasawa manifold

Example 6.1 Case III-(2). Let G be the matrix Lie group defined by

$$\begin{align*}G := \left\{ \left( \begin{array}{ccc} 1 & z^1 & z^3 \\ 0 & 1 & z^2 \\ 0 & 0 & 1 \end{array} \right) \in \mathrm{GL}(3;\mathbb{C}) \mid z^1,\,z^2,\,z^3 \in\mathbb{C} \right\}\cong \mathbb{C}^3, \end{align*}$$

where the product is the one induced by matrix multiplication. This is usually called the Heisenberg group. Consider the discrete subgroup $\Gamma $ defined by

$$\begin{align*}\Gamma := \left\{ \left( \begin{array}{ccc} 1 & \omega^1 & \omega^3 \\ 0 & 1 & \omega^2 \\ 0 & 0 & 1 \end{array} \right) \in G \mid \omega^1,\,\omega^2,\,\omega^3 \in\mathbb{Z}[\sqrt{-1}] \right\}. \end{align*}$$

The quotient $X=\Gamma \setminus G$ is called the Iwasawa manifold. A basis of $H^0(X,\Omega ^1)$ is given by

$$\begin{align*}\varphi^1 = d z^1,~ \varphi^2 = d z^2,~ \varphi^3 = d z^3-z^1\,d z^2, \end{align*}$$

and a dual basis $\theta ^1, \theta ^2, \theta ^3\in H^0(X,T_X^{1,0})$ is given by

$$\begin{align*}\theta^1=\frac{\partial}{\partial z^1},~\theta^2=\frac{\partial}{\partial z^2} + z^1\frac{\partial}{\partial z^3},~\theta^3=\frac{\partial}{\partial z^3}. \end{align*}$$

The structure equation in terms of the basis $\{\varphi ^1,\varphi ^2,\varphi ^3\}$ is given by

$$\begin{align*}d\varphi^1=0,\quad d\varphi^2=0,\quad d\varphi^3=-\varphi^{12}. \end{align*}$$

X is equipped with the Hermitian metric $\sum _{i=1}^3\varphi ^i\otimes \bar {\varphi }^i$ . The Beltrami differential of the Kuranishi family of X is

$$\begin{align*}\phi(t) = \sum_{i=1}^3\sum_{\lambda=1}^2t_{i\lambda}\theta^i\bar{\varphi}^{\lambda} - D(t)\theta^3\bar{\varphi}^{3},~\text{with}~D(t)=t_{11}t_{22}-t_{21}t_{12}, \end{align*}$$

and the Kuranishi space of X is

$$\begin{align*}\mathcal{B}=\{t=(t_{11}, t_{12}, t_{21}, t_{22}, t_{31}, t_{32})\in \mathbb{C}^6\mid |t_{i\lambda}|<\epsilon, i=1, 2, 3, \lambda=1,2 \}, \end{align*}$$

where $\epsilon>0$ is sufficiently small. Set

$$\begin{align*}\phi_1=\sum_{i=1}^3\sum_{\lambda=1}^2t_{i\lambda}\theta^i\bar{\varphi}^{\lambda},~ \phi_2 = -D(t)\theta^3\bar{\varphi}^{3}, \end{align*}$$

then by direct computations, we have

$$ \begin{align*} &\mathcal{L}_{\phi_1}^{1,0}\varphi^1 = \mathcal{L}_{\phi_1}^{1,0}\varphi^2=\mathcal{L}_{\phi_1}^{1,0}\bar{\varphi}^1=\mathcal{L}_{\phi_1}^{1,0}\bar{\varphi}^{2}=\mathcal{L}_{\phi_1}^{1,0}\bar{\varphi}^{3}=0,\\[5pt] &\mathcal{L}_{\phi_1}^{1,0}\varphi^3 = \sum_{\lambda=1}^2(t_{1\lambda}\varphi^2-t_{2\lambda}\varphi^1)\wedge\bar{\varphi}^{\lambda},~\mathcal{L}_{\phi_2}^{1,0}\varphi^i =\mathcal{L}_{\phi_2}^{1,0}\bar{\varphi}^{i}=0,~i=1,2,3. \end{align*} $$

In particular,

(6.1) $$ \begin{align} { \partial }\mathcal{L}_{\phi_1}^{1,0}\varphi^i={ \partial }\mathcal{L}_{\phi_1}^{1,0}\bar{\varphi}^i=\mathcal{L}_{\phi_2}^{1,0}\varphi^i =\mathcal{L}_{\phi_2}^{1,0}\bar{\varphi}^{i}=0,\quad i=1, 2, 3. \end{align} $$

The Bott–Chern/Aeppli cohomology of X are easily computed by using left invariant forms. Let us first consider canonical Aeppli deformations.

• • $H_{A}^{1,0}(X)=\mathbb {C}\{\varphi ^1,\varphi ^2,\varphi ^3\}$ ,  $H_{A}^{0,1}(X)=\mathbb {C}\{\varphi ^{\bar {1}},\varphi ^{\bar {2}},\varphi ^{\bar {3}}\}$ ;

• • $H_{A}^{2,0}(X)=\mathbb {C}\{\varphi ^{13},\varphi ^{23} \}$ ,  $H_{A}^{0,2}(X)=\mathbb {C}\{\varphi ^{\overline {13}},\varphi ^{\overline {23}}\}$ ;

• • $H_{A}^{1,1}(X)=\mathbb {C}\{\varphi ^{1\bar {1}},\varphi ^{1\bar {2}},\varphi ^{1\bar {3}},\varphi ^{2\bar {1}},\varphi ^{2\bar {2}},\varphi ^{2\bar {3}},\varphi ^{3\bar {1}},\varphi ^{3\bar {2}}\}$ ;

• • $H_{A}^{3,0}(X)=\mathbb {C}\{\varphi ^{123}\}$ ,  $H_{A}^{2,1}(X)=\mathbb {C}\{\varphi ^{13\bar {1}},\varphi ^{13\bar {2}},\varphi ^{13\bar {3}},\varphi ^{23\bar {1}},\varphi ^{23\bar {2}},\varphi ^{23\bar {3}}\}$ ;

• • $H_{A}^{1,2}(X)=\mathbb {C}\{\varphi ^{1\overline {13}},\varphi ^{1\overline {23}},\varphi ^{2\overline {13}},\varphi ^{2\overline {23}},\varphi ^{3\overline {13}},\varphi ^{3\overline {23}}\}$ ,  $H_{A}^{0,3}(X)=\mathbb {C}\{\varphi ^{\overline {123}}\}$ ;

• • $H_{A}^{3,1}(X)=\mathbb {C}\{\varphi ^{123\bar {1}},\varphi ^{123\bar {2}},\varphi ^{123\bar {3}}\}$ ,  $H_{A}^{2,2}(X)=\mathbb {C}\{\varphi ^{13\overline {23}},\varphi ^{23\overline {13}},\varphi ^{23\overline {23}},\varphi ^{13\overline {13}}\}$

• • $H_{A}^{1,3}(X)=\mathbb {C}\{\varphi ^{1\overline {123}},\varphi ^{2\overline {123}},\varphi ^{3\overline {123}}\}$

• • $H_{A}^{3,2}(X)=\mathbb {C}\{\varphi ^{123\overline {13}},\varphi ^{123\overline {23}}\}$ ,  $H_{A}^{2,3}(X)=\mathbb {C}\{\varphi ^{13\overline {123}},\varphi ^{23\overline {123}}\}$

