Abstract
For a compact complex manifold, we introduce holomorphic foliations associated with certain abelian subgroups of the automorphism group. If there exists a transverse Kähler structure on such a foliation, then we obtain a nice differential graded algebra which is quasiisomorphic to the de Rham complex and a nice differential bigraded algebra which is quasiisomorphic to the Dolbeault complex like the formality of compact Kähler manifolds. Moreover, under certain additional condition, we can develop Morgan’s theory of mixed Hodge structures as similar to the study on smooth algebraic varieties.
1 Introduction
When a connected Lie group H acts on a smooth manifold M local freely, we have a smooth foliation \({\mathcal {F}}\) whose leaves are Horbits. In addition, if M is a complex manifold, H is a complex Lie group and the Haction is holomorphic, then the foliation \({\mathcal {F}}\) is holomorphic. We are interested in transverse complex geometry on a foliated manifold \((M,{\mathcal {F}})\). Denote by \(\Omega ^{*}(M)\) the space of the differential forms on M. We say that \(\omega \in \Omega ^{*}(M) \) is basic if \(i_{X_{v}}\omega =0\) and \(L_{X_{v}}\omega =0\) for any \(v\in \mathfrak {h}\), where \(X_v\) denotes the fundamental vector field generated by \(v \in \mathfrak {h}\), and \(i_{X_{v}}\) and \(L_{X_{v}}\) are the interior product and the Lie derivation with \(X_v\), respectively. For a holomorphic foliation \({\mathcal {F}}\) on a complex manifold M with the complex structure J, a transverse Kähler structure on \({\mathcal {F}}\) is a closed real basic (1, 1)form \(\omega \) such that \(\omega _p (v,Jv) \ge 0\) for any \(v \in T_pM\) and \(p \in M\), and the equality holds if and only if v sits in the subspace \(T_p{\mathcal {F}}\) that consists of all vectors tangent to the leaf through p.
In this paper, we introduce an intrinsically defined holomorphic foliation for arbitrary compact complex manifold, that we call canonical foliation. Let M be a compact complex manifold. Let \(G_{M}\) be the identity component of the group of all biholomorphisms on M. Let T be a maximal compact torus of \(G_{M}\) and \({\mathfrak {t}}\) the Lie algebra of T. Let J be the complex structure on the Lie algebra of \(G_M\). Put
and denote by \(H_{M}\) the corresponding Lie subgroup of \(G_{M}\). Then \(H_{M}\) acts on M local freely (see [13]). Moreover, \(H_{M}\) is a central subgroup in \(G_M\) and \(H_{M}\) does not depend on the choice of T (see Lemma 2.1). By the local freeness, for any connected subgroup \(H\subset H_{M}\), we have the holomorphic foliation \({{\mathcal {F}}}_{H}\). We call \({\mathcal F}_{H}\) a central foliation associated with H and \({\mathcal F}_{H_M}\) the canonical foliation. If H is a compact complex torus, then the central foliation \({{\mathcal {F}}}_{H}\) associated with H gives a holomorphic principal Seifert bundle structure on a complex manifold M over the complex orbifold M / H (see [22]). Moreover, if the action of H is free, then such Seifert bundle is a holomorphic principal torus bundle over a complex manifold. Conversely, holomorphic principal torus bundle structure gives a holomorphic free complex torus action. Thus, a central foliation is a generalization of a holomorphic principal torus bundle.
The purpose of this paper is to study (nonKähler) complex manifolds admitting a transverse Kähler structure on a central foliation \({{\mathcal {F}}}_{H}\). In particular, we study the de Rham and Dolbeault complexes of such complex manifolds. Typical examples are holomorphic principal torus bundles over compact Kähler manifolds. A Calabi–Eckmann manifold is a holomorphic principal torus bundle over \(\mathbb {C}P^m \times \mathbb {C}P^n\), and its underlying smooth manifold is diffeomorphic to \(S^{2m+1} \times S^{2n+1}\). Extending Calabi–Eckmann’s construction, Meersseman constructed a large class of nonKähler compact complex manifolds. Such complex manifolds are called LVM manifolds (see [18, 19]). Every LVM manifold admits a transverse Kähler structure (see [18]). Among LVM manifolds with the canonical foliations, some are principal Seifert bundles. At that time the leaf space \(M/{\mathcal {F}}_{H_M}\) is a projective toric variety [19]. However, there are many LVM manifolds with the canonical foliations which are not principal Seifert bundles.
We notice that our object appears in a certain nonKäher Hermitian manifold. A Vaisman manifold is a nonKäher locally conformal Kähler manifold with the nonzero parallel Lee form. There are many important nonKähler manifolds which are Vaisman (e.g., Hopf manifolds, KodairaThurston manifolds). On any Vaisman manifold, there exists a complex onedimensional central foliation with a transverse Kähler structure which is canonically determined by its Vaisman structure.
For a complex manifold M with a holomorphic foliation \(\mathcal F\), we consider the basic de Rham complex \(\Omega ^{*}_{B}(M)\), basic Dolbeault complex \(\Omega _{B}^{*,*}(M)\), basic de Rham cohomology \(H^{*}_{B}(M)\) and basic Dolbeault cohomology \(H^{*,*}_{B}(M)\) for \({\mathcal {F}}\). If there exists a transverse Kähler form with respect to \({\mathcal {F}}\) and \({\mathcal {F}}\) is homologically oriented, then there is the Hodge decomposition
(see [8]).
Definition 1.1

For a manifold M, a (de Rham) model of M is a differential graded algebra (shortly DGA) \(A^{*}\) such that \(A^{*}\) is quasiisomorphic to the de Rham complex \(\Omega ^{*}(M)\), i.e., there exists a sequence of DGA homomorphisms
$$\begin{aligned} A^{*}\leftarrow C^{*}_{1}\rightarrow C^{*}_{2}\leftarrow \cdot \cdot \cdot \leftarrow C^{*}_{n}\rightarrow \Omega ^{*}(M) \end{aligned}$$such that all the morphisms are quasiisomorphisms (i.e., inducing cohomology isomorphisms).

For a complex manifold M, a Dolbeault model of M is a differential bigraded algebra (shortly DBA) \(B^{*,*}\) such that \(B^{*,*}\) is quasiisomorphic to the Dolbeault complex \(\Omega ^{*,*}(M)\), i.e., there exists a sequence of DBA homomorphisms
$$\begin{aligned} B^{*,*}\leftarrow C^{*,*}_{1}\rightarrow C^{*,*}_{2}\leftarrow \cdot \cdot \cdot \leftarrow C^{*,*}_{n}\rightarrow \Omega ^{*,*}(M) \end{aligned}$$such that all the morphisms are quasiisomorphisms.
On a compact Kähler manifold M, the de Rham cohomology \(H^{*}(M)\) with the trivial differential is a model of M (Formality [7]) and the Dolbeault cohomology \(H^{*,*}(M)\) with the trivial differential is a Dolbeault model of M (Dolbeault Formality [21]). In this paper we prove:
Theorem 1.2
(See also Theorem 4.13) Let M be a compact complex manifold. We assume that M admits a transverse Kähler structure on a complex kdimensional central foliation \({{\mathcal {F}}}_{H}\). Then there exists a model \(A^*\) of M with a differential d and a Dolbeault model \(B^{*,*}\) of M with a differential \({{\bar{\partial }}}\) satisfying the followings:

(1)
Let W be a real 2kdimensional vector space with a direct sum decomposition \(W\otimes \mathbb {C}=W^{1,0}\oplus W^{0,1}\) satisfying \(\overline{W^{1,0}}=W^{0,1}\). As graded algebras, \(A^* =H^*_B(M)\otimes \bigwedge W\). As bigraded algebras, \(B^{*,*} = H^{*,*}_B(M) \otimes \bigwedge (W^{1,0} \oplus W^{0,1})\). Here, the degree of an element in W is 1 and bidegree of an element in \(W^{1,0}\) (respectively, \(W^{0,1})\) is (1, 0) (respectively, (0, 1)).

