The heredity and bimeromorphic invariance of the $\partial\overline{\partial}$-lemma property

TL;DR

This paper provides a new proof regarding the stability of the $ ext{dd}^c$-lemma under blow-ups, and discusses its inheritance and invariance under bimeromorphic transformations, using explicit cohomology formulas.

Contribution

It introduces a simplified proof of the $ ext{dd}^c$-lemma's behavior under blow-ups and explores its heredity and invariance properties in complex geometry.

Findings

The $ ext{dd}^c$-lemma property is preserved under blow-up transformations.

Explicit formulas for Dolbeault cohomology facilitate understanding of the property.

The $ ext{dd}^c$-lemma is hereditary and bimeromorphically invariant in certain contexts.

Abstract

We give a simple proof of a result on the $\partial\bar{\partial}$-lemma property under a blow-up transformation by Deligne--Griffiths--Morgan--Sullivan's criterion. Here, we use an explicit blow-up formula for Dolbeault cohomology given in our previous work, which can be induced by a morphism expressed on the level of spaces of forms and currents. At last, we discuss the heredity and bimeromorphic invariance of the $\partial\bar{\partial}$-lemma property.

Equations

$$(\text{Bn+k})\Rightarrow (\text{Hn+k,k+1})\Rightarrow (\text{Hn}).$$

$$F^p{A}^k(X{)}_{\mathbb{C}}=\underset{\begin{array}{c}r+s=k\\ r\ge p\end{array}}{⨁}{A}^{r,s}(X),$$

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

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

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

$$K^{\bullet ,\bullet }=A^{\bullet ,\bullet }(X)\oplus \underset{i=1}{\overset{r-1}{⨁}}{A}^{\bullet -i,\bullet -i}(Y).$$

$$\psi :K^{\bullet ,\bullet }\to {\mathcal{D}}^{\prime \bullet ,\bullet }(\tilde{X})$$

$$(\alpha ,{\beta }^1,\dots ,{\beta }^{r-1})\mapsto {\pi }^{\ast }\alpha +\sum _{i=1}^{r-1}{i}_{E\ast }\left(t^{i-1}\wedge (\pi {|}_E{)}^{\ast }{\beta }^i\right),$$

$$H_{\bar{\partial }}^{\bullet ,\bullet }(X)\oplus \underset{i=1}{\overset{r-1}{⨁}}{H}_{\bar{\partial }}^{\bullet -i,\bullet -i}(Y)\tilde{\to }{H}_{\bar{\partial }}^{\bullet ,\bullet }(\tilde{X}),$$

$$F^{\bullet }{H}^k(X,\mathbb{C})\oplus \underset{i=1}{\overset{r-1}{⨁}}{F}^{\bullet -i}{H}^{k-2i}(Y,\mathbb{C})\tilde{\to }{F}^{\bullet }{H}^k(\tilde{X},\mathbb{C})$$

$$V^{p,q}(X)\oplus \underset{i=1}{\overset{r-1}{⨁}}{V}^{p-i,q-i}(Y)\tilde{\to }{V}^{p,q}(\tilde{X})$$

$$\varphi :H_{\bar{\partial }}^{\bullet ,\bullet }(\tilde{X})\tilde{\to }{H}_{\bar{\partial }}^{\bullet ,\bullet }(X)\oplus \underset{i=1}{\overset{r-1}{⨁}}{H}_{\bar{\partial }}^{\bullet -i,\bullet -i}(Y),$$

$$\alpha \mapsto ({\pi }_{\ast }\alpha ,{\beta }^1,\dots ,{\beta }^{r-1}),$$

$$\mu =\sum _{i=0}^{r-1}{t}^i\wedge {\pi }^{\ast }(\bullet ):\underset{i=0}{\overset{r-1}{⨁}}{A}^{\bullet -i,\bullet -i}(X)\to A^{\bullet ,\bullet }(\mathbb{P}(E))$$

Full text

1. Introduction

In non-Kähler geometry, the heredity and bimeromorphic invariance of the $\partial\bar{\partial}$-lemma property are two interesting problems, extensively studied in [1, 3, 6, 7, 12, 15, 16, 17] especially in the recent days. 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.