• • $H_{A}^{3,3}(X)=\mathbb {C}\{\varphi ^{123\overline {123}}\}.$

In fact, in view of (6.1), for any $\sigma _0\in \ker { \partial }{ \bar {\partial } }\cap A^{p,q}(X)$ , we have

$$\begin{align*}{ \partial } i_{\phi_1}{ \partial }\sigma_0={ \partial }\mathcal{L}_{\phi_1}^{1,0}\sigma_0=0 \end{align*}$$

and thus

$$\begin{align*}\sigma_1 = G_{A}({ \partial }\bar{{ \partial }})^{*} { \partial } i_{\phi_1}{ \partial } \sigma_0=0,\quad \sigma_2 = G_{A}({ \partial }\bar{{ \partial }})^{*} ({ \partial } i_{\phi_2}{ \partial } \sigma_0 + { \partial } i_{\phi_1}{ \partial } \sigma_1 )=0. \end{align*}$$

Similarly, there holds

$$\begin{align*}{ \partial }\sum_{j=1}^{k}\mathcal{L}_{\phi_j}^{1,0}\sigma_{k-j}=0 , \quad k\geq 1, \end{align*}$$

and $\sigma _k= G_{A}({ \partial }\bar {{ \partial }})^{*}\sum _{i+j=k} { \partial } i_{\phi _j} { \partial }\sigma _i=0$ for any $k\geq 1$ . Therefore, for each $(p,q)\in \mathbb {N}\times \mathbb {N}$ the Aeppli deformations of $(p,q)$ -forms are canonically unobstructed and $w_t^{p,q}=0$ for any $t\in \mathcal {B}$ .

Let us now consider canonical Bott–Chern deformations.

• • $H_{BC}^{2,0}(X)=\mathbb {C}\{\varphi ^{12},\varphi ^{13},\varphi ^{23} \}$ ;

• • $H_{BC}^{2,1}(X)=\mathbb {C}\{\varphi ^{12\bar {1}},\varphi ^{12\bar {2}},\varphi ^{13\bar {1}},\varphi ^{13\bar {2}},\varphi ^{23\bar {1}},\varphi ^{23\bar {2}}\}$ ;

• • $H_{BC}^{2,2}(X)=\mathbb {C}\{\varphi ^{12\overline {23}},\varphi ^{23\overline {23}},\varphi ^{13\overline {23}}, \varphi ^{12\overline {13}},\varphi ^{23\overline {13}},\varphi ^{13\overline {13}},\varphi ^{23\overline {12}},\varphi ^{13\overline {12}}\}$ .

For $(p,q)=(2,0)$ . Let $\sigma _0 = a_{12}\varphi ^{12} +a_{13}\varphi ^{13}+a_{23}\varphi ^{23}$ , we have

$$\begin{align*}{ \partial } i_{\phi_1}\sigma_0=(a_{13}t_{11}+a_{23}t_{21})\varphi^{12\bar{1}}+(a_{13}t_{12}+a_{23}t_{22})\varphi^{12\bar{2}} \end{align*}$$

is exact if and only if ${ \partial } i_{\phi _1}\sigma _0=0$ , i.e.,

(6.2) $$ \begin{align} \left\{\!\!\!\!\!\!\!\!\!\!\! \begin{array}{rr} &t_{21}a_{23} + t_{11}a_{13} = 0, \\[5pt] &t_{22}a_{23} + t_{12}a_{13} = 0, \end{array} \right. \end{align} $$

has solutions for $(a_{23}, a_{13})$ , and in this case the (degree $1$ term of) canonical deformation is given by

$$\begin{align*}\sigma_{1}=-G_{BC}({ \bar{\partial} }^{*}{ \partial }{ \partial }^{*}+{ \bar{\partial} }^{*})\partial i_{\phi_1}\sigma_0 =0. \end{align*}$$

On the other hand,

$$\begin{align*}\partial i_{\phi_2}\sigma_0 = 0 \Longrightarrow \sigma_{2}=-G_{BC}({ \bar{\partial} }^{*}{ \partial }{ \partial }^{*}+{ \bar{\partial} }^{*})\partial(i_{\phi_2}\sigma_0+i_{\phi_1}\sigma_1) =0, \end{align*}$$

and $\phi _k=0,~ k>2$ implies that $\sigma _{k}= G_{BC}({ \bar {\partial } }^{*}{ \partial }{ \partial }^{*}+{ \bar {\partial } }^{*})\sum _{i+j=k} { \partial } i_{\phi _j}\sigma _i=0$ for any $k>2$ .

Therefore, for $V=H_{BC}^{2,0}(X)$ , we have (see (4.1))

$$\begin{align*}V_{BC,t}^{2,0}=\{a_{12}\varphi^{12} + a_{23}\varphi^{23} + a_{13}\varphi^{13} \mid (a_{12}, a_{23}, a_{13})\in \mathbb{C}^3~\text{satisfy}~(6.2)\}\subseteq H_{BC}^{2,0}(X), \end{align*}$$

and

$$\begin{align*}v_t^{2,0}=\dim H_{BC}^{2,0}(X)-\dim V_{BC,t}^{2,0}= \left\{ \begin{array}{rcl} 0, &(t_{11},t_{12},t_{21},t_{22}) =0 \\[5pt] 1, &(t_{11},t_{12},t_{21},t_{22}) \neq 0,D(t)=0 \\[5pt] 2, &(t_{11},t_{12},t_{21},t_{22}) \neq 0,D(t)\neq 0. \end{array} \right. \end{align*}$$

For $(p,q)$ =(2,1). Let $\sigma _0 = \sum a_{ij{\bar {k}}}\varphi ^{ij{\bar {k}}}\in \mathcal {H}_{BC}^{2,1} $ , we have

$$ \begin{align*} { \partial } i_{\phi_1}\sigma_0&=(t_{11}a_{13\bar{2}}-t_{12}a_{13\bar{1}}+t_{21}a_{23\bar{2}}-t_{22}a_{23\bar{1}})\varphi^{12\overline{12}}\\ &=a(t){ \partial }{ \bar{\partial} }\varphi^{3\bar{3}}, \end{align*} $$

where $a(t):=-(t_{11}a_{13\bar {2}}-t_{12}a_{13\bar {1}}+t_{21}a_{23\bar {2}}-t_{22}a_{23\bar {1}})$ . In this case,

$$ \begin{align*} \sigma_{1}&=-G_{BC}({ \bar{\partial} }^{*}{ \partial }{ \partial }^{*}+{ \bar{\partial} }^{*}){ \partial } i_{\phi_1}\sigma_0\\ &=-a(t)G_{BC}({ \bar{\partial} }^{*}{ \partial }{ \partial }^{*}+{ \bar{\partial} }^{*}){ \partial }{ \bar{\partial} }\varphi^{3\bar{3}}\\ &=a(t)G_{BC}({ \bar{\partial} }^{*}{ \partial }{ \partial }^{*}{ \bar{\partial} }+{ \bar{\partial} }^{*}{ \bar{\partial} }){ \partial }\varphi^{3\bar{3}}\\ &=a(t)G_{BC}(\square_{BC}-({ \partial }{ \bar{\partial} })({ \partial }{ \bar{\partial} })^{*}){ \partial }\varphi^{3\bar{3}}\\ &=-a(t)G_{BC}(\square_{BC}-({ \partial }{ \bar{\partial} })({ \partial }{ \bar{\partial} })^{*})\varphi^{12\bar{3}}\\ &=-a(t)G_{BC}\square_{BC}\varphi^{12\bar{3}}\\ &=-a(t)(1-\mathcal{H}_{BC}^{2,1})\varphi^{12\bar{3}}\\ &=-a(t)\varphi^{12\bar{3}}. \end{align*} $$

On the other hand, ${ \partial } i_{\phi _2}\sigma _0=-\mathcal {L}_{\phi _2}^{1,0}\sigma _0=0$ and ${ \partial } i_{\phi _1}\sigma _1=-\mathcal {L}_{\phi _1}^{1,0}\sigma _1=0$ . Therefore,

$$\begin{align*}\sigma_{2}=-G_{BC}({ \bar{\partial} }^{*}{ \partial }{ \partial }^{*}+{ \bar{\partial} }^{*}){ \partial }(i_{\phi_2}\sigma_0+i_{\phi_1}\sigma_1) =0, \end{align*}$$

and $\phi _k=0,~ k>2$ implies that $\sigma _{k}=0,~ k>2$ . It follows that the Bott–Chern deformations of $(2,1)$ -forms are canonically unobstructed and $v_t^{2,1}=0$ for any $t\in \mathcal {B}$ .

