ON TAMING AND COMPATIBLE SYMPLECTIC FORMS

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1 ON TAMING AND COMPATIBLE SYMPLECTIC FORMS RICHARD HIND COSTANTINO MEDORI ADRIANO TOMASSINI Abstract. Let (X J be an almost complex manifold. The almost complex structure J acts on the space of 2-forms on X as an involution. A 2-form α is J-anti-invariant if Jα = α. We investigate the anti-invariant forms and their relation to taming and compatible symplectic forms. For every closed almost complex manifold in contrast to invariant forms e sho that the space of closed anti-invariant forms has finite dimension. If X is a closed almost-complex manifold ith a taming symplectic form then e sho that there are no non trivial exact anti-invariant forms. On the other hand e construct many examples of almost-complex manifolds ith exact anti-invariant forms hich are therefore not tamed by any symplectic form. In particular e use our analysis to give an explicit example of an almost-complex structure hich is locally almost-kähler but not globally tamed. The non-existence of exact anti-invariant forms hoever does not in itself imply that there exists a taming symplectic form. We sho ho to construct examples in all dimensions. Introduction Almost-complex structures on a manifold X can be categoried according to hether or not there exist taming or compatible symplectic forms. We recall that a symplectic form ω tames an almost-complex structure J if ω(v Jv > 0 for all nonero tangent vectors v and ω is compatible ith J if the formula g(v = ω(v J defines a Riemannian metric g on X. If ω is compatible ith J then the triple (X J ω is sometimes called an almost-kähler manifold. A Kähler manifold is an almost-kähler manifold ith J integrable. Let J = J (X be the set of almost-complex structures on X then e can define subsets J tame = J tame (X and J comp = J comp (X of J to be the almost-complex structures for hich there exists a taming or compatible symplectic form respectively. We also define a subset J loc.tame of J tame hich consists of locally tame almost-complex structures that is J J loc.tame if there exists an open cover {U i } of X such that J Ui J tame (U i for all i. Similarly e can define locally compatible almost-complex structures J loc.comp J comp. It is immediate that J loc.tame = J and that the set of (integrable complex structures I J loc.comp. In summary e have the folloing diagram of inclusions. Date: April Mathematics Subject Classification. 32Q60 53C5 58A2. Key ords and phrases. pure and full almost complex structure; J-invariant form; J-anti-invariant form. Partially supported by Fondaione Bruno Kessler-CIRM (Trento and by GNSAGA of INdAM.

2 2 ( I J loc.comp k j J loc.tame = J k 2 J comp i J tame When our manifold X has dimension 4 the map j is actually a surjection in other ords J = J loc.comp. For a complete proof of this see Lejmi [3 Theorem ]. It is a question of Donaldson [4 question 2] as to hether the map i is also a surjection. In this paper a key observation is the folloing. Proposition 0.. If X is closed (compact ithout boundary and J J tame then there are no non-ero exact J-anti-invariant 2 forms. Recall that a 2-form α is anti-invariant if Jα = α here Jα(v = α(jv J. In dimension 4 there are no non-ero exact anti-invariant forms ith respect to any J see Corollary.2 but in higher dimensions the existence of an exact anti-invariant form is an obstruction to the existence of a taming symplectic form. It is in fact quite easy to find examples of almost-complex structures admitting exact anti-invariant forms. The folloing is a consequence of Theorem.4. Theorem 0.2. Suppose that W 4n is a 4n dimensional manifold ith trivial tangent bundle. Then X = W S S has an almost-complex structure J for hich there exist non-ero exact anti-invariant 2-forms. The methods used to establish Theorem 0.2 are very topological they rely on Gromov s h-principle. Therefore e have little control on the almost-complex structure in particular it is difficult in this ay to find examples hich lie in J loc.comp. This issue is addressed in section 3. For example in section 3.3 e explicitly construct a nonintegrable almostcomplex structure on a 6-dimensional manifold hich is locally compatible yet admits a non-ero exact anti-invariant form and so lies in J loc.comp \ (J tame I. We remark hoever that the non-existence of exact anti-invariant forms is not a sufficient condition for an almost-complex structure to be tamed by a symplectic form. In dimension 4 since e never have any exact anti-invariant forms examples are given by any almostcomplex manifolds hich are not symplectic. In higher dimensions e can use a theorem of Peternell [7 Theorem.4] to imply the folloing. Theorem 0.3. A non-kähler Moišeon manifold has no non-ero exact anti-invariant forms but no taming symplectic form. In dimension 6 a simpler concrete example is the folloing. Theorem 0.4. The product of the Hopf surface and CP does not have non-ero exact anti-invariant forms or any symplectic forms at all. Cohomology properties can also be used to categorie almost-complex structures. Folloing [5] e can define subspaces H + J (X H J (X H2 (X R the second de Rham cohomology of X as follos. A class a H + J (X if there exists a 2 form α ith [α] = a and Jα = α. Similarly a class a H J (X if it has a representative α hich is antiinvariant ith respect to J. The almost-complex manifold (X J is called C -pure if

