Deformations of Higher-Page Analogues of $\partial\bar\partial$-Manifolds

We extend the notion of essential deformations from the case of the Iwasawa manifold, for which they were introduced recently by the first-named author, to the general case of page-$1$-$\partial\bar\partial$-manifolds that were jointly introduced very recently by all three authors. We go on to obtain an analogue of the unobstructedness theorem of Bogomolov, Tian and Todorov for page-$1$-$\partial\bar\partial$-manifolds. As applications of this discussion, we study the small deformations of certain Nakamura solvmanifolds and reinterpret the cases of the Iwasawa manifold and its $5$-dimensional analogue from this standpoint.

It turns out that for some complex parallelisable n-dimensional nilmanifolds X, such as the 5-dimensional analogue I (5) of the Iwasawa manifold I (3) , the non-complex parallelisable small deformations no longer coincide with those parametrised by E n−1, 1 2 (X) even when this space injects in a natural way into E n−1, 1 1 (X). Recall that, classically, E n−1, 1 1 (X) parametrises all the small deformations of X when there is no obstruction to deforming the complex structure of X.
We define the small essential deformations of a compact Calabi-Yau page-1-∂∂-manifold X as those small deformations of X that are parametrised by E n−1, 1 2 (X). However, in general, there is no canonical injection of E n−1, 1 2 (X) into E n−1, 1 1 (X), although there is always a canonical surjection from a canonical vector subspace E n−1, 1 1 (X) 0 of E n−1, 1 1 (X) to E n−1, 1 2 (X). Under the Calabi-Yau page-1-∂∂-assumption on X, we manage to associate in a natural, unique way a linear injection 3 leading up to it.) As a consequence of this discussion, we propose the following definition of small essential deformations of the complex structure of X induced by a given metric ω on X.
(II) On the other hand, we study in §.4 the possible unobstructedness of the small deformations, both essential and ordinary, of Calabi-Yau page-1-∂∂-complex structures. We get a generalisation of the following classical Bogomolov-Tian-Todorov theorem (see [Tia87], [Tod89]): The Kuranishi family of a compact Kähler Calabi-Yau manifold is unobstructed.
The Kähler assumption can be weakened to the ∂∂-assumption (and even to the E 1 = E ∞ assumption), as pointed out in [Tia87], [Ran92] and [Kaw92] -see also discussions by various other authors such as [Pop13, Theorem 1.2].
Our main result is the following statement to the effect that, under certain cohomological conditions, a similar phenomenon holds when the ∂∂-assumption is further weakened to the page-1-∂∂-assumption. See Definition 4.1 for the meaning of unobstructedness for the essential Kuranishi family. Meanwhile, for any bidegree (p, q), we let Z p, q r (X) stand for the vector space of smooth E rclosed (p, q)-forms on X. (These are the smooth (p, q)-forms on X that represent E r -cohomology classes on the r-th page of the Frölicher spectral sequence. See e.g. Proposition 2.3 in [PSU20b], reproduced as Proposition 2.8 below, for a description of them.) Theorem 1.2. Let X be a compact Calabi-Yau page-1-∂∂-manifold with dim C X = n. Fix a non-vanishing holomorphic (n, 0)-form u on X and suppose that ψ 1 (t) (ρ 1 (s) u) ∈ Z n−2, 2 2 (1) for all ψ 1 (t), ρ 1 (s) ∈ C ∞ 0, 1 (X, T 1, 0 X) such that ψ 1 (t) u, ρ 1 (s) u ∈ ker d ∪ Im ∂. (i) Then, the essential Kuranishi family of X is unobstructed.
This undertaking is justified by the fact that unobstructedness of the Kuranishi family occurs for some well-known compact complex manifolds that are not ∂∂-manifolds (so, the Bogomolov-Tian-Todorov theorem may not apply), but are page-1-∂∂-manifolds, such as I (3) and I (5) . The point we will make is that I (3) and I (5) are not isolated examples, but they are part of a pattern.
