The $\partial\bar{\partial}$-lemma under surjective maps

TL;DR

This paper investigates the behavior of the $ ext{d} ext{d}^c$-lemma in complex manifolds under surjective holomorphic maps and establishes a criterion for product manifolds to satisfy this lemma based on their components.

Contribution

It proves that the $ ext{d} ext{d}^c$-lemma is preserved under surjective maps and characterizes product manifolds satisfying the lemma via their factors.

Findings

The $ ext{d} ext{d}^c$-lemma is preserved under surjective holomorphic maps.

A product manifold satisfies the $ ext{d} ext{d}^c$-lemma iff all its factors do.

The result leverages Deligne-Griffiths-Morgan-Sullivan's theorem.

Abstract

We consider the $\partial\bar{\partial}$-lemma for complex manifolds under surjective holomorphic maps. Furthermore, using Deligne-Griffiths-Morgan-Sullivan's theorem, we prove that a product compact complex manifold satisfies the $\partial\bar{\partial}$-lemma if and only if so do all its components.

Equations

$$H_{dR}^k(K^{\bullet ,\bullet }):=H^k(K^{\bullet },d),$$

$$H_{{\partial }_1}^{p,q}(K^{\bullet ,\bullet }):=H^p(K^{\bullet ,q},{\partial }_1)\text{mbox}\text{mbox}and\text{mbox}{H}_{{\partial }_2}^{p,q}(K^{\bullet ,\bullet }):=H^q(K^{p,\bullet },{\partial }_2).$$

$$H_{BC}^{p,q}(K^{\bullet ,\bullet }):=\frac{\text{ker}{\partial }_1^{p,q}\cap \text{ker}{\partial }_2^{p,q}}{\text{im}{\partial }_1^{p-1,q}\circ {\partial }_2^{p-1,q-1}}\text{mbox}\text{mbox}and\text{mbox}{H}_A^{p,q}(K^{\bullet ,\bullet }):=\frac{\text{ker}{\partial }_1^{p,q+1}\circ {\partial }_2^{p,q}}{\text{im}{\partial }_1^{p-1,q}+\text{im}{\partial }_2^{p,q-1}}.$$

$$F^p{K}^k:=\underset{\begin{array}{c}r+s=k\\ r\ge p\end{array}}{⨁}{K}^{r,s}.$$

$$E_{\infty }^{p,k-p}(K^{\bullet ,\bullet })=F^p{H}_{dR}^k(K^{\bullet ,\bullet })/F^{p+1}{H}_{dR}^k(K^{\bullet ,\bullet }).$$

$$b_k(K^{\bullet ,\bullet })=\sum _{\begin{array}{c}p+q=k\end{array}}{\text{dim}}_{\mathbb{C}}{E}_{\infty }^{p,q}(K^{\bullet ,\bullet })\le \dots \le \sum _{\begin{array}{c}p+q=k\end{array}}{\text{dim}}_{\mathbb{C}}{E}_s^{p,q}(K^{\bullet ,\bullet })$$

$$\le$$

$$F^p{H}^k(X,\mathbb{C})=\{[\alpha ]\in H^k(X,\mathbb{C})|\alpha \in F^p{A}^k(X{)}_{\mathbb{C}}\text{mbox}andd\alpha =0\}.$$

$$H^k(X,\mathbb{C})=\underset{p+q=k}{⨁}{V}^{p,q}(X),$$

$$\overline{V^{p,q}(X)}=V^{q,p}(X),\text{mbox}foranyp+q=k.$$

$$\Delta (K^{\bullet ,\bullet }):=\sum _{p,q\ge 0}{h}_{BC}^{p,q}(K^{\bullet ,\bullet })+\sum _{p,q\ge 0}{h}_A^{p,q}(K^{\bullet ,\bullet })-2\sum _{k\ge 0}{b}_k(K^{\bullet ,\bullet }).$$

$$(f_{\ast }\circ (T\wedge ),\text{mbox}\text{pr}):A^{\bullet ,\bullet }(X)\to {\mathcal{D}}^{\bullet ,\bullet }(Y)\oplus A^{\bullet ,\bullet }(X)/f^{\ast }{A}^{\bullet ,\bullet }(Y)$$

$$A^{\bullet ,\bullet }(X){\simeq }_1{A}^{\bullet ,\bullet }(Y)\oplus A^{\bullet ,\bullet }(X)/f^{\ast }{A}^{\bullet ,\bullet }(Y).$$

$$f_{\ast }(T\wedge f^{\ast }\alpha )=c\cdot \alpha$$

$$0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\cdots H^q(A^{p,\bullet }(Y))\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$H^q(A^{p,\bullet }(X))\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$H^q(A^{\bullet ,p}(X))\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$b_k(X)=$$

$$\le$$

$$=$$

$$\alpha =c^{-1}\cdot f_{\ast }(T\wedge f^{\ast }\alpha )=\partial \bar{\partial }(c^{-1}\cdot f_{\ast }(T\wedge \gamma )).$$

$$\sum _{i=1}^r{\pi }^{\ast }(\bullet )\wedge t_i:\underset{i=1}{\overset{r}{⨁}}{A}^{\bullet ,\bullet }(X)[-u_i,-v_i]\to A^{\bullet ,\bullet }(E)$$

$${\tau }_{\bar{\partial }}^{p,q}=\sum _{i=1}^r{\pi }^{\ast }(\bullet )\cup [t_i{]}_{\bar{\partial }}:\underset{i=1}{\overset{r}{⨁}}{H}_{\bar{\partial }}^{p-u_i,q-v_i}(X)\to H_{\bar{\partial }}^{p,q}(E),$$

