ON COHOMOLOGICAL DECOMPOSITION OF ALMOST-COMPLEX

arXiv:0909.3604v1 [math.DG] 19 Sep 2009

ON COHOMOLOGICAL DECOMPOSITION OF ALMOST-COMPLEX MANIFOLDS AND DEFORMATIONS DANIELE ANGELLA AND ADRIANO TOMASSINI Abstract. While small deformations of K¨ ahler manifolds are K¨ ahler too, we prove that the cohomological property to be C ∞ -pure-and-full is not a stable condition under small deformations. This property, that has been recently introduced and studied by T.-J. Li and W. Zhang in [20] and T. Draghici, T.-J. Li and W. Zhang in [11, 12], is weaker than the K¨ ahler one and characterizes the almost-complex structures that induce a decomposition in cohomology. We also study the stability of this property along curves of almost-complex structures constructed starting from the harmonic representatives in special cohomology classes.

1. Introduction Let (M, J) be a compact almost-complex 2n-dimensional manifold and let ω be a symplectic form on M . Then J is said to be ω-tamed if ω (·, J·) > 0 and ωcompatible (or ω-calibrated ) if g (·, ·) := ω (·, J·) is a J-Hermitian metric. Define the tamed cone KJt as the open convex cone given by the projection in cohomology of the space of the symplectic forms taming J, namely def 2 KJt = [ω] ∈ HdR (M ; R) | J is ω-tamed , and the compatible cone KJc as its subcone given by the projection of the space of the symplectic forms compatible with J, namely def 2 [ω] ∈ HdR (M ; R) | J is ω-compatible . KJc = T.-J. Li and W. Zhang proved in [20, Theorem 1.2] that if J is integrable and KJc is non-empty then the following relation between the two cones holds: 2 (M ; R) ; (1) KJt = KJc + H∂2,0 (M ) ⊕ H∂0,2 (M ) ∩ HdR

they also proved that if J is integrable and 2n = 4 then KJt is empty if and only if KJc is empty: this gives a partial answer to a question of Donaldson’s, [10, Question 2]. In order to generalize (1) for an arbitrary almost-complex structure, T.-J. Li and W. Zhang introduced in [20] the concept of C ∞ -pure-and-full almost-complex structure. More precisely, an almost-complex structure J is said to be C ∞ -pureand-full if it induces the decomposition (1,1)

2 HdR (M ; R) = HJ

(2,0),(0,2)

(M )R ⊕ HJ

(M )R ,

2000 Mathematics Subject Classification. 53C55; 53C25; 32G05. Key words and phrases. pure and full almost complex structure; cohomology; deformation. This work was supported by the Project MIUR “Geometric Properties of Real and Complex Manifolds” and by GNSAGA of INdAM. 1

ON COHOMOLOGICAL DECOMPOSITION AND DEFORMATIONS

2

(1,1)

(2,0),(0,2)

(M )R (respectively, HJ (M )R ) is given by the projecwhere the group HJ tion in cohomology of the space ∧2,0 M ⊕ ∧0,2 M ∩ ∧2 M (respectively, ∧1,1 M ∩ ∧2 M ); more in general, J is said to be C ∞ -full if the equality (1,1)

2 HdR (M ; R) = HJ

(2,0),(0,2)

(M )R + HJ

(M )R

2 holds, namely if each cohomology class in HdR (M ; R) has at least one type of pure degree representatives. In [20, Theorem 1.3], T.-J. Li and W. Zhang proved that if J is C ∞ -full and if KJc is non-empty, then (2,0),(0,2)

KJt = KJc + HJ

(M )R ,

2 generalizes the group where ⊕ H∂0,2 (M ) ∩ HdR (M ; R) in (1). In [20] dual notions starting from the space of currents are also defined: we will recall in Section 2 what a pure-and-full almost-complex structure is. Further studies about C ∞ -pure-and-full almost-complex structures have been carried out in [11] and [16]. In particular, T. Draghici, T.-J. Li and W. Zhang proved in [11, Theorem 2.3] that every almost-complex structure on a compact 4-dimensional manifold is C ∞ -pureand-full. As a consequence of the last two quoted results, (see [20, Corollary 1.4]), if (M, J) is a compact almost complex 4-manifold such that KJc is non-empty, then (2,0),(0,2) (M )R HJ

H∂2,0 (M )

(2,0),(0,2)

KJt = KJc + HJ (1,1)

(M )R .

In particular, if b+ (M ) = dimR HJ (M )R = 1, then KJt = KJc . In real dimension greater than 4, things are different. Indeed, for example, there are almost-complex structures on compact 6-dimensional solvmanifolds which are not C ∞ -pure (see [16, Example 3.3]). This turns our attention to the 6-dimensional case. In this paper, we are interested in studying small deformations of C ∞ -pure-andfull complex structures. The celebrated theorem of K. Kodaira and D. C. Spencer, [18, Theorem 15], states that Kähler metrics on compact complex manifolds are stable under small deformations; L. Alessandrini and G. Bassanelli proved in [2] that this stability fails to be true for the class of p-K¨ ahler manifolds, where p ∈ {2, . . . , n − 1} (see [1] for the precise definition), e.g. for the class of balanced metrics, namely the J-Hermitian metrics on compact complex manifolds whose fundamental form ω satisfies d ω n−1 = 0. Since the C ∞ -pure-and-full condition is weaker than the Kähler one (more precisely, as a consequence of [20], see Theorem 7, every compact complex manifold verifying the ∂∂-Lemma is C ∞ -pure-and-full), it could be interesting to establish if the C ∞ pure-and-full complex structures are stable under small deformations. As hinted by T.-J. Li and W. Zhang in a previous version of [20], we study the stability of the standard complex structure on the Iwasawa manifold and try to deform a C ∞ -pureand-full almost-complex structure starting with J-anti-invariant forms as explained in [19]. In [21] I. Nakamura computed the small deformations of the Iwasawa manifold X, dividing them in three classes. Then, a direct computation shows that the complex structure on X is C ∞ -pure-and-full.


