1. Introduction
On a Kähler manifold
, the most fundamental local identity is perhaps the commutation relation between the exterior differential
d and the adjoint
to the Lefschetz operator,
where ★ denotes the Hodge star operator and
denotes the extension of
J to all forms.
This identity, due to A. Weil [
1], strongly depends on the Kähler condition,
, and in fact is true when removing the integrability condition
. So, it is valid for almost Kähler and also symplectic manifolds as well [
2,
3,
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
is not closed.
The main result is given in Theorem 1 below, which has several applications including the uniqueness of the Dirichlet problem
on any compact domain
in an almost complex manifold. This in turn implies that the Dolbeault cohomology introduced in [
8], for all almost complex manifolds, satisfies
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
-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
and the Hermitian metric allows one to consider the Laplacian operators associated to each of these components. In the compact case, the numbers
given by the kernel of
in bidegree
are finite by elliptic operator theory. When
J is integrable (and so
M is a complex manifold) the operator
vanishes and these are just the Hodge numbers
. In this case, the Hodge-to-de Rham spectral sequence gives inequalities
where
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
is the converse inequality
Of course, in the integrable Kähler case both inequalities are true and so one recovers the well-known consequence of the Hodge decomposition
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
2. Preliminaries
Let
denote the complex valued differential forms of an almost complex manifold
. For any Hermitian metric, define the associated Hodge-star operator
where
is the fundamental
-form, and
is the volume form determined by the Hermitian metric. Note
on
.
Define
, so that
on
. Similarly, consider the bidegree decomposition of the exterior differential
where the bidegree of each component is given by
We then let
for
and we have the bidegree decomposition
where
Let be the real -operator given by . Let . Then and . Let denote the primitive forms of total degree k.
It is well known that defines a representation of and induces the Lefschetz decomposition on forms:
Lemma 1. and this direct sum decomposition respects the bigrading.
Let
be the graded commutator, where
denotes the total degree of
A. This defines a graded Poisson algebra
The following is well known (e.g., [
10] Corollary 1.2.28):
Lemma 2. For all and By induction, and the fact that and L commute, we have:
Let be the extension of J to all forms as an algebra map with respect to wedge product, so that acts on by multiplication by . Then so that . Note that and ★ commute, and and commute for all . The following is a direct calculation.
Lemma 4. If an operator has bidegree , then The above result readily implies that
Finally, the following is well known (e.g., [
10] Proposition 1.2.31):
Lemma 5. If M is an almost Hermitian manifold of dimension , then for all and all , 4. Applications
On an almost Kähler manifold, using the bidegree decompositions of
d and
, one may derive from (
1) the relation
involving
,
∂ and the adjoint of
. For a non-Kähler Hermitian manifold there is an additional term
where
is the zero-order
torsion operator (see [
5,
6]). In the case of
-forms this gives
Next we use Theorem 1 to derive this local identity also in the non-integrable case.
Proposition 1. For all in an almost Hermitian manifold we have Proof. By bidegree reasons
is a primitive form and we have
where
are primitive. By expanding each term in the equality of Theorem 1 with respect to the bidegree decomposition
, in the case
, we obtain:
and
In particular, all terms decompose into sums of pure bidegrees
and
. Note as well that the remaining term
given in Theorem 1 has pure bidegree
, since
must have bidegree
. By putting together all terms of bidegree
we obtain the desired identity. ☐
Remark 2. The proof of Proposition 1 gives a second identity relating the operators Λ, μ and and their adjoints, which also contains the term . For forms in , this extra term vanishes by bidegree reasons, since . Then the second identity reads This corrects the identity known in the almost Kähler case for arbitrary forms (see [4]). The previous proposition can be used to give a uniqueness result for the Dirichlet problem on compact domains with a boundary.
Corollary 1. Let Ω be a compact domain in an almost complex manifold , with smooth boundary, and let , and be smooth. Then the Dirichlet problem, has at most one solution .
In particular, if is a compact connected almost complex manifold, and is a smooth map of almost complex manifolds, then f is constant.
Proof. It suffices to show the only solution to the homogenous equation with is a constant function.
In any coordinate chart
containing any maximum point, we pullback
J to
and consider the
J-preserving map
. The components of
d are natural with respect to this
J-preserving map and we use a compatible metric on
to define
and
. Then by Proposition 1 with
we obtain
on
. Note
is quadratic, self-adjoint, and positive, and
is first order since
is zeroth order, because
. Then the right hand side is zero, so the maximum principle due to E. Hopf applies [
11], showing
u is constant in a neighborhood of the maximum point and therefore, by connectedness,
u is constant.
The final claim follows taking , with empty boundary, , and noting the condition that f is a map of almost complex manifolds implies . ☐
Remark 3. In [8], we introduce a Dolbeault cohomology theory that is valid for all almost complex manifolds. The above corollary is key in showing that, for a compact connected almost complex manifold, this cohomology is well-behaved in lowest bidegree, in the sense that . Finally, we refer the reader to the work of Feehan and Leness [
9], where the relation of Proposition 1, for
, 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
-norm.