$$b_k(E)=\sum _{i=1}^r{b}_{k-u_i-v_i}(X)=\sum _{i=1}^r\sum _{p+q=k-u_i-v_i}{\text{dim}}_{\mathbb{C}}{E}_s^{p,q}(X)=\sum _{p+q=k}{\text{dim}}_{\mathbb{C}}{E}_s^{p,q}(E),$$

$${⨁}_{i=1}^r{H}_{\partial }^{p-u_i,q-v_i}(X)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$E_s^{\bullet ,\bullet }(X){\otimes }_{\mathbb{C}}{E}_s^{\bullet ,\bullet }(Y)\cong E_s^{\bullet ,\bullet }(X\times Y)$$

$$b_k(X\times Y)=$$

$$\le$$

$$=$$

$$H^k(X\times Y,\mathbb{C})=$$

Full text

\EditorNote

∗ The author is supported by the National Natural Science Foundation of China (Grant No. 12001500, 12071444), the Natural Science Foundation of Shanxi Province of China (Grant No. 201901D111141) and Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (Grant No. 2020L0290).

\AuthorMark

\TitleMark

Department of Mathematics, North University of China,

Taiyuan, Shanxi 030051, P. R. China

menglingxu@nuc.edu.cn

The $\partial\bar{\partial}$-Lemma Under Surjective Maps

Lingxu MENG

\Abstract

We consider the $\partial\bar{\partial}$-lemma for complex manifolds under surjective holomorphic maps. Furthermore, using Deligne-Griffiths-Morgan-Sullivan’s theorem, we prove that a product compact complex manifold satisfies the $\partial\bar{\partial}$-lemma if and only if so do all its components.

\Keywords

$\partial\bar{\partial}$-lemma; surjective holomorphic map; product complex manifold; fiber bundle; $E_{1}$-isomorphism

\MRSubClass

32Q99

1 Introduction

In this paper, all complex manifolds are assumed to be connected.

The $\partial\bar{\partial}$-lemma on a compact complex manifold $X$ refers to that for every pure-type $d$-closed form on $X$, the properties of $d$-exactness, $\partial$-exactness, $\bar{\partial}$-exactness and $\partial\bar{\partial}$-exactness are equivalent while a compact complex manifold is called a $\partial\bar{\partial}$-manifold if the $\partial\bar{\partial}$-lemma holds on it. The most well-known examples are compact Kähler manifolds. All complex manifolds in the Fujiki class $\mathscr{C}$ are $\partial\bar{\partial}$-manifolds [7] and not Kählerian in general. There also exist many examples which are not in $\mathscr{C}$, see [2, 3, 4, 10]. In particular, D. Angella, T. Suwa, N. Tardini and A. Tomassini [4] provided such an example which is simply-connected. R. Friedman [8] recently proved the $\partial\bar{\partial}$-lemma holds on general Clemens manifolds.

The real homotopy type of a $\partial\bar{\partial}$-manifold is a formal consequence of its de Rham cohomology ring [7], which is a strong topological obstruction of the $\partial\bar{\partial}$-lemma property. D. Angella, N. Tardini, A. Tomassini [5, 6] gave criterions of $\partial\bar{\partial}$-manifolds by non-Kählerness degrees. The $\partial\bar{\partial}$-lemma property is stable under small deformations of the complex structure [17, 18, 6]. Recently, an important result is that the blow-up of a compact complex manifold along a complex submanifold is a $\partial\bar{\partial}$-manifold if and only if the complex manifold and the complex submanifold are both $\partial\bar{\partial}$-manifolds [12, 19, 4, 13, 15], which implies the heredity and bimeromorphic invariance of the $\partial\bar{\partial}$-lemma property are equivalent.

Recently, J. Stelzig [13, 14, 15] deeply investigated the structure of bounded double complexes. The $E_{r}$-isomorphism between double complexes is a central notion in his works. In particular, the $E_{1}$-isomorphism is very useful for the study of various cohomologies and the $\partial\bar{\partial}$-lemma property. We consider the $E_{1}$-isomorphism under surjective holomorphic maps and apply it to the $\partial\bar{\partial}$-lemma property as follows.

Theorem 1.1**.**

Let $f:X\rightarrow Y$ be a holomorphic map between compact complex manifolds with $r=\textrm{dim}_{\mathbb{C}}X-\textrm{dim}_{\mathbb{C}}Y$. Assume that there exists a closed $(r,r)$-current $T$ on $X$ such that $f_{*}T\neq 0$.

$(1)$* If the Frölicher spectral sequence degenerates at $E_{s}$ for $X$, it degenerates at $E_{s}$ for $Y$.*

$(2)$* $\Delta(X)\geq\Delta(Y)$.*

$(3)$* Suppose that $X$ satisfies the $\partial\bar{\partial}$-lemma. Then $Y$ is a $\partial\bar{\partial}$-manifold.*

Some applications of Theorem 1.1 will be given in Sect. 3. Furthermore, we study the $\partial\bar{\partial}$-lemma on product complex manifolds.

Theorem 1.2**.**

Let $X$ and $Y$ be compact complex manifolds. Then $X\times Y$ satisfies the $\partial\bar{\partial}$-lemma, if and only if, $X$ and $Y$ do.