3 H + J (X H J (X = {0} and C -full if H + J (X+H J (X = H2 (X R. In [5 Theorem 2.3] Drahjici Li and Zhang sho that an almost complex structure on a compact 4-dimensional manifold is C -pure-and-full. Furthermore in [4 Theorem.3] Li and Zhang proved that if J is C -full and if the compatible cone K c J = { [ω] H 2 (X; R ω is compatible ith J } is non-empty then K t J = K c J + H J (X. Here e focus on H J (X and study Z J (X the real vector space of closed anti-invariant 2-forms. We have already seen that if X is closed and J J tame then there are no nonero exact anti-invariant forms and so the map Z J (X H J (X H2 (X R is an injection. This can be contrasted ith the case of invariant forms Z + J (X. At least if J is integrable then Z + J (X is alays infinite dimensional. In the case hen J J comp e can be more precise. Let g be the Riemannian metric associated to a compatible symplectic form. Then e sho in Proposition 2.2 that Z J (X H g(x the set of harmonic 2-forms ith respect to g. In other ords e have the folloing. Proposition 0.5. J-anti-invariant forms are harmonic ith respect to any Riemannian metric associated to a compatible symplectic form. It turns out that even if J / J comp the closed anti-invariant forms Z J lie in the kernel of a second order elliptic operator see Proposition 2.4. Hence e have an alternative proof of a theorem from [0] saying that anti-invariant forms satisfy a unique continuation principle see Proposition 2.6. The paper is organied as follos. After fixing some notation e establish some basic facts about anti-invariant forms in section and prove Proposition 0. and Theorem 0.2. To complement these results e also derive the examples of Theorems 0.3 and 0.4. In section 2 e discuss the relation beteen anti-invariant forms and harmonic forms and in particular prove the unique continuation theorem for anti-invariant forms. Finally in section 3 e construct our explicit examples. Acknoledgements. We ould like to thank Fondaione Bruno Kessler-CIRM (Trento for their support and very pleasant orking environment.. Anti-invariant forms on almost-complex manifolds. 3 We start by fixing some notation. Let (X J be a 2n-dimensional almost complex manifold and denote by Λ 2 (X the space of 2-forms on X. The almost complex structure acts on Λ 2 (X as an involution by setting Jα(u v = α(ju Jv. Folloing [5] α Λ 2 (X is said to be J-invariant or invariant if Jα = α and J-anti-invariant or anti-invariant if Jα = α. We denote by Λ + J (X and by Λ J (X the space of J-invariant J-antiinvariant forms respectively. Let Z(X be the space of closed 2-forms on X. We set Z ± J (X = Λ± J (X Z(X and H ± J (X = {a H2 (X; R a = [α] α Z ± J (X}.