(2)
The differentials d and \({{\bar{\partial }}}\) are trivial on \(H^*_B(M)\) and \(H^{*,*}_B(M)\) respectively. \(dW \subset H^2_B(M)\), \({{\bar{\partial }}} W^{1,0}\subset H^{1,1}_B(M)\) and \({{\bar{\partial }}} W^{0,1}\subset H^{0,2}_B(M)\).
In [25], Tanré constructed a Dolbeault model for a holomorphic principal torus bundle over a compact Kähler manifold. The theorem above slightly generalizes the result of Tanré.
More precisely, a vector space W as in the theorem is a 2kdimensional subspace of \(\Omega ^1(M)^H\) such that the bilinear map \({\mathfrak {h}} \times W \ni (v,w) \mapsto i_{X_v}w \in \mathbb {R}\) is nondegenerate, where \({\mathfrak {h}}\) is the Lie algebra of H and \(\Omega ^1(M)^H\) is the Hinvariant subspace of \(\Omega ^1(M)\). We can choose W to be closed under the complex structure of \(\Omega ^1(M)\). Then \(W^{1,0}\) and \(W^{0,1}\) are defined as (1, 0)part and (0, 1)part of \(W \otimes \mathbb {C}\), respectively. The differential \(W \rightarrow H^2_B(M)\) is given by \(w \mapsto [dw]_B\), where \([dw]_B\) denotes the basic cohomology class represented by \(dw \in \Omega ^2_B(M)\). Similarly, the differentials \(W^{1,0} \rightarrow H^{1,1}_B(M)\) and \(W^{0,1} \rightarrow H^{0,2}_B(M)\) are given by \(w \rightarrow [{{\bar{\partial }}} w]_B\), where \([{{\bar{\partial }}} w]_B\) denotes the basic Dolbeault cohomology class represented by \({{\bar{\partial }}} w \in \Omega ^2_B(M) \otimes \mathbb {C}\).
By these results, we can construct explicit de Rham and Dolbeault models of Vaisman manifolds (see Sect. 6.3). Recently, similar de Rham models are also constructed in [5].
Definition 1.3
A central foliation \({{\mathcal {F}}}_{H}\) is fundamental if for any \(w\in W\), \([dw]_{B}\in H^{2}_{B}(M)\) is represented by a closed basic (1, 1)form.
We prove:
Theorem 1.4
(See also Theorem 5.6) Let M be a compact complex manifold. We assume that M admits a transverse Kähler structure on a fundamental central foliation \({{\mathcal {F}}}_{H}\). Then the de Rham cohomology of M admits an \(\mathbb {R}\)mixed Hodge structure so that:

(1)
\(H^{1}(M,\mathbb {C})=H^{1}_{1,0}\oplus H^{1}_{0,1}\oplus H_{1,1}^{1}\)

(2)
\(H^{2}(M,\mathbb {C})=H^{2}_{2,0}\oplus H^{2}_{1,1}\oplus H_{0,2}^{2}\oplus H^{2}_{2,1}\oplus H^{2}_{1,2}\oplus H_{2,2}^{2}\)
and Sullivan’s minimal model of the complex valued de Rham complex admits the Morgan’s bigrading [20].
As a consequence of this result, we can say that not every finitely generated group can be the fundamental group of a compact complex manifold admitting a transverse Kähler structure on a fundamental central foliation.
2 Central foliations
Let M be a compact complex manifold. In this section we define the canonical foliation and central foliations on M. Let \(G_{M}\) be the identity component of the group of all biholomorphisms on M. \(G_{M}\) is a complex Lie group (see [3]). Denote by \(\mathfrak {g}_{M}\) the Lie algebra of \(G_{M}\) and by J the complex structure on \(\mathfrak {g}_M\). Let T be a maximal compact torus of \(G_M\) and \({\mathfrak {t}}\) the Lie algebra of T. Put
and denote by \(H_{M}\) the corresponding Lie subgroup of \(G_{M}\). Then \(H_{M}\) acts on M local freely (see [13, Proposition 3.3]).
Lemma 2.1
The following holds:

(1)
Elements in \({\mathfrak {h}}_{M}\) centralize \(\mathfrak {g}_{M}\).

(2)
\({\mathfrak {h}}_{M}\) does not depend on the choice of T.
Proof
Since T is compact, \(\mathfrak {g}_{M}\) is a unitary representation of T. In particular, \(\mathfrak {g}_{M}\) is a unitary representation of \(H_{M}\). However, \(H_{M}\) is a holomorphic subgroup of \(G_{M}\) and hence \(\mathfrak {g}_{M}\) is a holomorphic representation of \(H_{M}\). Therefore, \(\mathfrak {g}_{M}\) is a trivial representation of \(H_M\), showing Part (1).
Let \(T'\) be another maximal compact torus of \(G_{M}\). Then, there exists \(g \in G_{M}\) such that \(gTg^{1} = T'\) (see [12, Chapter XV, Section 3] for detail). Put
Then, it follows from \(gTg^{1} = T'\) that \({\text {Ad}}_g ({\mathfrak {h}}_{M}) = {\mathfrak {h}}'\). On the other hand, by (1), we have that \({\text {Ad}}_g \) is the identity on \({\mathfrak {h}}_{M}\). Therefore, \({\mathfrak {h}}_{M}\) does not depend on the choice of T, proving (2). \(\square \)
We remark that any \(\mathbb {C}\)subspace \({\mathfrak {h}}\) of \({\mathfrak {h}}_M\) defines a holomorphic foliation \({\mathcal {F}}_H\) on M. We call \({\mathcal {F}}_H\) a central foliation on M and \({\mathcal {F}}_{H_M}\) the canonical foliation on M. It follows from Lemma 2.1 that the canonical foliation does not depend on the choice of T, that is, the canonical foliation is intrinsic to compact complex manifolds.
3 Hirsch extensions and minimal models
In this section, DGAs are defined over \({{\mathbb {K}}}=\mathbb {Q}, \mathbb {R}\) or \(\mathbb {C}\), if we do not specify. Let \((A^{*},d_{A})\) be a DGA. Let k be an integer. For a linear map \(\beta :V \rightarrow A^{k+1}\) with \(d_{A}\circ \beta =0\), we define a Hirsch extension \((B,d_{B})\) of \(A^{*}\) in degree k such that \(B^{*}=A^{*}\otimes \bigwedge V\) with \(\deg (v)=k\) for any \(v\in V\), \(d_{B}=d_{A}\) on \(A^{*}\) and \(d_{B}=\beta \) on V. Defining the filtration on \(B^{*}\) by \(F^{p}(B^{*})=A^{*\ge p}\otimes \bigwedge V\), we have the spectral sequence \(E^{*,*}\) with \(E^{p,q}_{2}=H^{p}(A)\otimes \bigwedge ^{q}V\). Consider the composition \(q\circ \beta :V\rightarrow H^{k+1}(A^{*})\) where \(q:\ker d_{A}\rightarrow H^{*}(A^{*})\) is the quotient map. The DGA structure of \(B^{*}\) is determined by the map \(q\circ \beta \) (independent of a choice of \(\beta \)) [10, 10.2].
Lemma 3.1
Let \((A^{*}_{1},d_{A_{1}})\) and \((A^{*}_{2},d_{A_{2}})\) be DGAs and \(f:A_{1}^{*}\rightarrow A_{2}^*\) a quasiisomorphism. Then for a Hirsch extension \(A^{*}_{1}\otimes V\) (resp. \(A^{*}_{2}\otimes V\)), we have a Hirsch extension \(A^{*}_{2}\otimes V\) (resp. \(A^{*}_{1}\otimes V\)) and quasiisomorphism
Proof
In case \(A^{*}_{1}\otimes V\) (\(\beta _{1}:V\rightarrow A^{k+1}_{1} \)) is given. Consider the Hirsch extension \(A^{*}_{2}\otimes V\) given by \(\beta _{2}=f\circ \beta _{1}:V\rightarrow A^{*}_{2}\) and the homomorphism \(f\otimes \mathrm{id}: A^{*}_{1}\otimes V\rightarrow A^{*}_{2}\otimes V\). Then we can easily show that \(f\otimes \mathrm{id}\) induces an isomorphism on the \(E_{2}\)term of the spectral sequence. Hence \(f\otimes \mathrm{id}\) is a quasiisomorphism.
In case \(A^{*}_{2}\otimes V\) (\(\beta _{2}:V\rightarrow A^{k+1}_{2} \)) is given. Since f is a quasiisomorphism, we can take a linear map \(\beta _{1}:V\rightarrow A^{*}_{1}\) so that \(d\circ \beta _{1}=0\) and \(q\circ f\circ \beta _{1} =q\circ \beta _{2}\). By the same argument as above, the Hirsch extension of \(A^{*}_{2}\) given by \(\beta _{2}:V\rightarrow A^{k+1}_{1} \) is identified with the one given by \( f\circ \beta _{1}: V\rightarrow A^{k+1}_{1} \). Under this identification, we have the homomorphism \(f\otimes \mathrm{id}: A^{*}_{1}\otimes V\rightarrow A^{*}_{2}\otimes V\), and as in the first case, we can show that this homomorphism is a quasiisomorphism. \(\square \)
Definition 3.2
A DGA \({\mathcal {M}}^{*}\) is minimal if:

\({\mathcal {M}}^{0}={\mathbb {K}}\).

\({\mathcal {M}}^{*}=\bigcup {\mathcal {M}}^{*}_{i}\) for a sequence of subDGAs
$$\begin{aligned} {\mathbb {K}}={\mathcal {M}}^{*}_{0}\subset {\mathcal {M}}^{*}_{1}\subset \dots \end{aligned}$$such that \({\mathcal {M}}_{i+1}^{*}\) is a Hirsch extension of \({\mathcal {M}}_{i}^{*}\).