Question 1.1** (Heredity).**

Does any closed complex submanifold of an $n$-dimensional $\partial\bar{\partial}$-manifold still satisfy the $\partial\bar{\partial}$-lemma?

Question 1.2** (Bimeromorphic invariance).**

Does any compact complex manifold being bimeromorphic to an $n$-dimensional $\partial\bar{\partial}$-manifold satisfy the $\partial\bar{\partial}$-lemma?

Clearly, the heredity is true for the $\partial\bar{\partial}$-manifolds of dimensions $\leq 2$. Suppose that $\widetilde{X}$ is a modification of a compact complex manifold $X$. A. Parshin [11] and P. Deligne, Ph. Griffiths, J. Morgan, D. Sullivan [6] proved that if $\widetilde{X}$ is a $\partial\bar{\partial}$-manifold, then so is $X$. L. Alessandrini [1] posed a question in its inverse direction: if $X$ satisfies the $\partial\bar{\partial}$-lemma, so does $\widetilde{X}$? We can easily prove that, Question 1.2 is equivalent to Alessandrini’s one. It is true on complex surfaces by the classical results that each compact complex surface with even first Betti number is Kähler (see [5, 8] for a uniform proof) and the first Betti number is a bimeromorphic invariant, while the case of threefolds was first proved by S. Rao, S. Yang, X.-D. Yang [12] using a Dolbeault blow-up formula and S. Yang, X.-D. Yang [17] using a Bott-Chern blow-up formula. The general case is still open. For any nonnegative integer $k\leq n$, we weaken Question 1.1 as

Question 1.3** (Heredity for codimension $\geq k$).**

Does any closed complex submanifold of codimension $\geq k$ of an $n$-dimensional $\partial\bar{\partial}$-manifolds still satisfy the $\partial\bar{\partial}$-lemma?

For convenience, Questions 1.1-1.3 are denoted by (\textrm{H_{n}}), ($B_{n}$) and (\textrm{H_{n,k}}), respectively. Obviously, (\textrm{H_{n}})=(\textrm{H_{n,0}})\Leftrightarrow(\textrm{H_{n,1}}) and if $k_{1}\leq k_{2}$, then (\textrm{H_{n,k_{1}}})\Rightarrow(\textrm{H_{n,k_{2}}}).

P. Deligne et al. [6, (5.21)] gave an important result, which related the $\partial\bar{\partial}$-lemma property with Hodge filtration and the degeneracy of the Frölicher spectral sequence at $E_{1}$-page. S. Rao, S. Yang and X.-D. Yang [12, Theorem 1.6] investigated the bimeromorphic invariance of the degeneracy of Frölicher spectral sequence at $E_{1}$ by their Dolbeault blow-up formula and pointed out that these results are applicable to Question 1.2 in the remarks after [12, Question 1.2]. Subsequently, their [13, Theorem 1.2] gave an explicit expression of the isomorphism between Dolbeault cohomologies in the blow-up formula to implicitly obtain (\textbf{B_{n}})\Leftrightarrow(\textrm{H_{n,2}}) via Proposition 3.3 indeed. D. Angella, T. Suwa, N. Tardini and A. Tomassini [3, Theorem 13, Questions 22-24] also studied this equivalence by the C̆ech-Dolbeault cohomology with additional hypotheses and generalized their results to compact complex orbifolds. In his PhD thesis, by Angella–Tomassini’s characterization [4, Theorems A and B], J. Stelzig [15, Corollary F] claimed that the $\partial\bar{\partial}$-lemma property is a bimeromorphic invariant of compact complex manifolds if and only if every submanifold of a $\partial\bar{\partial}$-manifold is again a $\partial\bar{\partial}$-manifold. Inspired by them, we will prove the following theorem.

Theorem 1.4**.**

For any integer $k\in\{1,2,\cdots,n\}$, there holds the implication hierarchy

[TABLE]

Moreover, (\textrm{H_{n,2}})\Rightarrow(\textbf{B_{n}}).

Acknowledgements

The author would like to express his sincere gratitude to Prof. Sheng Rao for explaining the details in their original manuscript on (\textbf{B_{n}})\Leftrightarrow(\textrm{H_{n,2}}). The author is supported by the National Natural Science Foundation of China (Grant No. 12001500, 12071444) and the Natural Science Foundation of Shanxi Province of China (Grant No. 201901D111141).