4 According to [5] an almost complex structure J is said to be C -pure if H + J (X H J (X = {0} C -full if H 2 (X; R = H + J (X + H J (X. We begin our study of anti-invariant forms in dimension 4. Lemma.. Let (X J g be a 4-dimensional almost Hermitian manifold. Let α Λ 2 (X. Then α 2 = f Vol g here f : X R is a smooth non-negative function on X and Vol g denotes the Riemannian volume form. Proof. Let p X and let {v Jv v 2 Jv 2 } be a g-orthonormal positive basis of T p X. Then if {v Jv v 2 Jv 2 } denotes the dual basis of {v Jv v 2 Jv 2 } any J-anti-invariant 2-form at p can be ritten as Hence α(p = λ(v v 2 Jv Jv 2 + µ(v Jv 2 + Jv v 2. α 2 (p = 2(λ 2 + µ 2 (v Jv v 2 Jv 2. Corollary.2. Let (X J g be a compact 4-dimensional almost Hermitian manifold. Then there are no non-trivial exact anti-invariant forms on X. Proof. Let α Z J (X. By assumption α 0; assume by contradiction that α = dβ. Then 0 = d(β dβ = α 2 = f Vol g > 0 and this is absurd. X X We ill see in Theorem.4 and in section 3 that non-ero exact anti-invariant forms can exist on higher dimensional almost-complex manifolds but the folloing proposition rules this out if the almost-complex structure J J tame. Proposition.3. Let (X J be a compact 2n-dimensional almost complex manifold. If ω is a symplectic form taming J then there are no non-ero exact J-anti-invariant forms. Proof. By contradiction. Let ω be a symplectic form taming J. Let α Z J (X be exact α = dβ. Let 2k be the maximal rank of α(p for p X. The folloing claim from linear algebra is useful. Claim. A ske-symmetric anti-invariant 2-form η on a complex vector space V has rank 2r divisible by 4. Moreover η r generates the complex orientation on V/ ker η (ith its induced complex structure if r/2 is even and the opposite of the complex orientation if r/2 is odd. Proof of claim. First note that as η is anti-invariant ker(η is indeed a complex subspace of V and so the quotient W = V/ ker(η inherits a complex structure. Then η r is a volume form on W hich implies that Jη r = λη r for some λ > 0. As η is anti-invariant this in turn implies that r must be even and e can take λ =. No e choose a basis of W of the form e f e 2 f 2... e r f r such that if i is odd e have η(e i f i = and Je i = e i+ and Jf i = f i+. Then necessarily if i is even e have η(e i f i =. We may also assume that η(e i e j = η(f i f j = 0 for all i j and η(e i f j = 0 for all i j. X 4

5 Given this e compute η r (e e 2 f f 2... f r f r = ( r η(e f η(e 2 f 2... η(e r f r = ( r/2 and the claim follos. Returning to the proof e have that k is even and α k ω n k = ( k/2 fω n here f is a non-negative function on X. This is because ω gives the complex orientation on any complex subspace. The function f is positive exactly hen α has maximum rank. Therefore 0 = ( k/2 d(β (dβ k ω n k = ( k/2 α k ω n k = f ω n > 0 and this is absurd. X To complement the above proposition the folloing theorem shos that almost-complex structures admitting exact anti-invariant forms can be constructed under fairly general hypotheses. Theorem.4. Suppose that an orientable manifold M 4n+ admits a 2 form α of everyhere maximal rank 4n such that the quotient bundle T M/ ker α M has an almostcomplex structure for hich α is anti-invariant. Then there exists an almost-complex structure J on M S hich admits an exact nonero anti-invariant 2 form. We emphasie that the hypotheses of the theorem are purely topological in particular e do not need to assume that α is closed. The proof does not use the hypothesis that M has dimension 4n + only that the dimension is odd. Hoever e have seen above that the rank of an anti-invariant form is necessarily a multiple of 4. Proof. The result is a consequence of a theorem of McDuff see [5] and [7 Thm 0.4.] hich states (in a simple form that a 2-form of maximal rank on an odd dimensional manifold can be deformed through forms of maximal rank to an exact form. Hence e can find maximal rank 2-forms α t on M such that α 0 = α and α is exact. Fixing a Riemannian metric on M the 4n dimensional subbundles (ker α and (ker α are isomorphic as symplectic vector bundles ith forms α and α respectively. Hence (ker α also admits an almost-complex structure J anti-invariant ith respect to α. The corresponding orientation on (ker α together ith one on M determines a trivialiation of ker α. Hence e can extend J to an almost-complex structure on M S such that J maps ker α onto T S. Let us pull back α to a 2-form α on M S using the natural projection. Then α is also nonero and exact. Finally since α vanishes on the complex planes spanned by ker α and T S it is anti-invariant as required. To close this section e discuss our examples of complex manifolds hich have no exact anti-invariant 2-forms but still have no taming symplectic forms. First let X be a Moišeon manifold that is a compact complex manifold hich admits a proper modification from a projective manifold. Then the folloing result holds see Peternell [7 Thm..4] Theorem.5. Let X be a Moišeon manifold. Assume there exists a real ( -form ω and a real 2-form ϕ on X such that i ω is positive definite X X 5