3

We prove that (see Theorem 10 for the precise statement) the small deformations of classes (i) and (iii) are C ∞ -pure-and-full while those ones of class (ii) are not. Hence, as a corollary we get the following (see Section 3). Theorem 11. Compact complex C ∞ -pure-and-full (or C ∞ -pure or C ∞ -full or pureand-full or pure or full) manifolds are not stable under small deformations of the complex structure. As C ∞ -pure-and-full property is defined for an arbitrary almost-complex structure (even not integrable), we study its stability along curves of almost-complex structures {Jt }t∈(−ε,ε) , too. In [12] it is proved the semi-continuity property of h± J for an almost-complex structure on a compact 4-dimensional manifold. More precisely, if M is a compact 4-manifold with an almost-complex structure J such that KJc 6= ∅, then for any almost complex structure J ′ in a sufficiently small neighborhood of J the following holds • KJc ′ 6= ∅ + • h+ J (M ) ≤ hJ ′ (M ) − • h− J (M ) ≥ hJ ′ (M )

In [19], curves of almost-complex structures parametrized by real forms of pure degree (2, 0) + (0, 2) are constructed. Using this construction, that (see we prove Theorem 13 for the precise statement) there exists a family N 6 (c) c of compact cohomologically K¨ ahler manifolds with no Kähler metrics such that (i) N 6 (c) admits a C ∞ -pure-and-full almost-complex structure J, (ii) each harmonic form of type (2, 0) + (0, 2) gives rise to a curve {Jt }t∈(−ε,ε) of C ∞ -pure-and-full almost-complex structures on N 6 (c), (iii) furthermore, the map (2,0),(0,2) N 6 (c) R t 7→ dimR HJt is an upper-semicontinuous function at t = 0.

In particular, we get the upper-semicontinuity property of h− J for this 6-dimensional example. We recall that, for a suitable c ∈ R, the  cz e 0 0    −cz 0 e 0 def  Sol(3) =  0 0 1    0 0 0

completely solvable Lie group   x    y   ∈ GL(4; R) | x, y, z ∈ R z     1

admits a cocompact discrete subgroup Γ(c); we define M

def

= Γ(c) \Sol(3)

and def

N 6 (c) = M × M ; the manifold N 6 (c) first appeared in [5] as an example of a cohomologically Kähler manifold; M. Fern´ andez, V. Mu˜ noz and J. A. Santisteban proved in [15] that it has no K¨ ahler structures.


4

In Section 2, we fix the notation, recall the main results on C ∞ -pure-and-full almost-complex structures from [20], [11] and [16] and we give an example of 6dimensional (compact) non-K¨ ahler solvmanifold endowed with a C ∞ -pure-and-full and pure-and-full almost complex structure. In Section 3, we prove the instability Theorem 11. Finally, in Section 4, we recall how a cohomology class in (2,0),(0,2) (M )R gives rise to a curve of almost-complex structures on (M, J); we HJ provide several examples of curves of C ∞ -pure-and-full almost-complex structures on 4 and 6-dimensional compact manifolds. This work has been originally developed as partial fulfillment for the first author’s Master Degreee in Matematica Pura e Applicata at Universit` a di Parma under the supervision of the second author. Acknowledgments. We would like to thank Tedi Draghici, Tian-Jun Li and Weiyi Zhang for their very useful comments and for pointing us the reference [12]. 2. C ∞ -pure-and-full almost-complex structures Let (M, J) be an almost-complex compact 2n-manifold. The endomorphism J on T M ⊗ C, having eigenvalues i and − i, induces a decomposition of ∧• (M ; C) L p,q p,q p,q k through ∧J M :=: ∧ M , namely ∧ M = p+q=k ∧J M . We ask about when (p,q)

this decomposition holds in cohomology. Define HJ (M ) as the projection in de (p,q),(q,p) (M )R as the projection in Rham cohomology of the space ∧p,q M ; define HJ de Rham cohomology of the space (∧p,q M ⊕ ∧q,p M ) ∩ ∧p+q M . (As a matter of notation, bigraduation without further specification refers to complex forms, single graduation to real ones.) In other words: (p,q),(q,p) q,p p+q p+q M . (M )R = [α] ∈ HdR (M ; R) | α ∈ (∧p,q HJ J M ⊕ ∧J M ) ∩ ∧ If S is a set of pairs (p, q), we define likewise     M def • ∧p,q M HJS (M ) = [α] ∈ HdR (M ; C) | α ∈   (p,q)∈S

and

HJS (M )R

def

• = HJS (M ) ∩ HdR (M ; R) .

T.-J. Li and W. Zhang give the following. Definition 1 ([20, Definition 2.4, Definition 2.5, Lemma 2.6]). An almost-complex structure J on M is said to be: • C ∞ -pure if (2,0),(0,2)

HJ

(1,1)

(M )R ∩ HJ

(M )R = {[0]} ;

∞

• C -full if (2,0),(0,2)

HJ

(1,1)

(M )R + HJ

2 (M )R = HdR (M ; R) ;

• C ∞ -pure-and-full if it is both C ∞ -pure and C ∞ -full, i.e. if the following decomposition holds: (2,0),(0,2)

HJ

(1,1)

(M )R ⊕ HJ

2 (M )R = HdR (M ; R) .


