Positivity Cones under Deformations of Complex Structures

TL;DR

This paper explores the properties of positivity cones and the degeneration of the Fr"olicher spectral sequence in complex manifolds, introducing new concepts like the $h$-$ar ext{d}$-lemma and analyzing their behavior under deformations.

Contribution

It introduces a positivity cone in $E_2$-cohomology and an $h$-$ar ext{d}$-lemma analogue, advancing understanding of complex structure deformations.

Findings

Partial degeneration at $E_2$ of the Fr"olicher spectral sequence.

Positivity cone behaves lower semicontinuously under deformations.

The $h$-$ar ext{d}$-lemma property is deformation open.

Abstract

We investigate connections between the sGG property of compact complex manifolds, defined in earlier work by the second author and L. Ugarte by the requirement that every Gauduchon metric be strongly Gauduchon, and a possible degeneration of the Fr\"olicher spectral sequence. In the first approach that we propose, we prove a partial degeneration at $E_2$ and we introduce a positivity cone in the $E_2$-cohomology of bidegree $(n-2,\,n)$ of the manifold that we then prove to behave lower semicontinuously under deformations of the complex structure. In the second approach that we propose, we introduce an analogue of the $\partial\bar\partial$-lemma property of compact complex manifolds for any real non-zero constant $h$ using the partial twisting $d_h$, introduced recently by the second author, of the standard Poincar\'e differential $d$. We then show, among other things, that this $h$-$\partial\bar\partial$-property is deformation open.

Equations

$$\widehat{\cal S}_{X}=\bigg{\{}\bigg{[}[\Gamma^{n-2,\,n}]_{\bar{\partial}}\bigg{]}_{d_{1}}\,\mid\,\exists\,\omega\hskip 4.30554pt\mbox{Hermitian metric such that}\hskip 4.30554pt\partial\Gamma^{n-2,\,n}=-\bar{\partial}\omega^{n-1}\bigg{\}}\subset E_{2}^{n-2,\,n}(X)$$

$$d_h:=h\partial +\bar{\partial }:C_k^{\infty }(X,\mathbb{C})\to C_{k+1}^{\infty }(X,\mathbb{C}),k\in \{0,\dots ,2n\},$$

$$H^{k}_{d_{h}}(X,\,\mathbb{C})=\ker(d_{h}:C^{\infty}_{k}(X,\,\mathbb{C})\to C^{\infty}_{k+1}(X,\,\mathbb{C}))\bigg{/}\mbox{Im}\,(d_{h}:C^{\infty}_{k-1}(X,\,\mathbb{C})\to C^{\infty}_{k}(X,\,\mathbb{C})),$$

$${\theta }_h:{\Lambda }^{p,q}{T}^{\star }X\to {\Lambda }^{p,q}{T}^{\star }X,u\mapsto {\theta }_{h}u:=h^{p}u,$$

$$u\in \text{mbox}Imd⟹u\in \text{mbox}Im(\partial \bar{\partial }).$$

$$u\in \text{mbox}Imd⟺u\in \text{mbox}Im\partial ⟺u\in \text{mbox}Im\bar{\partial }⟺u\in \text{mbox}Im(\partial \bar{\partial }).$$

$$u\in \text{mbox}Im{d}_h⟺u\in \text{mbox}Im{d}_{-h^{-1}}⟺u\in \text{mbox}Imd⟺u\in \text{mbox}Im(d_h{d}_{-h^{-1}})=\text{mbox}Im(\partial \bar{\partial }).$$

$$\dots \stackrel{d_0}{⟶}{E}_0^{p,q}(X)\stackrel{d_0}{⟶}{E}_0^{p,q+1}(X)\stackrel{d_0}{⟶}\dots ,\text{mbox}where{d}_0=\bar{\partial }.$$

$$E_1^{p,q}(X)=H^{p,q}(X,\mathbb{C})=\ker (d_0:E_0^{p,q}(X)\to E_0^{p,q+1}(X))/\text{mbox}Im(d_0:E_0^{p,q-1}(X)\to E_0^{p,q}(X))$$

$$\dots \stackrel{d_1}{⟶}{E}_1^{p,q}(X)\stackrel{d_1}{⟶}{E}_1^{p+1,q}(X)\stackrel{d_1}{⟶}\dots$$

$$E_r^{p,q}(X)=\ker (d_{r-1}:E_{r-1}^{p,q}(X)\to E_{r-1}^{p+r-1,q-r+2}(X))/\text{mbox}Im(d_{r-1}:E_{r-1}^{p-r+1,q+r-2}(X)\to E_{r-1}^{p,q}(X)),$$

$$\dots \stackrel{d_2}{⟶}{E}_2^{p,q}(X)\stackrel{d_2}{⟶}{E}_2^{p+2,q-1}(X)\stackrel{d_2}{⟶}\dots ,$$

$$\bar{\partial }\alpha =0\text{mbox}and\partial \alpha \in \text{mbox}Im\bar{\partial },$$

$$d_{2}\bigg{(}\bigg{[}[\alpha]_{\bar{\partial}}\bigg{]}_{d_{1}}\bigg{)}=\bigg{[}[\partial u_{1}]_{\bar{\partial}}\bigg{]}_{d_{1}}\hskip 12.91663pt\mbox{where}\hskip 4.30554ptu_{1}\hskip 4.30554pt\mbox{is any form such that}\hskip 4.30554pt\partial\alpha=\bar{\partial}u_{1}.$$

$$\bigg{[}[\alpha]_{\bar{\partial}}\bigg{]}_{d_{1}}=0\iff\exists u\in C^{\infty}_{p-1,\,q}(X,\,\mathbb{C})\cap\ker\bar{\partial}\hskip 4.30554pt\mbox{and}\hskip 4.30554ptv\in C^{\infty}_{p,\,q-1}(X,\,\mathbb{C})\hskip 4.30554pt\mbox{such that}\hskip 4.30554pt\alpha=\partial u+\bar{\partial}v.$$

$$T:H^{2n-2}_{DR}(X,\,\mathbb{C})\longrightarrow E_{2}^{n-2,\,n}(X),\hskip 12.91663pt\{\alpha\}_{DR}\mapsto\bigg{[}[\alpha^{n-2,\,n}]_{\bar{\partial}}\bigg{]}_{d_{1}},$$

$$\text{mbox}ImT=\ker d_2^{n-2,n},$$

$$\partial {\alpha }^{n-1,n-1}+\bar{\partial }{\alpha }^{n,n-2}=0\text{mbox}and\bar{\partial }{\alpha }^{n-1,n-1}+\partial {\alpha }^{n-2,n}=0.$$

$${\alpha }^{n,n-2}=\partial {\beta }^{n-1,n-2}+\bar{\partial }{\beta }^{n,n-3},{\alpha }^{n-1,n-1}=\partial {\beta }^{n-2,n-1}+\bar{\partial }{\beta }^{n-1,n-2}\text{mbox}and{\alpha }^{n-2,n}=\partial {\beta }^{n-3,n}+\bar{\partial }{\beta }^{n-2,n-1}.$$

$$\partial {\alpha }^{n-2,n}=-\bar{\partial }{\Omega }^{n-1,n-1}=-\bar{\partial }({\Omega }^{n-1,n-1}-u)\text{mbox}and\partial {\Omega }^{n-1,n-1}=\partial u+\bar{\partial }v.$$

$$d\alpha =0\text{mbox}andT(\{\alpha {\}}_{DR})=[[{\alpha }^{n-2,n}{]}_{\bar{\partial }}{]}_{d_1}.$$

$$\partial {\alpha }^{n-1,n-1}+\bar{\partial }{\alpha }^{n,n-2}=0\text{mbox}and\partial {\alpha }^{n-2,n}+\bar{\partial }{\alpha }^{n-1,n-1}=0.$$

$$\partial {\alpha }^{n-2,n}=-\bar{\partial }{\Omega }^{n-1,n-1}.$$

$$\partial {\beta }^{n-2,n}=-\bar{\partial }\overline{{\Omega }^{n-1,n-1}}.$$

$${\Gamma }_1:=\overline{({\alpha }^{n-2,n}+{\beta }^{n-2,n})}+({\Omega }^{n-1,n-1}+\overline{{\Omega }^{n-1,n-1}})+({\alpha }^{n-2,n}+{\beta }^{n-2,n})$$

$$T(\{{\Gamma }_1{\}}_{DR})=[[{\alpha }^{n-2,n}+{\beta }^{n-2,n}{]}_{\bar{\partial }}{]}_{d_1}\in E_2^{n-2,n}(X).$$

$${\Gamma }_2:=(\overline{{\beta }^{n-2,n}}-\overline{{\alpha }^{n-2,n}})+({\Omega }^{n-1,n-1}-\overline{{\Omega }^{n-1,n-1}})+({\alpha }^{n-2,n}-{\beta }^{n-2,n})$$

$$T(\{{\Gamma }_2{\}}_{DR})=[[{\alpha }^{n-2,n}-{\beta }^{n-2,n}{]}_{\bar{\partial }}{]}_{d_1}\in E_2^{n-2,n}(X).$$

$$\frac{{\Gamma }_1+{\Gamma }_2}{2}=\overline{{\beta }^{n-2,n}}+{\Omega }^{n-1,n-1}+{\alpha }^{n-2,n}$$

$$T\bigg{(}\bigg{\{}\frac{\Gamma_{1}+\Gamma_{2}}{2}\bigg{\}}_{DR}\bigg{)}=\bigg{[}[\alpha^{n-2,\,n}]_{\bar{\partial}}\bigg{]}_{d_{1}}\in E_{2}^{n-2,\,n}(X).$$

Full text

Positivity Cones under Deformations of Complex Structures

Houda Bellitir and Dan Popovici

Abstract. We investigate connections between the sGG property of compact complex manifolds, defined in earlier work by the second author and L. Ugarte by the requirement that every Gauduchon metric be strongly Gauduchon, and a possible degeneration of the Frölicher spectral sequence. In the first approach that we propose, we prove a partial degeneration at $E_{2}$ and we introduce a positivity cone in the $E_{2}$-cohomology of bidegree $(n-2,\,n)$ of the manifold that we then prove to behave lower semicontinuously under deformations of the complex structure. In the second approach that we propose, we introduce an analogue of the $\partial\bar{\partial}$-lemma property of compact complex manifolds for any real non-zero constant $h$ using the partial twisting $d_{h}$, introduced recently by the second author, of the standard Poincaré differential $d$. We then show, among other things, that this $h$-$\partial\bar{\partial}$-property is deformation open.