2. Preliminaries

2.1. A criterion on the $\partial\bar{\partial}$-lemma

For a compact complex manifold $X$, a natural filtration on the complex $A^{\bullet}(X)_{\mathbb{C}}$ of $\mathbb{C}$-valued smooth forms on $X$ is defined as

[TABLE]

for all $k$, $p$, which give a spectral sequence $(E_{r}^{p,q},F^{p}H^{k}(X,\mathbb{C}))$, namely, the Frölicher spectral sequence of $X$. Then $E_{1}^{p,q}=H^{p,q}_{\bar{\partial}}(X)$ and

[TABLE]

Clearly, $F^{p}H^{k}(X,\mathbb{C})=0$ for $p<0$ or $p>k$. For convenience, we call $F^{\bullet}H^{k}(X,\mathbb{C})$ the Hodge filtration on $H^{k}(X,\mathbb{C})$. Set $V^{p,q}(X)=F^{p}H^{k}(X,\mathbb{C})\cap\overline{F}^{q}H^{k}(X,\mathbb{C})$ for $p+q=k$, where $\overline{F}^{q}H^{k}(X,\mathbb{C})$ is the complex conjugation of the complex subspace $F^{q}H^{k}(X,\mathbb{C})$ in $H^{k}(X,\mathbb{C})$. We say that the Hodge filtration gives a Hodge structure of weight $k$ on $H^{k}(X,\mathbb{C})$, if

[TABLE]

and

[TABLE]

P. Deligne, Ph. Griffiths, J. Morgan and D. Sullivan established the well-known criterion on the $\partial\bar{\partial}$-lemma as follows.

Theorem 2.1** ([6, (5.21)]).**

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

$(1)$* $X$ satisfies the $\partial\bar{\partial}$-lemma.*

$(2)$* $(a)$ The Frölicher spectral sequence of $X$ degenerates at $E_{1}$, 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*.*

For a compact complex manifold $X$, denote by $b_{k}(X)$, $h^{p,q}(X)$ the $k$-th Betti, $(p,q)$-th Hodge numbers respectively.

$(1)$ In general, $b_{k}(X)\leq\sum\limits_{\begin{subarray}{c}p+q=k\end{subarray}}h^{p,q}(X)$ for all $k$.

$(2)$ The statement of Theorem 2.1 $(2)(a)$ 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 $b_{k}(X)=\sum\limits_{\begin{subarray}{c}p+q=k\end{subarray}}h^{p,q}(X)$ for all $k$.

We refer to [3, Sect. 1.5] and [14, Sect. 2.3] for more discussions on the Frölicher spectral sequence and the Hodge structure.

2.2. Some notations

Assume that $X$ is a complex manifold with complex dimension $n$. Denote by $\mathcal{D}^{\prime p,q}(X)$ the space of $(p,q)$-currents on $X$, which is defined as the dual of the topological vector space $A^{n-q,n-q}(X)$ equipped with its natural topology. The operators $\partial$ and $\bar{\partial}$ on $A^{\bullet,\bullet}(X)$ naturally induce two differentials $\partial$ and $\bar{\partial}$ on $\mathcal{D}^{\prime\bullet,\bullet}(X)$. Evidently, $(A^{\bullet,\bullet}(X),\partial,\bar{\partial})$ and $(\mathcal{D}^{\prime\bullet,\bullet}(X),\partial,\bar{\partial})$ are both double complexes. Denote by $H^{q}(\mathcal{D}^{\prime p,\bullet}(X))$ the $q$-th cohomology of the complex $(\mathcal{D}^{\prime p,\bullet}(X),\bar{\partial})$. The natural inclusion $A^{p,\bullet}(X)\hookrightarrow\mathcal{D}^{\prime p,\bullet}(X)$ induces an isomorphism $\rho_{X}:H_{\bar{\partial}}^{p,q}(X)\tilde{\rightarrow}H^{q}(\mathcal{D}^{\prime p,\bullet}(X))$.

