Bott–Chern and Aeppli Cohomology for Almost-Complex Manifolds

Definitions and mathematical statement

Let $M$ be a smooth real manifold of dimension $2n$ with an almost-complex structure $J$, meaning a smooth bundle endomorphism $J:TM\to TM$ satisfying $J^2=-\mathrm{id}_{TM}$. Write $\Omega^{p,q}(M,J)$ for the smooth complex-valued forms of type $(p,q)$, where $0\leq p,q\leq n$. The exterior derivative decomposes according to type as \[ d=\mu+\partial+\bar\partial+\bar\mu, \] with respective bidegrees $(2,-1)$, $(1,0)$, $(0,1)$, and $(-1,2)$. Without integrability, $\partial^2$ and $\bar\partial^2$ need not vanish, so the classical formulas below cannot simply be assumed to define cohomology groups.

For comparison, when $J$ is integrable, one has $d=\partial+\bar\partial$, $\partial^2=\bar\partial^2=0$, and $\partial\bar\partial+\bar\partial\partial=0$. The classical groups are \[ H_{\bar\partial}^{p,q}(M)=\frac{\ker(\bar\partial:\Omega^{p,q}\to\Omega^{p,q+1})}{\bar\partial\Omega^{p,q-1}}, \] \[ H_{BC}^{p,q}(M)=\frac{\{\alpha\in\Omega^{p,q}:\partial\alpha=0,\ \bar\partial\alpha=0\}}{\partial\bar\partial\Omega^{p-1,q-1}}, \qquad H_A^{p,q}(M)=\frac{\ker(\partial\bar\partial:\Omega^{p,q}\to\Omega^{p+1,q+1})}{\partial\Omega^{p-1,q}+\bar\partial\Omega^{p,q-1}}. \] Here the form spaces in these formulas are on $M$, and form spaces with indices outside $0,\ldots,n$ are zero. Complex de Rham cohomology is \[ H_{\mathrm{dR}}^k(M;{\mathbb{C}})=\frac{\ker(d:\Omega^k(M;{\mathbb{C}})\to\Omega^{k+1}(M;{\mathbb{C}}))}{d\Omega^{k-1}(M;{\mathbb{C}})}. \] In the integrable case, representatives induce natural maps \[ H_{BC}^{p,q}(M)\longrightarrow H_{\bar\partial}^{p,q}(M)\longrightarrow H_A^{p,q}(M), \] and a $d$-closed representative gives a map $H_{BC}^{p,q}(M)\to H_{\mathrm{dR}}^{p+q}(M;{\mathbb{C}})$. Complex conjugation exchanges bidegrees $(p,q)$ and $(q,p)$ on Bott--Chern and Aeppli cohomology and squares to the identity. Wedge product makes Bott--Chern cohomology a bigraded algebra and Aeppli cohomology a bigraded module over it.

Construction problem. For an almost-complex manifold $(M,J)$, find suitable Bott--Chern-type and Aeppli-type cohomology groups which recover these classical groups when $J$ is integrable, complete the source's requested "vertical row" of the Dolbeault--de Rham diagram, respect the intended natural maps and spectral sequences, carry a real structure, and possess the expected algebra and module structures.

Unresolved qualifications. The supplied record does not display the diagram or identify its vertical row, its arrows, the nonintegrable Dolbeault theory to use, or the spectral sequences and compatibility conditions intended. It also does not specify the grading or precise axioms for the proposed groups, real structure, algebra, and module. The classical structures above give reference meanings, but do not determine a unique extension to the nonintegrable setting. No compactness, boundary, or auxiliary metric hypothesis is supplied. Consequently this is a construction request with unresolved specifications, rather than a fully determined existence assertion. The supplied literature assessment also mentions a canonical almost-complex $\partial\bar\partial$-lemma, but states no version of that lemma or requirement that it hold.

Short English statement

The problem asks for analogues of Bott--Chern and Aeppli cohomology on almost-complex manifolds that recover the integrable theory and fit the intended comparison maps, spectral sequences, real structure, and multiplication structures. The source omits the comparison diagram and several required axioms, so the precise construction target remains partly unspecified.

Sources

• Supplied local registry record: permanent UnsolvedMath ID $30004546$, titled "Bott--Chern and Aeppli Cohomology for Almost-Complex Manifolds," in the Oberwolfach Reports Open Problems collection. Exact source problem code: catalogue entry. The registry gives proposed year 2020 and names no individual proposer.
• Supplied source citation: Mini-Workshop: Almost Complex Geometry (2021), Oberwolfach Reports, pp. 1657--1691, DOI 10.4171/owr/2020/33. https://doi.org/10.4171/owr/2020/33. The supplied background describes these reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature source: Tardini--Tomassini, 2020. https://doi.org/10.1515/coma-2020-0006.
• Supplied literature source: Oberwolfach Report 33/2020. https://doi.org/10.4171/owr/2020/33.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that the usual Dolbeault, Bott--Chern, and Aeppli cohomologies are not generally defined for nonintegrable almost-complex structures, that harmonic replacements exist, and that a canonical almost-complex $\partial\bar\partial$-lemma and the requested vertical comparison remain unsettled. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.
• The supplied registry marks the statement corrected_verified and says its mathematical content was preserved while notation, typography, omitted definitions, or duplicate clauses were repaired. This records the registry's qualification and does not establish mathematical verification of this draft.