2 Preliminaries

2.1 Real structure

For an injective complex linear map $\phi:U\rightarrow V$ of complex vector spaces, $U$ can be viewed as a complex subspace (i.e., $\phi(U)$) of $V$ via $\phi$. Suppose that $U_{1}$, …, $U_{n}$, $V$ are complex vector spaces and $\phi:\bigoplus\limits_{i=1}^{n}U_{i}\rightarrow V$ or $V\rightarrow\bigoplus\limits_{i=1}^{n}U_{i}$ is an isomorphism, where $\bigoplus\limits_{i=1}^{n}U_{i}$ is an external direct sum of $U_{1}$, …, $U_{n}$. Then $V=\bigoplus\limits_{i=1}^{n}\phi(U_{i})$ or $V=\bigoplus\limits_{i=1}^{n}\phi^{-1}(U_{i})$ is an internal direct sum of subspaces. We simply write them as $V=\bigoplus\limits_{i=1}^{n}U_{i}$, which is said to be the direct sum of the subspaces $U_{1}$, …, $U_{n}$ via $\phi$.

If there exsists a real vector space $K$ such that $U=K\otimes_{\mathbb{R}}\mathbb{C}$, we say that $U$ is a complex vector space with real structure. A complex linear map $\phi:U\rightarrow V$ of complex vector spaces is said to be real, if it is the complexification of a real linear map of real vector spaces, namely, $\phi=\psi\otimes\textrm{id}:U=K\otimes_{\mathbb{R}}\mathbb{C}\rightarrow V=L\otimes_{\mathbb{R}}\mathbb{C}$ for a (real) linear map $\psi:K\rightarrow L$ of real vector spaces. For example, $H^{k}(X,\mathbb{C})=H^{k}(X,\mathbb{R})\otimes_{\mathbb{R}}\mathbb{C}$ is a complex vector space with real structure for a smooth manifold $X$ and the pull back $f^{*}:H^{k}(Y,\mathbb{C})\rightarrow H^{k}(X,\mathbb{C})$ is real for a smooth map $f:X\rightarrow Y$ of smooth manifolds, since it is the complexification of $f^{*}:H^{k}(Y,\mathbb{R})\rightarrow H^{k}(X,\mathbb{R})$. Assume that $\phi:U\rightarrow V$ is injective and real. We view $U$ as a complex subspace of $V$, while the complex conjugation $\overline{W}$ of the complex subspace $W\subseteq U$ can be viewed as the one of $\phi(W)$ in $V$ (i.e., $\overline{\phi(W)}$) via $\phi$, since $\overline{\phi(W)}=\phi(\overline{W})$. Suppose that $U$ and $V$ are complex vector spaces with real structures. Then $U\otimes_{\mathbb{C}}V$ has a natural real structure, under which the complex conjugation $\overline{\alpha\otimes\beta}$ of $\alpha\otimes\beta$ in $U\otimes_{\mathbb{C}}V$ is just $\bar{\alpha}\otimes\bar{\beta}$ for $\alpha\in U$ and $\beta\in V$.

2.2 Double complex and related structures

In this subsection, we refer to [15, Section 2] for more details.

Let $(K^{\bullet,\bullet},\partial_{1},\partial_{2})$ be a double complex with two endomorphisms $\partial_{1}$ , $\partial_{2}$ of bidegree $(1,0)$ and $(0,1)$, which satisfy that $\partial_{i}\circ\partial_{i}=0$ for $i=1$, $2$ and $\partial_{1}\circ\partial_{2}+\partial_{2}\circ\partial_{1}=0$. For convenience, we briefly write it as $K^{\bullet,\bullet}$ sometimes. A double complex $K^{\bullet,\bullet}$ is said to be bounded, if $K^{p,q}=0$ except for finitely many $(p,q)\in\mathbb{Z}^{2}$.

From now on, all the double complexes mentioned are assumed to be of vector spaces over $\mathbb{C}$ and bounded. For a double complex $K^{\bullet,\bullet}$, let $\partial^{p,q}_{1}:K^{p,q}\rightarrow K^{p+1,q}$ and $\partial^{p,q}_{2}:K^{p,q}\rightarrow K^{p,q+1}$ be the restrictions of $\partial_{1}$ and $\partial_{2}$ respectively. Recall the following constructions.

$\bullet$ The de Rham cohomology

[TABLE]

where $(K^{\bullet},d=\partial_{1}+\partial_{2})$ is the simple complex associated to $K^{\bullet,\bullet}$.

$\bullet$ The row and column cohomologies

[TABLE]

$\bullet$ The Bott-Chern and Aeppli cohomologies

[TABLE]

$\bullet$ The filtration by columns

[TABLE]

This filtration on $K^{\bullet}$ induce the one on $H_{dR}^{k}(K^{\bullet,\bullet})$, i.e., the (first) Hodge filtration on the de Rham cohomology. We get the so called (first) Frölicher spectral sequence for $K^{\bullet,\bullet}$, denoted by $(E^{p,q}_{s}(K^{\bullet,\bullet}),F^{p}H^{k}_{dR}(K^{\bullet,\bullet}))$. Set $b_{k}(K^{\bullet,\bullet}):=\textrm{dim}_{\mathbb{C}}H^{k}_{dR}(K^{\bullet,\bullet})$ and $h_{\partial_{2}}^{p,q}(K^{\bullet,\bullet}):=\textrm{dim}_{\mathbb{C}}H_{\partial_{2}}^{p,q}(K^{\bullet,\bullet})$. In particular, $E^{p,q}_{1}(K^{\bullet,\bullet})=H^{p,q}_{\partial_{2}}(K^{\bullet,\bullet})$ and

[TABLE]

In general,

[TABLE]

The Frölicher spectral sequence degenerates at $E_{s}$ (i.e., $E_{s}=E_{\infty}$) if and only if $b_{k}(K^{\bullet,\bullet})=\sum_{\begin{subarray}{c}p+q=k\end{subarray}}\textrm{dim}_{\mathbb{C}}E_{s}^{p,q}(K^{\bullet,\bullet})$ for all $k$.