Similarly, we can determine $v_t^{p,q}$ for all $(p,q)$ . It turns out there are only two cases for which $v_t^{p,q}\neq 0$ : $(p,q)=(2,0), (2,2)$ (see [Reference XiaXia21, Section 6] for the computations of $v_t^{2,2}$ ). As a result, we get the following table (Table 1).

Table 1:

$v^{\bullet ,\bullet }_t$ and $w^{\bullet ,\bullet }_t$ for cases $(i), (ii), and (iii).$

Note: We write $(i), (ii), and (iii)$ for the three cases when $(t_{11}, t_{12}, t_{21}, t_{22})=0$ , $(t_{11}, t_{12}, t_{21}, t_{22})\neq 0$ and $D(t)= 0$ , $D(t)\neq 0$ , respectively.

6.2 The complex parallelizable Nakamura manifold

Example 6.2 Let $X=\Gamma \setminus G$ be the solvable manifold constructed by Nakamura in Example III-(3b) of [Reference NakamuraNak75]. We will use an equivalent description of X given byFootnote ³ De Bartolomeis–Tomassini [Reference De Bartolomeis and TomassiniDBT06] (see also [Reference Angella and KasuyaAK17a, Reference Angella and KasuyaAK17b, Reference HasegawaHas10]). In fact, following the conventions in [Reference Angella and KasuyaAK17b, Section 4], let $G:=({ \mathbb {C} }^3,*)$ be the complex solvable Lie group with the multiplication $*$ defined by

$$\begin{align*}(z_1,z_2,z_3)*(w_1,w_2,w_3)=(z_1+w_1,e^{w_1}z_2+w_2,e^{-w_1}z_3+w_3). \end{align*}$$

In other words, G is the semi-direct product ${ \mathbb {C} }\ltimes _\varphi { \mathbb {C} }^2$ where the homomorphism $\varphi :{ \mathbb {C} }\to { \mathbb {C} }^2$ is given by

$$\begin{align*}\varphi(z)=\left ( \begin{matrix} e^{z} & 0\\ 0 & e^{-z}\\ \end{matrix} \right ),\quad \forall z\in{ \mathbb{C} }. \end{align*}$$

The lattice $\Gamma \subset G$ has the following generators:

$$ \begin{align*} T_1(z)&:=(z_1+\lambda,e^{\lambda}z_2,e^{-\lambda}z_3),\quad T_2(z):=(z_1+2\pi i,z_2,z_3),\\ T_3(z)&:=(z_1,z_2+1,z_3-\mu),\quad T_4(z):=(z_1,z_2+\mu,z_3+1),\\ T_5(z)&:=(z_1,z_2+2\pi i,z_3-2\pi i\mu),\quad T_6(z):=(z_1,z_2+2\pi i\mu,z_3+2\pi i), \end{align*} $$

where $\lambda =\log \frac {3+\sqrt {5}}{2}$ and $\mu =\log \frac {\sqrt {5}-1}{2}$ (c.f. [Reference De Bartolomeis and TomassiniDBT06]). We will consider the deformation family of X whose Beltrami differential is

$$\begin{align*}\phi(t)=\phi_1=t\frac{{ \partial }}{{ \partial } z^{1}}dz^{1}. \end{align*}$$

According to [Reference Angella and KasuyaAK17b, Corollary 2.21 and Theorem 2.22], there is a sub-double complex $C_\Gamma ^{\bullet ,\bullet }$ (explicitly determined) of $(A^{\bullet ,\bullet }(X), d={ \partial }+{ \bar {\partial } })$ such that

(6.3) $$ \begin{align} H^{\bullet,\bullet}_{{ \bar{\partial} }}(C_\Gamma^{\bullet,\bullet})\cong H^{\bullet,\bullet}_{{ \bar{\partial} }}(X),\quad H^{\bullet,\bullet}_{BC}(C_\Gamma^{\bullet,\bullet})\cong H^{\bullet,\bullet}_{BC}(X). \end{align} $$

Combining this with [Reference StelzigSte21, Corollary 13], we also have

(6.4) $$ \begin{align} H^{\bullet,\bullet}_{A}(C_\Gamma^{\bullet,\bullet})\cong H^{\bullet,\bullet}_{A}(X). \end{align} $$

Thanks to the works of Angella–Kasuya, $C_\Gamma ^{\bullet ,\bullet }$ has already been computed in [Reference Angella and KasuyaAK17b, Section 4]. In fact, we have