Let $f:X\rightarrow Y$ be a proper holomorphic map between complex manifolds. Set $r=\textrm{dim}_{\mathbb{C}}X-\textrm{dim}_{\mathbb{C}}Y$. The pushforward $f_{*}:\mathcal{D}^{\prime\bullet,\bullet}(X)\rightarrow\mathcal{D}^{\prime\bullet-r,\bullet-r}(Y)$ of the currents defines a morphism $f_{*}:H^{q}(\mathcal{D}^{\prime p,\bullet}(X))\rightarrow H^{q-r}(\mathcal{D}^{\prime p-r,\bullet}(Y))$ for any $p$, $q$. For convenience, we also denote by $f_{*}$ the morphism $\rho_{Y}\circ f_{*}\circ\rho_{X}^{-1}:H_{\bar{\partial}}^{p,q}(X)\rightarrow H_{\bar{\partial}}^{p-r,q-r}(Y)$.

3. The Hodge structures on blow-ups and projective bundles

3.1. Blow-up cases

Let $\pi:\widetilde{X}\rightarrow X$ be the blow-up of a compact complex manifold $X$ along a complex submanifold $Y$ and $E$ the exceptional divisor. Set $r=\textrm{codim}_{\mathbb{C}}Y\geq 2$ and assume that $i_{E}:E\rightarrow\widetilde{X}$ is the inclusion. Let $t\in\mathcal{A}^{1,1}(E)$ be a Chern form of the universal line bundle $\mathcal{O}_{E}(-1)$ on $E={\mathbb{P}(N_{Y/X})}$. Define a double complex

[TABLE]

and a morphism of bounded double complexes

[TABLE]

as

[TABLE]

where $\alpha\in A^{\bullet,\bullet}(X)$ and $\beta^{i}\in A^{\bullet-i,\bullet-i}(Y)$. By [9, Theorem 1.2], $\psi$ induces an isomorphism

[TABLE]

i.e., the isomorphism on $E_{1}$-pages between the spectral sequences associated to $K^{\bullet,\bullet}$ and $\mathcal{D}^{\prime\bullet,\bullet}(\widetilde{X})$. Hence $\psi$ induces an isomorphism $H^{k}(X,\mathbb{C})\oplus\bigoplus\limits_{i=1}^{r-1}H^{k-2i}(Y,\mathbb{C})\tilde{\rightarrow}H^{k}(\widetilde{X},\mathbb{C})$ with the isomorphism on the Hodge filtrations

[TABLE]

for any $k$. Moreover, $\psi$ induces an isomorphism

[TABLE]

for any $p$, $q$.

Lemma 3.1**.**

For a given $k$, the Hodge filtration gives a Hodge structure of weight $k$ on $H^{k}(\widetilde{X},\mathbb{C})$, if and only if, the Hodge filtrations give a Hodge structure of weight $k$ on $H^{k}(X,\mathbb{C})$ and a Hodge structure of weight $k-2i$ on $H^{k-2i}(Y,\mathbb{C})$ for all $1\leq i\leq r-1$.

By (3.1), (3.2) and Remark 2.2, we easily obtain

Lemma 3.2** ([12, Theorem 1.6]).**

The Frölicher spectral sequence of $\widetilde{X}$ degenerates at $E_{1}$, if and only if, so do those of $X$ and $Y$.

Combining Lemmas 3.1, 3.2 and Theorem 2.1, we get

Proposition 3.3**.**

Let $\widetilde{X}$ be the blow-up of a compact complex manifold $X$ along a complex submanifold $Y$ of complex codimension $\geq 2$. Then $\widetilde{X}$ satisfies the $\partial\bar{\partial}$-lemma, if and only if, $X$ and $Y$ do.

Remark 3.4*.*

S. Rao, S. Yang, X.-D. Yang [12, Theorem 1.6] [13, Theorem 1.2] first understood Proposition 3.3 from the viewpoint of Deligne–Griffiths–Morgan–Sullivan’s criterion for the $\partial\bar{\partial}$-lemma and S. Yang, X.-D. Yang [17, Theorem 1.3] studied it from the viewpoint of Angella–Tomassini’s characterization for the case of threefolds. Shortly, D. Angella, T. Suwa, N. Tardini, A. Tomassini [3, Theorem 13] also considered it by use of the C̆ech-Dolbeault cohomology under some additional assumptions. Eventually, J. Stelzig obtianed a blow-up formula for Bott-Chern cohomology and wrote this result out explicitly in [15, Corollary 1.40] [4, Theorems A and B].