Suppose that $X$ is a compact complex manifold. Denote by $A^{p}(X)_{\mathbb{C}}$, $A^{p,q}(X)$, $\mathcal{D}^{p,q}(X)$ and $H^{k}(X,\mathbb{C})$ the spaces of complex valued smooth $p$-forms, smooth $(p,q)$-forms, $(p,q)$-currents and the de Rham cohomology with complex coefficients on $X$, respectively. The double complexes $(A^{\bullet,\bullet}(X),\partial,\bar{\partial})$ and $(\mathcal{D}^{\bullet,\bullet}(X),\partial,\bar{\partial})$ are bounded. Then $b_{k}(X):=b_{k}(A^{\bullet,\bullet}(X))$ and $h^{p,q}(X):=h_{\bar{\partial}}^{p,q}(A^{\bullet,\bullet}(X))$ are just the $k$-th Betti and $(p,q)$-th Hodge numbers respectively. Denote by $(E_{s}^{p,q}(X),F^{p}H^{k}(X,\mathbb{C}))$ the Frölicher spectral sequence for $A^{\bullet,\bullet}(X)$. Then $E_{1}^{p,q}(X)=H^{p,q}_{\bar{\partial}}(X)$ and

[TABLE]

Denote by $\overline{F}^{q}H^{k}(X,\mathbb{C})$ the complex conjugation of the subspace $F^{q}H^{k}(X,\mathbb{C})$ in $H^{k}(X,\mathbb{C})$. Set $V^{p,q}(X):=F^{p}H^{k}(X,\mathbb{C})\cap\overline{F}^{q}H^{k}(X,\mathbb{C})$ for $p+q=k$. We say that the Hodge filtration gives a Hodge structure of weight $k$ on $H^{k}(X,\mathbb{C})$, if

[TABLE]

[TABLE]

P. Deligne, P. Griffiths, J. Morgan and D. Sullivan established the well-known theorem on the relationship between the $\partial\bar{\partial}$-lemma and the Frölicher spectral sequence as follows.

Theorem 2.1** ([7, (5.21)]).**

For a compact complex manifold $X$, the following statements are equivalent:

$(i)$* The $\partial\bar{\partial}$-lemma is true for $X$.*

$(ii)$* $(a)$ The Frölicher spectral sequence of $X$ degenerates at $E_{1}$ $($i.e., E_{1}=E_{\infty}$$), and*

*** $(b)$ the Hodge filtration gives a Hodge structure of weight $k$ on $H^{k}(X,\mathbb{C})$, for every $k\geq 0$.*

Remark 2.2**.**

In Theorem 2.1,

$(1)$* $(ii)(a)$ is equivalent to that $F^{p}H^{k}(X,\mathbb{C})/F^{p+1}H^{k}(X,\mathbb{C})\cong H^{p,k-p}_{\bar{\partial}}(X)$ for all $k$, $p$, and hence is equivalent to that $b_{k}(X)=\sum_{\begin{subarray}{c}p+q=k\end{subarray}}h^{p,q}(X)$ for all $k$.*

$(2)$* $(ii)(b)$ implies that $F^{p}H^{k}(X,\mathbb{C})=\bigoplus_{\begin{subarray}{c}r+s=k\\ r\geq p\end{subarray}}V^{r,s}(X)$.*

$(3)$* For a $\partial\bar{\partial}$-manifold $X$, $V^{p,q}(X)\cong H_{\bar{\partial}}^{p,q}(X)$.*

If a morphism of double complexes induces isomorphisms on all column cohomologies (i.e., induces an isomorphism at $E_{1}$ between Frölicher spectral sequences), then it also induce isomorphisms on all de Rham cohomologies. A morphism of double complexes is called an $E_{1}$-isomorphism, if it induces isomorphisms on all row and column cohomologies. We say $K^{\bullet,\bullet}\simeq_{1}L^{\bullet,\bullet}$, if there exists an $E_{1}$-isomorphism $K^{\bullet,\bullet}\rightarrow L^{\bullet,\bullet}$. Then $\simeq_{1}$ is an equivalence relation by [13, Lemma 1.24] (or [15, Proposition 11]). Set $h_{BC}^{p,q}(K^{\bullet,\bullet}):=\textrm{dim}_{\mathbb{C}}H^{p,q}_{BC}(K^{\bullet,\bullet})$, $h_{A}^{p,q}(K^{\bullet,\bullet}):=\textrm{dim}_{\mathbb{C}}H^{p,q}_{A}(K^{\bullet,\bullet})$ and

[TABLE]

By the definition, $\Delta(\bullet)$ is additive for direct sums. Assume that $K^{\bullet,\bullet}\simeq_{1}L^{\bullet,\bullet}$. Then $b_{k}(K^{\bullet,\bullet})=b_{k}(L^{\bullet,\bullet})$. By [15, Corollary 13], $h_{BC}^{p,q}(K^{\bullet,\bullet})=h_{BC}^{p,q}(L^{\bullet,\bullet})$ and $h_{A}^{p,q}(K^{\bullet,\bullet})=h_{A}^{p,q}(L^{\bullet,\bullet})$. Hence, $\Delta(K^{\bullet,\bullet})=\Delta(L^{\bullet,\bullet})$.