6 ii d(ω ϕ = 0 iii C ϕ = 0 for all curves C X. Then X is projective. This directly implies Theorem 0.3 as follos. Proposition.6. Any non-kähler Moišeon manifold X has no non-trivial d-exact antiinvariant 2-forms and no taming symplectic forms. Proof. First if α is a d-exact anti-invariant 2-form α on X then its pull back to a projective manifold is also exact and anti-invariant. By Proposition.3 this implies that α must be identically ero. The fact that X has no taming symplectic form has already been pointed out by Draghici and Zhang [6] but e give the argument here for completeness. Arguing by contradiction suppose that η is a taming symplectic form. We can rite η = ω ψ ψ 2 here ω is a real ( -form ψ is a real (2 0-form and ψ 2 is a real (0 2-form. Then ψ and ψ 2 vanish on complex lines and so since η is taming the form ω is positive definite. Setting ϕ = ψ + ψ 2 the remaining to conditions of Theorem.5 are clearly satisfied and so X must be projective a contradiction. Finally e give a proof of Theorem 0.4. Let Y be the Hopf surface that is Y = (C 2 \0/ 2 ith its induced complex structure. Proposition.7. The product X = CP Y does not have exact anti-invariant forms or any symplectic forms at all. Proof. The 6-manifold X is diffeomorphic to S 2 S 3 S and so has no cohomology classes a ith a 3 0. Therefore it admits no symplectic forms at all. There are to projections p p 2 : X CP. The first is just projection onto the first factor the second is induced by projection onto Y and then quotienting by C to get Y/C = (C 2 \ 0/C = CP. Therefore e can pull-back the Fubini-Study form using p and p 2 to get invariant 2-forms ω and ω 2 on X. Suppose that there exists a non-ero exact anti-invariant 2-form α on X. As e are orking in dimension 6 e have that α(x has rank 0 or 4 at all points x X. Observe that applying Stokes Theorem as in Proposition.3 gives a contradiction if there exists a closed 2-form Ω on X hich satisfies α 2 Ω 0 and α 2 Ω(x > 0 at least for some x X. When α 0 its kernel is a complex line. Therefore as ω and ω 2 are invariant e have α 2 (ω + ω 2 0 (for the complex orientation on X and hence by the argument above e must have α 2 (ω + ω 2 0. This implies that hen α(x 0 it s kernel is generated by r and ir here r is the radial or S direction in Y (coming from a suitably scaled radial vector in C 2 and ir is parallel to the Hopf fibration. Indeed if the kernel ere transverse to this plane the form ω + ω 2 ould evaluate nontrivially. Hence r and ir lie in ker(α(x for all x X and α is invariant under the vectorfields r and ir. These vectorfields generate a torus action on X hose projection onto the orbit space is just the holomorphic projection (p p 2 : X CP CP. Hence α is a pull-back of a form α on CP CP. As α is a closed anti-invariant form so is α. Furthermore as α is anti-invariant it must vanish hen restricted to both CP factors. Therefore it s cohomology class is trivial and so 6

7 α is exact. But by Corollary.2 the only exact anti-invariant forms on a 4-dimensional manifold are identically 0 and this completes our proof. 2. Hodge star operator for anti-invariant forms Let (X J g be an almost Hermitian manifold of dimension 2n and denote by ω the fundamental form of g. Then e have the folloing Proposition 2.. Let α be J-anti-invariant 2-form on (X J g. Then (2 α = (n 2! α ωn 2. For the sake of completeness e give the proof of (2. Proof. Let α be any J-anti-invariant form on (X J. Then α is a (2n 2-form. The Lefschet decomposition applied to Λ 2n 2 (X yields to Λ 2n 2 (X = i 0 L i (P 2(n i 2 (X 7 here L : Λ k (X Λ k+2 L(γ = γ ω is the Lefschet operator and P k (X is the space of primitive forms hich can be identified ith ker L n k+ Λ k (X (see e.g. [ Prop..2.30]. Therefore α = fω n + L n 2 (γ here f is a smooth function and γ P 2 (X. Then taking in the last formula by [ Prop..2.30] e get α = fω (n 2!Jγ. Since α is J-anti-invariant by the last formula f = 0 and γ = (n 2! α. Then (2 is proved. As a consequence e obtain the folloing Proposition 2.2. Let (X J g ω be a 2n-dimensional almost Kähler manifold. Then Z J (X H2 (X here H 2 (X denotes the space of 2-harmonic forms on X ith respect to the Hermitian metric g. Proof. Let α Z J (X. Then by formula (2 since α and ω are closed e get: d α = d (α = (n 2! d(α ωn 2 = 0 that is α co-closed. Since α is closed by assumption then α is harmonic. We record the folloing corollary hich of course also follos from Proposition.3. Corollary 2.3. If (X J g ω is a compact 2n-dimensional almost Kähler manifold then the natural map Z J (X H2 dr (X; R is an injection. In particular dim R (Z J (X b 2(X. Furthermore the map is an isomorphism if and only if H J (X = H2 dr (X; R.