5

For a complex manifold M , by saying that M is, for example, C ∞ -pure-and-full, we mean that the integrable almost-complex structure naturally associated with it is C ∞ -pure-and-full. We also use the following notations: • by saying that J is complex-C ∞ -pure we mean that the sum (2,0)

HJ

(1,1)

(M ) + HJ

(0,2)

(M ) + HJ

(M )

is direct; • by saying that J is complex-C ∞ -full we mean that the equality (2,0)

2 HdR (M ; C) = HJ

(1,1)

(M ) + HJ

(0,2)

(M ) + HJ

(M )

holds; • by saying that J is complex-C ∞ -pure-and-full we mean that J induces the decomposition (2,0)

2 (M ; C) = HJ HdR

(1,1)

(M ) ⊕ HJ

(0,2)

(M ) ⊕ HJ

(M ) .

Remark 2. While being complex-C ∞ -full is a stronger condition that being C ∞ full, one has to assume J to be integrable to have that complex-C ∞ -pure condition implies the C ∞ -pure one. Note also that if J is C ∞ -pure then (0,2) (2,0) (1,1) HJ (M ) ∩ HJ (M ) + HJ (M ) = {[0]} . Moreover, we say that J is C ∞ -pure-and-full at the k-th stage if J induces a dek composition of HdR (M ; R); for k = 2, we recover the previous definitions.

Using the complex of currents instead of the complex of forms and the de Rham homology instead of the de Rham cohomology, one can define analogous concepts dually. Recall that the space of currents of dimension k (or degree 2n − k) is the topological dual of ∧k M : we denote it with Dk M :=: D2n−k M ; we refer to [7], [9] as general references for the study of currents. Dually, the exterior differential d on ∧• M induces a differential on D• M , that we denote again as d; we call de Rham homology H• (M ; R) the cohomology of the differential complex (D• M, d); k we remember that HdR (M ; R) ≃ H2n−k (M ; R). As J induces a bigraduation on • ∧ (M ; C), so Dp,q M are defined. J J Therefore, let H(2,0),(0,2) (M )R (respectively, H(1,1) (M )R ) be the subspace of H2 (M ; R) given by the homology classes represented by a current of bidimension (2, 0) + (0, 2) (respectively, (1, 1)). We recall the following definition by T.-J. Li and W. Zhang (see [20]). Definition 3 ([20, Definition 2.15, Lemma 2.16]). An almost-complex structure J on M is said to be: • pure if J J (M )R = {[0]} ; (M )R ∩ H(1,1) H(2,0),(0,2) • full if J J (M )R = H2 (M ; R) ; (M )R + H(1,1) H(2,0),(0,2)

• pure-and-full if it is both pure and full, i.e. if the following decomposition holds: J J (M )R = H2 (M ; R) . (M )R ⊕ H(1,1) H(2,0),(0,2)


6

The relations between being C ∞ -pure-and-full and being pure-and-full are summarized in the following. Theorem 4 (see also [20, Proposition 2.30]). The following relations between C ∞ pure-and-full and pure-and-full concepts hold: +3 pure at the k-th stage C ∞ -full at the k-th stage +3 C ∞ -pure at the (2n − k) -th stage

full at the (2n − k) -th stage

Proof. First, we prove that if J is C ∞ -full at the k-th stage then it is also pure at the k-th stage; for the sake of simplicity, we assume k = 2. Let 2 h·, ··i : HdR (M ; R) → H2 (M ; R) J J the non-degenerate duality paring. Let c ∈ H(2,0),(0,2) (M )R ∩ H(1,1) (M )R , with c 6= [0]. Obviously, hc, ·i ⌊H (2,0),(0,2) (M)R = 0 and hc, ·i ⌊H (1,1) (M)R = 0; since J is J J C ∞ -full, it follows that c = [0]. The same argument works to prove that a full J is also C ∞ -pure. To conclude the proof, we have to prove the two vertical arrows, namely that ?

C ∞ -full at the k-th stage ⇒ full at the (2n − k) -th stage and that ?

pure at the k-th stage ⇒ C ∞ -pure at the (2n − k) -th stage . Recall that a form of degree k can be viewed as a current of dimension 2n − k (and degree k), by means of the map Z def ϕ∧·. T· : ∧k M → D2n−k M , ϕ 7→ Tϕ (·) = M (p,q) HJ (M )R

Holding Td · = d T· , this map induces the inclusion k being HdR (M ; R) ≃ H2n−k (M ; R), the statements follow.

J ֒→ H(n−p,n−q) (M )R ;

To prove that C ∞ -full ⇒ pure, see also [16, Theorem 3.7]. 2n−2 2 (M ; R) and HdR A link between HdR (M ; R) could provide further relations ∞ between C -pure-and-full and pure-and-full notions. This is the matter of the following results, proved in [16]. Theorem 5 ([16, Theorem 3.7]). Let g be a Hermitian metric on (M, J). If the 2 harmonic representatives of the classes in HdR (M ; R) are of pure degree, then J is ∞ both C -pure-and-full and pure-and-full. 2n−2 2 (M ; R) and HdR (M ; R) On a symplectic 2n-manifold (M, ω), a link between HdR could be provided if the Hard Lefschetz Condition holds; recall that (M, ω) is said to satisfy the Hard Lefschetz Condition (HLC) if, for every k ∈ {0, . . . , n}, the isomorphism k ≃ n−k n+k (HLC) ω : HdR (M ; R) −→ HdR (M ; R) .

holds.

Theorem 6 ([16, Theorem 4.1]). Let ω be a symplectic form on M satisfying (HLC) and let J an ω-compatible almost-complex structure on M . If J is C ∞ -pureand-full, then it is also pure-and-full.