$$ \begin{align*} C_{\Gamma}^{0,0}&={ \mathbb{C} }\{1\},\quad C_{\Gamma}^{3,0}=\mathbb{C}\{dz^{123}\},\quad C_{\Gamma}^{0,3}=\mathbb{C}\{dz^{\overline{123}}\},\quad C_{\Gamma}^{3,3}=\mathbb{C}\{dz^{123\overline{123}}\}\\ C_{\Gamma}^{1,0}&=\mathbb{C}\{dz^1,e^{-z^1}dz^2,e^{z^1}dz^3,e^{-z^{\bar{1}}}dz^2,e^{z^{\bar{1}}}dz^3\}\\ C_{\Gamma}^{0,1}&=\mathbb{C}\{dz^{\bar{1}},e^{-z^1}dz^{\bar{2}},e^{z^1}dz^{\bar{3}},e^{-z^{\bar{1}}}dz^{\bar{2}},e^{z^{\bar{1}}}dz^{\bar{3}}\}\\ C_{\Gamma}^{2,0}&=\mathbb{C}\{e^{-z^{1}}dz^{12},e^{z^{1}}dz^{13},dz^{23},e^{-z^{\bar{1}}}dz^{12},e^{z^{\bar{1}}}dz^{13} \}\\ C_{\Gamma}^{1,1}&=\mathbb{C}\{dz^{1\bar{1}},e^{-z^{1}}dz^{1\bar{2}},e^{z^{1}}dz^{1\bar{3}},e^{-z^{1}}dz^{2\bar{1}},e^{-2z^{1}}dz^{2\bar{2}},dz^{2\bar{3}},e^{z^{1}}dz^{3\bar{1}},dz^{3\bar{2}},\\ &e^{2z^{1}}dz^{3\bar{3}},e^{-z^{\bar{1}}}dz^{2\bar{1}},e^{-z^{\bar{1}}}dz^{1\bar{2}},e^{z^{\bar{1}}}dz^{1\bar{3}},e^{z^{1}}dz^{3\bar{1}},e^{-2z^{\bar{1}}}dz^{2\bar{2}},e^{2z^{\bar{1}}}dz^{3\bar{3}} \}\\ C_{\Gamma}^{0,2}&=\mathbb{C}\{e^{-z^{1}}dz^{\overline{12}},e^{z^{1}}dz^{\overline{13}},dz^{\overline{23}},e^{-z^{\bar{1}}}dz^{\overline{12}},e^{z^{\bar{1}}}dz^{\overline{13}}\}\\ C_{\Gamma}^{2,1}&=\mathbb{C}\{e^{-z^{1}}dz^{12\bar{1}},e^{-2z^{1}}dz^{12\bar{2}},dz^{12\bar{3}},e^{z^{1}}dz^{13\bar{1}},dz^{13\bar{2}},e^{2z^{1}}dz^{13\bar{3}},dz^{23\bar{1}},\\ &e^{-z^{1}}dz^{23\bar{2}},e^{z^{1}}dz^{23\bar{3}},e^{-z^{\bar{1}}}dz^{12\bar{1}},e^{z^{\bar{1}}}dz^{13\bar{1}},e^{-2z^{\bar{1}}}dz^{12\bar{2}},e^{-z^{\bar{1}}}dz^{23\bar{2}},e^{2z^{\bar{1}}}dz^{13\bar{3}},e^{z^{\bar{1}}}dz^{23\bar{3}} \}\\ C_{\Gamma}^{1,2}&=\mathbb{C}\{e^{-z^{\bar{1}}}dz^{1\overline{12}},e^{-2z^{\bar{1}}}dz^{2\overline{12}},dz^{3\overline{12}}, e^{z^{\bar{1}}}dz^{1\overline{13}},dz^{2\overline{13}},e^{2z^{\bar{1}}}dz^{3\overline{13}},dz^{1\overline{23}},\\ &e^{-z^{\bar{1}}}dz^{2\overline{23}},e^{z^{\bar{1}}}dz^{3\overline{23}},e^{-z^{1}}dz^{1\overline{12}},e^{z^{1}}dz^{1\overline{13}},e^{-2z^{1}}dz^{2\overline{12}},e^{-z^{1}}dz^{2\overline{23}},e^{2z^{1}}dz^{3\overline{13}},e^{z^{1}}dz^{3\overline{13}} \} \end{align*} $$

and

$$ \begin{align*} C_{\Gamma}^{3,1}&=\mathbb{C}\{dz^{123\bar{1}},e^{-z^{1}}dz^{123\bar{2}},e^{z^{1}}dz^{123\bar{3}},e^{-z^{\bar{1}}}dz^{123\bar{2}},e^{z^{\bar{1}}}dz^{123\bar{3}} \}\\ C_{\Gamma}^{2,2}&=\mathbb{C}\{e^{-2z^{1}}dz^{12\overline{12}},dz^{12\overline{13}},e^{-z^{1}}dz^{12\overline{23}},dz^{13\overline{12}},e^{2z^{1}}dz^{13\overline{13}},e^{z^{1}}dz^{13\overline{23}}, e^{-z^{1}}dz^{23\overline{12}},\\ &e^{z^{1}}dz^{23\overline{13}},dz^{23\overline{23}},e^{-2z^{\bar{1}}}dz^{12\overline{12}},e^{-z^{\bar{1}}}dz^{23\overline{12}},e^{-z^{\bar{1}}}dz^{12\overline{23}},e^{z^{\bar{1}}}dz^{13\overline{23}},e^{2z^{\bar{1}}}dz^{13\overline{13}}, e^{z^{\bar{1}}}dz^{23\overline{13}}\}\\ C_{\Gamma}^{1,3}&=\mathbb{C}\{dz^{1\overline{123}},e^{-z^{\bar{1}}}dz^{2\overline{123}},e^{z^{\bar{1}}}dz^{3\overline{123}},e^{-z^{1}}dz^{2\overline{123}},e^{z^{1}}dz^{3\overline{123}}\}\\ C_{\Gamma}^{3,2}&=\mathbb{C}\{e^{-z^{1}}dz^{123\overline{12}},e^{z^{1}}dz^{123\overline{13}},dz^{123\overline{23}},e^{-z^{\bar{1}}}dz^{123\overline{12}},e^{z^{\bar{1}}}dz^{123\overline{13}}\}\\ C_{\Gamma}^{2,3}&=\mathbb{C}\{e^{-z^{1}}dz^{12\overline{123}},e^{z^{1}}dz^{13\overline{123}},dz^{23\overline{123}},e^{-z^{\bar{1}}}dz^{12\overline{123}},e^{z^{\bar{1}}}dz^{13\overline{123}}\}. \end{align*} $$

By using (6.4), we easily get the following Aeppli cohomology:

$$ \begin{align*} H_{A}^{1,0}(X)&=\mathbb{C}\{dz^1,e^{-z^1}dz^2,e^{z^1}dz^3,e^{-z^{\bar{1}}}dz^2,e^{z^{\bar{1}}}dz^3\}\\ H_{A}^{0,1}(X)&=\mathbb{C}\{dz^{\bar{1}},e^{-z^1}dz^{\bar{2}},e^{z^1}dz^{\bar{3}},e^{-z^{\bar{1}}}dz^{\bar{2}},e^{z^{\bar{1}}}dz^{\bar{3}}\}\\ H_{A}^{1,1}(X)&=\mathbb{C}\{dz^{1\bar{1}},e^{-z^{1}}dz^{2\bar{1}},e^{-2z^{1}}dz^{2\bar{2}},dz^{2\bar{3}},e^{z^{1}}dz^{3\bar{1}},dz^{3\bar{2}},\\ &e^{2z^{1}}dz^{3\bar{3}},e^{-z^{\bar{1}}}dz^{1\bar{2}},e^{z^{\bar{1}}}dz^{1\bar{3}},e^{-2z^{\bar{1}}}dz^{2\bar{2}},e^{2z^{\bar{1}}}dz^{3\bar{3}}, \}\\ H_{A}^{2,1}(X)&=\mathbb{C}\{dz^{12\bar{3}},dz^{13\bar{2}},dz^{23\bar{1}},e^{-z^{1}}dz^{23\bar{2}},e^{z^{1}}dz^{23\bar{3}},e^{-2z^{\bar{1}}}dz^{12\bar{2}},e^{2z^{\bar{1}}}dz^{13\bar{3}},e^{-z^{\bar{1}}}dz^{23\bar{2}},e^{z^{\bar{1}}}dz^{23\bar{3}} \}\\ H_{A}^{1,2}(X)&=\mathbb{C}\{dz^{1\overline{23}}, e^{-2z^{1}}dz^{2\overline{12}},dz^{2\overline{13}},e^{-z^{1}}dz^{2\overline{23}},dz^{3\overline{12}},e^{2z^{1}}dz^{3\overline{13}},e^{z^{1}}dz^{3\overline{23}},e^{-z^{\bar{1}}}dz^{2\overline{23}},e^{z^{\bar{1}}}dz^{3\overline{23}} \}. \end{align*} $$

X is equipped with the Hermitian metric $h=\sum _{i=1}^3\varphi ^i\otimes \bar {\varphi }^i$ where

$$\begin{align*}\varphi^1=dz^1,\quad\varphi^2=e^{-z^1}dz^2,\quad\varphi^3=e^{z^1}dz^3. \end{align*}$$

Under this metric, the representatives of these cohomology classes listed as above are all harmonic.

Let us first consider canonical Aeppli deformations.

