---
title: Aeppli Cohomology in Complex Geometry
url: https://www.emergentmind.com/topics/aeppli-cohomology
type: topic
---

# Aeppli Cohomology in Complex Geometry

Aeppli cohomology is a fundamental secondary cohomology theory in complex geometry, defined for a complex manifold and deeply linked to the interplay between the Dolbeault, Bott–Chern, and de Rham cohomologies. Its algebraic and analytic properties provide a detailed measure of the failure of the $\partial\bar\partial$-lemma, and its duality structure forms a cornerstone in both compact and non-compact complex analytic contexts. Aeppli cohomology also provides the correct cohomological receptacle for various Hermitian metric characteristic classes, obstructs certain geometric structures, and appears naturally in deformation theory and in the study of geometric flows.

## 1. Definition and Algebraic Structure

On a complex manifold $X$ of dimension $n$, let $A^{p,q}(X)$ denote the space of smooth $(p,q)$-forms. The Dolbeault differentials,
\[
\partial: A^{p,q}(X) \to A^{p+1,q}(X), \qquad \bar\partial: A^{p,q}(X) \to A^{p,q+1}(X),
\]
satisfy $\partial^2=\bar\partial^2=0$, $\partial\bar\partial+\bar\partial\partial=0$. The Aeppli cohomology group in bidegree $(p,q)$ is then
\[
H_A^{p,q}(X) = \frac{\{\alpha\in A^{p,q}(X)\mid \partial\bar\partial\alpha=0\}}{\im(\partial:A^{p-1,q}\to A^{p,q}) + \im(\bar\partial:A^{p,q-1}\to A^{p,q})}.
\]
Its dimension is finite on compact $X$ by elliptic theory for the associated fourth-order Laplacian. These groups fit into a double complex with the Bott–Chern cohomology, Dolbeault cohomology, and de Rham cohomology, and their interrelations are mediated by exact sequences (Varouchas) and spectral sequences (Frölicher) [1507.07112].

## 2. Relations to Other Cohomology Theories and Duality

Aeppli cohomology is dual to Bott–Chern cohomology under an integral pairing:
\[
H_{BC}^{p,q}(X)\times H_A^{n-p,n-q}(X)\to\mathbb{C},\quad ([\alpha],[\beta])\mapsto \int_X\alpha\wedge\beta,
\]
which is non-degenerate on compact Hermitian manifolds [1507.07112]. There are natural maps, induced by the identity on forms, from
\[
H_{BC}^{p,q}(X)\to H_{\bar\partial}^{p,q}(X)\to H_A^{p,q}(X).
\]
Generic inequalities $\dim H_{BC}^{p,q}(X)\leq \dim H_{\bar\partial}^{p,q}(X)\leq \dim H_A^{p,q}(X)$ hold, with equality characterizing the $\partial\bar\partial$-lemma. The Serre-type duality $H_{BC}^{p,q}(X)\cong (H_A^{n-p,n-q}(X))^*$ provides a key structural instrument; on Stein manifolds or strongly convex non-compact spaces, this extends to perfect pairings with compact support [2601.03529].

On a manifold satisfying the $\partial\bar\partial$-lemma (e.g., compact Kähler), these cohomology groups coincide with Dolbeault and de Rham [1507.07112].

## 3. Quantitative Inequalities and the $\partial\bar\partial$-Lemma

Aeppli and Bott–Chern cohomologies satisfy strong dimension inequalities. For a compact complex $n$-fold $X$ and every integer $k$,
\[
\sum_{p+q=k} \left(\dim H^{p,q}_{BC}(X)+\dim H^{p,q}_A(X)\right) \geq 2\,b_k(X),
\]
where $b_k(X) = \dim H^k_{dR}(X;\mathbb{C})$ [1403.2298]. Equality in these Frölicher-type inequalities in all degrees precisely characterizes the validity of the $\partial\bar\partial$-lemma.

Aeppli and Bott–Chern numbers are related to the Hodge numbers via sharp upper bounds:
\[
h_A^k(X)\leq \min\{k+1,2n-k+1\}(h^k(X)+h^{k-1}(X)),
\]
with analogous statements for Bott–Chern cohomology [1507.07112].

These dimension formulas and inequalities encode fine-grained information about the geometry of $X$—the degree to which it fails to be Kähler, the presence of "zigzag" elements in the double complex, and the formality of the associated bigraded structure.