\(d{{\mathcal {M}}}^{*}\subset {{\mathcal {M}}}^{+}\cdot {{\mathcal {M}}}^{+}\) where \(M^{+}=\bigoplus _{j>0}{{\mathcal {M}}}^{j}\).
We say that a DGA \({\mathcal {M}}^{*}\) is kminimal if \(\mathcal M^{*}\) is minimal and \(\bigoplus _{j>k}{{\mathcal {M}}}^{j} \subset {{\mathcal {M}}}^{+}\cdot {{\mathcal {M}}}^{+}\). Equivalently, each extension in a sequence for \({\mathcal {M}}^{*}\) has degree at most k.
Definition 3.3
Let \(A^{*}\) be a DGA with \(H^{0}(A^{*})={\mathbb {K}}\).

A minimal DGA \({\mathcal {M}}^{*}\) is a minimal model of \(A^{*}\) if there is a quasiisomorphism \({\mathcal {M}}\rightarrow A^{*}\).

A kquasiisomorphism \({\mathcal {M}}^* \rightarrow A^*\) is a homomorphism of DGAs that induces an isomorphism \(H^{j}({\mathcal {M}}^{*})\cong H^{j}(A^{*})\) for \(j\le k\) and an injection \(H^{k+1}({\mathcal {M}}^{*})\hookrightarrow H^{k+1}(A^{*})\). A kminimal DGA \({\mathcal {M}}^{*}\) is the kminimal model of \(A^{*}\) if there is a kquasiisomorphism \({\mathcal {M}}^{*}\rightarrow A^{*}\).
Theorem 3.4
[24] For a DGA \(A^*\) with \(H^0(A^*) = {\mathbb {K}}\), a minimal model and a kminimal model exist, and each of them is unique up to DGA isomorphism.
The minimal models give the following “de Rham homotopy theory”. We shall state it but omit the details. See [7, 10, 20, 24] for the details.
Theorem 3.5
Let M be a compact smooth manifold. Consider the de Rham complex \(\Omega ^{*}(M)\) as a DGA. Then

The 1minimal model of \(\Omega ^{*}(M)\) is the dual to the Lie algebra of the nilpotent completion of \(\pi _1(M)\).

If M is simply connected, then the minimal model of \(\Omega ^{*}(M)\) determines the real homotopy type of M.
4 Models for transverse Kähler torus actions
In this section, we give a model and Dolbeault model of a complex manifold equipped with a central foliation.
4.1 Models for compact Lie group actions
The following result is wellknown (see [9] for example).
Proposition 4.1
Let M be a compact manifold and K a compact connected Lie group. Assume that K acts on M. Then the inclusion
induces a cohomology isomorphism.
Let M be a complex manifold. Let \((\Omega ^{*,*}(M),{{\bar{\partial }}})\) be the Dolbeault complex of M. Suppose that a group K acts on M as biholomorphisms. Then the space \(\Omega ^{*,*}(M)^{K}\) of Kinvariant differential forms is a subcomplex of \(\Omega ^{*,*}(M)\).
Proposition 4.2
Let M be a compact complex manifold and K a connected compact Lie group. Assume that K acts on M as biholomorphisms and the induced action on the Dolbeault cohomology is trivial. Then the inclusion
induces an isomorphism on Dolbeault cohomology.
Proof
Let \(d\mu \) be the normalized Haar measure of K. Define the linear map
Then I commutes with Dolbeault operator \({{\bar{\partial }}}\), that is, I induces a bigraded module homomorphism \(H^{*,*}(M) \rightarrow H(\Omega ^{*,*}(M)^K)\). Since I is the identity on \(\Omega ^{*,*}(M)^K\), the composition
of the homomorphisms induced by the inclusion and I is the identity. Therefore, the homomorphism \(H(\Omega ^{*,*}(M)^K) \rightarrow H^{*,*}(M)\) induced by the inclusion \(\Omega ^{*,*}(M)^{K}\subset \Omega ^{*,*}(M)\) is injective.
Since the induced action on the Dolbeault cohomology is trivial, for a \({{\bar{\partial }}}\)closed form \(\omega \in \Omega ^{*,*}(M)\) and any \(g\in K\), there exists \(\theta _{g}\in \Omega ^{*,*}(M)\) such that
By using Green operator, we can take \(\theta _{g} \) smoothly on K. Integrating by \(d\mu \), we have
Hence the inclusion
induces a surjection on Dolbeault cohomology. \(\square \)
Corollary 4.3
Let M be a compact complex manifold and K a connected compact Lie group acting on M as biholomorphisms. Let H be a dense Lie subgroup of K such that H is a complex Lie group and the restricted action of K to H on M is holomorphic. Then, the inclusion \(\Omega ^{*,*}(M)^K \subset \Omega (M)\) induces an isomorphism on Dolbeault cohomology.
Proof
By Proposition 4.2, we only need to know that the representation of K on \(H^{*,*}(M)\) is trivial under the assumptions of this proposition. Since K acts on M as biholomorphisms, the representation of K on \(H^{*,*}(M)\) is \(\mathbb {C}\)linear. Since K is compact, there exists a Hermitian inner product on \(H^{*,*}(M)\) that is invariant under K.
Consider the restricted representation \(H\rightarrow GL(H^{*,*}(M))\). Since H is a complex Lie group and the restricted action of K to H on M is holomorphic, this representation is holomorphic [16]. On the other hand, by the same argument as above, this representation is unitary. Therefore, the representation of H on \(H^{*,*}(M)\) is trivial. Since H is dense in K, the representation of K on \(H^{*,*}(M)\) is also trivial. The proposition is proved. \(\square \)
4.2 Models for torus actions
Let T be a compact torus and H a connected Lie subgroup (not necessary to be closed in T). Let M be a paracompact smooth manifold equipped with an action of T. In this section, we suppose that the restricted action of T to H on M is local free. Denote by \(\mathfrak {t}\) and \(\mathfrak {h}\) the Lie algebras of T and H respectively.
Lemma 4.4
There exists a \(\mathfrak {h}\)valued 1form \(\omega \) on M such that

(1)
\(i_{X_v}\omega = v\) for all \(v \in \mathfrak {h}\),

(2)
\(\omega \) is Tinvariant.
Proof
Since T is compact and M is paracompact, it follows from the slice theorem that there exists a locally finite open covering \({\mathcal {U}} = \{ U_\lambda \}_\lambda \) such that each \(U_\lambda \) is Tequivariantly diffeomorphic to \(T \times _{T_\lambda }V_\lambda \) via \(\varphi _\lambda \), where \(T_\lambda \) is a closed subgroup of T and \(V_\lambda \) is a representation space of \(T_\lambda \). Let \(\pi : T \times _{T_\lambda }V_\lambda \rightarrow T/T_\lambda \) be the map induced by the first projection \(T \times V_\lambda \rightarrow T\). Since the action of H on M is local free, we have that \(\mathfrak {h}\cap \mathfrak {t}_\lambda = 0\). Therefore, there exists a \(\mathfrak {h}\)valued 1form \(\omega _\lambda \) on \(T/T_\lambda \) that satisfies the conditions (1) and (2). Since \(\pi \) and \(\varphi _\lambda \) are Tinvariant, the pullback \((\pi \circ \varphi _\lambda )^*\omega _\lambda \) that is a \(\mathfrak {h}\)valued 1form on \(U_\lambda \) also satisfies the conditions (1) and (2).
Let \(\{\rho _\lambda \}\) be a partition of unity subordinate to the open covering \({\mathcal {U}}\). Averaging \(\rho _\lambda \) with the normalized Haar measure on T, we may assume that every \(\rho _\lambda \) is Tinvariant. Then the 1form
on M satisfies the condition (1) and (2), as required. \(\square \)
Since the Haction is local free, the Haction induces the foliation \({\mathcal {F}}\) whose leaves are Horbits of M. Denote by \(T^{\prime }\) the closure of H.
Lemma 4.5
\(\Omega ^{*}(M)^{T^{\prime }}=\Omega ^{*}(M)^{H}\).
Proof
Since \(H \subset T'\), we have the inclusion \(\Omega ^*(M)^{T'} \subset \Omega ^*(M)^{H}\). For \(g \in T'\), take a sequence \(\{g_i\}_{i =1,\dots }\) of elements in H so that \(\lim _{i \rightarrow \infty }g_i = g\). Then we have
for any \(\omega \in \Omega ^*(M)^{H}\), showing the opposite inclusion \(\Omega ^*(M)^{T'} \supset \Omega ^*(M)^{H}\). The lemma is proved. \(\square \)
Consider the basic forms
We want to construct a finitedimensional subspace \(W \subset \Omega ^1(M)^H\) such that