1 Introduction

In this paper, we explore a few connections between the metric geometry of compact complex manifolds and some Hodge-theoretic aspects related to the Frölicher spectral sequence (a classical object linking the differential and the complex structures, see e.g. a reminder of the definition in $\S.$2 below) of these manifolds.

It is well known that the existence of a Kähler metric implies the best possible degeneration (i.e. at the first page $E_{1}$) of this spectral sequence. However, compact complex manifolds admit only rarely Kähler metrics and no other metric property is currently known to imply the degeneration at some page of this spectral sequence. The following conjecture was proposed and solved in a special case in [Pop16].

Conjecture 1.1**.**

Let $X$ be a compact complex manifold. If an SKT metric (i.e. a $C^{\infty}$ positive definite $(1,\,1)$-form $\omega$ such that $\partial\bar{\partial}\omega=0$) exists on $X$, the Frölicher spectral sequence of $X$ degenerates at the second page $E_{2}$.

The general case of this conjecture, that will certainly have a role to play in the classification theory of compact complex manifolds, remains open. In this paper, we investigate a possible variant of it in which SKT metrics are replaced by strongly Gauduchon (sG) metrics and the degeneration issue is often confined to the cohomology of $X$ in a degree close to the maximal one. We take our cue from a result of Ceballos-Otal-Ugarte-Villacampa [COUV16, Theorem 5.6] asserting that the Frölicher spectral sequence of any $6$-real-dimensional nilmanifold endowed with an invariant complex structure and carrying an sG metric degenerates at $E_{2}$ and from a possible generalisation of this statement they wondered about in

Question 1.2**.**

([COUV16, Question 5.7]) Does the Frölicher spectral sequence of any $3$-dimensional compact complex manifold carrying an sG metric degenerate at $E_{2}$?

Let $X$ be a compact complex manifold with $\mbox{dim}_{\mathbb{C}}X=n$. Recall that Hermitian metrics, defined as $C^{\infty}$ positive definite $(1,\,1)$-forms $\omega$, always exist on $X$. Even Gauduchon metrics $\omega$, defined as Hermitian metrics satisfying the extra condition $\partial\bar{\partial}\omega^{n-1}=0$, always exist (cf. [Gau77]). However, strongly Gauduchon (sG) metrics, introduced in [Pop13] in the context of deformations of complex structures and defined by the stronger requirement that $\partial\omega^{n-1}$ be $\bar{\partial}$-exact, need not exist. Compact complex manifolds that admit sG metrics were called strongly Gauduchon (sG) manifolds and the notion covers a wide range of manifolds and considerably enlarges the class of compact Kähler manifolds and their bimeromorphic models ($=$ the so-called Fujiki class ${\cal C}$ manifolds).

Section 3 of this paper investigates elements of the $E_{2}$ degeneration of the Frölicher spectral sequence of sGG manifolds, a class of compact complex manifolds introduced and studied in [PU18a]. They are defined by the requirement that every Gauduchon metric be strongly Gauduchon. In particular, sG metrics exist on every sGG manifold. Moreover, thanks to [PU18a, Lemma 1.3], an $n$-dimensional compact complex manifold $X$ is sGG if and only if the following special case of the $\partial\bar{\partial}$-lemma holds on $X$:

for every $d$-closed $(n,\,n-1)$-form $\Gamma$ on $X$, if $\Gamma$ is $\partial$-exact, then $\Gamma$ is also $\bar{\partial}$-exact.

We exploit this fact in at least two ways in section 3:

$(1)$  by showing that every sGG manifold has the partial Frölicher $E_{2}$ degeneration property that the map $d_{2}$ acting in bidegree $(n-2,\,n)$ on the second page of its Frölicher spectral sequence vanishes identically (cf. Proposition 3.3);

$(2)$  by introducing three versions ${\cal S}_{X}\subset E_{2}^{n-2,\,n}(X)$, $\widetilde{\cal S}_{X}\subset H_{DR}^{2n-2}(X,\,\mathbb{R})$ (cf. Definition 3.5) and $\widehat{\cal S}_{X}\subset E_{2}^{n-2,\,n}(X)$ (cf. (15)) of a positivity cone that we call the $E_{2}sG$-cone of a given compact complex $n$-dimensional manifold $X$. The term $E_{2}sG$ refers to the fact that this cone consists of (double) cohomology classes of bidegree $(n-2,\,n)$ that arise on the second page of the Frölicher spectral sequence of $X$ and are thus a refinement of the Dolbeault cohomology.

The surprising aspect is that we thus effectively introduce a notion of positivity for cohomology classes of bidegree $(n-2,\,n)$ that runs counter to the familiar notions of positivity that exist in all the bidegrees $(p,\,p)$ (but not $(p,\,q)$ with $p\neq q$) in complex geometry. This is done by using strongly Gauduchon metrics that enable the existing notion of positivity in bidegree $(n-1,\,n-1)$ to carry over to the bidegree $(n-2,\,n)$ in a natural way. The $E_{2}sG$-cone is empty if the manifold $X$ is not strongly Gauduchon. It depends on the complex structure of $X$. We study this dependence by proving the following

Theorem 1.3**.**

Let $\pi:{\cal X}\longrightarrow\Delta$ be a holomorphic family of compact complex $n$-dimensional manifolds over a ball $\Delta\subset\mathbb{C}^{N}$ centred at the origin. Suppose that the fibre $X_{0}:=\pi^{-1}(0)$ is an sGG manifold.

Then, the De Rham $E_{2}sG$-cone $\widetilde{\cal S}_{X_{t}}\subset H_{DR}^{2n-2}(X,\,\mathbb{R})$ of the fibre $X_{t}:=\pi^{-1}(t)\subset{\cal X}$ varies in a lower semicontinuous way with $t\in\Delta$ varying in a small enough neighbourhood of $0\in\Delta$.

By $X$ we mean the smooth manifold that underlies all the fibres $X_{t}$ of the family (known to be $C^{\infty}$ trivial by the classical Ehresmann Theorem, but in general not holomorphically trivial), so the real De Rham cohomology group $H_{DR}^{2n-2}(X,\,\mathbb{R})$ of degree $2n-2$ is independent of the (complex structure of the) fibre $X_{t}$. The meaning of lower semicontinuous in connection with the dependence on $t$ of the De Rham $E_{2}sG$-cone $\widetilde{\cal S}_{X_{t}}$ is made precise in Theorem 3.9.

We also use the Serre-type duality (proved in the forthcoming joint paper [PU18b] of the second author with L. Ugarte by means of the pseudo-differential Laplacian introduced in [Pop16]) between any pair $(E_{2}^{p,\,q}(X),\,E_{2}^{n-p,\,n-q}(X))$ of vector spaces of complementary bidegrees featuring on the second page of the Frölicher spectral sequence of $X$. In this way, we prove the following analogue in our context of Lamari’s duality [Lam99, Lemma 3.3] between the pseudo-effective cone and the closure of the Gauduchon cone of any compact complex manifold.

Proposition 1.4**.**

Let $X$ be an $n$-dimensional sGG compact complex manifold on which an arbitrary Hermitian metric $\gamma$ has been fixed. The dual of the closure of the cone

[TABLE]

under the duality $E_{2}^{2,\,0}(X)\times E_{2}^{n-2,\,n}(X)\longrightarrow\mathbb{C}$ is the closed convex cone in $E_{2}^{2,\,0}(X)$ consisting of the $E_{2}$-classes $[[\theta^{2,\,0}]_{\bar{\partial}}]_{d_{1}}$ “representable” by $\gamma$-positive currents $\tau^{2,\,0}:C^{\infty}_{n-2,\,n}(X,\,\mathbb{R})_{\gamma}\longrightarrow\mathbb{R}$.

We refer to Proposition 3.16 for a more precise statement and to Definition 3.13 for the notion of $\gamma$-positive current of bidegree $(2,\,0)$. This is another extension of the classical notions of positivity in complex geometry to a bidegree $(p,\,q)$ with $p\neq q$. The bidegree $(2,\,0)$ is especially important on holomorphic symplectic (not necessarily Kähler) manifolds and this lead will hopefully be investigated in future work.