(III) Finally, we demonstrate in §.5 the role played by the small essential deformations by means of examples. Our main observation is the following

Background and preliminaries
We first review some general issues in the theory of deformations of complex structures, make an observation about some non-essential deformations and then review some specific properties of our class of page-1-∂∂-manifolds recently introduced in [PSU20a].

Small deformations of complex structures
Let X be a compact complex manifold with dim C X = n. Recall that small deformations of the complex structure of X over a base B may be described by smooth T 1, 0 X-valued (0, 1)-forms ψ(t) ∈ C ∞ 0, 1 (X, T 1, 0 X) depending on t ∈ B and satisfying the integrability condition (see e.g. [KS60]): In fact, given such a ψ, the space of (0, 1)-tangent vectors for the complex structure determined by ψ is given by (Id +ψ)T 0,1 X . Let t = (t 1 , ..., t m ) ∈ C m with m = dim C H 0, 1 (X, T 1, 0 X). Writing for the Taylor expansion of ψ around 0, (so each ψ ν (t) is a homogeneous polynomial of degree ν in the variables t = (t 1 , . . . , t m )), the integrability condition is easily seen to be equivalent tō ∂ψ 1 (t) = 0 and the following sequence of conditions: The Kuranishi family of X is said to be unobstructed if there exists a choice {β 1 , . . . , β m } of representatives of cohomology classes that form a basis {[β 1 ], . . . , [β m ]} of H 0, 1 (X, T 1, 0 X) such that the integrability condition is satisfied (i.e. all the equations (Eq. (ν)) are solvable) for any choice of parameters (t 1 , . . . , t m ) ∈ C m defining ψ 1 (t) = t 1 β 1 + · · · + t m β m . By the fundamental result of [Kur62], when this qualitative condition is satisfied, a convergent solution ψ(t) can be built for small t through an inductive construction of the ψ ν (t)'s from the given ψ 1 (t) by solving the equations (Eq. (ν)) and choosing at every step the solution with minimal L 2 norm for a pregiven Hermitian metric on X. The r.h.s. of each of these equations is∂-closed, so the only obstruction to solvability is the possible non-∂-exactness of the r.h.s. The resulting (germ of a) family (X t ) t∈∆ of complex structures on X is called the Kuranishi family of X. (It depends on the metric, but different choices of metrics yield isomorphic families.) If it is unobstructed, its base B is smooth and can be viewed as an open ball about 0 in the cohomology vector space H 0, 1 (X, T 1, 0 X).
This manifold is the 5-dimensional analogue of the 3-dimensional Iwasawa manifold I (3) . The following fact was observed in [Rol11].
Proof. It was given in [Rol11].

Cohomological triviality of complex parallelisable deformations of nilmanifolds
In this subsection, we pave the way for the later introduction in §.3 of the notion of small essential deformations by exhibiting the opposite type of objects: a class of superfluous (hence dispensable) small deformations. Specifically, we show that the complex parallelisable small deformations of a complex parallelisable nilmanifold have the same geometry (from the cohomological point of viewsee Corollary 2.7 -and as far as the universal cover is concerned -see Theorem 2.4) as the original manifold.
For a complex parallelisable nilmanifold X = G/Γ, where G is a simply connected nilpotent complex Lie group and Γ ⊂ G is a lattice, the Dolbeault cohomology can be computed by left invariant forms (cf. [Sak76]). In particular, one has (cf. [Nak75]): Furthermore, g is actually a complex Lie algebra and g 1,0 ⊂ g C is a complex subalgebra, where g C is the complexification of g. In fact, one has an identification of complex Lie algebras g ∼ = g 1,0 given by z → 1 2 (z − iJz). In what follows, we will always tacitly use the above identifications. Of particular interest are the cohomology classes in where Z(g) is the centre of g, (which coincides with the Lie algebra of the centre Z(G) of G since G is connected). The last inclusion is a consequence of the identification g ∼ = g 1,0 . Elements in H par (X) are called infinitesimally complex parallelisable deformations of X due to the following Theorem 2.3. ( [Rol11]) Let X = G/Γ be a complex parallelisable nilmanifold. Let µ ∈ H 0,1 (X, T 1,0 X). The following statements are equivalent.