For a compact complex manifold $X$, the inclusion $(A^{\bullet,\bullet}(X),\partial,\bar{\partial})\hookrightarrow(\mathcal{D}^{\bullet,\bullet}(X),\partial,\bar{\partial})$ is an $E_{1}$-isomorphism, i.e., $A^{\bullet,\bullet}(X)\simeq_{1}\mathcal{D}^{\bullet,\bullet}(X)$. Set $\Delta(X):=\Delta(A^{\bullet,\bullet}(X))$. D. Angella, A. Tomassini [6, Theorems A and B] and J. Stelzig [15, Theorem 9] proved that

Theorem 2.3**.**

For any bounded double complex $K^{\bullet,\bullet}$, $\Delta(K^{\bullet,\bullet})\geq 0$. Moreover, a compact complex manifold $X$ satisfies the $\partial\bar{\partial}$-lemma if and only if $\Delta(X)=0$.

3 An $E_{1}$-isomorphism under surjective maps

Consider the holomorphic map $f:X\rightarrow Y$ between compact complex manifolds with $r=\textrm{dim}_{\mathbb{C}}X-\textrm{dim}_{\mathbb{C}}Y\geq 0$ satisfying that

$(\star)$ * there exists a closed $(r,r)$-current $T$ on $X$ such that $f_{*}T\neq 0$.*

Notice that, any map satisfying $(\star)$ is surjective. Indeed, if not, by the proper mapping theorem, $f(X)$ is an analytic subset in $Y$ with dimension $<\textrm{dim}_{\mathbb{C}}Y$. For a general point $o\in f(X)$, $X_{o}=f^{-1}(o)$ has dimension $\textrm{dim}_{\mathbb{C}}X-\textrm{dim}_{\mathbb{C}}f(X)>r$. The natural map $H_{BC}^{p,q}(X)\rightarrow H^{p,q}_{BC}(\mathcal{D}^{\bullet,\bullet}(X))$ induced by the inclusion is an isomorphism, so there exists a closed smooth $(r,r)$-form $\Omega$ on $X$ such that $T=\Omega+\partial\bar{\partial}S$ for some $S\in\mathcal{D}^{r-1,r-1}(X)$. Then $f_{*}T=f_{*}\Omega=\int_{X_{o}}\Omega|_{X_{o}}=0$, which contradicts $(\star)$.

Example 3.1**.**

Let $f:X\rightarrow Y$ be a surjective holomorphic map between compact complex manifolds with $r=\textrm{dim}_{\mathbb{C}}X-\textrm{dim}_{\mathbb{C}}Y\geq 0$. The condition $(\star)$ was satisfied in the following cases:

$(1)$* $f$ is a Kähler morphism;*

$(2)$* $X$ is a $r$-Kähler manifold;*

$(3)$* $\textrm{dim}_{\mathbb{C}}X=\textrm{dim}_{\mathbb{C}}Y$, i.e., $r=0$;*

$(4)$* $X$ is a Kähler manifold;*

$(5)$* there exists an analytic subset $Z$ of $X$ with codimension $r$ such that $f(Z)=Y$.*

Recall that, a morphism $f:X\rightarrow Y$ of complex spaces is said to be a Kähler morphism, if there exists an open covering $\mathfrak{U}=\{U_{\alpha}\}$ of $X$ and a system of smooth functions $\{p_{\alpha}\in C^{\infty}(U_{\alpha})\}$ satisfying that the restriction $p_{\alpha}|_{U_{\alpha}\cap X_{y}}$ is a strictly plurisubharmonic function for every $y\in Y$ and $p_{\alpha}-p_{\beta}$ is a pluriharmonic function on $U_{\alpha}\cap U_{\beta}\cap X_{y}$ for every $\alpha$, $\beta$, where $X_{y}=f^{-1}(y)$ is the fiber of $f$ over $y$ (c.f. [9, Definition 4.1]). Evidently, $\{\sqrt{-1}\partial\bar{\partial}p_{\alpha}\}$ give a real closed $(1,1)$-form $\omega_{f}$ globally defined on $X$ and the restriction $\omega_{f}|_{X_{y}}$ is a Kähler form on $X_{y}$ for every $y$. We call such an $\omega_{f}$ the relative Kähler form of $f$. In $(1)$, the constant $f_{*}\omega_{f}^{r}=\int_{X_{y}}\omega_{f}^{r}|_{X_{y}}>0$ for any regular value $y$ of $f$. A complex manifold $X$ is called $p$-Kählerian, if it admits a closed transverse $(p,p)$-form $\Omega$ (c.f. [1, Definition 1.1, 1.3]). In such case, $\Omega|_{Z}$ is a volume form on $Z$ for any complex submanifold $Z$ of pure dimension $p$ of $X$. In $(2)$, there exists a closed transverse $(r,r)$-form $\Omega$ on $X$ and the constant $f_{*}\Omega=\int_{X_{y}}\Omega|_{X_{y}}>0$ for any regular value $y$ of $f$. Any complex manifold is [math]-Kählerian and any Kähler manifold is $p$-Kählerian for $p$ smaller than its dimension, so $(3)$ and $(4)$ are special cases of $(2)$. In $(5)$, denote by $T_{Z}$ the current defined by the integral on $Z$. Then $T_{Z}$ is a closed $(r,r)$-current on $X$ and $f_{*}T_{Z}=d>0$, where $d$ is the general multiplicity of $Z$ over $Y$.

We generalize [13, Lemma 1.38] (or [14, Lemma 7]) as follows.