Section 4 of this paper takes up the Frölicher degeneration issue and the variation of the complex structure from a different point of view that still ties in with the theory of sGG manifolds.

Let $X$ be a compact complex manifold with $\mbox{dim}_{\mathbb{C}}X=n$. In [Pop17], for every positive constant $h$, the differential operator $d=\partial+\bar{\partial}$ associated with the smooth structure of $X$ was modified to

[TABLE]

by rescaling its $(1,\,0)$-part in the splitting induced by the complex structure of $X$. Unlike $d$, the operators $d_{h}$ depend on the complex structure of $X$ while also sharing some properties with $d$. The most striking of these is that the $d_{h}$-cohomology of $X$, relying on the integrability property $d_{h}^{2}=0$ and defined for every $k\in\{0,\dots,2n\}$ by

[TABLE]

is isomorphic to the De Rham cohomology of $X$ via the isomorphism $H^{k}_{DR}(X,\,\mathbb{C})\ni\{u\}_{DR}\mapsto\{\theta_{h}u\}_{d_{h}}\in H^{k}_{d_{h}}(X,\,\mathbb{C})$ induced by the pointwise isomorphism

[TABLE]

at the level of pure-type differential forms on $X$. The operators $d_{h}$ capture in a certain sense the relationships between the smooth and the complex structures of $X$.

On the other hand, there exist in the literarure at least two notions of the $\partial\bar{\partial}$-lemma being satisfied by a compact complex manifold that capture the smooth structure-complex structure relationship.

(a)  An early notion that has been used by many authors appeared in [DGMS75, Lemmas 5.11 and 5.15]. It defines the fulfilment of the $\partial\bar{\partial}$-lemma (or of the equivalent $dd^{c}$-lemma) on a compact complex manifold by the requirement that any smooth differential form $u$ of any degree (but not necessarily of pure type) that is both $\partial$-closed and $\bar{\partial}$-closed satisfies the following implication:

[TABLE]

This implication is actually an equivalence since the reverse implication holds trivially on any manifold.

For example, this is what was meant by the $\partial\bar{\partial}$-lemma holding on a given manifold in [AT12].

(b)  In [Pop14], the term $\partial\bar{\partial}$-manifold was introduced to mean that a given compact complex manifold $X$ satisfies the $\partial\bar{\partial}$-lemma if for any $d$-closed pure-type form $u$ on $X$ the following exactness properties are equivalent:

[TABLE]

The last property trivially implies the others, so the above equivalences reduce to each of the other three forms of exactness implying $(\partial\bar{\partial})$-exactness. Since $u$ is of pure type, the $d$-closedness assumption on $u$ is equivalent to $u$ being assumed both $\partial$-closed and $\bar{\partial}$-closed.

On the face of it, condition (4) is more restrictive than (3), but (4) is only required to apply to pure-type forms. It is implicit in [DGMS75, Lemma 5.15] that version (a) of the $\partial\bar{\partial}$-lemma condition (required to hold on all, not necessarily pure-type forms) actually implies version (b).

For every constant $h\in\mathbb{R}\setminus\{0\}$, we introduce in this paper the notion of $h$-$\partial\bar{\partial}$-manifold that implies the above version (a) (hence also version (b)) of the $\partial\bar{\partial}$-condition. The idea is to use the operators $d_{h}$ to capture the interplay between the smooth structure and the complex structure.

Definition 1.5**.**

Let $h\in\mathbb{R}\setminus\{0\}$ be an arbitrary constant. A compact complex manifold $X$ with $\mbox{dim}_{\mathbb{C}}X=n$ is said to be an h-$\partial\bar{\partial}$-manifold if for every $k\in\{0,1,\dots,2n\}$ and every $k$-form $u\in\ker d_{h}\cap\ker d_{-h^{-1}}$, the following exactness conditions are equivalent:

[TABLE]

We prove in Corollary 4.19 that an equivalent property is obtained by the removal of the condition $u\in\mbox{Im}\,d$ from the above sequence of equivalences. Note that the forms $u$ are not required to be of pure type in Definition 1.5. Unlike $\partial$, $\bar{\partial}$ and $\partial\bar{\partial}$, the operators $d_{h}$ do not map pure-type forms to pure-type forms, so the proof of the implication “$(a)_{n}\Longrightarrow(b)_{n}$ “ in [DGMS75, Lemma 5.15] does not seem to adapt easily to yield a proof of the possible implication “version (a) of the $\partial\bar{\partial}$-property $\Longrightarrow$ the $h$-$\partial\bar{\partial}$-property“. Actually, we do not know whether this last implication holds. A priori, the $h$-$\partial\bar{\partial}$-property is stronger when $h\notin\{-1,\,1\}$.

Introducing $d_{h}$-analogues of the standard Bott-Chern and Aeppli cohomologies and following the pattern of Wu’s proof in [Wu06] of the stability under small deformations of the complex structure of the standard $\partial\bar{\partial}$-property, we prove that the analogous statement holds for our $h$-$\partial\bar{\partial}$-property.

Theorem 1.6**.**

Fix an arbitrary constant $h\in\mathbb{R}\setminus\{0\}$. The $h$-$\partial\bar{\partial}$-property of compact complex manifolds is open under deformations of the complex structure.

See Theorem 4.16 for a more precise statement.

One last explanation is in order about the choice of pairing $d_{h}$ with $d_{-h^{-1}}$, rather than with the more natural-looking $\overline{d}_{h}=h\,d_{h^{-1}}$. This choice is forced on us by a formula of the Bochner-Kodaira-Nakano type (the h-BKN identity) that we establish in Theorem 4.5 as a consequence of what we call $h$-commutation relations that we compute for the operators $d_{h}$ and for an arbitrary Hermitian metric in Lemma 4.1. The pair $(d_{h},\,d_{-h^{-1}})$ generalises the classical pair $(d,\,d^{c})$ since for $h=-1$, $d_{-1}$ is a constant multiple (which does not change either the kernel or the image) of $d^{c}=-JdJ=i(\bar{\partial}-\partial)=i\,d_{-1}$.

Acknowledgments. This paper is part of the first author’s PhD thesis under the co-supervision of Saïd Asserda and the second author. The first author is grateful to Professor Saïd Asserda for his constant support, patient guiding of her work throughout the duration of her thesis and careful reading of the paper. Both authors are grateful to Ahmed Zeriahi for fruitful discussions and for his interest in this work.

2 Preliminaries

Let $X$ be a compact complex manifold with $\mbox{dim}_{\mathbb{C}}X=n$.

Let $(E_{r},\,d_{r})$ be the Frölicher spectral sequence of $X$. It relates the De Rham cohomology of $X$ to its Dolbeault cohomology in the following way. The $0^{th}$ page features the (infinite-dimensional) $\mathbb{C}$-vector spaces $E_{0}^{p,\,q}=C^{\infty}_{p,\,q}(X,\,\mathbb{C})$ of $C^{\infty}$ forms of arbitrary bidegree $(p,\,q)$ with $0\leq p,q\leq n$ and the linear maps

[TABLE]

The $1^{st}$ page is defined as the cohomology of the $0^{th}$ page and consists of the (finite-dimensional) Dolbeault cohomology groups

[TABLE]

and the linear maps

[TABLE]

defined by $\partial$ in cohomology in the following way: $d_{1}([\alpha]_{\bar{\partial}})=[\partial\alpha]_{\bar{\partial}}$ for all $[\alpha]_{\bar{\partial}}\in E_{1}^{p,\,q}$.

We then continue by induction and define the $r^{th}$ page as the cohomology of the $(r-1)^{st}$ page, namely

[TABLE]

where the linear maps $d_{r}$ on each page $r$ are of bidegree $(r,\,-r+1)$. In particular, the $2^{nd}$ page is of the form

[TABLE]

where each space $E_{2}^{p,\,q}(X)$ consists of double cohomology classes \bigg{[}[\alpha]_{\bar{\partial}}\bigg{]}_{d_{1}} of smooth $(p,\,q)$-forms $\alpha$ satisfying the conditions

[TABLE]

while each map $d_{2}:E_{2}^{p,\,q}(X)\longrightarrow E_{2}^{p+2,\,q-1}(X)$ is defined by

[TABLE]

The definition of $d_{2}$ is independent of the choice of the $\bar{\partial}$-potential $u_{1}$.

It is easy to check that the vanishing condition for an arbitrary element \bigg{[}[\alpha]_{\bar{\partial}}\bigg{]}_{d_{1}}\in E_{2}^{p,\,q}(X) is

[TABLE]

3 The $E_{2}sG$ cone

Let $X$ be a compact complex manifold with $\mbox{dim}_{\mathbb{C}}X=n$.

3.1 Complex-valued cohomology

We shall first deal essentially with the De Rham cohomology of $X$ with values in $\mathbb{C}$ and the standard space $E_{2}^{n-2,\,n}(X)$ on the second page of the Frölicher spectral sequence.

Proposition 3.1**.**

The following canonical linear map

[TABLE]

is well defined and its image is given by

[TABLE]

where $d_{2}^{n-2,\,n}:E_{2}^{n-2,\,n}(X)\to E_{2}^{n,\,n-1}(X)$ is the $d_{2}$-map acting in bidegree $(n-2,\,n)$.