7

We give now a class of examples of C ∞ -pure-and-full and pure-and-full manifolds. Clearly, compact K¨ ahler manifolds are C ∞ -pure-and-full at every stage (and then also pure-and-full at every stage). Furthermore, the following theorem holds. Theorem 7 (see [20],[11]). Let M be a compact complex manifold; if the HodgeFr¨ olicher spectral sequence degenerates at the first step, then M is C ∞ -pure-and-full at every stage, so it is also pure-and-full at every stage. The proof of Theorem 7 can be obtained using the same argument true for a compact complex surface as in [11]. For the sake of completeness, we will give it in the general case. Since the Frölicher spectral sequence degenerates at E1 , then H∂p,q (M ) =

F p (H p+q (M ; C)) , F p+1 (H p+q (M ; C))

where Fp and

  H k (M ; C) = [α] | α ∈ 

M

′

′

∧p ,q (M ) | dα = 0

p′ +q′ =k, p′ ≥p

H k (M ; C) =

M

′

H p,q ,

 

,



p+q=k

where ′

H p,q = F p H k (M ; C) ∩ F q (H k (M ; C))

(see e.g. [4]). Then, the same proof of Lemma 2.15 in [11] gives ′

H p,q = HJp,q (M ) .

Therefore, H k (M ; C) =

M

p+q=k

′

H p,q =

M

HJp,q (M ) ,

p+q=k

i.e. J is complex-C ∞ -pure-and-full and, consequently, it is C ∞ -pure-and-full (remark 2). As a consequence of the last theorem, we have that (1) the compact complex surfaces, (2) the compact complex manifolds satisfying the ∂∂-Lemma (i.e., the compact complex manifolds for which every ∂-closed, ∂-closed and d-exact form is also ∂∂-exact) (3) and the compact complex manifolds admitting a Kähler structure are C ∞ -pure-and-full and pure-and-full manifolds. Indeed, for (1) we have that the Hodge-Frölicher spectral sequence degenerates at the first step by [4, Theorem 2.6], while for (2) it degenerates at the first step by [8, §5.21]; finally, for (3) we have that a compact complex manifold admitting a K¨ ahler metric satisfies the ∂∂-Lemma, see [8, §5.11]. Actually, T. Draghici, T.-J. Li and W. Zhang proved the following. Theorem 8 ([11, Theorem 2.3]). Every almost-complex structure on a compact 4-manifold is C ∞ -pure-and-full as well as pure-and-full. This turns our attention to the 6-dimensional case.


Example 9. Let G be the Lie group  x e 1 0  0 e−x1   0 0 A=  0 0   0 0 0 0

8

of matrices of the following form  0 x3 x2 ex1 0 0 x2 e−x1 0 x4   x1 e 0 0 x5   0 x6  0 e−x1  0 0 0 x1  0 0 0 1

for x1 , . . . , x6 ∈ R. Then G is a 6-dimensional simply-connected completely solvable Lie group. According to [14], there exists a uniform discrete subgroup Γ ⊂ G, so that M = Γ\G is a 6-dimensional compact solvmanifold. The following 1-forms on G e1 = dx1 ,

e2 = dx2 ,

e3 = exp(−x1 ) (dx3 − x2 dx5 )

e4 = exp(x1 ) (dx4 − x2 dx6 ) , e5 = exp(−x1 )dx5 , e6 = exp(x1 )dx6 give rise to 1-forms on M . We immediately obtain that d e1 = 0 , (2)

d e2 = 0 ,

d e3 = −e1 ∧ e3 − e2 ∧ e5 ,

d e4 = e1 ∧ e4 − e2 ∧ e6 , d e5 = −e1 ∧ e5 , d e6 = e1 ∧ e5

Since G is completely solvable, in view of the Hattori theorem (see [17]), we easily obtain by (2), that (3) H 2 (M ; R) = spanR e1 ∧ e2 , e5 ∧ e6 , e3 ∧ e6 + e4 ∧ e5 ; Therefore, setting

 1 ϕ = e1 + i e2   ϕ2 = e3 + i e4   3 ϕ = e5 + i e6

,

we have that the almost complex structure J whose complex forms of type (1, 0) are ϕ1 , ϕ2 , ϕ3 is C ∞ -full. Indeed, 1 1 (1,1) 1 3 ϕ ∧ ϕ1 , − 2i ϕ ∧ ϕ3 , HJ (M )R = spanR − 2i (2,0),(0,2)

HJ

(M )R

= spanR

1

2i

ϕ2 ∧ ϕ3 − ϕ2 ∧ ϕ3

.

According to theorem 5, since the harmonic representatives are of pure type, J is both C ∞ -pure-and-full and pure-and-full. 3. Instability along curves of complex structures In this section, we will show that the condition to be C ∞ -pure-and-full for a complex structure is not stable under small deformations. In order to do this, we will consider the Iwasawa manifold, showing that there are curves of complex structures that are not C ∞ -pure-and-full. We first recall the definition of the Iwasawa manifold and some of its properties, see e. g. [21], [13]. On C3 , consider the product ∗ defined as def

(z1 , z2 , z3 ) ∗ (w1 , w2 , w3 ) = (z1 + w1 , z2 + w2 , z3 + z1 w2 + w3 ) .


9

C3 , ∗ is a nilpotent Lie group isomorphic to   z1 z3  1 z2  ∈ GL (3; C) | z1 , z2 , z3 ∈ C .  0 1 3 We have that (Z [i]) ⊂ C3 is a cocompact discrete subgroup of C3 , ∗ . The Iwasawa manifold X is defined as the manifold def 3 X = (Z [i]) C3 , ∗ .