Lemma 3.2**.**

Let $f:X\rightarrow Y$ be a holomorphic map between compact complex manifolds with $r=\textrm{dim}_{\mathbb{C}}X-\textrm{dim}_{\mathbb{C}}Y$. Assume that there exists a closed $(r,r)$-current $T$ on $X$ such that $f_{*}T\neq 0$. Then

[TABLE]

is an $E_{1}$-isomorphism. Moreover,

[TABLE]

Proof Fix an integer $p$. Denote by $A^{p,\bullet}(X)$, $A^{\bullet,p}(X)$, $\mathcal{D}^{p,\bullet}(X)$ and $\mathcal{D}^{\bullet,p}(X)$ the complex $(A^{p,\bullet}(X),\bar{\partial})$, $(A^{\bullet,p}(X),\partial)$, $(\mathcal{D}^{p,\bullet}(X),\bar{\partial})$ and $(\mathcal{D}^{\bullet,p}(X),\partial)$, respectively. By the assumption, $c=f_{*}T$ is a nonzero constant. By the projection formula,

[TABLE]

for every $\alpha\in A^{\bullet,\bullet}(Y)$, which implies that $f^{*}$ is injective. Hence, there is a short exact sequence

[TABLE]

of double complexes, which induces a long exact sequence

[TABLE]

for any $p$. By (3.2), the composition

[TABLE]

is the left inverse map of $f^{*}$. For all $p$, $q$, we obtain split exact sequences

[TABLE]

from (3.3), which implies that (3.1) induces isomorphisms on column cohomologies. Similarly,

[TABLE]

are split exact sequences for all $p$, $q$ and the composition

[TABLE]

is the left inverse map of $f^{*}$, which implies that (3.1) induces isomorphisms on row cohomologies. So (3.1) is an $E_{1}$-isomorphism. Since $A^{\bullet,\bullet}(Y)\simeq_{1}\mathcal{D}^{\bullet,\bullet}(Y)$, we get the second conclusion. $\Box$

Now, we give a proof of Theorem 1.1 as follows.

$\mathbf{Proof\mbox{ }of\mbox{ }Theorem}$ 1.1 Assume that the Frölicher spectral sequence degenerates at $E_{s}$ for $X$. Then $b_{k}(X)=\sum_{p+q=k}\textrm{dim}_{\mathbb{C}}E_{s}^{p,q}(X)$ for all $k$. We have

[TABLE]

So $b_{k}(Y)=\sum_{p+q=k}\textrm{dim}_{\mathbb{C}}E_{s}^{p,q}(Y)$ for all $k$, which implies that the Frölicher spectral sequence degenerates at $E_{s}$ for $Y$. By Lemma 3.2, $\Delta(X)=\Delta(Y)+\Delta(A^{\bullet,\bullet}(X)/f^{*}A^{\bullet,\bullet}(Y))$. By Theorem 2.3, $(2)$ follows. Suppose that $X$ is a $\partial\bar{\partial}$-manifold. By $(2)$ and Theorem 2.3, we easily get $(3)$. Here, we give a more direct proof. Let $\alpha$ be a $(p,q)$-form on $Y$ such that $\alpha=d\beta$ for some $\beta\in A^{p+q-1}(Y)$. Then $f^{*}\alpha\in A^{p,q}(X)$ satisfies that $f^{*}\alpha=d(f^{*}\beta)$. Since $X$ is a $\partial\bar{\partial}$-manifold, $f^{*}\alpha=\partial\bar{\partial}\gamma$ for some $\gamma\in A^{p-1,q-1}(X)$. For the nonzero constant $f_{*}\Omega$, set $c=f_{*}\Omega$. By the projection formula,

[TABLE]

The natural map $H_{BC}^{p,q}(Y)\rightarrow H^{p,q}_{BC}(\mathcal{D}^{\bullet,\bullet}(Y))$ induced by the inclusion is an isomorphism, so $\alpha$ is $\partial\bar{\partial}$-exact as a smooth form on $Y$. Therefore, $Y$ is a $\partial\bar{\partial}$-manifold. ∎

We study the $\partial\bar{\partial}$-lemma on fiber bundles.

Proposition 3.3**.**

Let $\pi:E\rightarrow X$ be a holomorphic fiber bundle over a compact complex manifold $X$, whose fibers are compact complex manifolds. Assume that there exist closed forms $t_{1}$, $\ldots$, $t_{r}$ of pure bidegrees satisfying that the restrictions of Dolbeault classes $[t_{1}]_{\bar{\partial}}$, $\ldots$, $[t_{r}]_{\bar{\partial}}$ to $E_{x}$ is a linear basis of $H_{\bar{\partial}}^{\bullet,\bullet}(E_{x})=\bigoplus_{p,q\geq 0}H_{\bar{\partial}}^{p,q}(E_{x})$ for every $x\in X$.

$(1)$* The Frölicher spectral sequence degenerates at $E_{s}$ for $E$, if and only if, it degenerates at $E_{s}$ for $Y$.*

$(2)$* If the $\partial\bar{\partial}$-lemma holds on $E$, then it also holds on $X$.*

$(3)$* Suppose that the restrictions of Dolbeault classes $[\bar{t}_{1}]_{\bar{\partial}}$, $\ldots$, $[\bar{t}_{r}]_{\bar{\partial}}$ to $E_{x}$ is also a linear basis of $H_{\bar{\partial}}^{\bullet,\bullet}(E_{x})$. Denote by $(u_{1},v_{1})$, …, $(u_{r},v_{r})$ the degrees of $t_{1}$, $\ldots$, $t_{r}$ respectively. Then*

[TABLE]

is an $E_{1}$-isomorphism and $\Delta(E)=r\cdot\Delta(X)$. Moreover, the $\partial\bar{\partial}$-lemma holds on $E$ if and only if it holds on $X$.

Proof Shortly write (3.4) as $\tau:K^{\bullet,\bullet}\rightarrow A^{\bullet,\bullet}(E)$. Between the two double complexes, $\tau$ induces isomorphisms at $E_{1}$-pages

[TABLE]

by [11, Theorem 1.3], hence induces isomorphisms $\bigoplus_{i=1}^{r}E_{s}^{p-u_{i},q-v_{i}}(X)\tilde{\rightarrow}E_{s}^{p,q}(E)$ at $E_{s}$-pages and isomorphisms $\bigoplus_{i=1}^{r}H^{k-u_{i}-v_{i}}(X,\mathbb{C})\tilde{\rightarrow}H^{k}(E,\mathbb{C})$ at $H$-pages.