Proof. Let $\{\alpha\}_{DR}\in H^{2n-2}_{DR}(X,\,\mathbb{C})$ be an arbitrary class and let $\alpha=\alpha^{n,\,n-2}+\alpha^{n-1,\,n-1}+\alpha^{n-2,\,n}$ be an arbitrary representative, where the $\alpha^{p,\,q}$’s are the components of $\alpha$ of types $(p,\,q)$. The condition $d\alpha=0$ is equivalent to

[TABLE]

Since $\bar{\partial}\alpha^{n-2,\,n}=0$ and $\partial\alpha^{n-2,\,n}=-\bar{\partial}\alpha^{n-1,\,n-1}\in\mbox{Im}\,\bar{\partial}$, $\alpha^{n-2,\,n}$ defines a class $[[\alpha^{n-2,\,n}]_{\bar{\partial}}]_{d_{1}}$ in $E_{2}^{n-2,\,n}(X)$.

To show well-definedness for $T$, we still have to show that the definition is independent of the choice of representative $\alpha$ of the De Rham class $\{\alpha\}_{DR}$. This is equivalent to showing that $T$ maps $0\in H^{2n-2}_{DR}(X,\,\mathbb{C})$ to $0\in E_{2}^{n-2,\,n}(X)$. Let $\alpha\in C^{\infty}_{2n-2}(X,\,\mathbb{C})$ be $d$-exact. Then, there exists $\beta=\beta^{n,\,n-3}+\beta^{n-1,\,n-2}+\beta^{n-2,\,n-1}+\beta^{n-3,\,n}$ a $(2n-3)$-form such that $\alpha=d\beta$. This amounts to

[TABLE]

Since $\bar{\partial}\beta^{n-3,\,n}=0$ for bidegree reasons, the last identity in (9) shows, thanks to (7), that $[[\alpha^{n-2,\,n}]_{\bar{\partial}}]_{d_{1}}=0$ in $E_{2}^{n-2,\,n}(X)$.

Let us now prove the inclusion $\ker d_{2}^{n-2,\,n}\subset\mbox{Im}\,T$ in (8). Let $[[\alpha^{n-2,\,n}]_{\bar{\partial}}]_{d_{1}}\in E_{2}^{n-2,\,n}(X)$ such that $d_{2}([[\alpha^{n-2,\,n}]_{\bar{\partial}}]_{d_{1}})=0$. Thanks to (5), (6) and (7), there exist forms $\Omega^{n-1,\,n-1}\in C^{\infty}_{n-1,\,n-1}(X,\,\mathbb{C})$, $u\in C^{\infty}_{n-1,\,n-1}(X,\,\mathbb{C})$ with $u\in\ker\bar{\partial}$ and $v\in C^{\infty}_{n,\,n-2}(X,\,\mathbb{C})$ such that

[TABLE]

If we put $\alpha:=\alpha^{n-2,\,n}+(\Omega^{n-1,\,n-1}-u)-v$, we see that

[TABLE]

Let us now prove the reverse inclusion $\ker d_{2}^{n-2,\,n}\supset\mbox{Im}\,T$ in (8). Let $[[\alpha^{n-2,\,n}]_{\bar{\partial}}]_{d_{1}}\in\mbox{Im}\,T$. This means that $\alpha^{n-2,\,n}$ is the $(n-2,\,n)$-component of a $d$-closed $(2n-2)$-form $\alpha=\alpha^{n,\,n-2}+\alpha^{n-1,\,n-1}+\alpha^{n-2,\,n}$. As already noticed, the condition $d\alpha=0$ is equivalent to

[TABLE]

On the other hand, $d_{2}([[\alpha^{n-2,\,n}]_{\bar{\partial}}]_{d_{1}})=-[[\partial\alpha^{n-1,\,n-1}]_{\bar{\partial}}]_{d_{1}}=[[\bar{\partial}\alpha^{n,\,n-2}]_{\bar{\partial}}]_{d_{1}}=0\in E_{2}^{n,\,n-1}$ since even $[\bar{\partial}\alpha^{n,\,n-2}]_{\bar{\partial}}=0\in H^{n,\,n-1}_{\bar{\partial}}(X,\,\mathbb{C})$. $\Box$

As a consequence, we obtain the following criterion for partial degeneration at $E_{2}$ of the Frölicher spectral sequence of $X$.

Corollary 3.2**.**

The canonical map $T$ defined in Proposition 3.1 is surjective if and only if the map $d_{2}$ vanishes identically in bidegree $(n-2,\,n)$.

We now show that the sGG assumption on the ambient manifold $X$ (see [PU18a] for the definition and a study of the class of sGG manifolds) suffices to guarantee the partial degeneration property mentioned above.

Proposition 3.3**.**

Let $X$ be a compact complex manifold with $\mbox{dim}_{\mathbb{C}}X=n$. If $X$ is sGG, the map $d_{2}^{n-2,\,n}:E_{2}^{n-2,\,n}(X)\longrightarrow E_{2}^{n,\,n-1}(X)$ on the $2^{nd}$ page of the Frölicher spectral sequence of $X$ vanishes identically (equivalently, the canonical linear map $T:H^{2n-2}_{DR}(X,\,\mathbb{C})\longrightarrow E_{2}^{n-2,\,n}(X)$ of Proposition 3.1 is surjective).

Proof. To prove that $T$ is surjective, let $[[\alpha^{n-2,\,n}]_{\bar{\partial}}]_{d_{1}}\in E_{2}^{n-2,\,n}(X)$ and let $\alpha^{n-2,\,n}$ be an arbitrary $(n-2,\,n)$-form representing this double class. Then $\partial\alpha^{n-2,\,n}$ is $\bar{\partial}$-exact, so there exists an $(n-1,\,n-1)$-form $\Omega^{n-1,\,n-1}$ such that

[TABLE]

Now, the sGG assumption on $X$ implies that $\bar{\partial}\overline{\Omega^{n-1,\,n-1}}$ is $\partial$-exact. Indeed, this is equivalent to $\partial\Omega^{n-1,\,n-1}$ being $\bar{\partial}$-exact. Meanwhile, $\partial\Omega^{n-1,\,n-1}$ is a $d$-closed and $\partial$-exact $(n,\,n-1)$-form, so (iii) of Lemma 1.3. in [PU18a] implies that $\partial\Omega^{n-1,\,n-1}$ is also $\bar{\partial}$-exact (thanks to $X$ being sGG).

Thus, there exists $\beta^{n-2,\,n}\in C^{\infty}_{n-2,\,n}(X,\,\mathbb{C})$ such that

[TABLE]

Consequently, $\partial(\alpha^{n-2,\,n}+\beta^{n-2,\,n})=-\bar{\partial}(\Omega^{n-1,\,n-1}+\overline{\Omega^{n-1,\,n-1}})$, hence

[TABLE]

is a $(2n-2)$-form such that $d\Gamma_{1}=0$ and

[TABLE]

Note that $\beta^{n-2,\,n}$ defines indeed an $E_{2}$-class since it is $\bar{\partial}$-closed (for bidegree reasons) and $\partial\beta^{n-2,\,n}$ is $\bar{\partial}$-exact (by construction).

We also get $\partial(\alpha^{n-2,\,n}-\beta^{n-2,\,n})=-\bar{\partial}(\Omega^{n-1,\,n-1}-\overline{\Omega^{n-1,\,n-1}})$, hence

[TABLE]

is a $(2n-2)$-form such that $d\Gamma_{2}=0$ and

[TABLE]

Putting (12) and (13) together, we finally get a $d$-closed $(2n-2)$-form

[TABLE]

satisfying the condition

[TABLE]

Thus, $T$ is surjective. $\Box$

When the manifold $X$ is sGG, we can take the surjectivity of the canonical linear map $T:H^{2n-2}_{DR}(X,\,\mathbb{C})\longrightarrow E_{2}^{n-2,\,n}(X)$ of Proposition 3.1 further by showing that every Hermitian metric $\omega$ on $X$ defines a natural injection of $E_{2}^{n-2,\,n}(X)$ into $H^{2n-2}_{DR}(X,\,\mathbb{C})$ that is a section of $T$. We will need the following Laplace-type pseudo-differential operator

[TABLE]

induced in every bidegree $(p,\,q)$ by any fixed Hermitian metric $\omega$ on $X$. (All the formal adjoints are computed w.r.t. the $L^{2}$ inner product defined by $\omega$ and so is the orthogonal projection $p^{\prime\prime}=p^{\prime\prime}_{\omega}$ onto the harmonic space $\ker\Delta^{\prime\prime}$, where $\Delta^{\prime\prime}:=\bar{\partial}\bar{\partial}^{\star}+\bar{\partial}^{\star}\bar{\partial}$ is the usual $\bar{\partial}$-Laplacian induced by $\omega$.) This operator was introduced in [Pop16] where it was shown that every double class $[[\alpha^{p,\,q}]_{\bar{\partial}}]_{d_{1}}\in E_{2}^{p,\,q}(X)$ has a unique representative lying in the kernel of $\widetilde{\Delta}=\widetilde{\Delta}_{\omega}$ (cf. [Pop16, Theorem 1.1]).

