**1. Introduction**

On a Kähler manifold (*<sup>M</sup>*, *J*, *<sup>ω</sup>*), the most fundamental local identity is perhaps the commutation relation between the exterior differential *d* and the adjoint Λ to the Lefschetz operator,

$$[\Lambda, d] = \star \mathbb{I}^{-1} \, d \, \mathbb{I} \star \, \tag{1}$$

where denotes the Hodge star operator and I denotes the extension of *J* to all forms.

This identity, due to A. Weil [1], strongly depends on the Kähler condition, *dω* = 0, and in fact is true when removing the integrability condition *NJ* ≡ 0. So, it is valid for almost Kähler and also symplectic manifolds as well [2–4]. On the other hand, there is also a generalization of the Kähler identities in the Hermitian setting (see [5,6]), which strongly uses integrability.

When the manifold is only almost Hermitian, then the above local identity does not hold in general, as noticed implicitly in [7]. The purpose of this short note is to show precisely how the above Kähler identity (1) becomes modified when the form *ω* is not closed.

The main result is given in Theorem 1 below, which has several applications including the uniqueness of the Dirichlet problem

$$
\partial \mathfrak{H} = \mathfrak{g} \qquad \text{with} \quad \mathfrak{u}|\_{\partial \Omega} = \mathfrak{q},
$$

on any compact domain Ω in an almost complex manifold. This in turn implies that the Dolbeault cohomology introduced in [8], for all almost complex manifolds, satisfies *H*0,0 Dol(*M*) ∼= C for a compact connected almost complex manifold.

Another application of the almost Hermitian identities of Theorem 1 appears in forthcoming work by Feehan and Leness [9]. There the fundamental relation of Proposition 1 is used to show that the moduli spaces of unitary anti-self-dual connections over any almost Hermitian 4-manifold is almost Hermitian, whenever the Nijenhuis tensor has sufficiently small *C*0-norm. This generalizes a well known result for Kähler manifolds that was exploited in Donaldson's work in the 1980s, and is expected to have consequences for the topology of almost complex 4-manifolds which are of so-called Seiberg–Witten simple type.

When *M* is compact, local identities lead to consequences in cohomology, often governed by geometric-topological inequalities. Indeed, the exterior differential inherits a bidegree decomposition into four components *d* = *μ*¯ + *∂* + *∂* + *μ* and the Hermitian metric allows one to consider the Laplacian operators associated to each of these components. In the compact case, the numbers

$$\ell^{p,q} := \dim \text{Ker} \left( \Delta\_{\mathfrak{d}} + \Delta\_{\mu} \right)|\_{\left(p,q\right)}$$

given by the kernel of <sup>Δ</sup>¯*∂* + <sup>Δ</sup>*μ* in bidegree (*p*, *q*) are finite by elliptic operator theory. When *J* is integrable (and so *M* is a complex manifold) the operator <sup>Δ</sup>*μ* vanishes and these are just the Hodge numbers *p*,*q* = *hp*,*q*. In this case, the Hodge-to-de Rham spectral sequence gives inequalities

$$\sum\_{p+q=k} \ell^{p,q} \ge b^k \mu$$

where *bk* denotes the *k*-th Betti number. On the other hand, as shown in [4], one main consequence of the local identity (1) in the almost Kähler case *dω* = 0 is the converse inequality

$$\sum\_{p+q=k} \ell^{p\mathcal{A}} \le b^k \text{.}$$

Of course, in the integrable Kähler case both inequalities are true and so one recovers the well-known consequence of the Hodge decomposition

$$\sum\_{p+q=k} \ell^{p\mathcal{A}} = b^k \text{.}$$

The local identities of [5,6] for complex non-Kähler manifolds include other algebra terms which lead to further Laplacian operators, leading also to various inequalities relating the geometry with the topology of the manifold.

With this note, we aim to further understand the origin of these inequalities by means of the correct version of (1) for almost Hermitian manifolds for which, a priori, the only geometric-topological inequality in the compact case is given by

$$\sum\_{p+q=k} \dim \text{Ker} \left( \Delta\_{\tilde{\mu}} + \Delta\_{\tilde{\varrho}} + \Delta\_{\tilde{\varrho}} + \Delta\_{\mu} \right) \vert\_{(p,q)} \le b^k.$$