(3) tµ induces a 1-parameter family of complex parallelisable manifolds for t small enough.
We will show that the cohomology is the same for all the complex parallelisable small deformations of a given complex parallelisable nilmanifold X = G/Γ. This will be a consequence of the following Theorem 2.4. Let X = G/Γ be a complex parallelisable nilmanifold, where G is a simply connected nilpotent complex Lie group and Γ ⊂ G is a lattice. The universal cover of any complex parallelisable small deformation of X is isomorphic to G as a complex Lie groups.
Proof. It is known that, for any left-invariant complex structure J making X into a complex parallelisable nilmanifold X = G/Γ, all the small deformations of J are again left-invariant (cf. [Rol11,sect. 4]). In particular, they are again of the form G/Γ, but now with a possibly different, yet still left invariant, complex structure. Thus, differentiably, the universal cover is always G, which is determined entirely by g through the Lie-group/Lie-algebra correspondence. However, the complex structure on G varies with µ but since it remains left-invariant, it is determined by the splitting g C = g 0,1 µ ⊕ g 1,0 µ into i-and (−i)-eigenspaces, which can be computed from the complex structure of the central fibre via the equalities g 0,1 µ = (Id +µ)g 0,1 and g 1,0 µ = (Id +μ)g 1,0 .
Claim 2.5. The linear map of vector spaces defined as (Id +µ) on g 0,1 0 and as (Id +µ) on g 1,0 0 , is an isomorphism of Lie algebras.
Proof of Claim 2.5. Since µ is small, α is an isomorphism of vector spaces and the point is to show that it is also a morphism of Lie algebras. We use [X,Ȳ ] = 0 for all X ∈ g 1,0 andȲ ∈ g 0,1 . Since Regarding the last-but-one equality, recall Cartan's formula (∂η)(X,Ȳ ) = −η([X,Ȳ ]) that holds for any left-invariant (0, 1)-formη and that µ ∈ ker∂ ∩ A 0,1 Finally, for all X ∈ g 1, 0 and allȲ ∈ g 0,1 , we have: Summing up, α is an isomorphism of Lie algebras. Thus, we get an induced isomorphism G → G which by construction is compatible with the complex structures corresponding to 0 resp. µ.
This finishes the proof of Claim 2.5 and that of Theorem 2.4.
Remark 2.6. Note that Theorem 2.4 does not state that the corresponding small deformations of X are themselves biholomorphic. For example, when X is a torus, we only recover the fact that the universal cover of each small deformation is C n (while, of course, the lattice changes).
For the last two cohomologies mentioned in the following statement, see Definition 3.4. in [PSU20b], recalled as Definition 2.10 below.
Corollary 2.7. Let X be a complex parallelisable small deformation of a complex parallelisable nilmanifold X. Then, there exists an isomorphism between the double complexes of left invariant forms on X and X .
In particular, there exist isomorphisms H(X) ∼ = H(X ), where H stands for any cohomology of one of the following types: Dolbeault, Frölicher E r , De Rham, Bott-Chern, Aeppli and higher-page Bott-Chern and Aeppli.
Proof. The first statement follows from Claim 2.5, since the double complex of left invariant forms can be computed in terms of the Lie-algebra with its complex structure, while the second follows from [Ste21, Prop. 12] and the fact that for any nilmanifold X = G/Γ, the inclusion of the double complex of left-invariant forms on G into all forms on X is an E 1 -isomorphism. (This is conjectured to hold for all complex nilmanifolds and it is known for complex parallelisable ones, see [Sak76]).

Brief review of page-r-∂∂-manifolds
Let X be a compact complex manifold with dim C X = n.

∂∂-manifolds
Recall that X is said to be a ∂∂-manifold if for any d-closed pure-type form u on X, the following exactness properties are equivalent: The ∂∂-property of X is equivalent to X admitting a canonical Hodge decomposition. This implies that X admits a canonical Hodge symmetry as well. These are properties of the Dolbeault cohomology groups H p, q ∂ (X, C) = E p, q 1 (X) of X. They lie on the first page of the Frölicher spectral sequence (FSS) of X. The ∂∂-property of X can also be characterised in terms of the Bott-Chern and Aeppli cohomologies. (See Theorem and Definition 2.11 below in the case r = 1 for precise statements of these properties.)