Proposition 3.4**.**

Let $X$ be a compact complex sGG manifold with $\mbox{dim}_{\mathbb{C}}X=n$ and let $\omega$ be an arbitrary Hermitian metric on $X$. For any class $[[\alpha^{n-2,\,n}]_{\bar{\partial}}]_{d_{1}}\in E_{2}^{n-2,\,n}(X)$, let $\alpha^{n-2,\,n}_{\omega}$ be the $\widetilde{\Delta}_{\omega}$-harmonic* representative of $[[\alpha^{n-2,\,n}]_{\bar{\partial}}]_{d_{1}}$, let $\Omega^{n-1,\,n-1}_{\omega}\in C^{\infty}_{n-1,\,n-1}(X,\,\mathbb{C})$ be the minimal $L^{2}_{\omega}$-norm solution of the equation $\bar{\partial}\Omega^{n-1,\,n-1}=-\partial\alpha^{n-2,\,n}_{\omega}$ (cf. (10)) and let $\beta^{n-2,\,n}_{\omega}\in C^{\infty}_{n-2,\,n}(X,\,\mathbb{C})$ be the minimal $L^{2}_{\omega}$-norm solution of the equation $\partial\beta^{n-2,\,n}=-\bar{\partial}\,\overline{\Omega^{n-1,\,n-1}_{\omega}}$ (cf. (11)).*

The linear map

[TABLE]

is injective and $T\circ j_{\omega}$ is the identity map of $E_{2}^{n-2,\,n}(X)$.

Proof. Suppose that $j_{\omega}([[\alpha^{n-2,\,n}]_{\bar{\partial}}]_{d_{1}})=0\in H^{2n-2}_{DR}(X,\,\mathbb{C})$ for some $[[\alpha^{n-2,\,n}]_{\bar{\partial}}]_{d_{1}}\in E_{2}^{n-2,\,n}(X)$. Then, there exists a smooth $(2n-3)$-form $u=u^{n,\,n-3}+u^{n-1,\,n-2}+u^{n-2,\,n-1}+u^{n-3,\,n}$ such that $\overline{\beta^{n-2,\,n}_{\omega}}+\Omega^{n-1,\,n-1}_{\omega}+\alpha^{n-2,\,n}_{\omega}=du$. This implies that $\alpha^{n-2,\,n}_{\omega}=\partial u^{n-3,\,n}+\bar{\partial}u^{n-2,\,n-1}$. Since $\bar{\partial}u^{n-3,\,n}=0$, this further implies that $[[\alpha^{n-2,\,n}]_{\bar{\partial}}]_{d_{1}}=[[\alpha^{n-2,\,n}_{\omega}]_{\bar{\partial}}]_{d_{1}}=0$. Thus, $j_{\omega}$ is injective.

The equality $T\circ j_{\omega}=\mbox{Id}_{E_{2}^{n-2,\,n}(X)}$ follows immediately from the definitions. $\Box$

3.2 Cohomology and sG metrics

We now introduce strongly Gauduchon (sG) metrics into our discussion.

Definition 3.5**.**

Let $X$ be a compact complex manifold with $\mbox{dim}_{\mathbb{C}}X=n$.

(a)  For every strongly Gauduchon metric (if any) $\omega>0$ on $X$, $\bar{\partial}\omega^{n-1}$ is $\partial$-exact. Let us denote by $\Gamma^{n-2,\,n}_{\omega}\in C^{\infty}_{n-2,\,n}(X,\,\mathbb{C})$ the (unique) solution of minimal $L^{2}_{\omega}$-norm of the equation

[TABLE]

Since $\bar{\partial}\Gamma^{n-2,\,n}_{\omega}=0$ (for bidegree reasons) and $\partial\Gamma^{n-2,\,n}_{\omega}\in\mbox{Im}\,\bar{\partial}$, $\Gamma^{n-2,\,n}_{\omega}$ defines an element in $E_{2}^{n-2,\,n}(X)$.

We consider the following subset

[TABLE]

that we call the $E_{2}sG$-cone of $X$.

The real $(2n-2)$-form $\Gamma_{\omega}:=\overline{\Gamma^{n-2,\,n}_{\omega}}+\omega^{n-1}+\Gamma^{n-2,\,n}_{\omega}$ is $d$-closed, so it defines a real De Rham cohomology class $\{\Gamma_{\omega}\}_{DR}$. We consider the following subset

[TABLE]

that we call the De Rham $E_{2}sG$-cone of $X$.

(b)  We also define the following variant of the $E_{2}sG$-cone of $X$ by dropping the $L^{2}_{\omega}$-norm minimality requirement on the solution $\Gamma^{n-2,\,n}$ of equation (14):

[TABLE]

Every metric $\omega$ involved in (15) is sG, so we obviously have ${\cal S}_{X}\subset\widehat{\cal S}_{X}$. We do not know whether the reverse inclusion holds.

The manifold $X$ is strongly Gauduchon if and only if ${\cal S}_{X}$ is non-empty.

We shall now prove that ${\cal S}_{X}$ (resp. $\widetilde{\cal S}_{X}$) is indeed a cone (i.e. a subset that is stable under multiplications by positive scalars) in $E_{2}^{n-2,\,n}(X)$ (resp. $H_{DR}^{2n-2}(X,\,\mathbb{R})$). We need a few preliminaries.

Lemma 3.6**.**

Let $X$ be a compact complex manifold with $\mbox{dim}_{\mathbb{C}}X=n$.

$(i)$*  For any Hermitian metric $\omega$ on $X$ and any positive real $\lambda$, the formal adjoints of $\bar{\partial}$ w.r.t. the metrics $\lambda\,\omega$ and $\omega$, as well as the corresponding $\bar{\partial}$-Laplacians, are related by the formulae*

[TABLE]

in all bidegrees.

(ii)  For any strongly Gauduchon metric $\omega$ on $X$ and any positive real $\lambda$, the forms $\Gamma^{n,\,n-2}_{\lambda\,\omega}:=\overline{\Gamma^{n-2,\,n}_{\lambda\,\omega}}$ and $\Gamma^{n,\,n-2}_{\omega}:=\overline{\Gamma^{n-2,\,n}_{\omega}}$ (see Definition 3.5) are related by the formula

[TABLE]

Consequently, we also have $\Gamma_{\lambda\,\omega}=\lambda^{n-1}\,\Gamma_{\omega}$ for any sG metric $\omega$ on $X$.

Proof. (i)  Let us fix an arbitrary bidegree $(p,\,q)$. For any forms $\alpha,\beta$ of respective bidegrees $(p,\,q-1)$ and $(p,\,q)$, we have

[TABLE]

and

[TABLE]

Since $\langle\langle\bar{\partial}\alpha,\,\beta\rangle\rangle_{\lambda\,\omega}=\langle\langle\alpha,\,\bar{\partial}^{\star}_{\lambda\,\omega}\beta\rangle\rangle_{\lambda\,\omega}$, the above formulae imply

[TABLE]

for all forms $\alpha$ and $\beta$. This proves the first formula in (16).

Since $\Delta^{\prime\prime}=\bar{\partial}\bar{\partial}^{\star}+\bar{\partial}^{\star}\bar{\partial}$ (when $\Delta^{\prime\prime}$ and $\bar{\partial}^{\star}$ are computed w.r.t. the same metric), the latter formula in (16) follows immediately from the former.

(ii)  By Definition 3.5, $\Gamma^{n,\,n-2}_{\omega}$ is the minimal $L^{2}_{\omega}$-norm solution of equation $\bar{\partial}\Gamma^{n,\,n-2}=-\partial\omega^{n-1}$. Consequently, the Neumann formula spells

[TABLE]

Thus, we get:

[TABLE]

where we used the analogue of (17) for $\lambda\,\omega$ to get the first identity and (17) again to get the third identity. $\Box$

Lemma 3.7**.**

Let $X$ be a compact complex manifold with $\mbox{dim}_{\mathbb{C}}X=n$. The sets ${\cal S}_{X}$ and $\widehat{\cal S}_{X}$ are cones in the $\mathbb{C}$-vector space $E_{2}^{n-2,\,n}(X)$, while the set $\widetilde{\cal S}_{X}$ is a cone in $H_{DR}^{2n-2}(X,\,\mathbb{R})$.

Moreover, the cone $\widehat{\cal S}_{X}$ is convex.

Proof. Let $[[\Gamma^{n-2,\,n}_{\omega}]_{\bar{\partial}}]_{d_{1}}\in{\cal S}_{X}$ and $\mu>0$ be arbitrary. Let $\lambda>0$ be the unique positive real such that $\lambda^{n-1}=\mu$. We have

[TABLE]

where we used (ii) of Lemma 3.6 to get the last identity. Now, $\lambda\,\omega$ is a strongly Gauduchon metric if $\omega$ is one, so $[[\Gamma^{n-2,\,n}_{\lambda\,\omega}]_{\bar{\partial}}]_{d_{1}}\in{\cal S}_{X}$.

Consequently, ${\cal S}_{X}$ is stable under multiplications by positive scalars, hence it is a cone. The same goes for $\widetilde{\cal S}_{X}$ since (ii) of Lemma 3.6 also applies to $\Gamma_{\omega}$. That $\widehat{\cal S}_{X}$ is a cone is trivial.