It is immediate to check that   1 def  0 H(3) =  0

X is a compact complex 3-dimensional nilmanifold; by [13], it follows that X is not formal; hence, it has no K¨ ahler metrics, see [8, Main Theorem]; nevertheless, there exists a balanced metric on X. We will need the following results on the cohomology of solvmanifolds. The Hattori-Nomizu theorem states that if M = Γ\G is a compact nilmanifold (or, more in general, a compact completely solvable solvmanifold, i.e. a compact solvmanifold such that, for every ξ in the Lie algebra g of G, all the eigenvalues of adξ are real) then (4)

• (M ; R) ≃ H • (∧• g∗ , d) HdR

(see [22], [17]), where the Chevalley-Eilenberg cohomology H • (∧• g∗ , d) is the cohomology of the complex ∧• g∗ endowed with the differential inherited from ∧• M ; equivalently, H • (∧• g∗ , d) is the cohomology of the complex of the left-invariant forms. A similar result holds for the Dolbeault cohomology of nilmanifolds. More precisely, for a compact complex nilmanifold M that is holomorphically parallelizable (i.e., with trivial holomorphic tangent bundle) or whose integrable almostcomplex structure J is rational (i.e., such that J [gQ ] ⊆ gQ , where gQ is a rational Lie subalgebra of g such that g = gQ ⊗ R) or whose J is obtained as a small deformation of a rational one, the following isomorphism holds: ∗ (5) H∂p,q (M ) ≃ H q ∧p,• gC , ∂ (see [23], [6]). In particular, (4) and (5) hold for the Iwasawa manifold and for its small deformations. Let z i i∈{1,2,3} be the standard complex coordinate system on C3 ; the following 3

(1, 0)-forms on C3 are invariant for the action (on the left) of (Z [i]) , so they give rise to a global coframe for T ∗ 1,0 X:  def  ϕ1 = d z 1    def . ϕ2 = d z 2     3 def ϕ = d z3 − z1 d z2 The structure equations are therefore  d ϕ1 = 0   d ϕ2 = 0   d ϕ3 = −ϕ1 ∧ ϕ2

.

By Hattori-Nomizu theorem, we compute the real cohomology algebra of X (for ¯ simplicity, we list the harmonic representative instead of its class and write ϕAB


10

for ϕA ∧ ϕ¯B and so on): 1 , HdR (X; R) = spanR ϕ1 + ϕ¯1 , i ϕ1 − ϕ¯1 , ϕ2 + ϕ¯2 , i ϕ2 − ϕ¯2 n ¯¯ ¯¯ ¯¯ 2 HdR (X; R) = spanR ϕ13 + ϕ13 , i ϕ13 − ϕ13 , ϕ23 + ϕ23 ,

o ¯¯ ¯ ¯ ¯ ¯ ¯ ¯ i ϕ23 − ϕ23 , ϕ12 − ϕ21 , i ϕ12 + ϕ21 , i ϕ11 , i ϕ22 ,

n ¯¯¯ ¯¯¯ ¯ ¯¯ 3 HdR (X; R) = spanR ϕ123 + ϕ123 , i ϕ123 − ϕ123 , ϕ131 + ϕ113 ,

¯ ¯¯ ¯ ¯¯ ¯ ¯¯ ¯ ¯¯ i ϕ131 − ϕ113 , ϕ132 + ϕ213 , i ϕ132 − ϕ213 , ϕ231 + ϕ123 , o ¯ ¯¯ ¯ ¯¯ ¯ ¯¯ . i ϕ231 − ϕ123 , ϕ232 + ϕ223 , i ϕ232 − ϕ223

Note that each harmonic representative is of pure degree. The Betti numbers of X are b0 = 1 , b1 = 4 , b2 = 8 , b3 = 10 . I. Nakamura in [21] computed the small deformations {Xt }t ∈ ∆(0, ε) ⊆ C6 of the Iwasawa manifold X: by [21, page 95], a local system of complex coordinates for the complex structure at t = (t11 , t12 , t21 , t22 , t31 , t32 ) ∈ C6 is given by  1 P2 1 j   ζt = z + j=1 t1j z¯  P 2 ζt2 = z 2 + j=1 t2j z¯j   P  3 ζt = z 3 + 2j=1 t3j + t2j z 1 z¯j + A(¯z) − D(t) z¯3 where

A(¯ z) D(t)

def

=

def

=

1 t11 t21 z¯1 z¯1 + 2 t11 t22 z¯1 z¯2 + t12 t22 z¯2 z¯2 , 2

t11 t22 − t12 t21 .

Nakamura also computed the numerical characters of these deformations, dividing them into three classes according to their Hodge diamond:

(i)

h1,0 3

h0,1 2

h2,0 3

h1,1 6

h0,2 2

h3,0 1

h2,1 6

h1,2 6

h0,3 1

(ii)

2

2

2

5

2

1

5

5

1

(iii)

2

2

1

5

2

1

4

4

1

More exactly, the classes are characterized by the following values of the parameters: class (i): t11 = t12 = t21 = t22 = 0; class (ii): D (t) = 0 but (t11 , t12 , t21 , t22 ) 6= (0, 0, 0, 0); class (iii): D (t) 6= 0.


11

Note that the Hodge diamond of the deformations of the class (i) is the same of the Iwasawa manifold, while deformations of the class (iii) have the HodgeFrölicher spectral sequence that degenerates at the first step. Note also that the table above proves that the Hodge numbers are not stable under small deformations, [21, Theorem 2], in contrast with the Kähler case. Equivalently, Xt could be viewed as C3 Γt where Γt is the group generated by the transformations  P2  ¯j ζ 1 7→ ζ 1 + ω1 + j=1 t1j ω    P ζ 2 7→ ζ 2 + ω2 + 2j=1 t2j ω ¯j  P   2  ζ 3 7→ ζ 3 + ω + P2 t ω 2 + A (¯ ω ) − D(t) ω ¯3 t ω ¯ + ω ζ + ¯ 2j j 1 3j j 3 j=1 j=1 3

varying ω :=: (ω1 , ω2 , ω3 ) ∈ (Z[i]) . In the sequel, J will denote the integrable almost-complex structure associated to X and Jt will denote the one associated to Xt . Now, we can prove the following. Theorem 10. Let X := (Z [i])3 C3 , ∗ be the Iwasawa manifold. Then: • X is C ∞ -pure-and-full at every stage as well as pure-and-full at every stage; • the small deformations of X of the classes (i) and (iii) are C ∞ -pure-and-full and pure-and-full at every stage; • the small deformations of X of the class (ii) are neither C ∞ -pure nor C ∞ full nor pure nor full.

