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Higher-page Bott–Chern and Aeppli cohomologies and applications

Dan Popovici, Jonas Stelzig and Luis Ugarte ORCID logo

Abstract

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)

r+1p,qrp,q1p,qCp,q(X)

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

kerΔ~(r)=rp,q,r>0,

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. (i)

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

    kerΔ~(r+1)=(ker(prDr-1)ker(Dr-1pr))
    (ker(pr-1Dr-2)ker(Dr-2pr-1))
    (ker(p1)ker(p1))(ker¯ker¯).

  2. (ii)

    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:

    (A.1)Cp,q(X)=kerΔ~(r+1)(Im¯+Im(p1)+Im(D1p2)++Im(Dr-1pr))
    (Im¯+Im(p1)+Im(p2D1)++Im(prDr-1)),

    where

    kerΔ~(r+1)(Im¯+Im(p1)+Im(D1p2)++Im(Dr-1pr))
    =ker¯ker(p1)ker(p2D1)ker(prDr-1)

    and

    kerΔ~(r+1)(Im¯+Im(p1)+Im(p2D1)++Im(prDr-1))
    =ker¯ker(p1)ker(D1p2)ker(Dr-1pr).

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:

(A.2)Cp,q(X)=rp,q(Im¯+(¯,r-1p-1,q))((¯,r-1p+1,q)+Im¯),

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:

Im¯+(¯,r-1p-1,q)=𝒞rp,q𝑎𝑛𝑑(¯,r-1p+1,q)+Im¯=𝒞rp,q.

Moreover, we have

𝒵rp,q=rp,q(Im¯+(¯,r-1p-1,q))=rp,q𝒞rp,q,
𝒵rp,q=rp,q((¯,r-1p+1,q)+Im¯)=rp,q𝒞rp,q,

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

Acknowledgements

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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