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Higher-page Bott–Chern and Aeppli cohomologies and applications

Dan Popovici, Jonas Stelzig and Luis Ugarte ORCID logo


For every positive integer r, we introduce two new cohomologies, that we call Er-Bott–Chern and Er-Aeppli, on compact complex manifolds. When r=1, they coincide with the usual Bott–Chern and Aeppli cohomologies, but they are coarser, respectively finer, than these when r2. They provide analogues in the Bott–Chern–Aeppli context of the Er-cohomologies featuring in the Frölicher spectral sequence of the manifold. We apply these new cohomologies in several ways to characterise the notion of page-(r-1)-¯-manifolds that we introduced very recently. We also prove analogues of the Serre duality for these higher-page Bott–Chern and Aeppli cohomologies and for the spaces featuring in the Frölicher spectral sequence. We obtain a further group of applications of our cohomologies to the study of Hermitian-symplectic and strongly Gauduchon metrics for which we show that they provide the natural cohomological framework.

Funding statement: This work has been partially supported by the projects MTM2017-85649-P (AEI/FEDER, UE), and E22-17R “Álgebra y Geometría” (Gobierno de Aragón/FEDER).

A Appendix

We refer to the appendix of [19] for the details of the inductive construction of a Hodge theory for the pages Er, with r3, of the Frölicher spectral sequence. Here, we will only remind the reader of the conclusion of that construction and will point out a reformulation of it that was used in the present paper.

Let X be an n-dimensional compact complex manifold. We fix an arbitrary Hermitian metric ω on X. As recalled in Section 2.2, for every bidegree (p,q), the ω-harmonic spaces (also called Er-harmonic spaces)


were constructed by induction on r>0 in [20, Section 3.2] such that every rp,q is isomorphic to the corresponding space Erp,q(X) featuring on the r-th page of the Frölicher spectral sequence of X.

Moreover, pseudo-differential “Laplacians” Δ~(r+1):rp,qrp,q were inductively constructed in the appendix to [19] such that


where Δ~(1)=Δ′′=¯¯+¯¯ is the usual ¯-Laplacian.

The conclusion of the construction in the appendix to [19] was the following statement. It gives a 3-space orthogonal decomposition of each space Cp,q(X), for every fixed r>0, that parallels the standard decomposition Cp,q(X)=kerΔ′′Im¯Im¯ for r=1.

Proposition A.1 ([19, Corollary 4.6]).

Let (X,ω) be a compact complex n-dimensional Hermitian manifold. For every rN>0, put Dr-1:=((Δ~(1))-1¯)((Δ~(r-1))-1¯) and D0=Id.

  1. For all r>0 and all (p,q), the kernel of Δ~(r+1):Cp,q(X)Cp,q(X) is given by


  2. For all r>0 and all (p,q), the following orthogonal 3 -space decomposition (in which the sums inside the big parentheses need not be orthogonal or even direct) holds:






For each rN>0, pr=prp,q stands for the Lω2-orthogonal projection onto Hrp,q.

We will now cast the 3-space decomposition (A.1) in the terms used in the present paper. Recall that in the proof of Lemma 3.3, we defined the following vector spaces for every r>0 and every bidegree (p,q) based on the terminology introduced in (iv) of Definition 3.1:

,rp,q:={αCp,q(X)α reaches 0 in at most r steps},
¯,rp,q:={βCp,q(X)¯β reaches 0 in at most r steps}.

When a Hermitian metric ω has been fixed on the manifold X and the adjoint operators and ¯ with respect to ω have been considered, we define the analogous subspaces ,rp,q,¯,rp,q of Cp,q(X) by replacing with and ¯ with ¯ in the definitions of ,rp,q,¯,rp,q.

Part (ii) of Proposition A.1 can be reworded as follows.

Proposition A.2.

Let (X,ω) be a compact complex n-dimensional Hermitian manifold. For every rN>0 and for all p,q{0,,n}, the following orthogonal 3-space decomposition (in which the sums inside the big parentheses need not be orthogonal or even direct) holds:


where Hrp,q is the Er-harmonic space induced by ω (see Section 2.2 and earlier in this appendix) and the next two big parentheses are the spaces of Er-exact (p,q)-forms, respectively Er-exact (p,q)-forms:


Moreover, we have


where Zrp,q and Zrp,q are the spaces of smooth Er-closed and Er-closed (p,q)-forms, respectively.


We are very grateful to the referee for their useful comments that helped us to improve the presentation of the paper.


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Received: 2020-07-09
Revised: 2021-02-26
Published Online: 2021-04-29
Published in Print: 2021-08-01

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