\(dW\subset \Omega ^{2}_{B}(M)\),

the bilinear map \(\mathfrak {h}\times W\ni (v,w)\mapsto i_{X_v}w\in \mathbb {R}\) is nondegenerate.
To do this, take a \(\mathfrak {h}\)valued 1form \(\omega \) as in Lemma 4.4. For a basis \(v_1,\dots , v_k\) of \(\mathfrak {h}\), we may write \(\omega =\sum _{i=1}^k w_i\otimes v_i\) with 1forms \(w_1,\dots , w_k\). We claim that \(dw_i \in \Omega _B^2(M)\). Since \(w_i\) is Tinvariant, by Cartan formula we have
for \(v \in \mathfrak {h}\) because \(i_{X_v}w_i\) is constant on M. By Cartan formula again,
This together with \(i_{X_v}dw_i =0\) yields that \(dw_i \in \Omega ^*_B(M)\). Then \(W=\langle w_{1},\dots , w_k\rangle \) is a desired space.
Proposition 4.6
We have the decomposition
Proof
For \(\omega \in \Omega ^{*}_{B}(M)\), the condition \(L_{X_v}\omega =0\) for all \(v \in \mathfrak {h}\) implies that \(\omega \in \Omega ^{*}(M)^H\). Since \(W\subset \Omega ^{1}(M)^{H}\) and \(\mathfrak {h}\times W\ni (v,w)\mapsto i_{X_v}w\in \mathbb {R}\) is nondegenerate, we have the inclusion
We will show that \(\Omega ^{*}(M)^{H}\subset \Omega ^{*}_{B}(M)\otimes \bigwedge W\). We say that \(\omega \in \Omega ^{*}(M)^{H}\) is of qtype if for any \(v_{1},\dots ,v_{q}\in \mathfrak {h}\) we have
If \(\omega \in \Omega ^{*}(M)^{H}\) is of 1type, \(\omega \in \Omega ^{*}_{B}(M)\). Suppose that \(\omega \in \Omega ^{*}(M)^{H}\) is of qtype for some \(q\ge 2\). Then for any \(v, v_1,\dots , v_{q1} \in \mathfrak {h}\), we have that
and
Therefore, we have that
Take a basis \(v_{1},\dots v_{k}\) of \(\mathfrak {h}\) and the dual basis \(w_{1},\dots ,w_{k}\) of W given by \(W\subset \Omega ^{1}(M)^{H}\) and \(\mathfrak {h}\times W\ni (v,w)\mapsto i_{X_v}w\in \mathbb {R}\). Then for \(\omega \in \Omega ^{*}(M)^{H}\) of qtype, we can see that the form
is of \((q1)\)type. It turns out that \(\omega \omega ' \in \Omega _B^*(M)\otimes \bigwedge ^{q1}W\). Since \(\omega '\) is of \((q1)\)type, applying the same argument eventually, we have that
showing the inclusion \(\Omega ^*(M)^H \subset \Omega _B^*(M)\otimes \bigwedge W\). The proposition is proved. \(\square \)
By Propositions 4.1, 4.6 and Lemma 4.5, we have the following result.
Corollary 4.7
The inclusion
induces a cohomology isomorphism.
Proposition 4.8
Suppose that \(\dim M = n+k\). Then, \(H^{n+k}(M) \cong H_B^n(M)\). In particular, \({\mathcal {F}}\) is homologically oriented if M is compact and oriented.
Proof
By Proposition 4.1 and Lemma 4.5, we can choose a representative \(\alpha \) of an element in \(H^{n+k}(M)\) so that \(\alpha \) sits in \(\Omega ^{n+k}(M)^H\). By Proposition 4.6, there uniquely exists \(\beta \in \Omega _B^n(M)\) such that \(\alpha = \beta \wedge w_1\wedge \dots \wedge w_k\). Conversely, for \(\beta \in \Omega _B^n(M)\), \(\alpha := \beta \wedge w_1\wedge \dots \wedge w_k \in \Omega ^n(M)^H\). Thanks to the degrees, \(\alpha \) and \(\beta \) both are automatically closed. Therefore, it suffices to show that \(\alpha \) is exact if and only if \(\beta \) is exact (in the sense of basic).
Let \(\alpha ' \in \Omega ^{n+k1}(M)^H\) such that \(d\alpha ' =\alpha \). By Proposition 4.6, we can write
with \(\beta ' \in \Omega _B^{n1}(M)\) and \(\beta _i \in \Omega _B^n(M)\) for \(i=1,\dots , k\). Then, it follows from \(dw_j \in \Omega _B^2(M)\) that \(\alpha = d\alpha ' = d\beta '\wedge w_1\wedge \dots \wedge w_k\). In particular, \(\beta = d\beta '\).
To see the converse, let \(\beta ' \in \Omega ^{n1}_B(M)\) such that \(d\beta ' = \beta \). Then
Since \(\beta ' \wedge dw_j =0\) by the degree, we have that
showing the equivalence of exactness between \(\alpha \) and \(\beta \). The proposition is proved. \(\square \)
4.3 Models for transverse Kähler torus actions
Let M be a compact complex manifold and T a compact torus acting on M as biholomorphisms. Let H be a dense Lie subgroup of T such that H is a complex Lie group and the restricted action of T to H on M is holomorphic and local free. Then we have a holomorphic central foliation \({\mathcal {F}}\) on M whose leaves are Horbits. As before, let \(\Omega ^*_B(M)\) denote the space of basic differential forms with respect to \({\mathcal {F}}\). Since M is a complex manifold and the Haction is holomorphic, \(\Omega ^1(M)^H\) and \(\Omega ^1_B(M)\) both are complex vector spaces.
Proposition 4.9
There exists a \(\mathbb {C}\)subspace W of \(\Omega ^1(M)^H\) such that

\(dW \subset \Omega ^2_B(M)\) and

\(\mathfrak {h}\times W \ni (v,w) \mapsto i_{X_v}w \in \mathbb {R}\) is nondegenerate.
Proof
By Lemma 4.4, there exists a \({\mathfrak {h}}\)valued 1form \(w \in \Omega ^1(M)\otimes {\mathfrak {h}}\) on M such that \(i_{X_v}w = v \) for all \(v \in {\mathfrak {h}}\) and Hinvariant. Let \(v_1,\dots , v_k\) be a \(\mathbb {C}\)basis of \({\mathfrak {h}}\) and \(J_{\mathfrak {h}}\) the complex structure on \({\mathfrak {h}}\). Then \(v_1,\dots , v_k, J_{\mathfrak {h}}v_1,\dots , J_{\mathfrak {h}}v_k\) form a \(\mathbb {R}\)basis of \({\mathfrak {h}}\). There exist \(w_1,\dots , w_k, w_{k+1},\dots , w_{2k} \in \Omega ^1(M)^H\) such that
For \(i=1,\dots , k\), we define \(w_i' \in \Omega ^1(M)^H\) to be \(w_i' = w_i\circ J\), where J denotes the complex structure on M. We define an Hinvariant \({\mathfrak {h}}\)valued 1form
It follows that \(i_{X_v}w' = v\) for all \(v \in {\mathfrak {h}}\) by definition of \(w'\) immediately. The subspace \(W= \langle w_1,\dots , w_k, w_1', \dots , w_k'\rangle \) of \(\Omega ^1(M)^H\) is closed under J. It follows from the Cartan formula that \(dW \subset \Omega _B^2(M)\) immediately. Therefore, W is a desired space, proving the proposition. \(\square \)
Let W be a \(\mathbb {C}\)subspace of \(\Omega ^1_B(M)\) as in Proposition 4.9. Then \(W\otimes \mathbb {C}\) is decomposed into (1, 0)part \(W^{1,0}\) and (0, 1)part \(W^{0,1}\). By tensoring \(\mathbb {C}\) with \(\Omega ^*(M)\otimes \bigwedge W\), we have the DBA
with the Dolbeault operator \({{\bar{\partial }}}\).
By Propositions 4.2, 4.6 and Lemma 4.5, we have the following result.
Corollary 4.10
We have an injection
which induces a cohomology isomorphism.
We consider the bigraded bidifferential algebra (BBA) \((\Omega ^{*,*}_{B}(M),\partial _{B},{{\bar{\partial }}}_{B})\). Put \(d^{c}=\sqrt{1}({{\bar{\partial }}}_B\partial _B)\). Then \(d^{c}\) is a differential on \(\Omega ^{*}_{B}(M)\). We say that the \(\partial _{B}{{\bar{\partial }}}_{B}\)lemma holds if
If the \(\partial _{B}{{\bar{\partial }}}_{B}\)lemma holds, then we have the quasiisomorphisms
and
(see [7]).
Proposition 4.11
Suppose that the \(\partial _{B}{{\bar{\partial }}}_{B}\)lemma holds. Then there exist quasiisomorphisms
and
Here, \({{\bar{\partial }}}'\) is a differential such that \(({{\bar{\partial }}}' {{\bar{\partial }}})w\) is \(\partial _B\)exact for any \(w \in W^{1,0}\oplus W^{0,1}\) and \({{\bar{\partial }}}'\alpha = {{\bar{\partial }}}\alpha \) for any \(\alpha \in \ker \partial _B\).
Proof
This follows from Lemma 3.1 immediately. \(\square \)
Theorem 4.12
(see [7, 8]) Let M be a compact manifold with a homologically oriented (that is, \(H_{B}^{{\text {codim}}{\mathcal {F}}}(M)\not =0\) ) transversely Kähler foliation \({\mathcal {F}}\). Then for the BBA \((\Omega ^{*,*}_{B}(M),\partial _{B},{{\bar{\partial }}}_{B})\), the \(\partial _{B}{{\bar{\partial }}}_{B}\)lemma holds.
This together with Propositions 4.8 and 4.11 implies the following result.
Theorem 4.13
Assume that the central foliation \({\mathcal {F}}\) admits a transversely Kähler structure. Then the DGAs \(\Omega ^{*}(M)\) and \(H_{B}^{*}(M)\otimes \bigwedge W\) (resp. DBAs \(\Omega ^{* ,*}(M)\) and \(H_{B}^{*,*}(M)\otimes \bigwedge (W^{1,0}\oplus W^{0,1})\)) are quasiisomorphic.
5 Mixed Hodge structures
The purpose of this section is to show that the cohomology and minimal model of a complex manifold equipped with a special transverse Kähler structure on a central foliation admits a certain bigrading. We begin with basic notions and facts.
5.1 Mixed Hodge structures
Let V be an \(\mathbb {R}\)vector space. An \(\mathbb {R}\)Hodge structure of weight n on an \(\mathbb {R}\)vector space V is a finite decreasing filtration \(F^{*}\) on \(V_\mathbb {C}= V \otimes \mathbb {C}\) such that
for each p. Equivalently, there exists a finite bigrading
such that
An \(\mathbb {R}\)mixed Hodge structure on V is a pair \((W_{*},F^{*})\) such that:

