Hodge theory for twisted differentials

We study cohomologies and Hodge theory for complex manifolds with twisted differentials. In particular, we get another cohomological obstruction for manifolds in class $\mathcal{C}$ of Fujiki. We give a Hodge-theoretical proof of the characterization of solvmanifolds in class $\mathcal{C}$ of Fujiki, first proven by D. Arapura.

From this, we get another cohomological obstruction for complex manifolds belonging to class C of Fujiki. We recall that a compact complex manifold (M, J) is said to be in class C of Fujiki [19]   The previous results can be adapted to compact complex orbifolds of global-quotient type, namely, quotients of compact complex manifolds by finite groups of biholomorphisms, see Theorem 2.8 and Corollary 2.9.
The second author studied in [24] the property of satisfying the (θ 1 , θ 2 )-Hodge decomposition for any θ 1 , θ 2 ∈ H 1,0 BC (M ). In [24,Theorem 1.7], he proved that a solvmanifold admitting hyper-strong-Hodgedecomposition admits a Kähler metric. Therefore, we get a more direct proof of the characterization of solvmanifolds in class C of Fujiki. This result was firstly proven by D. Arapura [7], by using the fact that "the classes of fundamental groups of compact [manifolds in class C of Fujiki] and compact Kähler manifolds coincide", which is proven by Hironaka elimination of indeterminacies (see [10,Lemma 2.1]). But our proof does not rely on the Hironaka elimination of indeterminacies.
1. Twisted differentials and cohomologies on complex manifolds For φ ∈ A r (M ) K , we define the operator If φ is a d-closed 1-form, then the operator Note that d φ satisfies the following Leibniz rule: The cochain complex (A • (M ) K , d φ ) is considered as the de Rham complex with values in the topologically trivial flat bundle M × K with the connection form φ. Hence the structure of the cochain complex (A • (M ) K , d φ ) is determined by the character ρ φ : π 1 (M ) → GL 1 (K) given by ρ φ (γ) = exp γ φ . In particular, the cochain complex (A • (M ) K , d φ ) is determined by the cohomology class [φ] ∈ H 1 (M ; K).
Note also that the associated total cochain complex is

1.2.
Hodge theory with twisted differentials. Let (M, J) be a compact complex manifold of complex dimension n. Take a J-Hermitian metric g on M . We consider the (R-linear, possibly C-anti-linear) Hodge- * -operator * : Consider the adjoint operators d * , ∂ * , and ∂ * of the operators d, ∂, and ∂, respectively, with respect to (·, ··). Then one has For φ ∈ A r (M ) K , consider the operator L φ and define its adjoint operator with respect to (·, ··): .

The identity induces natural maps
holds.
For the classical theory of self-adjoint elliptic differential operators, see, e.g., [25, page 450], we get the following corollaries. Corollary 1.4. Let (M, J) be a compact complex manifold endowed with a Hermitian metric g. Take with respect to the inner product induced by g. Hence there is an isomorphism depending on the metric. • There is an orthogonal decomposition with respect to the inner product induced by g. Hence there is an isomorphism depending on the metric.

1.5.
Hodge theory on Kähler manifolds with twisted differentials. Consider the case of a compact complex manifold endowed with a Kähler metric. Thanks to the Kähler identities for twisted differentials, we have the following analogue of the classical Hodge decomposition theorem. Proposition 1.6. Let (M, J) be a compact complex manifold endowed with a Kähler metric g. Take and Proof. For the sake of completeness, we detail the proof.
. By the Hodge decomposition theorem for compact Kähler manifolds, one has that the identity induces the isomorphism H 1,0 Firstly, note that, by the Kähler identities for twisted differentials, we have: and, by conjugation, where ω denotes the (1, 1)-form associated to g. Therefore Hence we have to show that Indeed, by using the Kähler identities, we have Analogously, θ2) . These identities conclude the proof.
In particular, it follows that, on compact Kähler manifolds, all the above cohomologies are isomorphic.
of Z-graded vector spaces.
Since the isomorphisms in [31, page 22] and Corollary 1.4 depend on the Kähler metric, also the isomorphisms in Corollary 1.7, a priori, depend on the Kähler metric. In fact, the following result holds, analogous to the ∂∂-Lemma. Theorem 1.8. Let (M, J) be a compact complex manifold endowed with a Kähler metric g. Take θ 1 , θ 2 ∈ H 1,0 BC (M ). Then the natural map induced by the identity is injective.
Proof. We detail the proof, which follows the argument in [16, pages 266-267]. We prove that θ2) .