8 In general on an almost Hermitian manifold (X J g of dimension 2n define a generalied co-differential on the space of 2-forms Γ(Λ 2 (X d : Γ(Λ 2 (X Γ(Λ (X by setting d (α = d (α + (n 2! (α d(ωn 2 here d denotes the usual co-differential on (X g. By formula (2 it follos that d vanishes on Z (X. 2 Let E be the differential operator on Γ(Λ 2 (X defined as E = (α + (n 2! d( (α d(ωn 2 Proposition 2.4. The differential operator E is a second order elliptic operator the principal part is the Hodge-de Rham laplacian and Z J (X ker(e. Proof. By the definition of E for any α Z J (X e have: E(α = (α + (n 2! d( (α d(ωn 2 = dd (α + d d(α + (n 2! d( (α d(ωn 2 = dd (α + (n 2! d( (α d(ωn 2 = dd (α = 0. Corollary 2.5. If (X J is a compact 2n-dimensional almost complex manifold then dim Z J (X < +. In contrast Z + J (X has infinite dimension if J is integrable because for any smooth function f : X R e have dd c f Z + J (X. We can no give another proof of the analytic continuation property for closed antiinvariant 2-forms (see [0 Thm.4.] Proposition 2.6. Let X be a 2n-dimensional connected almost complex manifold. Let α Z J (X be vanishing at infinite order at some point p X. Then α is identically ero. Proof. By Proposition 2.4 α is a solution of an elliptic PDE hose leading term is the Laplacian. Hence by [] (see also [2] the form α has strong unique continuation. In contrast this is false for Z + J (X for the same reason as before. 3. Computations of Z 2 (X In this section e ill do some explicit computations on the space of anti-invariant forms on complex manifolds to contrast ith the indirect Theorem.4. In section 3. e give an example of a complex manifold ith dim R Z J (X > dim R H J (X. By Corollary 2.3 this implies that the manifold is not almost Kähler. Indeed by Proposition.3 there is not even a taming symplectic form. Another such example is given in section 3.2. Finally in section 3.3 e construct an almost-complex manifold for hich e can rite don explicitly a compatible symplectic form on small open sets. Hoever it also admits a non-ero exact anti-invariant form and so by Proposition.3 has no globally defined taming symplectic form. 8

9 9 3.. Iasaa manifold. On C 3 consider the product defined as ( 2 3 ( 2 3 = ( It is immediate to check that ( C 3 is a nilpotent Lie group isomorphic to H(3 = GL (3; C 2 3 C. 0 0 We have that (Z [i] 3 C 3 is a cocompact discrete subgroup of ( C 3. The Iasaa manifold X is defined as the manifold X = (Z [i] 3 ( C 3. It is a compact complex 3-dimensional nilmanifold; by [8] it follos that X is not formal; hence it has no Kähler metrics see [3 Main Theorem]; nevertheless there exists a balanced metric on X. Let ( i i {23} be the standard complex coordinate system on C3 ; the folloing ( 0-forms on C 3 are invariant for the action (on the left of (Z [i] 3 so they give rise to a global coframe for T 0 X: ϕ = d ϕ 2 = d 2 ϕ 3 = d 3 d 2. The structure equations are therefore d ϕ = 0 d ϕ 2 = 0 d ϕ 3 = ϕ ϕ 2. By Hattori-Nomiu theorem e compute the real cohomology group HdR 2 (X; R of X (for simplicity e list the harmonic representative instead of its class and rite ϕ A B for ϕ A ϕ B : { ( HdR 2 (X; R = span R ϕ 3 + ϕ 3 i ϕ 3 ϕ 3 ϕ 23 + ϕ 2 3 ( ( } i ϕ 23 ϕ 2 3 ϕ 2 ϕ 2 i ϕ 2 + ϕ 2 i ϕ i ϕ 2 2 Note that each harmonic representative is of pure degree and hence the complex structure is C -pure and full. The Betti numbers of X are b 0 = b = 4 b 2 = 8 b 3 = 0. Then 2 (ϕ2 ϕ 3 + ϕ 2 ϕ 3 2i (ϕ2 ϕ 3 ϕ 2 ϕ 3 2 (ϕ ϕ 2 + ϕ ϕ 2 2i (ϕ ϕ 2 ϕ ϕ 2 2 (ϕ ϕ 3 + ϕ ϕ 3 2i (ϕ ϕ 3 ϕ ϕ 3 are J-anti-invariant closed 2-forms on X and consequently dim R Z J (X > dim R H J (X.