To prove the convexity of $\widehat{\cal S}_{X}$, it suffices to show that $\widehat{\cal S}_{X}$ is stable under additions. This is immediate since if $\partial\Gamma^{n-2,\,n}_{i}=-\bar{\partial}\omega_{i}^{n-1}$ for $i\in\{1,\,2\}$ and $\omega_{i}$ Hermitian metrics on $X$, then $\partial(\Gamma^{n-2,\,n}_{1}+\Gamma^{n-2,\,n}_{2})=-\bar{\partial}\omega_{0}^{n-1}$, where $\omega_{0}>0$ is the unique positive definite $C^{\infty}$ $(1,\,1)$-form on $X$ such that $\omega_{0}^{n-1}=\omega^{n-1}_{1}+\omega^{n-1}_{2}>0$. Therefore, $[[\Gamma^{n-2,\,n}_{1}+\Gamma^{n-2,\,n}_{2}]_{\bar{\partial}}]_{d_{1}}\in\widehat{\cal S}_{X}$. $\Box$

The De Rham $E_{2}sG$-cone $\widetilde{\cal S}_{X}$ depends on the complex structure of $X$ and we now show this dependence to be lower semicontinuous in the sense described below in families of sGG manifolds. It actually suffices to assume that one fibre is sGG as all the nearby fibres are then sGG by the deformation openness of the sGG property (cf. [PU18, Corollary 1.7]).

Definition 3.8**.**

Let $\pi:{\cal X}\longrightarrow\Delta$ be a holomorphic family of compact complex $n$-dimensional manifolds over a ball $\Delta\subset\mathbb{C}^{N}$ centred at the origin. Suppose that the fibre $X_{0}:=\pi^{-1}(0)$ is strongly Gauduchon. Let $X$ stand for the $C^{\infty}$ manifold that underlies the fibres $X_{t}$ with $t\in\Delta$.

With every sG metric $\omega$ on $X_{0}$, we associate a local section $\tau_{\omega}$ of the constant real vector bundle ${\cal H}^{2n-2}_{\mathbb{R}}\longrightarrow\Delta$ whose fibre is the real De Rham cohomology space $H^{2n-2}_{DR}(X,\,\mathbb{R})$ as follows.

As in Definition 3.5, we let $\Gamma_{\omega}:=\overline{\Gamma^{n-2,\,n}_{\omega}}+\omega^{n-1}+\Gamma^{n-2,\,n}_{\omega}$ be the real $d$-closed $(2n-2)$-form defined by the minimal $L^{2}_{\omega}$-norm solution $\Gamma^{n-2,\,n}_{\omega}$ of the equation $\partial\Gamma^{n-2,\,n}_{\omega}=-\bar{\partial}\omega^{n-1}$. (We put $\partial:=\partial_{0}$ and $\bar{\partial}:=\bar{\partial}_{0}$.) The component $(\Gamma_{\omega})^{n-1,\,n-1}_{t}$ of $\Gamma_{\omega}$ of type $(n-1,\,n-1)$ for the complex structure of $X_{t}$ is positive definite if $t$ is close enough to [math], by the continuity of the dependence on $t$ of $(\Gamma_{\omega})^{n-1,\,n-1}_{t}$ and the positivity of $(\Gamma_{\omega})^{n-1,\,n-1}_{0}=\omega^{n-1}>0$. Hence, for every $t$ close to [math], there exists a unique positive definite smooth $(1,\,1)$-form $\omega_{t}$ on $X_{t}$ such that $\omega_{t}^{n-1}=(\Gamma_{\omega})^{n-1,\,n-1}_{t}>0$. In particular, $\omega_{0}=\omega$.

Since $d\Gamma_{\omega}=0$, the form $\bar{\partial}_{t}\omega_{t}^{n-1}$ is $\partial_{t}$-exact (so $\omega_{t}$ is an sG metric on $X_{t}$). We let $\Gamma_{\omega_{t}}^{n-2,\,n}\in C^{\infty}_{n-2,\,n}(X_{t},\,\mathbb{C})$ be the minimal $L^{2}_{\omega_{t}}$-norm solution of the equation

[TABLE]

and we consider the real $d$-closed $(2n-2)$-form on $X$ defined as

[TABLE]

for $t$ close to [math]. In particular, $\Gamma_{\omega}(0)=\Gamma_{\omega}$. Finally, we put

[TABLE]

for all $t$ in a sufficiently small neighbourhood $U$ (depending on $\omega$) of [math] in $\Delta$.

The lower semicontinuity result for the De Rham $E_{2}sG$-cone $\widetilde{\cal S}_{X}\subset H^{2n-2}_{DR}(X,\,\mathbb{R})$ when the complex structure of $X$ varies is the following

Theorem 3.9**.**

Let $\pi:{\cal X}\longrightarrow\Delta$ be a holomorphic family of compact complex $n$-dimensional manifolds over a ball $\Delta\subset\mathbb{C}^{N}$ centred at the origin. Suppose that the fibre $X_{0}:=\pi^{-1}(0)$ is an sGG manifold.

For every sG metric $\omega$ on $X_{0}$, the section $\tau_{\omega}$ of the constant real vector bundle ${\cal H}^{2n-2}_{\mathbb{R}}\longrightarrow\Delta$ on a small neighbourhood of [math] in $\Delta$ constructed in Definition 3.8 is $C^{\infty}$.

In particular, every element $\{\Gamma_{\omega}\}_{DR}$ of the De Rham $E_{2}sG$-cone $\widetilde{\cal S}_{X_{0}}$ of $X_{0}$ extends to a $C^{\infty}$ family of elements $\{\Gamma_{\omega}(t)\}_{DR}$ of the De Rham $E_{2}sG$-cones $\widetilde{\cal S}_{X_{t}}$ of the nearby fibres $X_{t}$. Moreover, there is such an extension for every representative $\Gamma_{\omega}$ of the given De Rham class $\{\Gamma_{\omega}\}_{DR}$ defined by an sG metric $\omega$. So, in this sense, the De Rham $E_{2}sG$-cone $\widetilde{\cal S}_{X_{0}}$ of $X_{0}$ can only be “smaller” than the De Rham $E_{2}sG$-cones $\widetilde{\cal S}_{X_{t}}$ of the nearby fibres $X_{t}$.

Proof. By the well-known Neumann formula, the minimal $L^{2}_{\omega_{t}}$-norm solution of equation (18) is

[TABLE]

where $(\partial_{t})^{\star}_{\omega_{t}}$ is the formal adjoint of $\partial_{t}$ w.r.t. the $L^{2}$ inner product induced by the metric $\omega_{t}$, while $\Delta^{\prime}_{\omega_{t}}=\partial_{t}(\partial_{t})^{\star}_{\omega_{t}}+(\partial_{t})^{\star}_{\omega_{t}}\partial_{t}$ is the $\partial$-Laplacian induced by $\omega_{t}$ and $\Delta_{\omega_{t}}^{{}^{\prime}-1}$ stands for its Green operator.

Now, the $(n-1,\,n)$-form $\bar{\partial}_{t}\omega_{t}^{n-1}$ varies in a $C^{\infty}$ way with $t$ and so do the differential operators $\Delta^{\prime}_{\omega_{t}}$ and $(\partial_{t})^{\star}_{\omega_{t}}$. Moreover, the classical Kodaira-Spencer theory (cf. [KS60]) applied to the $C^{\infty}$ family $(\Delta^{\prime}_{\omega_{t}})_{t\in\Delta}$ of elliptic differential operators acting in bidegree $(n-1,\,n)$ ensures that the family $(\Delta_{\omega_{t}}^{{}^{\prime}-1})_{t\in\Delta}$ of their Green operators is again $C^{\infty}$ if the dimensions of the kernels $\ker\Delta^{\prime}_{\omega_{t}}$ (that are isomorphic to the $\partial$-cohomology spaces $H^{n-1,\,n}_{\partial}(X_{t},\,\mathbb{C})$ by the Hodge isomorphism) are independent of $t$. However, $H^{n-1,\,n}_{\partial}(X_{t},\,\mathbb{C})$ is $\mathbb{C}$-anti-linearly isomorphic to $H^{n,\,n-1}_{\bar{\partial}}(X_{t},\,\mathbb{C})$ by conjugation, while the latter vector space is Serre-dual to $H^{0,\,1}_{\bar{\partial}}(X_{t},\,\mathbb{C})$, so its dimension equals the Hodge number $h^{0,\,1}_{\bar{\partial}}(t)$ of the fibre $X_{t}$ for every $t$.

Here is where the sGG assumption on the fibre $X_{0}$ comes in. By [PU18, Corollary 1.7], it ensures that the Hodge numbers $h^{0,\,1}_{\bar{\partial}}(t)$ are independent of $t$ when $t$ varies in a small enough neighbourhood of [math]. Thus, the Green operators $\Delta_{\omega_{t}}^{{}^{\prime}-1}$ in bidegree $(n-1,\,n)$, hence also the $(n-2,\,n)$-forms $\Gamma_{\omega_{t}}^{n-2,\,n}$, vary in a $C^{\infty}$ way with $t$ near [math]. Since so also do (for trivial reasons) the $(n-1,\,n-1)$-forms $\omega_{t}^{n-1}$, we infer that the smooth $(2n-2)$-forms

[TABLE]

vary in a $C^{\infty}$ way with $t$ in a small enough neighbourhood of [math]. The application of the De Rham cohomology class being a smooth operation, we conclude that $\tau_{\omega}(t):=\{\Gamma_{\omega}(t)\}_{DR}$ depends in a $C^{\infty}$ way on $t$ varying in a small enough neighbourhood of $0\in\Delta$. $\Box$

3.3 Real-valued cohomology

We shall now deal with the real version of some of the objects introduced in $\S.$3.1 for the sake of enhanced flexibility.