Terminology
Recall that the Frölicher spectral sequence (FSS) of X is a finite collection of complexes (called pages) that inductively refine the Dolbeault cohomology of X until it becomes (non-canonically) isomorphic to the De Rham cohomology. The first page, whose spaces are denoted by E •, • 1 , is defined as the Dolbeault complex (with spaces E p, • 1 of bidegrees (p, •) for every fixed p), while for every integer r ≥ 2 the r-th page E •, • r is defined as the cohomology of the previous page. In [PSU20b,§.2.2], we introduced the following terminology. Fix r ∈ N and a bidegree (p, q) with p, q ∈ {0, . . . , n}. A smooth C-valued (p, q)-form α on X is said to be E r -closed if it represents an E r -cohomology class, denoted by {α} Er ∈ E p, q r (X), on the r-th page of the Frölicher spectral sequence of X. Meanwhile, α is said to be E r -exact if it represents the zero E r -cohomology class, i.e. if {α} Er = 0 ∈ E p, q r (X). The C-vector space of C ∞ E r -closed (resp. E r -exact) (p, q)-forms will be denoted by Z p, q r (X) (resp. C p, q r (X)). Of course, C p, q r (X) ⊂ Z p, q r (X) and E p, q r (X) = Z p, q r (X)/C p, q r (X). These notions are explicitly characterised as follows. (See [PSU20b, Proposition 2.3.].) . . , r − 1} satisfying the following tower of r equations: We say in this case that∂α = 0 and ∂α runs at least (r − 1) times.
with ξ arbitrary and ζ satisfying the following tower of (r − 1) equations: We say in this case that∂ζ reaches 0 in at most (r − 1) steps.
(iii) The following inclusions hold in every bidegree (p, q): Moreover, we say that a form α ∈ C ∞ p, q (X) is E r -closed ifᾱ is E r -closed and we say that α is E r -exact ifᾱ is E r -exact.
(ii) We refer to the properties of α in the two towers of (r − 1) equations under (i) by saying that ∂α, resp.∂α, runs at least (r − 1) times.
(iii) We say that a form α ∈ C ∞ p, q (X) is E r E r -exact if there exist smooth forms ζ, ξ, η such that and such that ζ and η further satisfy the following conditions. There exist smooth forms v r−3 , . . . , v 0 and u r−3 , . . . , u 0 such that the following two towers of r − 1 equations are satisfied: (iv) We refer to the properties of ζ, resp. η, in the two towers of (r − 1) equations under (iii) by saying that∂ζ, resp. ∂η, reaches 0 in at most (r − 1) steps.
To unify the definitions, we will also say that a form As with E r and E r , it follows at once from Definition 2.9 that the E r E r -closedness condition becomes stronger and stronger as r increases, while the E r E r -exactness condition becomes weaker and weaker as r increases. In other words, the following inclusions of vector spaces hold: The main notions introduced and studied in [PSU20b] are the following higher-page analogues of the Bott-Chern and Aeppli cohomologies. (i) The E r -Bott-Chern cohomology group of bidegree (p, q) of X is defined as the following quotient complex vector space: (ii) The E r -Aeppli cohomology group of bidegree (p, q) of X is defined as the following quotient complex vector space: When r = 1, the above groups coincide with the standard Bott-Chern, respectively Aeppli, cohomology groups. Moreover, for every (p, q), one has a sequence of canonical linear surjections: and a sequence of subspaces of H p, q A (X): It can be shown that the representatives of E r -Bott-Chern classes can be alternatively described as the forms that are simultaneously E r -closed and E r -closed, while the E r -Aeppli-exact forms can be alternatively described as those forms lying in C p, q r + C p, q r .