Remark 3.5*.*

S. Rao, S. Yang, X.-D. Yang [13, Theorem 1.2] gave an isomorphism for blow-up in the inverse direction of $\psi$ as

[TABLE]

[TABLE]

where $i_{E}^{*}\alpha=\sum\limits_{i=0}^{r-1}h^{i}\cup(\pi|_{E})^{*}\beta^{i}$ for unique $\beta^{i}\in H_{\bar{\partial}}^{\bullet-i,\bullet-i}(Y)$, $0\leq i\leq r-1$ and $h=[t]_{\bar{\partial}}\in H_{\bar{\partial}}^{1,1}(E)$. Actually, $\phi$ can also be lifted to a morphism between complexes of the spaces of forms and currents, see [10, Lemma 6.5]. Using this morphism, we can also give the relationship between $V^{p,q}(X)$, $V^{p,q}(Y)$ and $V^{p,q}(\tilde{X})$ by above progress.

As we know, the exceptional divisor for the blow-up $\widetilde{X}$ of $X$ along $Y$ is biholomorphic to the projective bundle of the normal bundle over $Y$ in $X$. By Proposition 3.3 and the following Proposition 3.9, we easily get

Corollary 3.6**.**

Let $\widetilde{X}$ be a blow-up of a complex manifold $X$ along a smooth center with the exceptional divisor $E$. Then $\widetilde{X}$ is a $\partial\bar{\partial}$-manifold, if and only if, $X$ and $E$ are both $\partial\bar{\partial}$-manifolds.

3.2. Projective bundle cases

Let $\pi:\mathbb{P}(E)\rightarrow X$ be the projective bundle associated to a holomorphic vector bundle $E$ of rank $r$ over a compact complex manifold $X$. Denote by $t\in\mathcal{A}^{1,1}(\mathbb{P}(E))$ a Chern form of $\mathcal{O}_{\mathbb{P}(E)}(-1)$. Define a morphism

[TABLE]

of bounded double complexes. Then $\mu$ induces an isomorphism on $E_{1}$-pages of the spectral sequences, see [12, Proposition 3.3], [3, Proposition 11] or [9, Corollary 3.2]. With the similar arguments as Sect. 3.1, we can prove following results

Lemma 3.7**.**

For a given $k$, the Hodge filtration gives a Hodge structure of weight $k$ on $H^{k}(\mathbb{P}(E),\mathbb{C})$, if and only if, the Hodge filtration gives a Hodge structure of weight $k-2i$ on $H^{k-2i}(X,\mathbb{C})$.

Lemma 3.8**.**

The Frölicher spectral sequence of $\mathbb{P}(E)$ degenerates at $E_{1}$, if and only if, so does that of $X$.

Proposition 3.9**.**

Let $\mathbb{P}(E)$ be the projective bundle associated to a holomorphic vector bundle $E$ on a compact complex manifold $X$. Then $\mathbb{P}(E)$ is a $\partial\bar{\partial}$-manifold, if and only if, $X$ is a $\partial\bar{\partial}$-manifold.

Remark 3.10*.*

The part of “ if ” in Proposition 3.9 was also proved by D. Angella et al. [3, Corollary 12] in a different way.

4. A proof of Theorem 1.4

Proof.

Here we just prove (\textrm{H_{n+k,k+1}})\Rightarrow(\textrm{H_{n}}) and the others are the direct corollary of Proposition 3.3 and the weak factorization theorem [2, Theorem 0.3.1].

Let $X$ be a $\partial\bar{\partial}$-manifold and $Y$ arbitrary closed complex submanifold of codimension $\geq 1$ in $X$. Note that $X\times\mathbb{C}P^{k}$ is the projective bundle associated to the trivial bundle $X\times\mathbb{C}^{k+1}$ over $X$ and thus satisfies the $\partial\bar{\partial}$-lemma by Proposition 3.9. Denote by $\{\textrm{pt}\}$ a set consisting of a single point in $\mathbb{C}P^{k}$. Then $Y\cong Y\times\{\textrm{pt}\}$ has the codimension $\geq k+1$ in $X\times\mathbb{C}P^{k}$ and satisfies the $\partial\bar{\partial}$-lemma by (\textrm{H_{n+k,k+1}}).

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