Definition 3.10**.**

Let $X$ be a compact complex manifold with $\mbox{dim}_{\mathbb{C}}X=n$.

(a)  For any element $[[\alpha^{n-2,\,n}]_{\bar{\partial}}]_{d_{1}}\in E_{2}^{n-2,\,n}(X)$ and any representative $\alpha^{n-2,\,n}\in C^{\infty}_{n-2,\,n}(X,\,\mathbb{C})$ of this double class, we know from (5) that $\bar{\partial}\alpha=0$ (trivial here for bidegree reasons) and that there exists a (not necessarily real and non-unique) form $\Omega^{n-1,\,n-1}\in C^{\infty}_{n-1,\,n-1}(X,\,\mathbb{C})$ such that $\partial\alpha^{n-2,\,n}=-\bar{\partial}\Omega^{n-1,\,n-1}$.

We refer to any such form $\Omega^{n-1,\,n-1}$ as an $(n-1,\,n-1)$-potential* of $\alpha^{n-2,\,n}$.*

(b)  We define the real part $E_{2}^{n-2,\,n}(X)_{\mathbb{R}}$ of the $\mathbb{C}$-vector space $E_{2}^{n-2,\,n}(X)$ by selecting the classes representable by forms admitting a real $(n-1,\,n-1)$-potential:

[TABLE]

By definition, $E_{2}^{n-2,\,n}(X)_{\mathbb{R}}$ is a real vector subspace of $E_{2}^{n-2,\,n}(X)$.

We shall now consider the real version of the map $T$ introduced in $\S.$3.1.

Lemma 3.11**.**

Let $X$ be a compact complex manifold with $\mbox{dim}_{\mathbb{C}}X=n$.

(i)  The following inclusion holds: $E_{2}^{n-2,\,n}(X)_{\mathbb{R}}\subset\ker d_{2}^{n-2,\,n}$.

(ii)  The restriction to $H_{DR}^{2n-2}(X,\,\mathbb{R})$ of the map $T$ defined in Proposition 3.1, namely the map

[TABLE]

assumes its values in the real space $E_{2}^{n-2,\,n}(X)_{\mathbb{R}}$ and is surjective.

Proof. (i)  Let $[[\alpha^{n-2,\,n}]_{\bar{\partial}}]_{d_{1}}\in E_{2}^{n-2,\,n}(X)_{\mathbb{R}}$ with $\partial\alpha^{n-2,\,n}=-\bar{\partial}\Omega^{n-1,\,n-1}$ for some real $(n-1,\,n-1)$-form $\Omega^{n-1,\,n-1}$. Thanks to (6), we have

[TABLE]

because by conjugating the identity defining $\Omega^{n-1,\,n-1}$ and using the fact that $\Omega^{n-1,\,n-1}$ is real, we get $\partial\Omega^{n-1,\,n-1}=-\bar{\partial}\overline{\alpha^{n-2,\,n}}$.

Therefore, $[[\alpha^{n-2,\,n}]_{\bar{\partial}}]_{d_{1}}\in\ker d_{2}^{n-2,\,n}$.

(ii)  Let $\{\alpha\}_{DR}\in H_{DR}^{2n-2}(X,\,\mathbb{R})$ and pick a real representative $\alpha=\alpha^{n,\,n-2}+\alpha^{n-1,\,n-1}+\alpha^{n-2,\,n}$. Since $\alpha$ is real, $\alpha^{n-1,\,n-1}$ is real. Since $\alpha$ is $d$-closed, $\partial\alpha^{n-2,\,n}=-\bar{\partial}\alpha^{n-1,\,n-1}$. Thus, $\alpha^{n-1,\,n-1}$ is a real $(n-1,\,n-1)$-potential of $\alpha^{n-2,\,n}$, so $T(\{\alpha\}_{DR})=[[\alpha^{n-2,\,n}]_{\bar{\partial}}]_{d_{1}}\in E_{2}^{n-2,\,n}(X)_{\mathbb{R}}$.

To prove that $T$ is surjective, let $[[\alpha^{n-2,\,n}]_{\bar{\partial}}]_{d_{1}}\in E_{2}^{n-2,\,n}(X)_{\mathbb{R}}$ with $\partial\alpha^{n-2,\,n}=-\bar{\partial}\Omega^{n-1,\,n-1}$ for some real $(n-1,\,n-1)$-form $\Omega^{n-1,\,n-1}$. Then, the $(2n-2)$-form

[TABLE]

is real, $d$-closed and $T(\{\alpha\}_{DR})=[[\alpha^{n-2,\,n}]_{\bar{\partial}}]_{d_{1}}\in E_{2}^{n-2,\,n}(X)_{\mathbb{R}}$. $\Box$

We are now in a position to see that the real space $E_{2}^{n-2,\,n}(X)_{\mathbb{R}}$ contains the $E_{2}sG$-cone $\widehat{\cal S}_{X}$ of $X$ defined in $\S.$3.1 as an open cone.

Lemma 3.12**.**

Let $X$ be a compact complex manifold with $\mbox{dim}_{\mathbb{C}}X=n$. The inclusions ${\cal S}_{X}\subset\widehat{\cal S}_{X}\subset E_{2}^{n-2,\,n}(X)_{\mathbb{R}}$ hold and the cone $\widehat{\cal S}_{X}$ is open in $E_{2}^{n-2,\,n}(X)_{\mathbb{R}}$.

Proof. The inclusion ${\cal S}_{X}\subset\widehat{\cal S}_{X}$ is obvious and has already been noticed. Let $[[\Gamma^{n-2,\,n}]_{\bar{\partial}}]_{d_{1}}\in\widehat{\cal S}_{X}$ be arbitrary. So, there exists a representative $\Gamma^{n-2,\,n}\in C^{\infty}_{n-2,\,n}(X,\,\mathbb{C})$ of this $E_{2}$-class and an sG metric $\omega$ on $X$ such that $\partial\Gamma^{n-2,\,n}=-\bar{\partial}\omega^{n-1}$. Since $\omega^{n-1}$ is real, $[[\Gamma^{n-2,\,n}]_{\bar{\partial}}]_{d_{1}}\in E_{2}^{n-2,\,n}(X)_{\mathbb{R}}$. This proves the inclusion $\widehat{\cal S}_{X}\subset E_{2}^{n-2,\,n}(X)_{\mathbb{R}}$.

Let $[[\alpha^{n-2,\,n}]_{\bar{\partial}}]_{d_{1}}\in E_{2}^{n-2,\,n}(X)_{\mathbb{R}}$ and let $\varepsilon>0$. Then, there exists a real form $\Omega^{n-1,\,n-1}\in C^{\infty}_{n-1,\,n-1}(X,\,\mathbb{C})$ such that $\bar{\partial}\Omega^{n-1,\,n-1}=-\partial\alpha^{n-2,\,n}$. We get

[TABLE]

On the other hand, the $(n-1,\,n-1)$-form $\omega^{n-1}+\varepsilon\,\Omega^{n-1,\,n-1}$ is real for every $\varepsilon$ and is positive definite if $\varepsilon>0$ is small enough. Therefore, for every small $\varepsilon>0$, there exists a unique positive definite $(1,\,1)$-form $\rho_{\varepsilon}>0$ such that $\rho_{\varepsilon}^{n-1}=\omega^{n-1}+\varepsilon\,\Omega^{n-1,\,n-1}$. We get $\partial\rho_{\varepsilon}^{n-1}=-\bar{\partial}(\overline{\Gamma^{n-2,\,n}}+\varepsilon\,\overline{\alpha^{n-2,\,n}})$, hence $\partial\rho_{\varepsilon}^{n-1}$ is $\bar{\partial}$-exact, so $\rho_{\varepsilon}$ is a strongly Gauduchon metric on $X$. Consequently,

[TABLE]

for every small $\varepsilon>0$. This proves that $\widehat{\cal S}_{X}$ is open in $E_{2}^{n-2,\,n}(X)_{\mathbb{R}}$. $\Box$

3.4 Duality of positive cones in the $E_{2}$ cohomology

It was proved in [PU18b] (by means of the pseudo-differential Laplacian $\widetilde{\Delta}$ introduced in [Pop16] that gives a Hodge theory for the second page of the Frölicher spectral sequence) that for any compact complex $n$-dimensional manifold $X$ and every $p,q\in\{0,\dots,n\}$, the canonical bilinear pairing

[TABLE]

is well defined (i.e. independent of the choices of representatives of the $E_{2}$-cohomology classes involved) and non-degenerate. Hence, it defines a Serre-type duality between $E_{2}^{p,\,q}(X)$ and $E_{2}^{n-p,\,n-q}(X)$.

Under this duality, the closure of our $E_{2}sG$ cone $\widehat{\cal S}_{X}\subset E_{2}^{n-2,\,n}(X)$, consisting of those $E_{2}$-classes of type $(n-2,\,n)$ that are “positive” in the sense of Definition 3.5, has a dual cone in $E_{2}^{2,\,0}(X)$ that we will now describe. To this end, we will introduce ad hoc notions of real and positive $(2,\,0)$-forms and currents that run counter to the standard definitions of real and positive forms and currents of bidegree $(p,\,p)$, but propose a not so far-fetched analogue thereof in this bidegree that is relevant to holomorphic symplectic geometry. A possible extension of this geometry on sGG manifolds is one of our motivations and will hopefully be attempted in future work.