(1)
\(W_{*}\) is an increasing filtration which is bounded below,

(2)
\(F^{*}\) is a decreasing filtration on \(V_{\mathbb {C}}\) such that the filtration on \(Gr_{n}^{W} V_{\mathbb {C}}\) induced by \(F^{*}\) is an \(\mathbb {R}\)Hodge structure of weight n.
We call \(W_{*}\) the weight filtration and \(F^{*}\) the Hodge filtration. If there exists a finite bigrading
satisfying
\(W_{n}(V_{\mathbb {C}})=\bigoplus _{p+q\le n} V_{p,q} \) and \(F_{r}(V_{\mathbb {C}})=\bigoplus _{p\ge r} V_{p,q} \) for any n, p, q, r, then we say that an \(\mathbb {R}\)mixed Hodge structure \((W_{*},F^{*})\) is \(\mathbb {R}\)split.
Even if an \(\mathbb {R}\)mixed Hodge structure \((W_{*},F^{*})\) is not \(\mathbb {R}\)split, we can obtain a canonical bigrading of \((W_{*},F^{*})\).
Proposition 5.1
[20, Proposition 1.9] Let \((W_{*},F^{*})\) be an \(\mathbb {R}\)mixed Hodge structure on an \(\mathbb {R}\)vector space V. Define \(V_{p,q}=R_{p,q}\cap L_{p,q}\) where \(R_{p,q}=W_{p+q}(V_{\mathbb {C}})\cap F^{p}(V_{\mathbb {C}})\) and \(L_{p,q}=W_{p+q}(V_{\mathbb {C}})\cap \overline{F^{q}(V_{\mathbb {C}})}+\sum _{i\ge 2} W_{p+qi}(V_{\mathbb {C}})\cap \overline{F^{qi+1}(V_{\mathbb {C}})}\). Then we have the bigrading \(V_{\mathbb {C}}=\bigoplus V_{p,q}\) such that \(\overline{V_{p,q}}=V_{q,p}\) modulo \(\bigoplus _{r+s<p+q} V_{r,s}\), \(W_{n}(V_{\mathbb {C}})=\bigoplus _{p+q\le n}V_{p,q}\) and \(F^{r}(V_{\mathbb {C}})=\bigoplus _{p\ge r} V_{p,q}\).
We say that the bigrading in this proposition is the canonical bigrading of an \(\mathbb {R}\)mixed Hodge structure \((W_{*},F^{*})\).
We notice that this bigrading gives an equivalence of the category of \(\mathbb {R}\)mixed Hodge structures on V and bigradings \(V_{\mathbb {C}}=\bigoplus V_{p,q}\) such that \((\bigoplus _{p+q\le i}V_{p,q})\cap V\) is a real structure of \(\bigoplus _{p+q\le i}V_{p,q}\) and \(\overline{V_{p,q}}=V_{q,p}\) modulo \(\bigoplus _{r+s<p+q} V_{r,s}\) (see [20, Proposition 1.11]).
5.2 Morgan’s mixed Hodge diagrams
In [6], Deligne proves that the real cohomology of a smooth algebraic variety over \(\mathbb {C}\) admits a canonical \(\mathbb {R}\)mixed Hodge structure. The following is Morgan’s reformulation of Deligne’s technique for studying the mixed Hodge theory on Sullivan’s minimal models.
Definition 5.2
[20, Definition 3.5] An \(\mathbb {R}\)mixed Hodge diagram is a pair of filtered \(\mathbb {R}\)DGA \((A^{*}, W_{*})\) and bifiltered \(\mathbb {C}\)DGA \((E^{*}, W_{*},F^{*})\) and filtered DGA map \(\phi :(A^{*}_{\mathbb {C}},W_{*})\rightarrow (E^{*},W_{*})\) such that:

(1)
\(\phi \) induces an isomorphism \(\phi ^{*}:\,_{W}E^{*,*}_{1}(A^{*}_{\mathbb {C}})\rightarrow \,_{W}E^{*,*}_{1}(E^{*})\) where \( \,_{W}E_{*}^{*,*}(\cdot )\) is the spectral sequence for the decreasing filtration \(W^{*}=W_{*}\).

(2)
The differential \(d_{0}\) on \(\,_{W}E^{*,*}_{0}(E^{*})\) is strictly compatible with the filtration induced by F.

(3)
The filtration on \(\,_{W}E_{1}^{p,q}(E^{*})\) induced by F is an \(\mathbb {R}\)Hodge structure of weight q on \(\phi ^{*}(\,_{W}E^{*,*}_{1}(A^{*}))\).
Now, Deligne’s \(\mathbb {R}\)mixed Hodge structure is described by the following way.
Theorem 5.3
[20, Theorem 4.3] Let \(\{(A^{*}, W_{*}), (E^{*}, W_{*},F^{*}),\phi \}\) be an \(\mathbb {R}\)mixed Hodge diagram. Define the filtration \(W^{\prime }_{*}\) on \(H^{r}(A^{*})\) (resp. \(H^{r}(E^{*}))\) as \(W^{\prime }_{i}H^{r}(A^{*})=W_{ir}(H^{r}(A^{*}))\) (resp. \(W^{\prime }_{i}H^{r}(E^{*})=W_{ir}(H^{r}(E^{*}))\)). Then the filtrations \(W^{\prime }_{*}\) and \(F^{*}\) on \(H^{r}(E^{*})\) give an \(\mathbb {R}\)mixed Hodge on \(\phi ^{*}(H^{r}(A^{*}))\).
Example 5.4
Let \(H^{*}\) be a graded commutative \(\mathbb {R}\)algebra. We suppose that for any p, q, \(H^{p}\) admits an \(\mathbb {R}\)Hodge structure \(H^{p}\otimes \mathbb {C}=\bigoplus _{s+t=p} H^{s,t}\) of weight p and the multiplication \(H^{p}\times H^{q}\rightarrow H^{p+q}\) is a morphism of Hodge structures. Let V be an \(\mathbb {R}\)vector space with a linear map \(\beta :V\rightarrow H^{2}\). We suppose that V admits an \(\mathbb {R}\)Hodge structure \(V\otimes \mathbb {C}=\bigoplus _{s+t=2} V^{s,t}\) of weight 2 and \(\beta :V\rightarrow H^{2}\) is a morphism of Hodge structure. (e.g., \( \beta (V)\subset H^{1,1}\).)
Under these assumptions, regarding \(H^{*}\) as a DGA with trivial differential, we consider the Hirsch extension \(A^{*}=H^{*}\otimes \bigwedge V\). Define the increasing filtration \(W_{*}A^{*}\) as
and decreasing filtration \(F^{*}A^{*}_{\mathbb {C}}\) as the Hodge filtration for the Hodge structure on \((H^{p}\otimes \mathbb {C})\otimes \bigwedge ^{q} (V\otimes \mathbb {C})\). Then for any p, q, we have:

\(\,_{W}E^{p,q}_{0}(A^{*}_{\mathbb {C}})=(H^{q2p}\otimes \mathbb {C})\otimes \bigwedge ^{p} (V\otimes \mathbb {C})\) and \(d_{0}\) is trivial.

\(\,_{W}E^{p,q}_{1}(A^{*}_{\mathbb {C}})=(H^{q2p}\otimes \mathbb {C})\otimes \bigwedge ^{p} (V\otimes \mathbb {C})\) and clearly F induces the Hodge structure of weight q.
Thus \(\{(A^{*},W_{*}), (A^{*}_{\mathbb {C}},W_{*},F^{*}), {\text {id}} :A^{*}_{\mathbb {C}}\rightarrow A^{*}_{\mathbb {C}}\}\) is an \(\mathbb {R}\)mixed Hodge diagram.
We can easily check that for the canonical bigrading \(H^{r}(A^{*}_{\mathbb {C}})=\bigoplus H^{r}_{p,q}\) of the \(\mathbb {R}\)mixed Hodge structure as in Theorem 5.3, we have

(1)
\(H^{1}(A^{*}_{\mathbb {C}})=H^{1}_{1,0}\oplus H^{1}_{0,1}\oplus H_{1,1}^{1}\).