Proof. We divide the proof in various steps. Step 1: X is a C ∞ -pure-and-full manifold at every stage. • (X; R) are of pure degree, the stateSince harmonic representatives in HdR ment follows from Theorem 5. Step 2: Small deformations of the class (i) remain C ∞ -pure-and-full at every stage. A coframe of (1, 0)-forms invariant for the action of Γt on C3 is given by  def  = d ζt1 ϕ1t    def . = d ζt2 ϕ2t     3 def ϕt = d ζt3 − ζt1 d ζt2 Hence, ϕ1t , ϕ2t , ϕ3t satisfies the same structure equations as ϕ1 , ϕ2 , ϕ3 . Therefore, the same argument in Step 1 applies to deformations of such a class. Step 3: Computation of the structure equations for small deformations of the class (ii). Consider the system of complex coordinates given by  P2 def  ζt1 = z 1 + λ=1 t1λ z¯λ    P2 def . ζt2 = z 2 + λ=1 t2λ z¯λ    P  3 def 2 z λ + A (¯ z) ζt = z 3 + λ=1 (t3λ + t2λ z 1 )¯


12

A straightforward computation gives (

z1

=

z2

=

γ ζt1 + λ1 ζ¯t1 + λ2 ζt2 + λ3 ζ¯t2 α µ0 ζ 1 + µ1 ζ¯1 + µ2 ζ 2 + µ3 ζ¯2 t

t

t

t

where α, β, γ, λi (for i ∈ {1, 2, 3}), µj (for j ∈ {0, 1, 2, 3}) are complex constants depending only on t = (t11 , t12 , t21 , t22 , t31 , t32 ) ∈ C6 and defined as follows:                                                                       

1

def

=

α

2

1 − |t22 | − t21 t¯12 t21 t¯11 + t22 t¯21

def

=

β

1 1 − |t11 | − αβ (t11 t¯12 + t12 t¯22 ) − t12 t¯21 2 −t11 1 + αt¯12 t21 + α |t22 |

def

=

γ

2

def

=

λ1

def

α (t11 t¯12 + t12 t¯22 ) 2 −t12 1 + αt¯12 t21 + α |t22 |

=

λ2

def

=

λ3

def

=

µ0 µ1 µ2

βγ

def

=

λ1 βγ − t21

def

=

1 + λ2 βγ

def

λ3 βγ − t22

=

µ3

.

Consider the (1, 0)-forms invariant for the action of Γt on C3 given by   ϕ1    t    

def

=

d ζt1

ϕ2t

def

=

d ζt2

ϕ3t

def

d ζt3 − z 1 d ζt2 − t21 z¯1 + t22 z¯2 d ζt1

=

;

we could now easily compute the structure equations:  d ϕ1t      d ϕ2 t  d ϕ3t    

=

0

=

0

=

σ12 ϕ1t ∧ ϕ2t + +σ1¯1 ϕ1t ∧ ϕ¯1t + σ1¯2 ϕ1t ∧ ϕ¯2t + σ2¯1 ϕ2t ∧ ϕ¯1t + σ2¯2 ϕ2t ∧ ϕ¯2t


where σ12 , σ1¯1 , σ1¯2 , σ2¯1 , σ2¯2 are given by  def ¯ 3 γ¯ + t22 α  ¯µ ¯3  σ12 = −γ + t21 λ       def 2  ¯  t α + |t | α γ 1 + t = t σ ¯ 12 22 21 21  11     def ¯12 α + |t22 |2 α = t σ t γ 1 + t ¯ 22 21 1 2      def  2  ¯ σ = −t γ 1 + t t α + |t | α  ¯ 11 21 12 22 21       def 2  σ2¯2 = −t12 γ 1 + t21 t¯12 α + |t22 | α

13

.

Note that, for small deformations of the class (ii), one has σ12 6= 0 and (σ1¯1 , σ1¯2 , σ2¯1 , σ2¯2 ) 6= (0, 0, 0, 0). This ends Step 3. Step 4: The small deformations of the class (ii) are neither C ∞ -pure nor full. Note that h i 1¯ 1 1¯ 2 2¯ 1 2¯ 2 = σ ϕ σ12 ϕ12 + σ ϕ + σ ϕ + σ ϕ 6= [0] ¯ ¯ ¯ ¯ 11 t 12 t 21 t 22 t t 2 in HdR (X; C). Therefore, (1,1) (2,0) (0,2) HJt (Xt ) ∩ HJt (Xt ) + HJt (Xt ) 6= {[0]} ,

and in particular Xt is not complex-C ∞ -pure. It follows from the fact observed at page 5 that Xt cannot be C ∞ -pure; from Theorem 4 it follows that Xt cannot be even full. Step 5: The small deformations of the class (ii) are neither pure nor C ∞ -full. For a fixed small t, choose two positive constants A and B such that (A σ1¯2 − B σ1¯1 , A σ2¯2 − B σ2¯1 ) 6= (0, 0) ; ¯ ¯ 3 3 computing − d A ϕ13 + B ϕ23 , note that t t

h i ¯ ¯ ¯ ¯¯ 1¯ 3 2¯ 3 1¯ 2 (A σ2¯1 − B σ1¯1 ) ϕ12 + (A σ2¯2 − B σ1¯2 ) ϕ12 − Aσ ¯12 ϕ13 −Bσ ¯12 ϕt2312 = t t t

i h ¯ 1 123¯ 2 6= [0] , = (A σ ¯1¯2 − B σ ¯1¯1 ) ϕ123 + (A σ ¯ − B σ ¯ ) ϕ ¯ ¯ 22 21 t t

4 in HdR (X; C). As before, it follows that Xt is not C ∞ -pure at the 4th stage, and consequently it is not even pure nor C ∞ -full, by Theorem 4. Step 6: Small deformations of the class (iii) remain C ∞ -pure-and-full at every stage. As already observed, for deformations of the class (iii) the Hodge-Frölicher spectral sequence degenerates at the first step, so that the statement follows from Theorem 7.