Suppose that the Frölicher spectral sequence degenerates at $E_{s}$ for $X$. Then

[TABLE]

which implies that the Frölicher spectral sequence degenerates at $E_{s}$ for $E$. A half of $(1)$ is proved.

Denote by $u$ the complex dimension of the fiber of $E$. Fix a point $x\in X$. By the hypothesis, there is some $t_{i}$ such that $[t_{i}]_{\bar{\partial}}|_{E_{x}}$ is the generator of $H^{u,u}_{\bar{\partial}}(E_{x})\cong\mathbb{C}$. Then $u_{i}=v_{i}=u$ and $\int_{E_{x}}t_{i}|_{E_{x}}\neq 0$, i.e., $\pi_{*}t_{i}\neq 0$. By Theorem 1.1 (1) (3), the other half of $(1)$ and $(2)$ follows.

Consider the commutative diagram

[TABLE]

where $\tau^{p,q}_{\partial}=\sum_{i=1}^{r}\pi^{*}(\bullet)\cup[t_{i}]_{\partial}$, $\bar{\tau}^{q,p}_{\bar{\partial}}=\sum_{i=1}^{r}\pi^{*}(\bullet)\cup[\bar{t}_{i}]_{\bar{\partial}}$ and the two vertical maps are both the complex conjugation, i.e., mapping $[\alpha]_{\partial}$ to $[\bar{\alpha}]_{\bar{\partial}}$. The complex conjugation is a real isomorphism and $\bar{\tau}^{q,p}_{\bar{\partial}}$ is an (complex) isomorphism by [11, Theorem 1.3]. By (3.6), $\tau^{p,q}_{\partial}$ is a real isomorphism and then an (complex) isomorphism. Combining the isomorphism $\tau^{p,q}_{\bar{\partial}}$ (i.e., (3.5)), we proved that $\tau$ is an $E_{1}$-isomorphism. Since $\Delta(A^{\bullet,\bullet}(X)[-u_{i},-v_{i}])=\Delta(X)$, $\Delta(E)=r\cdot\Delta(X)$. By Theorem 2.3, we obtain the last conclusion. $\Box$

Remark 3.4**.**

In the previous work, we proved Proposition 3.3 (3) under a stronger assumption, see [11, Theorem 5.5].

Remark 3.5**.**

The flag bundles satisfy the assumptions in Proposition 3.3 (3), see [11, Corollary 5.7].

Remark 3.6**.**

Fix an integer $n$. The following two statements are equivalent:

$(1)$* The $\partial\bar{\partial}$-lemma property is bimeromorphic invariant in the category of compact complex manifolds of dimension $n$.*

$(2)$* The $\partial\bar{\partial}$-lemma holds on compact complex manifolds of dimension $n$ which are meromorphic image of $\partial\bar{\partial}$-manifolds of dimension $n$.*

Assume that $(1)$ holds. Suppose that $X$ satisfies the $\partial\bar{\partial}$-lemma and $f:X\dashrightarrow Y$ is a surjective meromorphic map between compact complex manifolds of dimension $n$. Any desingularization $Z$ of the graph of $f$ is a $\partial\bar{\partial}$-manifold, since it is bimeromorphic to $X$. The map from $Z$ onto $Y$ satisfies $(\star)$ (see Example 3.1 $(3)$), so $Y$ satisfies the $\partial\bar{\partial}$-lemma by Theorem 1.1 $(3)$. Hence $(2)$ holds. The other part obviously follows.

In general cases, the bimeromorphic invariance of the $\partial\bar{\partial}$-lemma property is still an open problem, see [12, 19, 4, 13, 15]. S. Yang and X.-D. Yang [19, Theorem 1.3] proved that the $\partial\bar{\partial}$-lemma property is bimeromorphic invariant in the category of threefolds. Hence, the $\partial\bar{\partial}$-lemma holds on the threefolds which are meromorphic images of $\partial\bar{\partial}$-threefolds.

4 The $\partial\bar{\partial}$-lemma on product complex manifolds

Lemma 4.1**.**

Supposet that $X$ and $Y$ are compact complex manifolds. For any $s\geq 1$, the Frölicher spectral sequence degenerates at $E_{s}$ for $X\times Y$, if and only if, it degenerates at $E_{s}$ for both $X$ and $Y$.

Proof As those in the proof of Proposition 3.3 (3), it is easy to prove that the external product $A^{\bullet,\bullet}(X)\otimes_{\mathbb{C}}A^{\bullet,\bullet}(Y)\rightarrow A^{\bullet,\bullet}(X\times Y)$ is an $E_{1}$-isomorphism by algebraic Künneth theorem and Künneth theorem for Dolbeault cohomology, where $A^{\bullet,\bullet}(X)\otimes_{\mathbb{C}}A^{\bullet,\bullet}(Y)$ is the double complex associated to $A^{\bullet,\bullet}(X)$ and $A^{\bullet,\bullet}(Y)$. So, for any $s\geq 1$, $E^{\bullet,\bullet}_{s}(A^{\bullet,\bullet}(X)\otimes_{\mathbb{C}}A^{\bullet,\bullet}(Y))\cong E^{\bullet,\bullet}_{s}(X\times Y)$, and then

[TABLE]

by [15, Lemma 15]. We have

[TABLE]

Then $b_{k}(X\times Y)=\sum_{p+q=k}\textrm{dim}_{\mathbb{C}}E_{s}^{p,q}(X\times Y)$ for all $k$, if and only if, $b_{k}(X)=\sum_{a+b=k}\textrm{dim}_{\mathbb{C}}E_{s}^{a,b}(X)$ and $b_{k}(Y)=\sum_{c+d=k}\textrm{dim}_{\mathbb{C}}E_{s}^{c,d}(Y)$ for all $k$, which implies the conclusion. $\Box$

Now, we give a proof of Theorem 1.2 as follows.