We consider the inclusion
Then we have the following result.
where H is given by the projection A • (M ) C → ker∆ and G is given by the inverse of the restriction of ∆ on A • (M ) C ∩ (ker ∆) ⊥ , such that Since ∆ is self-adjoint, G and H are also self-adjoint. We can define the operators ∆, G, and H on This completes the proof.

Modifications and cohomologies with twisted differentials.
Consider the pull-back f * : ). Since f * commutes with ∂ and ∂, then is a morphism of differential Z-graded complexes, and is a morphism of bi-differential Z-graded complexes. In particular, f induces the maps :

2.3.
Hodge decomposition and ∂∂-Lemma with twisted differentials. As in [16], we consider the following definitions in the case of twisted differentials.
We say that (M, J): i.e., if the natural map induced by the identity is injective; (ii) admits the (θ 1 , θ 2 )-Hodge decomposition if the natural maps and induced by the identity are isomorphisms.
By [16,Lemma 5.15], see also [6,Lemma 1.4], we have the following equivalent characterizations of ∂ (θ1,θ2) ∂ (θ1,θ2) -Lemma. (In case of double complex, a further characterization is proven in [5].) induced by the identity is injective; • the natural map induced by the identity is an isomorphism; • the natural maps and induced by the identity are injective; • the natural maps and θ2) ; ∂ (θ1,θ2) , ∂ (θ1,θ2) ) induced by the identity are surjective. Furthermore, they imply the following conditions: induced by the identity in injective; • the natural map induced by the identity in surjective.
In the Kähler case, we can summarize Theorem 1.8 and Theorem 1.9 in the following.

2.7.
Complex orbifolds of global-quotient type. I. Satake introduce in [29] the notion of orbifold, also called V-manifold ; see also [8,9]. It is a singular complex space whose singularities are locally isomorphic to quotient singularities C n /G , for finite subgroups G ⊂ GL n (C). In particular, we are interested in compact complex orbifolds of global-quotient type, namely, compact complex orbifolds given byM = M /G where M is a compact complex manifold and G is a finite group of biholomorphisms of M . See [3] and the references therein for motivations.

Solvmanifolds
In this section, we consider solvmanifolds, i.e., compact quotients Γ\G where G is a connected simplyconnected solvable Lie group and Γ is a co-compact discrete subgroup.

Solvmanifolds and
As regards solvmanifolds in class C of Fujiki, they are characterized in [7, Theorem 9] by D. Arapura. More precisely, it follows from [7, Theorem 3, Theorem 9] that, for solvmanifolds endowed with complex structures, the properties of admitting Kähler metrics and of belonging to class C of Fujiki are equivalent. The proof is sketched at [7, page 136], and is based on the fact that a finitely-presented group is a Fujiki group if and only if it is a Kähler group, see also [10, Theorem 1.1] by G. Bharali, I. Biswas, and M. Mj. In fact, their result founds on the Hironaka elimination of indeterminacies, [10, §2]. By using the results by the second author in [24] and the above results, we can provide a different and more direct proof, of cohomological flavour. Proof. Take any θ 1 , θ 2 ∈ H 1,0 BC (M ). By Corollary 2.7, the manifold (M, J) admits the (θ 1 , θ 2 )-Hodge decomposition. In [24], the property of satisfying the Hodge-decomposition with respect to any θ 1 , θ 2 ∈ H 1,0 BC (M ) is called hyper-strong-Hodge-decomposition. The second author proved in [24, Theorem 1.7] that a solvmanifold admitting hyper-strong-Hodge-decomposition admits a Kähler metric.