Higher-page analogues of the ∂∂-property
In [PSU20a, Theorem and Definition 1.2.] and [PSU20b, Theorem 1.3.], we generalised the notion of ∂∂-manifold to the higher pages of the Frölicher spectral sequence in the following form that also features the higher-page Bott-Chern cohomology and the higher-page Aeppli cohomology of X introduced in [PSU20b].
Theorem and Definition 2.11. Let X be a compact complex manifold with dim C X = n. Fix an arbitrary positive integer r. The following statements are equivalent.
(1) For every bidegree (p, q), every class {α p, q } Er ∈ E p, q r (X) can be represented by a d-closed (p, q)-form and for every k, the linear map is well-defined by means of d-closed pure-type representatives and bijective.
In this case, X is said to have the E r -Hodge Decomposition property.
(2) The Frölicher spectral sequence of X degenerates at E r and the De Rham cohomology of X is pure.
A compact complex manifold X that satisfies any of the equivalent conditions (1)-(5) is said to be a page-(r − 1)-∂∂-manifold.
The relations among these notions are captured in Observation 2.12. ([PSU20a, Corollary 2.11.) Let X be a compact complex manifold.
That the E r -Hodge Decomposition property implies the higher-page analogue of the Hodge Symmetry property was proved in [PSU20a, Corollary 2.10.].
Corollary 2.13. Let r be a positive integer. Any page-(r − 1)-∂∂-manifold X has the E r -Hodge Symmetry property in the following sense.
For all p, q ∈ {0, . . . , n}, (a) every class {α p, q } Er ∈ E p, q r (X) contains a d-closed representative α p, q ; (b) the linear map E p, q r (X) {α p, q } Er → {α p, q } Er ∈ E q, p r (X) is well-defined (in the sense that it does not depend on the choice of d-closed representative α p, q of the class {α p, q } Er ) and bijective.

Essential deformations of Calabi-Yau manifolds
The notion of essential deformations was introduced in [Pop18] in the special case of the Iwasawa manifold I (3) . We now extend it to the class of all Calabi-Yau page-1-∂∂-manifolds X. The idea is to keep only those small deformations of X that are parametrised by E n−1, 1 2 (X) instead of the larger-dimensional E n−1, 1 1 (X). However, E n−1, 1 2 (X) need not inject canonically into E n−1, 1 1 (X), so we will have to work with injections defined by a background Hermitian metric on X.
Let X be a compact complex manifold with dim C X = n. Recall that, for every integer r ≥ 1 and every bidegree (p, q), the vector space of smooth E r -closed (resp. E r -exact) (p, q)-forms on X is denoted by Z p, q r (X) (resp. C p, q r (X)). We now define the following vector subspace of E p, q 1 (X): In other words, E p, q 1 (X) 0 = ker d 1 consists of the E 1 -cohomology classes (i.e. Dolbeault cohomology classes) representable by E 2 -closed forms of type (p, q), where d 1 : E p, q 1 (X) −→ E p+1, q 1 (X), d 1 ([u]∂) = [∂u]∂, is the differential in bidegree (p, q) on the first page of the FSS.
Lemma 3.1. For all p, q, the canonical linear map is well defined and surjective. Its kernel consists of the E 1 -cohomology classes representable by E 2 -exact forms of type (p, q).
In particular, P p, q is injective (hence an isomorphism) if and only if C p, q 1 (X) = C p, q 2 (X). Proof. Well-definedness means that P p, q ({α} E 1 ) is independent of the choice of representative of the class {α} E 1 ∈ E p, q 1 (X) 0 . This follows from the inclusion C p, q 1 (X) ⊂ C p, q 2 (X). The other three statements are obvious.
Since the map P p, q : E p, q 1 (X) 0 → E p, q 2 (X) is surjective, there exist injective linear maps J p, q : E p, q 2 (X) → E p, q 1 (X) 0 such that P p, q • J p, q = Id E p, q 2 (X) . However, there is no unique or even canonical choice for such a map J p, q . Indeed, there may exist different representatives α 1 and α 2 of a same class {α} E 2 ∈ E p, q 2 (X) that represent distinct classes {α 1 } E 1 = {α 2 } E 1 in E p, q 1 (X) even if α 1 and α 2 are d-closed. Choosing a section J p, q : E p, q 2 (X) → E p, q 1 (X) 0 for P p, q amounts to choosing a representative α in each class {α} E 2 ∈ E p, q 2 (X).