As a corollary of the last theorem, we obtain the following theorem of instability. Theorem 11. Compact complex C ∞ -pure-and-full (or C ∞ -pure or C ∞ -full or pureand-full or pure or full) manifolds are not stable under small deformations of the complex structure.


14

Remark 12. By the proof of Theorem 10, it follows that, the numbers def

(1,1)

h+ Jt (X) = dimR HJt

(X)R ,

def

(2,0),(0,2)

h− Jt (X) = dimR HJt

(X)R ,

where Jt is a small deformation of class (iii), are not invariant. Indeed, − h+ J (X) = hJ (X) = 4 ,

and h+ Jt (X) = 5 ,

h− Jt (X) = 3 ,

for t 6= 0. This is in contrast with the complex deformations of a complex structure J on a − 4-dimensional compact manifold M , for which h+ J (M ) and hJ (M ) are topological invariants (see [12]). 4. Stability along curves of almost-complex structures The C ∞ -pure-and-full property makes sense for an arbitrary almost-complex structure, even not integrable. In this section, we will study the stability of this property along curves of almost-complex structures. Let J be an almost-complex structure on M 2n . Recall the following local result, see [3]: a curve {Jt }t∈I⊆R of almost-complex structures through J could be written, for t ∈ (−ε, ε) with ε > 0 sufficiently small, as (6)

Jt = (id − Lt ) J (id − Lt )

−1

,

where Lt is an endomorphism of the tangent bundle such that Lt J + J Lt = 0 ; we could write Lt =: t L+o(t); recall also that: if J is compatible with a symplectic form ω, then the curves made up of ω-compatible almost-complex structures Jt are exactly those ones for which Lt = L. For several examples of families constructed in such a way, see [16]. We begin with studying curves through the standard Kähler structure on the complex 2-torus, T2C , J0 , ω0 . Let   ℓ   0  , L =    −ℓ 0 where ℓ ∈ C ∞ (R4 ; R) is a Z4 -periodic function. Defining (for small t) (7)

Jt, ℓ

def

= (id − t L) J (id − t L)−1 ,

we get the ω0 -compatible almost-complex structure  tℓ − 11 + −tℓ  Jt, ℓ =   1+tℓ 1−tℓ

1

Set

def

α :=: α(t, ℓ) =



−1   . 

1 + tℓ . 1 − tℓ


15

A coframe for the holomorphic cotangent bundle is given by ( 1 ϕt := d x1 + i α d x3 , ϕ2t := d x2 + i d x4 from which we compute (

d ϕ1t = i d α ∧ d x3

; d ϕ2t = 0 note that taking ℓ = ℓ x1 , x3 , the corresponding almost-complex structure Jt, ℓ is integrable, in fact (Jt, ℓ , ω0 ) is a Kähler structure on T2C . Remember that Jt, ℓ has to be C ∞ -pure-and-full, T2C being a 4-dimensional manifold. For the sake of simplicity, we assume ℓ = ℓ x2 not constant. Setting v1

v2 w1 w2 w3 w4

def

=

d x1 ∧ d x2 − α d x3 ∧ d x4 ,

def

=

d x1 ∧ d x4 − α d x2 ∧ d x3 ,

def

=

α d x1 ∧ d x3 ,

def

=

d x2 ∧ d x4 ,

def

=

d x1 ∧ d x2 + α d x3 ∧ d x4 ,

def

d x1 ∧ d x4 + α d x2 ∧ d x3 ,

=

the condition in order that an arbitrary Jt, ℓ -anti-invariant real 2-form ψ = A v1 + B v2 is closed is expressed by  A3 − B1 α = 0      A4 − B2 = 0 (8)    −A1 α − B3 = 0   −B4 α − A2 α − A α2 = 0

(here and later on, we write, for example, A3 instead of obtain ψ = B v2

with

∂A ∂x3 ).

By solving (8), we

B∈R.

Therefore, for small enough t 6= 0, according to [12], we have the upper-semicontinuity property (2,0),(0,2) (2,0),(0,2) T2C R , T2C R ≤ 1 < 2 = dimR HJ dimR HJt, ℓ from which we get the lower one (1,1) (1,1) dimR HJt, ℓ T2C R ≥ 5 > 4 = dimR HJ T2C R .

Now, we turn our attention to the case of dimension greater than 4. We construct curves through the standard Kähler structure on the complex 3-torus, T3C , J0 , ω0 .


Let



∞

6

   L =    

16



ℓ 0 0 −ℓ 0 0 6

    ,   

where ℓ ∈ C (R ; R) is a Z -periodic function. As before, defining Jt, ℓ (for small t) as in (7), we get the ω0 -compatible almost-complex structure   tℓ − 11 + −tℓ   −1     −1  .  Jt, ℓ =  1 + t ℓ    1−tℓ   1 1

As before, setting

1 + tℓ , 1 − tℓ it follows that a coframe for the holomorphic cotangent bundle is given by  1 ϕ := d x1 + i α d x4   t ϕ2t := d x2 + i d x5 ;   3 3 6 ϕt := d x + i d x def

α :=: α(t, ℓ) =

therefore

 d ϕ1t = i d α ∧ d x4   d ϕ2t = 0   d ϕ3t = 0

;

note that ℓ = ℓ x1 , x4 gives rise to integrable almost-complex structures, in fact ∞ to K¨ ahler structures; therefore, in such a case Jt, ℓ is C -pure-and-full. Again, we 3 assume ℓ = ℓ x not constant. The Jt, ℓ -anti-invariant real closed 2-forms are ψ = D d x16 − α d x34 + E d x23 − d x56 + F d x26 − d x35 where D, E, F ∈ R. So, for small t 6= 0, we have the upper-semicontinuity property (2,0),(0,2) (2,0),(0,2) dimR HJt, ℓ T3C R . T3C R ≤ 3 < 6 = dimR HJ (1,1) Unfortunately, the explicit computation of HJt, ℓ T3C R seems to be not so simple. In particular, it is not clear if Jt, ℓ remains C ∞ -full (note that Jt, ℓ is C ∞ -pure by [11, Proposition 2.7], see also [16, Proposition 3.2]).