For $(p,q)=(1,0)$ . Let

$$\begin{align*}\sigma_0 = a_{1}dz^1+a_{2}e^{-z^1}dz^2+a_{3}e^{z^1}dz^3+a_{4}e^{-z^{\bar{1}}}dz^2+a_{5}e^{z^{\bar{1}}}dz^3\in\mathcal{H}_{A}^{1,0}(X), \end{align*}$$

we have

$$ \begin{align*} { \partial }\mathcal{L}_{\phi_1}^{1,0}\sigma_0&={ \partial }[-a_1{ \partial } i_{\phi_{1}}dz^{1}+a_{2}i_{\phi_{1}}{ \partial }(e^{-z^{1}}dz^{2})+a_{3}i_{\phi_{1}}{ \partial }(e^{z^{1}}dz^{3})]\\ &={ \partial }[-a_{1}{ \partial }(tdz^{\bar{1}})-a_{2}e^{-z^1}i_{\phi_{1}}dz^{12}+a_{3}e^{z^1}i_{\phi_{1}}dz^{13}]\\ &={ \partial }[a_2te^{-z^{1}}dz^{2\bar{1}}-a_{3}te^{z^{1}}dz^{3\bar{1}}]\\ &=-a_{2}te^{-z^{1}}dz^{12\bar{1}}-a_{3}te^{z^{1}}dz^{13\bar{1}}, \end{align*} $$

which is ${ \partial }{ \bar {\partial } }$ -exact if and only if ${ \partial }\mathcal {L}_{\phi _1}^{1,0}\sigma _0=0$ , i.e.,

(6.5) $$ \begin{align} a_{2}t=a_{3}t=0 \end{align} $$

and in this case the canonical Aeppli deformation is given by

$$\begin{align*}\sigma_1 = G_{A}({ \partial }\bar{{ \partial }})^{*} { \partial } i_{\phi_1}{ \partial } \sigma_0=0,\quad \sigma_2 = G_{A}({ \partial }\bar{{ \partial }})^{*} ({ \partial } i_{\phi_2}{ \partial } \sigma_0 + { \partial } i_{\phi_1}{ \partial } \sigma_1 )=0. \end{align*}$$

Similarly, there holds

$$\begin{align*}{ \partial }\sum_{j=1}^{k}\mathcal{L}_{\phi_j}^{1,0}\sigma_{k-j}=0 , \quad k\geq 2, \end{align*}$$

and $\sigma _k= G_{A}({ \partial }\bar {{ \partial }})^{*}\sum _{i+j=k} { \partial } i_{\phi _j} { \partial }\sigma _i=0$ for any $k\geq 2$ . Therefore, for $W=\mathcal {H}_{A}^{1,0}(X)$ , we have (see Definition 4.1)

$$ \begin{align*} W_{t}^{1,0}= &\Big\{ \sigma_0\in W \mid \sigma(t)\in\ker { \partial }{ \bar{\partial} }_{\phi(t)},\\ &\text{where}~\sigma (t)=\sum_{k} \sigma_k~\text{with}~\sigma_k=G_{A}({ \partial }\bar{{ \partial }})^{*}\sum_{i+j=k} { \partial } i_{\phi_j} { \partial }\sigma_i,~\forall k>0 \Big\}\\ &=\Big\{ \sigma_0\in W \mid (6.5)~\text{is~satisfied} \Big\}, \end{align*} $$

and

$$\begin{align*}w_t^{1,0}=\dim H_{A}^{1,0}(X) - \dim W_{t}^{1,0}= \left\{\!\!\! \begin{array}{rcl} 0, &t =0 \\[5pt] 2, &t\neq 0. \end{array} \right. \end{align*}$$

For $(p,q)=(0,1)$ . Let

$$\begin{align*}\sigma_0 = a_1dz^{\bar{1}}+a_2e^{-z^1}dz^{\bar{2}}+a_3e^{z^1}dz^{\bar{3}}+a_4e^{-z^{\bar{1}}}dz^{\bar{2}}+a_5e^{z^{\bar{1}}}dz^{\bar{3}}\in\mathcal{H}_{A}^{0,1}(X), \end{align*}$$

we have

$$ \begin{align*} { \partial }\mathcal{L}_{\phi_1}^{1,0}\sigma_0&={ \partial }[a_2 i_{\phi_{1}}{ \partial }(e^{-z^{1}}dz^{\bar{2}})+a_3 i_{\phi_{1}}{ \partial }(e^{z^{1}}dz^{\bar{3}})]\\ &={ \partial }(-a_2te^{-z^{1}}dz^{\overline{12}}+a_3te^{z^{1}}dz^{\overline{13}})\\ &=a_2te^{-z^{1}}dz^{1\overline{12}}+a_3te^{z^{1}}dz^{1\overline{13}} \end{align*} $$

is ${ \partial }{ \bar {\partial } }$ -exact if and only if ${ \partial }\mathcal {L}_{\phi _1}^{1,0}\sigma _0=0$ , i.e.,

(6.6) $$ \begin{align} a_2t=a_3t=0 \end{align} $$

and in this case $\sigma _k= G_{A}({ \partial }\bar {{ \partial }})^{*}\sum _{i+j=k} { \partial } i_{\phi _j} { \partial }\sigma _i=0$ for any $k\geq 1$ . Therefore, for $W=\mathcal {H}_{A}^{0,1}(X)$ , we have

$$ \begin{align*} W_{t}^{0,1}=\Big\{ \sigma_0\in W \mid (6.6)~\text{is~satisfied} \Big\}, \end{align*} $$

and

$$\begin{align*}w_t^{0,1}=\dim H_{A}^{0,1}(X) - \dim W_{t}^{0,1}= \left\{\!\!\! \begin{array}{rcl} 0, &t =0 \\[5pt] 2, &t\neq 0. \end{array} \right. \end{align*}$$

For $(p,q)=(1,1)$ . Let

$$ \begin{align*} \sigma_0 &= a_1dz^{1\bar{1}}+a_2e^{-z^{1}}dz^{2\bar{1}}+a_3e^{-2z^{1}}dz^{2\bar{2}}+a_4dz^{2\bar{3}}+a_5e^{z^{1}}dz^{3\bar{1}}+\\ &a_6dz^{3\bar{2}}+a_7e^{2z^{1}}dz^{3\bar{3}}+a_8e^{-z^{\bar{1}}}dz^{1\bar{2}}+a_9e^{z^{\bar{1}}}dz^{1\bar{3}}+a_{10}e^{-2z^{\bar{1}}}dz^{2\bar{2}}+a_{11}e^{2z^{\bar{1}}}dz^{3\bar{3}}\in\mathcal{H}_{A}^{1,1}(X), \end{align*} $$

we have

$$ \begin{align*} { \partial }\mathcal{L}_{\phi_1}^{1,0}\sigma_0&={ \partial }[a_3 i_{\phi_{1}}{ \partial }(e^{-2z^{1}}dz^{2\bar{2}})+a_7 i_{\phi_{1}}{ \partial }(e^{2z^{1}}dz^{3\bar{3}})]\\ &={ \partial }(2a_{3}te^{-2z^{1}}dz^{2\overline{12}}-2a_{7}te^{2z^{1}}dz^{3\overline{13}})\\ &=-4a_{3}te^{-2z^{1}}dz^{12\overline{12}}-4a_{7}te^{2z^{1}}dz^{13\overline{13}} \end{align*} $$

is ${ \partial }{ \bar {\partial } }$ -exact if and only if ${ \partial }\mathcal {L}_{\phi _1}^{1,0}\sigma _0=0$ , i.e.,

