For every positive integer r, we introduce two new cohomologies, that we call -Bott–Chern and -Aeppli, on compact complex manifolds. When , they coincide with the usual Bott–Chern and Aeppli cohomologies, but they are coarser, respectively finer, than these when . They provide analogues in the Bott–Chern–Aeppli context of the -cohomologies featuring in the Frölicher spectral sequence of the manifold. We apply these new cohomologies in several ways to characterise the notion of page---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).
We refer to the appendix of  for the details of the inductive construction of a Hodge theory for the pages , with , 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 , the ω-harmonic spaces (also called -harmonic spaces)
were constructed by induction on in [20, Section 3.2] such that every is isomorphic to the corresponding space featuring on the r-th page of the Frölicher spectral sequence of X.
Moreover, pseudo-differential “Laplacians” were inductively constructed in the appendix to  such that
where is the usual -Laplacian.
The conclusion of the construction in the appendix to  was the following statement. It gives a 3-space orthogonal decomposition of each space , for every fixed , that parallels the standard decomposition for .
Proposition A.1 ([19, Corollary 4.6]).
Let be a compact complex n-dimensional Hermitian manifold. For every , put and .
For all and all , the kernel of is given by
For all and all , the following orthogonal 3 -space decomposition (in which the sums inside the big parentheses need not be orthogonal or even direct) holds:(A.1)
For each , stands for the -orthogonal projection onto .
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 and every bidegree based on the terminology introduced in (iv) of Definition 3.1:
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 of by replacing with and with in the definitions of .
Part (ii) of Proposition A.1 can be reworded as follows.
Let be a compact complex n-dimensional Hermitian manifold. For every and for all , the following orthogonal 3-space decomposition (in which the sums inside the big parentheses need not be orthogonal or even direct) holds:
where is the -harmonic space induced by ω (see Section 2.2 and earlier in this appendix) and the next two big parentheses are the spaces of -exact -forms, respectively -exact -forms:
Moreover, we have
where and are the spaces of smooth -closed and -closed -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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