(2)
\(H^{2}(A^{*}_{\mathbb {C}})=H^{2}_{2,0}\oplus H^{2}_{1,1}\oplus H_{0,2}^{2}\oplus H^{2}_{2,1}\oplus H^{2}_{1,2}\oplus H_{2,2}^{2}\).
Morgan’s result on Sullivan’s minimal models of \(\mathbb {R}\)mixed Hodge diagrams is the following.
Theorem 5.5
[20, Sections 6, 8] Let \(\{(A^{*}, W_{*}), (E^{*}, W_{*},F^{*}),\phi \}\) be an \(\mathbb {R}\)mixed Hodge diagram. Then the minimal model (resp. 1minimal model) \({\mathcal {M}}^{*}\) of the DGA \(E^{*}\) with a quasiisomorphism (resp. 1quasiisomorphism) \(\phi :{\mathcal {M}}^{*}\rightarrow E^{*}\) satisfies the following conditions:

\({\mathcal {M}}^{*}\) admits a bigrading
$$\begin{aligned} {\mathcal {M}}^{*}=\bigoplus _{p,q\ge 0}{\mathcal {M}}^{*}_{p,q} \end{aligned}$$such that \({\mathcal {M}}^{*}_{0,0}={\mathcal {M}}^{0}=\mathbb {C}\) and the product and the differential are of type (0, 0).

For some real structure of \({\mathcal {M}}^{*}\), the bigrading \(\bigoplus _{p,q\ge 0}{\mathcal {M}}^{*}_{p,q}\) induces an \(\mathbb {R}\)mixed Hodge structure.

Consider the canonical bigrading \(H^{r}(E^{*})=\bigoplus V_{p,q}\) for the \(\mathbb {R}\)mixed Hodge structure as in Theorem 5.3. Then \(\phi ^{*}:H^{r}({\mathcal {M}}^{*})\rightarrow H^{r}(E^{*})\) sends \(H^{r}({\mathcal {M}}^{*}_{p,q})\) to \(V_{p,q}\).
5.3 Mixed Hodge diagrams for transverse Kähler structures on central foliations
Let M be a compact complex manifold. We assume that M admits a transverse Kähler structure on a central foliation \({\mathcal F}_{H}\). Let \((\Omega ^{*,*}_{B}(M),\partial _{B},{{\bar{\partial }}}_{B})\) be the BBA of basic differential forms associated with \({\mathcal {F}}_H\). The basic BottChern cohomology\(H^{*,*}_{B,BC}(M)\) is defined to be
Then we have \(\overline{H^{p,q}_{B, BC}(M)}=H^{q,p}_{B, BC}(M)\) and the natural algebra homomorphisms
and
By \(\partial _{B}{{\bar{\partial }}}_{B}\)Lemma, these maps are isomorphisms (see [7, Remark 5.16]). Thus, we have the Hodge decomposition
and
We remark that this decomposition does not depend on the choice of a transverse Kähler structure.
Under the assumptions as in Theorem 4.13, we consider the model \({{\mathcal {A}}}^{*}=H_{B}^{*}(M)\otimes \bigwedge W\) as in Theorem 4.13. We suppose that \({{\mathcal {F}}}_{H}\) is fundamental as in Definition 1.3. we can obtain the mixed Hodge diagram \(\{(A^{*},W_{*}), (A^{*}_{\mathbb {C}},W_{*},F^{*}), {\text {id}} :A^{*}_{\mathbb {C}}\rightarrow A^{*}_{\mathbb {C}}\}\) as in Example 5.4. Finally we obtain the following statement.
Theorem 5.6
Let M be a compact complex manifold. We assume that M admits a transverse Kähler structure on a fundamental central foliation \({{\mathcal {F}}}_{H}\). Consider the minimal model \({\mathcal {M}}\) (resp. 1minimal model) of \(A^{*}_{\mathbb {C}}(M)\) with a quasiisomorphism (resp. 1quasiisomorphism) \(\phi :{\mathcal {M}}\rightarrow A_{\mathbb {C}}^{*}(M)\). Then we have:

(1)
For each r, the real de Rham cohomology \(H^{r}(M,\mathbb {R})\) admits an \(\mathbb {R}\)mixed Hodge structure such that

\(H^{1}(M,\mathbb {C})=H^{1}_{1,0}\oplus H^{1}_{0,1}\oplus H_{1,1}^{1}\)

\(H^{2}(M,\mathbb {C})=H^{2}_{2,0}\oplus H^{2}_{1,1}\oplus H_{0,2}^{2}\oplus H^{2}_{2,1}\oplus H^{2}_{1,2}\oplus H_{2,2}^{2}\)
where \(H^{r}(M,\mathbb {C})=\bigoplus H^{r}_{p,q}\) is the canonical bigrading.


(2)
\({\mathcal {M}}^{*}\) admits a bigrading
$$\begin{aligned} {\mathcal {M}}^{*}=\bigoplus _{p,q\ge 0}{\mathcal {M}}^{*}_{p,q} \end{aligned}$$such that \({\mathcal {M}}^{*}_{0,0}={\mathcal {M}}^{0}=\mathbb {C}\) and the product and the differential are of type (0, 0).

(3)
For some real structure of \({\mathcal {M}}^{*}\), the bigrading \(\bigoplus _{p,q\ge 0}{\mathcal {M}}^{*}_{p,q}\) induces an \(\mathbb {R}\)mixed Hodge structure.

(4)
The induced map \(\phi ^{*}:H^{r}({\mathcal {M}}^{*})\rightarrow H^{r}(M,\mathbb {C})\) sends \(H^{r}({\mathcal {M}}^{*}_{p,q})\) to \(H^{r}_{p,q}\).
In this theorem, for the 1minimal model \({\mathcal {M}}\) with a 1quasiisomorphism \(\phi :{\mathcal {M}}\rightarrow A_{\mathbb {C}}^{*}(M)\), we have:

\(H^{1}({\mathcal {M}}^{*})=H^{1}({\mathcal {M}}^{*}_{1,0})\oplus H^{1}({\mathcal {M}}^{*}_{0,1})\oplus H^{1}({\mathcal {M}}^{*}_{1,1})\)

\(H^{2}({\mathcal {M}}^{*})=H^{2}({\mathcal {M}}^{*}_{2,0})\oplus H^{2}({\mathcal {M}}^{*}_{1,1})\oplus H^{2}({\mathcal {M}}^{*}_{0,2})\oplus H^{2}({\mathcal {M}}^{*}_{2,1})\oplus H^{2}({\mathcal {M}}^{*}_{1,2})\oplus H^{2}({\mathcal {M}}^{*}_{2,2})\).
By Theorem 3.5, we can translate this condition to certain condition on the Lie algebra of the nilpotent completion of the fundamental group \(\pi _{1}(M)\) as [20, Theorem 9.4] . We obtain:
Theorem 5.7
Let M be a compact complex manifold. We assume that M admits a transverse Kähler structure on a fundamental central foliation \({{\mathcal {F}}}_{H}\). Then the Lie algebra of the nilpotent completion of the fundamental group \(\pi _{1}(M)\) is isomorphic to \({{\mathcal {F}}}(H)/{{\mathcal {I}}}\) such that

H is a \(\mathbb {C}\)vector space with a bigrading \(H=H_{1,0}\oplus H_{0,1}\oplus H_{1,1}\)