10 3.2. Nakamura manifold. The Nakamura manifold is the compact quotient X = Γ\G of G by a uniform discrete subgroup Γ. By [2 Corollary 4.2] e have { HdR 2 (X; R = span R [e 4 ] [e 26 e 35 ] [e 23 e 56 ] [cos(2x 4 (e 23 + e 56 sin(2x 4 (e 26 + e 35 ] } [sin(2x 4 (e 23 + e 56 cos(2x 4 (e 26 + e 35 ] i.e. in this case the de Rham cohomology of M is not isomorphic to H (g. The previous representatives are all harmonic forms. The complex structure on the solvmanifold X can be defined in term of ( 0-forms as follos: We have that the real forms ϕ = e + ie 4 ϕ 2 = e 2 + ie 5 ϕ 3 = e 3 + ie 6 2 (ϕ2 ϕ 3 + ϕ 2 ϕ 3 2i (ϕ2 ϕ 3 ϕ 2 ϕ 3 2 (ϕ ϕ 2 + ϕ ϕ 2 2i (ϕ ϕ 2 ϕ ϕ 2 2 (ϕ ϕ 3 + ϕ ϕ 3 2i (ϕ ϕ 3 ϕ ϕ 3 are anti-invariant and closed. Therefore dim Z J (X > b 2(X and by Corollary 2.3 the complex structure J does not admit any compatible Kähler metric. This can be also derived by complex Hodge theory since ϕ 2 is a non closed holomorphic -form. It has also to be remarked that as a consequence of a result by Hasegaa (see [9 main theorem] X does not admit any Kähler structure Locally almost-kähler non globally almost Kähler manifold. In this section e ill provide a family of 6-dimensional almost complex (non complex manifolds (N J hich are locally almost Kähler but not globally. We first recall the construction of N (see [6] and [2]. Let A SL(2 Z ith to distinct real eigenvalues e λ and e λ here λ > 0. Let Q GL(2 R such that QAQ = Λ = ( e λ 0 0 e λ On C 2 ith coordinates ( let be defined by ( ( ( ( ( m + 2πin = + Q m 2 + 2πin 2 here m m 2 n n 2 Z. Then C 2 / is a complex torus T 2 C and [( ] [ ( ] Λ = Λ.

11 ( ( is a ell defined automorphism of T 2 C. Indeed if ( ( Λ = Λ ( = Λ ( so that Λ For example take + ΛQ ( m + 2πin + QA m 2 + 2πn 2 ( Λ. Then λ = log and e can choose (3 P = ( m + 2πin = m 2 + 2πn 2 A = ( 2 ( 5 2 ( = Λ. 5 2 then ( m + 2πin + Q m 2 + 2πn 2 Set λ = log µ =. 2 2 Let x x 3 x 4 x 5 x 6 denote coordinates on R 6 and according to the previous notation set = x 3 + ix 5 = x 4 + ix 6. Consider the folloing transformation of R 5 : T (x x 3 x 4 x 5 x 6 = (x + λ e λ x 3 e λ x 4 e λ x 5 e λ x 6. We set here N = R x 2 2πZ R x R 4 x 3 x 4 x 5 x 6 /Γ T (x Γ = Span Z ( µ 0 0 t ( µ 0 0 t (0 0 2π 2πµ t (0 0 2πµ 2π t and T (x denotes the subgroup of transformations generated by T (x so that T 2 C R 4 x 3 x 4 x 5 x 6 /Γ. Then N is a compact 6-dimensional manifold. The folloing six -forms on R 6 e = dx e 2 = dx 2 e 3 = exp( x dx 3 e 4 = exp(x dx 4 e 5 = exp( x dx 5 e 6 = exp(x dx 6.