$\mathbf{Proof\mbox{ }of\mbox{ }Theorem\mbox{ }\ref{product0}}$ Let $\textrm{pr}_{1}$ and $\textrm{pr}_{2}$ be the projections of $X\times Y$ onto $X$ and $Y$ respectively.

Suppose that $X\times Y$ is a $\partial\bar{\partial}$-manifold. Fix a point $y\in Y$. Clearly, $\textrm{pr}_{1}$ has a multisection $X\times\{y\}$, so it satisfies $(\star)$ (see Example 3.1 (5)). By Theorem 1.1 $(3)$, $X$ satisfies the $\partial\bar{\partial}$-lemma. Similarly, the $\partial\bar{\partial}$-lemma holds on $Y$.

Suppose that $X$ and $Y$ are both $\partial\bar{\partial}$-manifolds. By Theorem 2.1 and Lemma 4.1, it is sufficient to check

Claim. The Hodge filtration gives a Hodge structure of weight $k$ on $H^{k}(X\times Y,\mathbb{C})$, for every $k\geq 0$.

Denote by $\phi$ both the external products $\textrm{pr}_{1}^{*}(\bullet)\wedge\textrm{pr}_{2}^{*}(\bullet):A^{\bullet}(X)_{\mathbb{C}}\otimes_{\mathbb{C}}A^{\bullet}(Y)_{\mathbb{C}}\rightarrow A^{\bullet}(X\times Y)_{\mathbb{C}}$ and $\textrm{pr}_{1}^{*}(\bullet)\cup\textrm{pr}_{2}^{*}(\bullet):H^{\bullet}(X,\mathbb{C})\otimes_{\mathbb{C}}H^{\bullet}(Y,\mathbb{C})\rightarrow H^{\bullet}(X\times Y,\mathbb{C})$. Via $\phi$, we have the direct sum decomposition

[TABLE]

For $a+b=p$, $h+l=k$, $F^{a}A^{h}(X)_{\mathbb{C}}\otimes_{\mathbb{C}}F^{b}A^{l}(Y)_{\mathbb{C}}$ is mapped into $F^{p}A^{k}(X\times Y)_{\mathbb{C}}$ by $\phi$, which implies that $F^{a}H^{h}(X,\mathbb{C})\otimes_{\mathbb{C}}F^{b}H^{l}(Y,\mathbb{C})\subseteq F^{p}H^{k}(X\times Y,\mathbb{C})$ via $\phi$ by (2.2). For convenience, $F^{p}H^{k}(X\times Y,\mathbb{C})$ will be written as $F^{p}H^{k}$ shortly. Then

[TABLE]

All $V^{r,s}(X)\otimes_{\mathbb{C}}V^{u,v}(Y)$ are direct summands in the decomposition (4.2) of $H^{k}$, so the last sum in (4.3) is a direct sum. Hence

[TABLE]

For any $p$, we have an (non canonical) isomorphism

[TABLE]

and then an (non canonical) isomorphism

[TABLE]

By (4.4) and (4.5), $F^{p}H^{k}=\bigoplus\limits_{\begin{subarray}{c}r+u\geq p\\ r+s+u+v=k\end{subarray}}V^{r,s}(X)\otimes_{\mathbb{C}}V^{u,v}(Y)$. Since $\phi$ is real,

[TABLE]

For $p+q=k$,

[TABLE]

From (4.2) and (4.6), the sum $\sum_{{p+q=k}}V^{p,q}(X\times Y)$ of complex subspaces in $H^{k}(X\times Y,\mathbb{C})$ is just the direct sum $\bigoplus_{p+q=k}V^{p,q}(X\times Y)$ and

[TABLE]

Clearly, $\phi$ is real, so $\overline{V^{p,q}(X\times Y)}=V^{q,p}(X\times Y)$ by (4.6). We proved the claim. ∎

Remark 4.2**.**

For the the part of “only if”, we can also give a direct proof. Suppose that $X\times Y$ satisfies the $\partial\bar{\partial}$-lemma. Denote $m=\textrm{dim}_{\mathbb{C}}Y$ and let $\Omega_{Y}$ be a closed smooth $(m,m)$-form on $Y$ such that its Dolbeault class is the inverse image of $1$ under the isomorphism $\int_{Y}:H_{\bar{\partial}}^{m,m}(Y)\tilde{\rightarrow}\mathbb{C}$. Set $\Omega=\textrm{pr}_{2}^{*}\Omega_{Y}$. Clearly, $\Omega$ is $d$-closed. The constant

[TABLE]

for any $x\in X$. By Theorem 1.1 (3), $X$ is a $\partial\bar{\partial}$-manifold. Similarly, $Y$ satisfies the $\partial\bar{\partial}$-lemma.

Remark 4.3**.**

By Proposition 3.3 (3), we gave another proof in the previous work [11, Corollary 5.8]

Acknowledgements.

The author warmly thanks Sheng Rao, Song Yang, Xiang-Dong Yang and Jonas Stelzig [16] for their useful discussions. The author would like to express gratitude to referees for their helpful suggestions and careful reading of the manuscript.