(6.7) $$ \begin{align} a_3t=a_7t=0 \end{align} $$

and in this case $\sigma _k= G_{A}({ \partial }\bar {{ \partial }})^{*}\sum _{i+j=k} { \partial } i_{\phi _j} { \partial }\sigma _i=0$ for any $k\geq 1$ . Therefore, for $W=\mathcal {H}_{A}^{1,1}(X)$ , we have

$$ \begin{align*} W_{t}^{1,1}=\Big\{ \sigma_0\in W \mid (6.7)~\text{is~satisfied} \Big\}, \end{align*} $$

and

$$\begin{align*}w_t^{1,1}=\dim H_{A}^{1,1}(X) - \dim W_{t}^{1,1}= \left\{\!\!\! \begin{array}{rcl} 0, &t =0 \\[5pt] 2, &t\neq 0. \end{array} \right. \end{align*}$$

For $(p,q)=(2,1)$ . Let

$$ \begin{align*} \sigma_0 &= a_1dz^{12\bar{3}}+a_2dz^{13\bar{2}}+a_3dz^{23\bar{1}}+a_4e^{-z^{1}}dz^{23\bar{2}}+\\ &a_5e^{z^{1}}dz^{23\bar{3}}+a_6e^{-2z^{\bar{1}}}dz^{12\bar{2}}+a_7e^{2z^{\bar{1}}}dz^{13\bar{3}}+a_8e^{-z^{\bar{1}}}dz^{23\bar{2}}+a_9e^{z^{\bar{1}}}dz^{23\bar{3}}\in\mathcal{H}_{A}^{2,1}(X), \end{align*} $$

we have

$$ \begin{align*} { \partial }\mathcal{L}_{\phi_1}^{1,0}\sigma_0&={ \partial }[a_4i_{\phi_{1}}{ \partial }(e^{-z^{1}}dz^{23\bar{2}})+a_5i_{\phi_{1}}{ \partial }(e^{z^{1}}dz^{23\bar{3}})\\ &\quad -a_6{ \partial } i_{\phi_{1}}(e^{-2z^{\bar{1}}}dz^{12\bar{2}})-a_7{ \partial } i_{\phi_{1}}(e^{2z^{\bar{1}}}dz^{13\bar{3}})]\\ &={ \partial }(-a_4te^{-z^{1}}dz^{23\overline{12}}+a_5te^{z^{1}}dz^{23\overline{13}})\\ &=a_4te^{-z^{1}}dz^{123\overline{12}}+a_5te^{z^{1}}dz^{123\overline{13}} \end{align*} $$

is ${ \partial }{ \bar {\partial } }$ -exact if and only if ${ \partial }\mathcal {L}_{\phi _1}^{1,0}\sigma _0=0$ , i.e.,

(6.8) $$ \begin{align} a_{4}t=a_{5}t=0 \end{align} $$

and in this case $\sigma _k= G_{A}({ \partial }\bar {{ \partial }})^{*}\sum _{i+j=k} { \partial } i_{\phi _j} { \partial }\sigma _i=0$ for any $k\geq 1$ . Therefore, for $W=\mathcal {H}_{A}^{2,1}(X)$ , we have

$$ \begin{align*} W_{t}^{2,1}=\Big\{ \sigma_0\in W \mid (6.8)~\text{is~satisfied} \Big\}, \end{align*} $$

and

$$\begin{align*}w_t^{2,1}=\dim H_{A}^{2,1}(X) - \dim W_{t}^{2,1}= \left\{\!\!\! \begin{array}{rcl} 0, &t =0 \\[5pt] 2, &t\neq 0. \end{array} \right. \end{align*}$$

For $(p,q)=(1,2)$ . Let

$$ \begin{align*} \sigma_0 & = a_1dz^{1\overline{23}}+a_2 e^{-2z^{1}}dz^{2\overline{12}}+a_3dz^{2\overline{13}}+a_4e^{-z^{1}}dz^{2\overline{23}}+\\ &a_5dz^{3\overline{12}}+a_6e^{2z^{1}}dz^{3\overline{13}}+a_7e^{z^{1}}dz^{3\overline{23}}+a_8e^{-z^{\bar{1}}}dz^{2\overline{23}}+a_9e^{z^{\bar{1}}}dz^{3\overline{23}}\in\mathcal{H}_{A}^{1,2}(X), \end{align*} $$

we have

$$ \begin{align*} { \partial }\mathcal{L}_{\phi_1}^{1,0}\sigma_0&={ \partial }[a_4i_{\phi_{1}}{ \partial }(e^{-z^{1}}dz^{2\overline{23}})+a_7i_{\phi_{1}}{ \partial }(e^{z^{1}}dz^{3\overline{23}})]\\ &={ \partial }(a_4te^{-z^{1}}dz^{2\overline{123}}-a_7te^{z^{1}}dz^{3\overline{123}})\\ &=-a_4te^{-z^{1}}dz^{12\overline{123}}-a_7te^{z^{1}}dz^{13\overline{123}} \end{align*} $$

is ${ \partial }{ \bar {\partial } }$ -exact if and only if ${ \partial }\mathcal {L}_{\phi _1}^{1,0}\sigma _0=0$ , i.e.,

(6.9) $$ \begin{align} a_{4}t=a_{7}t=0 \end{align} $$

and in this case $\sigma _k= G_{A}({ \partial }\bar {{ \partial }})^{*}\sum _{i+j=k} { \partial } i_{\phi _j} { \partial }\sigma _i=0$ for any $k\geq 1$ . Therefore, for $W=\mathcal {H}_{A}^{1,2}(X)$ , we have

$$ \begin{align*} W_{t}^{1,2}=\Big\{ \sigma_0\in W \mid (6.9)~\text{is~satisfied} \Big\}, \end{align*} $$

and

$$\begin{align*}w_t^{1,2}=\dim H_{A}^{1,2}(X) - \dim W_{t}^{1,2}= \left\{ \begin{array}{rcl} 0, &t =0 \\[5pt] 2, &t\neq 0. \end{array} \right. \end{align*}$$

Now we consider canonical Bott–Chern deformations.