\({{\mathcal {I}}}\) is a Homogeneous ideal of the free bigraded Lie algebra generated by H such that \({{\mathcal {I}}}\) has generators of types \((\,1,\,1)\), \((\,1,\,2)\), \((\,2,\,1)\) and \((\,2,\,2)\) only.
As a consequence, the Lie algebra of the nilpotent completion of the fundamental group \(\pi _{1}(M)\) is determined by \(\pi _{1}(M)/\Gamma _{5}\) where \(\Gamma _{5}\) is the fifth term of the lower central series of \(\pi _{1}(M)\) [20, Corollary 9.5]. Thus, we can say that not every finitely generated group can be the fundamental group of a compact complex manifold with transverse Kähler structure on a fundamental central foliation.
6 Examples and applications
6.1 Simple examples
Example 6.1
Consider the product \(S^{1,2n1}=S^{1}\times S^{2n1}\) of a circle and a \((2n1)\)dimensional sphere equipped with a complex structure so that there exists a special transverse Kähler structure on a onedimensional central foliation \({{\mathcal {F}}}_{H}\). Then, by our results, \(\Omega ^{*}(S^{1,2n1})\) is quasiisomorphic to the DGA \( A^{*}=H_{B}^{*}(S^{1,2n1})\otimes \bigwedge W\). By \(\dim H^{1}(S^{1,2n1})=1\) and \(H^{1}(S^{1,2n1},\mathbb {C})=H^{1,0}_{B}(S^{1,2n1})\oplus H^{0,1}_{B}(S^{1,2n1})\oplus \ker d\vert _{W}\), we have \(H^{1,0}_{B}(S^{1,2n1})\oplus H^{0,1}_{B}(S^{1,2n1})=0\) and \(\dim \ker d\vert _{W}=1\). By \(\dim H^{2}(S^{1,2n1})=0\), the differential \(d:W\rightarrow H^{2}_{B}(S^{1,2n1})\) is surjective and hence \(\dim H^{2}_{B}(S^{1,2n1})=1\). Take \(W=\langle x,y\rangle \) so that \(dx\not =0\) in \(H^{2}_{B}(S^{1,2n1})\) and \(dy=0\). We have \(H^{2}_{B}(S^{1,2n1})=\langle dx\rangle \). Since \( dx\in H^{2}_{B}(S^{1,2n1})\) must contain transverse Kähler form, we have \((dx)^{i}\not =0\) for any \(i\le n1\). Inductively we can easily compute \( H^{2i}_{B}(S^{1,2n1})=\langle (dx)^{i}\rangle \) and \( H^{2i1}_{B}(S^{1,2n1})=0\) for \(2\le i\le n1\).
Consider the Hodge decomposition
Then we have \(H^{i,i}_{B}(S^{1,2n1})=\langle (dx)^{i}\rangle \) for any \(i\le n1\) and \(H^{p,q}_{B}(S^{1,2n1})=0\) for \(p\not =q\). Take the decomposition \(W\otimes \mathbb {C}=W^{1,0}\oplus W^{0,1}\) with \(W^{1,0}=\langle z\rangle \). Then we have \(dz=cdx \) for some \(c\in \mathbb {C}\). Thus we have \({{\bar{\partial }}} z= cdx \) and \({{\bar{\partial }}} \bar{z}= 0 \). Hence \(\Omega ^{* ,*}(S^{1,2n1})\) is quasiisomorphic to the DBA
Thus every complex structure on \(S^{1,2n1}\) with a transverse Kähler structure on a onedimensional fundamental central foliation \({{\mathcal {F}}}_{H}\) has same basic Betti, basic Hodge and Hodge numbers. There are many such complex structures; see Example 6.9.
Example 6.2
Consider the product \(S^{3,3}=S^{3}\times S^{3}\) of two threedimensional spheres equipped with a complex structure so that there exists a transverse Kähler structure on a onedimensional central foliation \({{\mathcal {F}}}_{H}\). Then, by our results, \(\Omega ^{*}(S^{3,3})\) is quasiisomorphic to the DGA \( A^{*}=H_{B}^{*}(S^{3,3})\otimes \bigwedge W\). By \(H^{1}(S^{3,3})=0\) and \(H^{2}(S^{3,3})=0\), we have \(H^{1}_{B}(S^{3,3})=0\) and the differential \(d:W\rightarrow H^{2}_{B}(S^{3,3})\) is bijective. Take \(W=\langle x,y\rangle \). Then \(H^{2}_{B}(S^{3,3})=\langle dx, dy\rangle \). By \(\dim H^{3}(S^{3,3})=2\), just two of the elements
are equal to 0. Take x, y so that \(dx\wedge dx=dy\wedge dy=0\) and \(dx\wedge dy\not =0\). Since the codimension of \({{\mathcal {F}}}_{H}\) is 4, we have \(\dim H^{4}_{B}(S^{3,3})=1\) and thus \(H^{4}_{B}(S^{3,3})= \langle dx\wedge dy\rangle \). Thus we have
Consider the Hodge decomposition
Then, by \(H^{1,1}_{B}(S^{3,3})\not =0\) and \(\dim H^{2}_{B}(S^{3,3})=2\), we have that \(H^{2,0}_{B}(S^{3,3})=H^{0,2}_{B}(S^{3,3})=0\). Thus \(H^{1,1}_{B}(S^{3,3})=\mathbb {C}\langle dx, dy\rangle \). Take the decomposition \(W\otimes \mathbb {C}=W^{1,0}\oplus W^{0,1}\) with \(W^{1,0}=\langle \alpha +\sqrt{1}\beta \rangle \). Now we have
and
By \(\langle x, y\rangle =\langle \alpha , \beta \rangle \), we have
Hence \(\Omega ^{* ,*}(S^{3,3})\) is quasiisomorphic to the DBA
We compute
and
Thus every complex structure on \(S^{3,3}\) with a transverse Kähler structure on a onedimensional central foliation \({{\mathcal {F}}}_{H}\) has same basic Betti, basic Hodge and Hodge numbers. Such complex manifolds are constructed as LVM manifolds associated with complex numbers \((\lambda _{1},\dots ,\lambda _{5}) \) with certain conditions (see [19, Section 5]).
Example 6.3
Consider the product \(S^{1,3} = S^1 \times S^3\) (resp. \(S^{3,3} = S^3 \times S^3\)) equipped with a complex structure so that there exists a transverse Kähler structure on a onedimensional central foliation \({\mathcal {F}}_{H_1}\) (resp. \({\mathcal {F}}_{H_2}\)). Then the product \(S^{1,3} \times S^{1,3}\) has the natural complex structure so that there exists a special transverse Kähler structure on a twodimensional central foliation \({\mathcal {F}}_{H_1 \times H_1}\). The Künneth formula allows us to compute the basic Betti, basic Hodge and Hodge numbers. By Künneth formula we have
and
Now we consider the complex onedimensional torus \(S^{1,1} = S^1 \times S^1\) and the central foliation \({\mathcal {F}}_{S^{1,1}}\) on \(S^{1,1}\). Then the product \(S^{1,1} \times S^{3,3}\) has the natural complex structure so that there exists a transverse Kähler structure on a twodimensional central foliation \({\mathcal {F}}_{S^{1,1} \times H_2}\). By Künneth formula we have
but
for some p, q. Indeed, \(\dim H^{1,0} (S^{1,1} \times S^{3,3}) = 1\) but \( \dim H^{1,0}(S^{1,3} \times S^{1,3}) = 0\). Thus, in general, the Hodge numbers depend on a complex structure.
6.2 Nilmanifolds
Let N be a simply connected nilpotent Lie group. We suppose that N admits a lattice \(\Gamma \), i.e., cocompact discrete subgroup. A compact homogeneous space \(\Gamma \backslash N\) is called a nilmanifold. It is known that a nilmanifold admits a Kähler structure if and only if it is a torus (see [2, 11]).
Denote by \(\mathfrak {n}\) the Lie algebra of N. Let J be an endomorphism of \(\mathfrak {n}\) satisfying \(J\circ J={\text {id}}\) and \([JA, JB]=[A,B]\) for any \(A, B\in \mathfrak {n}\). Then J induces a complex structure on \(\Gamma \backslash N\). Such complex structure is called abelian. We assume that \(\mathfrak {n}\) is nonabelian and 2step, i.e., \([\mathfrak {n},[\mathfrak {n},\mathfrak {n}]]=0\). Let C be the center of N and \(\psi :N\rightarrow N/C\) the quotient map. Then we have the holomorphic principal torus bundle
where T and M are complex tori \( \Gamma \cap C\backslash C\) and \(M=\psi (\Gamma )\backslash \psi (N)\) respectively. Let \(\mathfrak {c}\) be the subalgebra of \(\mathfrak {n}\) corresponding to C. Consider the complex \(\bigwedge {\mathfrak {n}}^{*}\) of leftNinvariant differential forms. Take \(W\subset \bigwedge ^1 {\mathfrak {n}}^{*}\) which is dual to \(\mathfrak {c}\). Then we have \(dW\subset \Omega ^{1,1}(\Gamma \backslash N)\). Thus, in this case, \(\Gamma \backslash N\) admits a transverse Kähler structure on the fundamental central foliation \(\mathcal F_{C}\).
We study the properties of nilmanifolds admitting special transverse Kähler structures on fundamental central foliations.
Proposition 6.4
Let \(\Gamma \backslash N\) be a nilmanifold with a (not necessarily leftinvariant) complex structure J. We assume that M admits a transverse Kähler structure on a kdimensional central foliation \({{\mathcal {F}}}_{H}\). Suppose that \({{\mathcal {F}}}_{H}\) is regular, i.e., H is compact and the Haction is free. Then \(\Gamma \backslash N\) is biholomorphic to a holomorphic principal torus bundle over a complex torus. In particular, \(\Gamma \backslash N\) is 2step nilmanifold (see [23]).
Proof
By the assumption, \(\Gamma \backslash N\) admits a holomorphic principal torus H bundle structure \(\Gamma \backslash N\rightarrow B\) so that the base space is a compact Kähler manifold. Since \(\Gamma \backslash N\) is an aspherical manifold with \(\pi _{1}(\Gamma \backslash N)\cong \Gamma \), B is a compact aspherical manifold such that \(\pi _{1}(B)\) is a finitely generated nilpotent group. By results in [1, 2, 11], B is a complex torus. Thus \(\Gamma \backslash N\) is a holomorphic principal torus bundle over a complex torus. \(\square \)
We are interested in the nonregular case.
Proposition 6.5
Let \(\Gamma \backslash N\) be a nilmanifold with a (not necessarily leftinvariant) complex structure J. We assume that \(\Gamma \backslash N\) admits a transverse Kähler structure on a fundamental central foliation \({{\mathcal {F}}}_{H}\). If H is complex onedimensional, then \(\Gamma \backslash N\) is diffeomorphic to a 2step nilmanifold.
Proof
Let M be a compact complex ndimensional manifold which admits a special transverse Kähler structure on a kdimensional central foliation \({{\mathcal {F}}}_{H}\). Then we have an isomorphism
Hence, for the mixed Hodge structure as in Theorem 5.6, \(H^{2n}(M,\mathbb {C})\) is generated by elements of bidegree \((n+k, n+k)\).
Consider nilmanifold \(\Gamma \backslash N\). Then the DGA \(\bigwedge \mathfrak {n}^{*}\) is the minimal model of \(\Omega ^{*}(\Gamma \backslash N)\) (see [11]). If \(\Gamma \backslash N\) admits a special transverse Kähler structure on a central foliation \({\mathcal F}_{H}\), then by Theorem 5.6, the minimal model \(\bigwedge \mathfrak {n}^{*}_{\mathbb {C}}\) of \(\Omega ^{*}(\Gamma \backslash N)\) admits a bigrading \(\bigwedge \mathfrak {n}^{*}_{\mathbb {C}}=\bigoplus {\mathcal {M}}_{p,q}^{*}\). Denote \({{\mathcal {M}}}^{*}_{w}=\bigoplus _{p+q=w} {\mathcal {M}}_{p,q}^{*}\) and \(m_{\omega }=\dim {\mathcal M}^{1}_{w}\). Since \(\dim {\mathcal {M}}^{1}=\dim \mathfrak {n}^{*}_{\mathbb {C}}=2n\), we have \(\sum _{W\ge 1}m_{W}=2n\). Since we have \(H^{2n}(M,\mathbb {C})=\bigwedge ^{2n}\mathfrak {n}^{*}_{\mathbb {C}} =\bigwedge ^{2n}\bigoplus _{W}{{\mathcal {M}}}^{1}_{W}\), we have \(\sum _{w\ge 1}wm_{w}=2n+2k\). Let \(k=1\). Then \(\sum _{w\ge 2}(w1)m_{w}=2\) and hence we have \(m_{2}=2\) and \(m_{i}=0\) for \(i\le 3\), or \(m_{2}=0\), \(m_{3}=1\) and \(m_{i}=0\) for \(i\le 4\). We can say \(d{{\mathcal {M}}}_{1}=0\) and \(\bigwedge \mathfrak {n}^{*}_{\mathbb {C}}=\bigwedge {{\mathcal {M}}}_{1}^{1}\otimes \bigwedge V\) with \(dV\subset \bigwedge ^{2} {{\mathcal {M}}}_{1}^{1}\). This implies that \(\mathfrak {n}\) is 2step. \(\square \)
We suggest the following problem.
Problem 6.6
For \(s\ge 3\) and \(k\ge 2\), does there exist a sstep nilmanifold admitting a special transverse Kähler structure on a kdimensional nonregular central foliation \({{\mathcal {F}}}_{H}\)?
6.3 Vaisman manifolds
Let (M, J) be a compact complex manifold with a Hermitian metric g. We consider the fundamental form \(\omega =g(,J)\) of g. The metric g is locally conformal Kähler (LCK) if we have a closed 1form \(\theta \) (called the Lee form) such that \(d\omega =\theta \wedge \omega \). It is known that if \(\theta \not =0\) and \(\theta \) is nonexact, then (M, J) does not admit a Kähler structure. Let \(\nabla \) be the Levi–Civita connection of g. A LCK metric g is a Vaisman metric if \( \nabla \theta =0\).
If g is Vaisman, then the following holds (see [27, 28]):