12 induce -forms on the manifold N. Therefore e immediately get de = 0 (4 de 2 = 0 de 3 = e e 3 de 4 = e e 4 de 5 = e e 5 de 6 = e e 6. The dual global frame {e... e 6 } on N is given by e = x e 2 = x 2 e 4 = exp( x e 5 = exp(x x 4 x 5 e 3 = exp(x x 3 e 6 = exp( x x 6 Let f = f(x 2 be a never vanishing Z-periodic function; let us define the almost complex structure J on N as Je = e 2 Je 2 = e Je 3 = f(x 2 e 5 Je 4 = e 6 Je 5 = f(x 2 e 3 Je 6 = e 4. Then it can be checked that J is integrable if and only if f is constant. We sho that J is locally almost Kähler. Indeed let ω be the local non degenerate and closed 2-form defined as ω = dx dx 2 + dx 3 dx 5 + dx 4 dx 6 ; then since J x = x 2 J x 4 = x 6 J x 2 = x J x 5 = f(x 2 x 3 J x 3 J x 6 = f(x 2 x 5 = x 4 e immediately get that Jω = ω and ω(j > 0 for any non-ero tangent vector i.e. J is locally almost Kähler. No e prove that J cannot be globally Kähler and more generally that there is no global taming symplectic form. In vie of Proposition.3 it is sufficient to find a nonero J-anti-invariant exact form. To this purpose let α = cos(x 2 e 2 e 4 + sin(x 2 e e 4 sin(x 2 e 2 e 6 + cos(x 2 e e 6 ; then according to (4 and to definition of J e have that α = d(sin(x 2 e 4 + cos(x 2 e 6 and that Jα = α i.e. α is a J-anti-invariant exact 2-form. References [] N. Aronsajn A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order J. Math. Pure Appl. 36 ( [2] P. de Bartolomeis A. Tomassini On solvable generalied Calabi-Yau manifolds Ann. Inst. Fourier 56 ( [3] P. Deligne P. Griffiths J. Morgan D. Sullivan Real homotopy theory of Kähler manifolds Invent. Math. 29 (975 no [4] S. K. Donaldson To-forms on four-manifolds and elliptic equations Inspired by S. S. Chern Nankai Tracts Math. vol. World Sci. Publ. Hackensack NJ 2006 pp

13 3 [5] T. Drǎghici T.-J. Li W. Zhang Symplectic forms and cohomology decomposition of almost complex four-manifolds Int. Math. Res. Not. IMRN 200 (200 no. 7. [6] T. Draghici W. Zhang A note on exact forms on almost complex manifolds Math. Res. Lett. 9 ( [7] Y. Eliashberg N. Mishashev Introduction to the h-principle Graduate Studies in Mathematics 48. American Mathematical Society Providence RI xviii+206 pp.. [8] M. Fernánde A. Gray The Iasaa manifold Differential geometry Peñíscola 985 Lecture Notes in Math. vol. 209 Springer Berlin [9] K. Hasegaa A note on compact solvmanifolds ith Kähler structures Osaka J. Math. 43 ( [0] R. Hind C. Medori A. Tomassini On non pure forms on almost complex manifolds to appear in Proc. Amer. Math. Soc.. [] D. Huybrechts Complex Geometry An Introduction Springer Berlin [2] J.L. Kadan Unique continuation in geometry Comm. Pure Appl. Math. 4 ( [3] M. Lejmi Strictly nearly Kähler 6-manifolds are not compatible ith symplectic forms C. R. Math. Acad. Sci. Paris 343 ( [4] T.-J. Li W. Zhang Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds Comm. Anal. Geom. 7 ( [5] D. McDuff Application of convex integration to symplectic and contact geometry Ann. Inst. Fourier (Grenoble 37 ( [6] I. Nakamura Complex parallelisable manifolds and their small deformations J. Differential Geometry 0 ( [7] T. Peternell Algebraicity Criteria for Compact Complex Manifolds Math. Ann. 275 ( Department of Mathematics University of Notre Dame Notre Dame IN address: hind.@nd.edu Dipartimento di Matematica e Informatica Università di Parma Parco Area delle Sciene 53/A 4324 Parma Italy address: costantino.medori@unipr.it address: adriano.tomassini@unipr.it