By using (6.3), we get

$$ \begin{align*} H_{BC}^{2,0}(X)&=\mathbb{C}\{e^{-z^{1}}dz^{12},e^{z^{1}}dz^{13},dz^{23} \}\\ H_{BC}^{1,1}(X)&=\mathbb{C}\{dz^{1\bar{1}},e^{-z^{1}}dz^{1\bar{2}},e^{z^{1}}dz^{1\bar{3}},dz^{2\bar{3}},dz^{3\bar{2}},e^{-z^{\bar{1}}}dz^{2\bar{1}},e^{z^{\bar{1}}}dz^{3\bar{1}} \}\\ H_{BC}^{2,1}(X)&=\mathbb{C}\{e^{-z^{1}}dz^{12\bar{1}},e^{-2z^{1}}dz^{12\bar{2}},dz^{12\bar{3}},e^{z^{1}}dz^{13\bar{1}},dz^{13\bar{2}},e^{2z^{1}}dz^{13\bar{3}},dz^{23\bar{1}},e^{-z^{\bar{1}}}dz^{12\bar{1}},e^{z^{\bar{1}}}dz^{13\bar{1}} \}\\ H_{BC}^{1,2}(X)&=\mathbb{C}\{ e^{-z^{1}}dz^{1\overline{12}},e^{z^{1}}dz^{1\overline{13}},dz^{1\overline{23}},dz^{2\overline{13}},dz^{3\overline{12}},e^-{z^{\bar{1}}}dz^{1\overline{12}}, e^{z^{\bar{1}}}dz^{1\overline{13}},e^{-2z^{\bar{1}}}dz^{2\overline{12}},e^{2z^{\bar{1}}}dz^{3\overline{13}} \}\\ H_{BC}^{3,1}(X)&=\mathbb{C}\{dz^{123\bar{1}},e^{-z^{1}}dz^{123\bar{2}},e^{z^{1}}dz^{123\bar{3}} \}\\ H_{BC}^{2,2}(X)&=\mathbb{C}\{e^{-2z^{1}}dz^{12\overline{12}},dz^{12\overline{13}},e^{-z^{1}}dz^{12\overline{23}},dz^{13\overline{12}},e^{2z^{1}}dz^{13\overline{13}},e^{z^{1}}dz^{13\overline{23}},\\ &dz^{23\overline{23}},e^{-2z^{\bar{1}}}dz^{12\overline{12}},e^{2z^{\bar{1}}}dz^{13\overline{13}},e^{-z^{\bar{1}}}dz^{23\overline{12}}, e^{z^{\bar{1}}}dz^{23\overline{13}}\}\\ H_{BC}^{3,2}(X)&=\mathbb{C}\{e^{-z^{1}}dz^{123\overline{12}},e^{z^{1}}dz^{123\overline{13}},dz^{123\overline{23}},e^{-z^{\bar{1}}}dz^{123\overline{12}},e^{z^{\bar{1}}}dz^{123\overline{13}}\}\\ H_{BC}^{2,3}(X)&=\mathbb{C}\{e^{-z^{1}}dz^{12\overline{123}},e^{z^{1}}dz^{13\overline{123}},dz^{23\overline{123}},e^{-z^{\bar{1}}}dz^{12\overline{123}},e^{z^{\bar{1}}}dz^{13\overline{123}}\}. \end{align*} $$

For $(p,q)=(2,0)$ . Let $\sigma _0 = a_1e^{-z^{1}}dz^{12}+a_2e^{z^{1}}dz^{13}+a_3dz^{23}\in \mathcal {H}_{BC}^{2,0}(X)$ , we have

$$ \begin{align*} { \partial } i_{\phi_1}\sigma_0&=-a_1t{ \partial }(e^{-z^{1}}dz^{2\bar{1}})-a_2t{ \partial }(e^{z^{1}}dz^{3\bar{1}})\\ &=a_1te^{-z^{1}}dz^{12\bar{1}}-a_2te^{z^{1}}dz^{13\bar{1}} \end{align*} $$

is ${ \bar {\partial } }$ -exact if and only if $ { \partial } i_{\phi _1}\sigma _0=0$ , i.e.,

(6.10) $$ \begin{align} a_1t=a_2t=0 \end{align} $$

and in this case

$$\begin{align*}\sigma_{1}=-G_{BC}({ \bar{\partial} }^{*}{ \partial }{ \partial }^{*}+{ \bar{\partial} }^{*})\partial i_{\phi_1}\sigma_0 =0. \end{align*}$$

On the other hand,

$$\begin{align*}\partial i_{\phi_2}\sigma_0 = 0 \Longrightarrow \sigma_{2}=-G_{BC}({ \bar{\partial} }^{*}{ \partial }{ \partial }^{*}+{ \bar{\partial} }^{*})\partial(i_{\phi_2}\sigma_0+i_{\phi_1}\sigma_1) =0, \end{align*}$$

and $\phi _k=0,~ k\geq 2$ implies that $\sigma _{k}= G_{BC}({ \bar {\partial } }^{*}{ \partial }{ \partial }^{*}+{ \bar {\partial } }^{*})\sum _{i+j=k} { \partial } i_{\phi _j}\sigma _i=0$ for any $k\geq 2$ .

Therefore, for $V=\mathcal {H}_{BC}^{2,0}(X)$ , we have (see (4.1))

$$\begin{align*}V_{BC,t}^{2,0}=\{a_1e^{-z^{1}}dz^{12}+a_2e^{z^{1}}dz^{13}+a_3dz^{23} \mid (a_{1},a_2.a_3)\in \mathbb{C}^3~\text{satisfy}~(6.10)\}, \end{align*}$$

and

$$\begin{align*}v_t^{2,0}=\dim H_{BC}^{2,0}(X)-\dim V_{BC,t}^{2,0}= \left\{ \begin{array}{rcl} 0, &t =0 \\[5pt] 2, &t \neq 0. \end{array} \right. \end{align*}$$

For $(p,q)=(1,1)$ . Let

$$\begin{align*}\sigma_0 = a_1dz^{1\bar{1}}+a_2e^{-z^{1}}dz^{1\bar{2}}+a_3e^{z^{1}}dz^{1\bar{3}}+a_4dz^{2\bar{3}}+ a_5dz^{3\bar{2}}+a_6e^{-z^{\bar{1}}}dz^{2\bar{1}}+a_7e^{z^{\bar{1}}}dz^{3\bar{1}}\in\mathcal{H}_{BC}^{1,1}(X), \end{align*}$$

we have

$$ \begin{align*} { \partial } i_{\phi_1}\sigma_0&=a_2t{ \partial }(e^{-z^{1}}dz^{\overline{12}})+a_3t{ \partial }(e^{z^{1}}dz^{\overline{13}})\\ &=-a_2te^{-z^{1}}dz^{1\overline{12}}+a_3te^{z^{1}}dz^{1\overline{13}} \end{align*} $$

is ${ \bar {\partial } }$ -exact if and only if $ { \partial } i_{\phi _1}\sigma _0=0$ , i.e.,

(6.11) $$ \begin{align} a_2t=a_3t=0 \end{align} $$

and in this case $\sigma _{k}= G_{BC}({ \bar {\partial } }^{*}{ \partial }{ \partial }^{*}+{ \bar {\partial } }^{*})\sum _{i+j=k} { \partial } i_{\phi _j}\sigma _i=0$ for any $k\geq 1$ .

Therefore, for $V=\mathcal {H}_{BC}^{1,1}(X)$ , we have

$$ \begin{align*} V_{BC,t}^{1,1}&=\{a_1dz^{1\bar{1}}+a_2e^{z^{1}}dz^{1\bar{2}}+a_3e^{-z^{1}}dz^{1\bar{3}}+a_4dz^{2\bar{3}}+a_5dz^{3\bar{2}}+a_6e^{z^{\bar{1}}}dz^{2\bar{1}}\\ &\quad +a_7e^{-z^{\bar{1}}}dz^{3\bar{1}} \mid (a_{1},\ldots,a_7)\in \mathbb{C}^7~\text{satisfy}~(6.11)\}, \end{align*} $$

and

$$\begin{align*}v_t^{1,1}=\dim H_{BC}^{1,1}(X)-\dim V_{BC,t}^{1,1}= \left\{ \begin{array}{rcl} 0, &t =0 \\[5pt] 2, &t \neq 0. \end{array} \right. \end{align*}$$

For $(p,q)=(2,1)$ . Let

$$ \begin{align*} \sigma_0 &= a_1e^{-z^{1}}dz^{12\bar{1}}+a_2e^{-2z^{1}}dz^{12\bar{2}}+a_3dz^{12\bar{3}}+a_4e^{z^{1}}dz^{13\bar{1}}+a_5dz^{13\bar{2}}+\\ &a_6e^{2z^{1}}dz^{13\bar{3}}+a_7dz^{23\bar{1}}+a_8e^{-z^{\bar{1}}}dz^{12\bar{1}}+a_9e^{z^{\bar{1}}}dz^{13\bar{1}}\in\mathcal{H}_{BC}^{2,1}(X), \end{align*} $$

we have

$$ \begin{align*} { \partial } i_{\phi_1}\sigma_0&=-a_2t{ \partial }(e^{-2z^{1}}dz^{2\overline{12}})-a_6t{ \partial }(e^{2z^{1}}dz^{3\overline{13}})\\ &=2a_2te^{-2z^{1}}dz^{12\overline{12}}-2a_6te^{2z^{1}}dz^{13\overline{13}} \end{align*} $$

is ${ \bar {\partial } }$ -exact if and only if ${ \partial } i_{\phi _1}\sigma _0=0$ , i.e.,

(6.12) $$ \begin{align} a_2t=a_6t=0 \end{align} $$

and in this case $\sigma _{k}= G_{BC}({ \bar {\partial } }^{*}{ \partial }{ \partial }^{*}+{ \bar {\partial } }^{*})\sum _{i+j=k} { \partial } i_{\phi _j}\sigma _i=0$ for any $k\geq 1$ .