In this subsection, we will establish an $E_{2}$ analogue for the bidegrees $(2,\,0)$ and $(n-2,\,n)$ of Lamari’s duality (cf. [Lam99, lemme 3.3]) between Demailly’s pseudo-effective cone ${\cal E}(X)\subset H^{1,\,1}_{BC}(X,\,\mathbb{R})$ (consisting of the Bott-Chern cohomology classes of all the $d$-closed, positive $(1,\,1)$-currents $T\geq 0$ on $X$, see [Dem92]) and the closure of the Gauduchon cone ${\cal G}_{X}\subset H^{n-1,\,n-1}_{A}(X,\,\mathbb{R})$ introduced in [Pop15] (consisting of the Aeppli cohomology classes of all the Gauduchon metrics $\omega^{n-1}>0$ on $X$).

We will assume throughout this subsection that $X$ is an sGG manifold. This will guarantee that every Gauduchon metric is actually strongly Gauduchon (see [PU18a]).

Definition 3.13**.**

Let $X$ be an sGG compact complex $n$-dimensional manifold.

(i)  We consider the following sets:

[TABLE]

Thus, $V\subset E_{\mathbb{R}}\subset E\subset C^{\infty}_{n-2,\,n}(X,\,\mathbb{C})$ and $E$ consists of the smooth $(n-2,\,n)$-forms that are $E_{2}$-closed (their $\bar{\partial}$-closedness is automatic for bidegree reasons), while $E_{\mathbb{R}}$ consists of the real (in this ad hoc sense) such forms and $V$ consists of the positive (in this ad hoc sense) such forms. Note that any metric $\omega$ featuring in the definition of $V$ is automatically strongly Gauduchon (or, equivalently, Gauduchon since $X$ is assumed sGG).

(ii)  Fix an arbitrary Hermitian metric $\gamma$ on $X$. Let $p_{Im\,\bar{\partial}}^{(\gamma)}:C^{\infty}_{n-2,\,n}(X,\,\mathbb{C})\longrightarrow\mbox{Im}\,\bar{\partial}$ be the orthogonal projection w.r.t. the $L^{2}_{\gamma}$-inner product onto the closed subspace of $\bar{\partial}$-exact $(n-2,\,n)$-forms, induced by the standard Hodge-theoretical $L^{2}_{\gamma}$-orthogonal $3$-space decomposition

[TABLE]

We consider the following sets:

[TABLE]

$C^{\infty}_{n-2,\,n}(X,\,\mathbb{R})_{\gamma}=$* *

\bigg{\{}\alpha^{n-2,\,n}\in C^{\infty}_{n-2,\,n}(X,\,\mathbb{C})\,\mid\,\exists\,\beta^{n-1,\,n-1}\hskip 4.30554pt\mbox{{\bf real} form such that}\hskip 4.30554ptp_{Im\,\bar{\partial}}^{(\gamma)}(\partial\alpha^{n-2,\,n})=-\bar{\partial}\beta^{n-1,\,n-1}\bigg{\}}.**

Thus, $U_{\gamma}\subset C^{\infty}_{n-2,\,n}(X,\,\mathbb{R})_{\gamma}\subset C^{\infty}_{n-2,\,n}(X,\,\mathbb{C})$ and $U_{\gamma}$ consists of the $\gamma$-positive (in this ad hoc sense) smooth $(n-2,\,n)$-forms, while $C^{\infty}_{n-2,\,n}(X,\,\mathbb{R})_{\gamma}$ consists of the $\gamma$-real (in this ad hoc sense) such forms. Unlike the sets defined under (i), these sets are not subjected to any $E_{2}$-closedness condition. In particular, the following inclusions hold:

[TABLE]

for every Hermitian metric $\gamma$ on $X$.

(iii)  In the context of (ii), by a $\gamma$-real current of bidegree $(2,\,0)$* on $X$ we mean any continuous $\mathbb{R}$-linear form*

[TABLE]

By such a current being $\gamma$-positive we mean that $\tau^{2,\,0}$ evaluates non-negatively on every element of $U_{\gamma}$. (So, in particular, the zero current $\tau^{2,\,0}=0$ is $\gamma$-positive.)

By a $\gamma$-real current $\tau^{2,\,0}$ of bidegree $(2,\,0)$ being $E_{2}$-exact* we mean that $\tau^{2,\,0}$ vanishes identically on the $\mathbb{R}$-vector space $E_{\mathbb{R}}$ of “real” $E_{2}$-closed $(n-2,\,n)$-forms defined under (i). 222This last notion is in keeping with the usual duality according to which a current is exact (w.r.t. a given cohomology) if and only if it vanishes identically on the closed (w.r.t the same cohomology) $C^{\infty}$ forms of complementary bidegree.*

The following properties of the above sets are immediate to check.

Lemma 3.14**.**

(a)  The set $E$ is a closed $\mathbb{C}$-vector subspace of $C^{\infty}_{n-2,\,n}(X,\,\mathbb{C})$, the sets $E_{\mathbb{R}}$ and $C^{\infty}_{n-2,\,n}(X,\,\mathbb{R})_{\gamma}$ are closed $\mathbb{R}$-vector subspaces of $C^{\infty}_{n-2,\,n}(X,\,\mathbb{C})$, while $V$ and $U_{\gamma}$ are open convex cones in $E_{\mathbb{R}}$, respectively $C^{\infty}_{n-2,\,n}(X,\,\mathbb{R})_{\gamma}$.

(b)  The following identity holds:

[TABLE]

Proof. (a)  The closedness conclusion for $E$ follows from the well-known fact (itself a consequence of standard elliptic theory on compact manifolds) that $\mbox{Im}\,\bar{\partial}$ is closed in the space of $C^{\infty}$ forms in which it lies.

To see that $E_{\mathbb{R}}$ is closed in $C^{\infty}_{n-2,\,n}(X,\,\mathbb{C})$, we need one further step. Let $\Gamma_{j}^{n-2,\,n}\rightarrow\Gamma^{n-2,\,n}\in C^{\infty}_{n-2,\,n}(X,\,\mathbb{C})$ in the $C^{\infty}$ topology as $j\rightarrow+\infty$, where $\Gamma_{j}^{n-2,\,n}\in E_{\mathbb{R}}$ for every $j\in\mathbb{N}$. For every $j$, let $\Gamma_{j}^{n-1,\,n-1}=-\Delta^{{}^{\prime\prime}-1}_{\gamma}\bar{\partial}^{\star}_{\gamma}\partial\Gamma_{j}^{n-2,\,n}\in C^{\infty}_{n-1,\,n-1}(X,\,\mathbb{C})$ be the solution of minimal $L^{2}_{\gamma}$-norm of the equation $\bar{\partial}\Gamma_{j}^{n-1,\,n-1}=-\partial\Gamma_{j}^{n-2,\,n}$. (So, the set of all the solutions is the affine subspace $\Gamma_{j}^{n-1,\,n-1}+\ker\bar{\partial}\subset C^{\infty}_{n-1,\,n-1}(X,\,\mathbb{C})$ and the hypothesis $\Gamma_{j}^{n-2,\,n}\in E_{\mathbb{R}}$ means that $(\Gamma_{j}^{n-1,\,n-1}+\ker\bar{\partial})\cap C^{\infty}_{n-1,\,n-1}(X,\,\mathbb{R})\neq\emptyset$, where $C^{\infty}_{n-1,\,n-1}(X,\,\mathbb{R})\subset C^{\infty}_{n-1,\,n-1}(X,\,\mathbb{C})$ is the real vector subspace of real forms.) Then $\mbox{Im}\,\bar{\partial}\ni\partial\Gamma_{j}^{n-2,\,n}\rightarrow\partial\Gamma^{n-2,\,n}$ in the $C^{\infty}$ topology as $j\rightarrow+\infty$, so $\partial\Gamma^{n-2,\,n}\in\mbox{Im}\,\bar{\partial}$ since $\mbox{Im}\,\bar{\partial}$ is closed. Moreover, $\Gamma^{n-1,\,n-1}=-\Delta^{{}^{\prime\prime}-1}_{\gamma}\bar{\partial}^{\star}_{\gamma}\partial\Gamma^{n-2,\,n}\in C^{\infty}_{n-1,\,n-1}(X,\,\mathbb{C})$ is the solution of minimal $L^{2}_{\gamma}$-norm of the equation $\bar{\partial}\Gamma^{n-1,\,n-1}=-\partial\Gamma^{n-2,\,n}$, so $\Gamma_{j}^{n-1,\,n-1}\rightarrow\Gamma^{n-1,\,n-1}$ in the $C^{\infty}$ topology as $j\rightarrow+\infty$ because the restriction to $\mbox{Im}\,\bar{\partial}$ of the operator $\Delta^{{}^{\prime\prime}-1}_{\gamma}\bar{\partial}^{\star}_{\gamma}$ is continuous in the $C^{\infty}$ topology. Since $(\Gamma_{j}^{n-1,\,n-1}+\ker\bar{\partial})\cap C^{\infty}_{n-1,\,n-1}(X,\,\mathbb{R})\neq\emptyset$