Since we are concerned with small deformations of X, we assume henceforth that (p, q) = (n, n − 1), although most of the following arguments apply in any bidegree. Let us fix a Hermitian metric ω on X. By the Hodge theory for the E 2 -cohomology introduced in [Pop16] (and used e.g. in [PSU20b]) and the standard Hodge theory for the Dolbeault cohomology, there are Hodge isomorphisms: and ∆ = ∆ ω = ∂p ∂ + ∂ p ∂ + ∆ is the pseudo-differential Laplacian introduced in [Pop16], ∆ =∂∂ +∂ ∂ is the standard∂-Laplacian, both associated with the metric ω, while p is the L 2 ω -orthogonal projection onto ker ∆ .
Under our page-1-∂∂-assumption on X, every class {α} E 2 ∈ E n−1, 1 2 (X) can be represented by a d-closed (n − 1, 1)-form. However, the ∆-harmonic (= the minimal L 2 -norm) representative need not be d-closed. Nevertheless, for a reason that will become apparent later on, we need to work with d-closed representatives. In order to make the choice of such a representative unique (once a metric ω has been fixed on X), we will modify the ∆-harmonic representative to a d-closed one in a unique way imposed by ω.
We will need the following result from an earlier work in our current special case r = 2. Fix an arbitrary integer r ≥ 2. The following properties are equivalent.
The preliminary observation that produces a unique d-closed representative of any class {α} E 2 ∈ E n−1, 1 2 (X) naturally associated with a given metric is the following Lemma 3.3. Let X be a compact complex page-1-∂∂-manifold with dim C X = n. Let ω be an arbitrary Hermitian metric on X.
In particular, if a canonical metric ω 0 exists on X (in the sense that ω 0 depends only on the complex structure of X with no arbitrary choices involved in its definition), the associated map J n−1, 1 ω 0 constitutes a canonical injection of E n−1, 1 2 (X) into E n−1, 1 1 (X).
(ii) Suppose that X carries a canonical Hermitian metric ω 0 . The space of small essential deformations of X is defined as the image in E n−1, 1 1 (X) of the canonical injection J n−1, 1 If the page-1-∂∂-assumption on X is replaced by a more general one (for example, the page-r-∂∂-assumption for some r ≥ 2 or merely the E r (X) = E ∞ (X) assumption for a specific r ≥ 2), one can define a version of essential deformations using higher pages than the second one. The most natural choice is the degenerating page E r = E ∞ of the FSS if r > 2. Since at the moment we are mainly interested in page-1-∂∂-manifolds, we confine ourselves to E 2 .
Example 3.6. (The Iwasawa manifold) If α, β, γ are the three canonical holomorphic (1, 0)forms induced on the complex 3-dimensional Iwasawa manifold X = G/Γ by dz 1 , dz 2 , dz 3 − z 1 dz 2 from C 3 (the underlying complex manifold of the Heisenberg group G), it is well known that α and β are d-closed, while dγ = ∂γ = −α ∧ β = 0. It is equally standard that the Dolbeault cohomology group of bidegree (2, 1) is generated as follows: In particular, we see that every E 1 -class of bidegree (2, 1) can be represented by a d-closed form.
Since every pure-type d-closed form is also E 2 -closed (and, indeed, E r -closed for every r), we get E 2, 1 1 (X) = E 2, 1 1 (X) 0 . It is equally standard that the E 2 -cohomology group of bidegree (2, 1) is generated as follows: It identifies canonically with the vector subspace as parametrising the essential deformations defined there for the Iwasawa manifold X.
Note that the hypotheses of Theorem 1.2, all of which are satisfied by X = I (3) and X = I (5) , have the advantage of being cohomological in nature, hence fairly general and not restricted to the class of nilmanifolds. Indeed, there is no mention of any structure equations in Theorem 1.2.