Let A and B be the dual vector fields of 1forms \(\theta \) and \(\theta \circ J\) with respect to g, respectively. Then \(A=JB\), \(L_{A}J=0\), \(L_{B}J=0\), \(L_{A}g=0\), \(L_{B}g=0\) and [A, B]=0.

The holomorphic vector field \(B\sqrt{1}A\) gives a holomorphic foliation \({\mathcal {F}}\).

The basic form \(d(\theta \circ J)\) is a transverse Kähler structure.

We denote by \({\text {Aut}}_{0}(M, g)\) the identity component of the group of holomorphic isometries, by \(\mathfrak {h}\) the abelian subalgebra \(\langle A, B\rangle \) of the Lie algebra of \({\text {Aut}}_{0}(M, g)\) and by H the connected Lie subgroup of \({\text {Aut}}_{0}(M, g)\) which corresponds to \(\mathfrak {h}\). Let T be the closure of H in \({\text {Aut}}_{0}(M, g)\). Then T is a torus.
Thus a compact Vaisman manifold M admits a transverse Kähler structure on the onedimensional fundamental central foliation \({{\mathcal {F}}}_{H}\). Hence, taking \(W=\langle \theta ,\theta \circ J\rangle \) our results can be applied to a compact Vaisman manifold. The cohomology of the DGA
is isomorphic to the de Rham cohomology of M and the cohomology of DBA
is isomorphic to the Dolbeault cohomology of M. We can easily compute
This implies a wellknown fact that the first Betti number of a compact Vaisman manifold is odd (see [27]). We have the mixed Hodge structure
with \(\dim H^{1}_{1,1}=1\) as in Theorem 5.6. We notice that Vaisman metrics are closely related to Sasakian structures. We can also obtain nice de Rham models of Sasakian manifolds like the above DGA (see [26]) and we can develop Morgan’s mixed Hodge theory on Sasakian manifolds (see [15]).
Since we have \({{\bar{\partial }}} (\theta +\sqrt{1}\theta \circ J)=\sqrt{1}d(\theta \circ J)\) and \({{\bar{\partial }}} (\theta \sqrt{1}\theta \circ J)=0\), we can easily obtain an isomorphism of DGA
Hence, by Theorem 4.13, we have the following (cf. [27, Theorem 3.5]).
Corollary 6.7
Let M be a compact complex manifold. We suppose that M admits a Vaisman metric. Then the two DGAs \((\Omega ^{*}(M)\otimes \mathbb {C},d)\) and \((\Omega ^{*}(M)\otimes \mathbb {C},{{\bar{\partial }}})\) are quasiisomorphic. In particular, there exists an isomorphism between the complex valued de Rham cohomology and the Dolbeault cohomology.
Remark 6.8
On compact Kähler manifold M, by the \(\partial {{\bar{\partial }}}\)lemma, two DGAs \((\Omega ^{*}(M)\otimes \mathbb {C},d)\) and \((\Omega ^{*}(M)\otimes \mathbb {C},{{\bar{\partial }}})\) are quasiisomorphic (see [21]).
Example 6.9
Let \(\Lambda =(\lambda _{1},\dots , \lambda _{n})\) be complex numbers so that \(0<\vert \lambda _{n}\vert \le \dots \le \vert \lambda _{1}\vert <1\). A primary Hopf manifold\(M_{\Lambda }\) is the quotient of \(\mathbb {C}^{n}\{0\}\) by the group generated by the transformation \((z_{1},\dots ,z_{n})\mapsto (\lambda _{1}z_{1},\dots ,\lambda _{n}z_{n})\). It is known that any \(M_{\Lambda }\) admits a Vaisman metric (see [14]). For any \(\Lambda \), \(M_{\Lambda }\) is diffeomorphic to \(S^{1,2n1}=S^{1}\times S^{2n1}\). On the other hand, the complex structure on \(M_{\Lambda }\) varies. If \(\lambda _{n}= \dots = \lambda _{1}\), then \(M_{\Lambda }\) is a holomorphic principal torus bundle over \(\mathbb {C}P^{n1}\). Otherwise, any holomorphic principal torus bundle structure over \(\mathbb {C}P^{n1}\) does not exist on \(M_{\Lambda }\). By Example 6.1 and the above arguments, we can obtain explicit representatives of de Rham, Dolbeault, basic de Rham and Basic Dolbeault cohomologies of \(M_{\Lambda }\) by using a Vaisman metric on \(M_{\Lambda }\).
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The first author was supported by JSPS GrantinAid for Young Scientists (B) 16K17596.
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Ishida, H., Kasuya, H. Transverse Kähler structures on central foliations of complex manifolds. Annali di Matematica 198, 61–81 (2019). https://doi.org/10.1007/s1023101807628
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DOI: https://doi.org/10.1007/s1023101807628
Keywords
 Transverse Kähler structure
 Central foliation
 Basic cohomology
 Basic Dolbeault cohomology
 Mixed Hodge structure