Now, we recall how to construct curves of almost-complex structure via a J-antiinvariant real 2-form, as in [19]. Let (M, J) be an almost-complex manifold; take g a J-Hermitian metric and γ a real 2-form in ∧2,0 M ⊕ ∧0,2 M . Define V to be the endomorphism of the tangent bundle such that (9)

γ (·, ·) =: g (V ·, ·) ;


17

a direct computation shows that V J + J V = 0. Therefore, setting def 1 L = VJ, 2 we have L J + J L = 0. At this point, for small t, define Jt, γ as in (7): Jt, γ

def

−1

= (id − t L) J (id − t L)

;

therefore, {Jt, γ }t∈(−ε,ε) is a curve of almost-complex structures naturally associated with γ. We give an example of a C ∞ -pure-and-full structure on a non-K¨ ahler manifold such that the stability property holds along a curve constructed in such a way. Let   cz  e 0 0 x       −cz  0 e 0 y def  ∈ GL(4; R) | x, y, z ∈ R .  Sol(3) =   0 0 1 z       0 0 0 1 Then Sol(3) is a completely solvable Lie group. For a suitable c ∈ R, there exists a cocompact discrete subgroup Γ(c) such that M

def

= Γ(c) \Sol(3)

is a compact 3-dimensional solvmanifold. Define def

N 6 (c) = M × M . The manifold N 6 (c) has been studied in [5] as an example of a cohomologically K¨ ahler manifold; M. Fern´ andez, V. Mu˜ noz and J. A. Santisteban proved in [15] that it has no K¨ ahler structures, although it is formal and it has a symplectic structure satisfying (HLC). In [16] a family of C ∞ -pure-and-full structures on N 6 (c) is provided. Now, we will construct a curve of C ∞ -pure-and-full almost-complex 6 structures i on N (c). Let e i∈{1,...,6} be a coframe for N 6 (c); the structure equations are  d e1 = c e1 ∧ e3       d e2 = −c e2 ∧ e3      d e3 = 0 .  d e4 = c e4 ∧ e6      5 5 6   d e = −c e ∧ e    d e6 = 0 Let J be the almost-complex structure given by  −1  1   −1 J =   1  



1

    .   −1 

By Hattori-Nomizu theorem one computes immediately 2 N 6 (c); R = spanR e1 ∧ e2 , e3 ∧ e6 − e4 ∧ e5 , e3 ∧ e6 + e4 ∧ e5 ; HdR


18

hence N 6 (c) is a C ∞ -pure-and-full and pure-and-full manifold, the harmonic representatives being of pure degree. Note that (2,0),(0,2)

HJ

N 6 (c) R = spanR e3 ∧ e6 + e4 ∧ e5 .

Put γ := e3 ∧ e6 + e4 ∧ e5 ; then the linear map V representing γ as in (9) is   0   0     −1  ,  V =   −1     1 1 and then it is immediate to compute  0  0   2L =    



    1   

−1 1 −1

and



Jt := Jt, Φ

Set

    =    

−1 1 4−t2 − 4+t 2

4−t2 4+t2

4t − 4+t 2 4t − 4+t 2

4t 4+t2 4t 4+t2

def

α :=: α(t) =

4 − t2 , 4 + t2

4−t2 4+t2

def

β :=: β(t) =

2

− 4−t 4+t2



     .   

4t . 4 + t2

A coframe for the Jt -holomorphic cotangent bundle is given by

The closed 2-forms 1 1¯1 ϕ , 2i t

 1 ϕ = e1 + i e2   t ϕ2t = e3 + i α e4 + β e6   3 ϕt = e5 + i −β e4 + α e6

.

1 ¯ 1 2¯2 ¯ ¯ + ϕ3t 3 β ϕt + α ϕ2t 3 − ϕ23 t 2i 2i

1 3¯3 α ϕ − d e5 , 2i t c

generates three different cohomology classes; hence, for small t 6= 0, we get 2 N 6 (c); R HdR

(1,1)

= HJ t

N 6 (c) R ;


19

this implies that J is C ∞ -full as well as pure. A straightforward computation yields ( α α 1 1¯1 ¯ 4 6 ϕt , ∗g ϕ3t 3 − d e5 + d e125 , HdR N (c); R = spanR ∗g 2i c c β αβ α 12¯1¯3 ¯ ¯¯ 1¯ 2 + d e125 ϕt + ϕ1t 213 + ϕ12 t 4 4 c (2,2)

= HJ t

)

=

N 6 (c) R .

Therefore N 6 (c) is also C ∞ -full at the 4th stage and, consequently, it is full as well as C ∞ -pure. Summarizing, we have proved the following. Theorem 13. There exists a compact manifold N 6 (c) such that: (i) N 6 (c) admits a C ∞ -pure-and-full almost-complex structure J; (ii) each harmonic form of type (2, 0) + (0, 2) gives rise to a curve {Jt }t∈(−ε,ε) of C ∞ -pure-and-full almost-complex structures on N 6 (c); (iii) furthermore, the map (2,0),(0,2) N 6 (c) R t 7→ dimR HJt is an upper-semicontinuous function at t = 0.

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