## 4. Deformation Theory and Jumping Phenomena

The dimensions of Aeppli cohomology can jump under small deformations of the complex structure. For a family $\pi: \mathcal{X}\to B$, the variation of $\dim H_A^{p,q}(\mathcal{X}_t)$ is controlled by explicit obstruction classes, computed via the Kodaira–Spencer map and higher-order cohomological invariants [1403.0285],[2506.12288]. The jumping formula
\[
\dim H_A^{p,q}(X) = \dim H_A^{p,q}(X_t) + v_t^{n-p,n-q} + w_t^{n-p-1,n-q-1}
\]
quantifies the necessary and sufficient conditions for the constancy of Aeppli numbers under deformation, expressed in terms of (un)obstructed Bott–Chern and Aeppli deformations [2506.12288].

On mild forms of the $\partial\bar\partial$-lemma (e.g., weak or page-$(r-1)$ versions), constancy of these numbers is guaranteed [2506.12288].

## 5. Metric Geometry, Obstructions, and Functional Characterizations

Aeppli cohomology naturally classifies Hermitian metrics modulo exact deformations. For Hermitian–symplectic metrics, Aeppli classes $[\omega]_A\in H^{1,1}_A(X,\mathbb{R})$ parametrize the space of such metrics. Fundamental works have introduced energy/volume functionals on these classes, with critical points corresponding to Kähler metrics and generalized volume invariants extending the Kähler case. Notably, the minimal completion $e=\tau_\omega+\omega+\overline{\tau_\omega}$ (where $\tau_\omega$ is the unique torsion form) yields a closed form whose de Rham class computes these invariants [2007.10647].

A distinguished cohomological obstruction, the $E_2$-torsion class $[\overline{\tau_\omega}]_{E_2} \in E_2^{0,2}(X)$, must vanish for a Hermitian-symplectic Aeppli class to contain a Kähler metric, providing a fine test for Kählerness beyond Bott–Chern and Dolbeault [2007.10647]. Analytically, the Euler–Lagrange equations associated with the volume functional relate to Monge–Ampère equations, leading to canonical metrics in Aeppli classes [2007.10647],[1310.3685].

In the context of Gauduchon metrics, the $(n-1,n-1)$ Aeppli classes parameterize the Gauduchon cone, and a Monge–Ampère–type equation assigns a unique metric representative to each admissible class, underlining the deep metric-cohomology correspondence [1310.3685].

## 6. Generalizations, Non-Compact Contexts, and the $L^2$ Theory

Aeppli cohomology generalizes beyond integrable complex structures to various settings: almost complex manifolds via $d, d^c$ operators [2303.17449], generalized complex structures using appropriate doubles $(d', d'')$ [1311.4667], and symplectic manifolds via $(d, d^\Lambda)$ [1403.2298]. In these settings, the generalized $d'd''$-lemma is characterized by the dimension equalities $h_{BC}^{p,q} = h_A^{p,q}$, ensuring the simultaneous degeneration of the associated spectral sequences [1311.4667].

On non-compact (especially Stein or strongly convex) manifolds, the $L^2$ theory of Aeppli cohomology is well-developed. Essential self-adjointness of the relevant operators, compactness properties of kernels, and the existence of perfect duality with Bott–Chern cohomology with compact support are established under suitable convexity and topological finiteness hypotheses [2601.03529],[2310.08993]. The $L^2$–Aeppli Laplacian admits Hodge isomorphisms and, in the compact case, coincides with the classical (finite-dimensional) Aeppli groups [2310.08993].

## 7. Applications and Explicit Computations

Explicit computations of Aeppli cohomology, especially in low dimensions and on explicitly constructed examples, provide valuable insights into the structure of non-Kähler metrics, flows, and manifold classification:

- On certain Lie groups, all Aeppli, Dolbeault, and Bott–Chern cohomology classes arise from left-invariant forms, with $\dim H^{1,1}_A(\mathrm{SU}(3))=1$, showing uniqueness of the Bismut-flat class and implying global stability of the pluriclosed flow [2202.13199].
- For 2-dimensional theta toroidal groups, Aeppli cohomology is computed explicitly, exhibiting Kähler-type behavior even in the non-compact case, as the $\partial\bar\partial$-lemma holds [2509.06384].
- In the study of the heterotic string moduli, Aeppli classes serve as natural local moduli coordinates, demonstrating their utility in both geometric and physical deformation problems [2401.05331].
- Obstructions derived from Aeppli cohomology provide sharp tests for the existence of astheno-Kähler metrics and other special Hermitian structures [2210.14126].

Aeppli cohomology thus serves as a central organizational tool for measuring non-Kählerianity, controlling deformations, dualities, and obstructions, and bridging metric geometry with the algebraic structure of complex manifolds.

Source: https://www.emergentmind.com/topics/aeppli-cohomology