Therefore, for $V=\mathcal {H}_{BC}^{2,1}(X)$ , we have

$$ \begin{align*} V_{BC,t}^{2,1}&=\{a_1e^{z^{1}}dz^{12\bar{1}}+a_2e^{2z^{1}}dz^{12\bar{2}}+a_3dz^{12\bar{3}}+a_4e^{-z^{1}}dz^{13\bar{1}}+a_5dz^{13\bar{2}}+a_6e^{-2z^{1}}dz^{13\bar{3}}\\ &\quad +a_7dz^{23\bar{1}}+a_8e^{z^{\bar{1}}}dz^{12\bar{1}}+a_9e^{-z^{\bar{1}}}dz^{13\bar{1}} \mid (a_{1},\ldots,a_9)\in \mathbb{C}^9~\text{satisfy}~(6.12)\}, \end{align*} $$

and

$$\begin{align*}v_t^{2,1}=\dim H_{BC}^{2,1}(X)-\dim V_{BC,t}^{2,1}= \left\{ \begin{array}{rcl} 0, &t =0 \\[5pt] 2, &t \neq 0. \end{array} \right. \end{align*}$$

For $(p,q)=(3,1)$ . Let

$$\begin{align*}\sigma_0 = a_1dz^{123\bar{1}}+a_2e^{-z^{1}}dz^{123\bar{2}}+a_3e^{z^{1}}dz^{123\bar{3}}\in\mathcal{H}_{BC}^{3,1}(X), \end{align*}$$

we have

$$ \begin{align*} { \partial } i_{\phi_1}\sigma_0&=a_2t{ \partial }(-e^{z^{1}}dz^{23\overline{12}})+a_3t{ \partial }(e^{z^{1}}dz^{23\overline{13}})\\ &=-a_2te^{-z^{1}}dz^{123\overline{12}}+a_3te^{z^{1}}dz^{123\overline{13}} \end{align*} $$

is ${ \bar {\partial } }$ -exact if and only if ${ \partial } i_{\phi _1}\sigma _0=0$ , i.e.,

(6.13) $$ \begin{align} a_2t=a_3t=0 \end{align} $$

and in this case $\sigma _{k}= G_{BC}({ \bar {\partial } }^{*}{ \partial }{ \partial }^{*}+{ \bar {\partial } }^{*})\sum _{i+j=k} { \partial } i_{\phi _j}\sigma _i=0$ for any $k\geq 1$ .

Table 2:

$v^{\bullet ,\bullet }_t$ and $w^{\bullet ,\bullet }_t$ for $t\neq 0.$

Therefore, for $V=\mathcal {H}_{BC}^{3,1}(X)$ , we have

$$ \begin{align*} V_{BC,t}^{3,1}=\{a_1dz^{123\bar{1}}+a_2e^{z^{1}}dz^{123\bar{2}}+a_3e^{-z^{1}}dz^{123\bar{3}} \mid (a_{1},a_2,a_3)\in \mathbb{C}^3~\text{satisfy}~(6.13)\}, \end{align*} $$

and

$$\begin{align*}v_t^{3,1}=\dim H_{BC}^{3,1}(X)-\dim V_{BC,t}^{3,1}= \left\{ \begin{array}{rcl} 0, &t =0 \\[5pt] 2, &t \neq 0. \end{array} \right. \end{align*}$$

For $(p,q)=(2,2)$ . Let

$$ \begin{align*} \sigma_0 &= a_1e^{-2z^{1}}dz^{12\overline{12}}+a_2dz^{12\overline{13}}+a_3e^{-z^{1}} dz^{12\overline{23}}+a_4dz^{13\overline{12}}+a_5e^{2z^{1}}dz^{13\overline{13}}+a_6e^{z^{1}}dz^{13\overline{23}}+\\ &a_7dz^{23\overline{23}}+a_8e^{-2z^{\bar{1}}}dz^{12\overline{12}}+a_9e^{2z^{\bar{1}}} dz^{13\overline{13}}+a_{10}e^{-z^{\bar{1}}}dz^{23\overline{12}}+a_{11}e^{z^{\bar{1}}} dz^{23\overline{13}}\in\mathcal{H}_{BC}^{2,2}(X), \end{align*} $$

we have

$$ \begin{align*} { \partial } i_{\phi_1}\sigma_0&=-a_3t{ \partial }(e^{-z^{1}}dz^{2\overline{123}})-a_6t{ \partial }(e^{z^{1}}dz^{3\overline{123}})\\ &=a_3te^{-z^{1}}dz^{12\overline{123}}-a_6te^{z^{1}}dz^{13\overline{123}} \end{align*} $$

is ${ \bar {\partial } }$ -exact if and only if ${ \partial } i_{\phi _1}\sigma _0=0$ , i.e.,

(6.14) $$ \begin{align} a_3t=a_6t=0 \end{align} $$

and in this case $\sigma _{k}= G_{BC}({ \bar {\partial } }^{*}{ \partial }{ \partial }^{*}+{ \bar {\partial } }^{*})\sum _{i+j=k} { \partial } i_{\phi _j}\sigma _i=0$ for any $k\geq 1$ .

Therefore, for $V=\mathcal {H}_{BC}^{2,2}(X)$ , we have

$$ \begin{align*} V_{BC,t}^{2,2}&=\{a_1e^{2z^{1}}dz^{12\overline{12}}+a_2dz^{12\overline{13}}+a_3e^{z^{1}}dz^{12\overline{23}}+a_4dz^{13\overline{12}}+a_5e^{-2z^{1}}dz^{13\overline{13}}\\ &\quad +a_6e^{-z^{1}}dz^{13\overline{23}}+a_7dz^{23\overline{23}}+a_8e^{2z^{\bar{1}}}dz^{12\overline{12}}+a_9e^{-2z^{\bar{1}}}dz^{13\overline{13}}+a_{10}e^{z^{\bar{1}}}dz^{23\overline{12}}\\ &\quad +a_{11}e^{-z^{\bar{1}}}dz^{23\overline{13}} \mid (a_{1},\ldots,a_{11})\in \mathbb{C}^{11}~\text{satisfy}~(6.14)\}, \end{align*} $$

and

$$\begin{align*}v_t^{2,2}=\dim H_{BC}^{2,2}(X)-\dim V_{BC,t}^{2,2}= \left\{\!\!\! \begin{array}{rcl} 0, &t =0 \\[5pt] 2, &t \neq 0. \end{array} \right. \end{align*}$$

Similarly, we can determine $w_t^{p,q}$ and $v_t^{p,q}$ for all $(p,q)$ . Collecting computations in this subsection all together, we get the following table (Table 2):