Finally, for the reader's convenience, we recall the following classical result (cf. Lemma 3.1. in [Tia87], Lemma 1.2.4. in [Tod89]) that will be made a key use of in the proof of Theorem 1.2.
The proof of Theorem 1.2 shows that it suffices to check condition (1) on a small subset of all forms. For instance, we have Remark 4.5. Suppose there is a sub-double-complex C ⊂ A X := (C ∞ · , · (X, C), ∂,∂) such that Then, for the conclusion of Theorem 1.2 to hold, it suffices to check that the forms in T satisfy condition (1) in Theorem 1.2.
Proof. By property (1), we have E n−1,1 2 (X) = E n−1,1 2 (C). We may therefore start the proof of Theorem 1.2 by picking η 1 ∈ C. By property (2), ψ 1 ∈ T . Again by property (1), the inclusion C → A X induces isomorphisms in Bott-Chern cohomology and higher pages of the Frölicher spectral sequence of C and A X [Ste21, Cor. 13], whenever a form in C is exact in any way (w.r.t. ∂,∂, ∂∂, d r ,...) in A X , one is able to find a primitive in C. Using this and property (2), whenever we use the Calabi-Yau isomorphism, we see that, at each step in the proof of Theorem 1.2, one may take the solutions to (Eq. (ν)) to lie in C.

Examples, applications and explicit computations
In this section, we apply our results to certain classes of compact complex paralellisable solvmanifolds of complex dimension 3 that were studied by Nakamura in [Nak75]. For the reader's convenience, we start by giving a brief rundown of the background by following Hasegawa's more recent treatment of these manifolds in [Has10], where Nakamura's discussion was expanded.
Let X be a compact complex paralellisable solvmanifold with dim C X = n. It is standard that any such X arises as a quotient X = G/Γ, where G is a simply connected solvable complex Lie group and Γ ⊂ G is a co-compact lattice (i.e. a discrete subgroup). Any such G is unimodular (i.e. the left-invariant Haar measure of G is also right-invariant, a fact that is equivalent to | det Ad g | = 1 for every g ∈ G). This is equivalent to the Lie algebra g of G being unimodular (i.e. tr (ad ξ ) = 0 for every ξ ∈ g). Now, let n = 3. Fix a C-basis {X, Y, Z} of g. The unimodular solvable complex Lie algebras g of complex dimension 3 are classified into the following types ( [Nak75], [Has10]): Note that in this case we have: g 1 := [g, g] = Z , so g 2 := [[g, g], g] = [ Z , X, Y, Z ] = 0, hence g is 2-step nilpotent. Note that in this case we have: g 1 := [g, g] = Y, Z , so g 2 := [[g, g], g] = Y, Z = g 1 , so g is indeed non-nilpotent. However, g (2) := [[g, g], [g, g]] = [ Y, Z , Y, Z ] = 0, so g is indeed solvable.
The solvmanifolds X = G/Γ corresponding to Lie groups G whose Lie algebras g are of this type (3) are usually called Nakamura manifolds. They are not nilmanifolds.
The lattices Γ ⊂ G of the simply connected solvable complex Lie groups G whose Lie algebras g belong to the respective above classes are completely determined as follows. (See ( [Nak75], [Has10]).) (1) If g is Abelian, then G = (C 3 , +) and any lattice Γ ⊂ G is Z-generated by an R-basis of C 3 R 6 . Hence, X := G/Γ is a complex torus.
(3) If g is non-nilpotent, then G is the semi-direct product G = C 2 φ C with This means that the group operation on G = C 2 φ C is defined by (a, b, c) (x, y, z) := (a + x, (b, c) + φ(a)(y, z)) = (a + x, b + e a y, c + e −a z).
5.1 Case of the Nakamura manifolds of type (3b) We now prove Proposition 1.3 by showing that the Nakamura manifolds of type (3b) constitute an example of manifolds satisfying the conditions of that statement. However, we cannot immediately apply Theorem 1.2 because of Observation 5.1. The Nakamura manifolds of class (3b) do not satisfy hypothesis (1) of Theorem 1.2.