We present smooth simply connected closed 4 k -dimensional manifolds N := N_k , for each k ∈ {2, 3, ⋯}, with distinct symplectic deformation equivalence classes [[ ω_i ]], i = 1, 2. To distinguish [[ ω_i ]]’s, we used the symplectic Z invariant in [4] which depends only on the symplectic deformation equivalence class. We have computed that Z ( N , [[ ω ₁ ]]) = ∞ and Z ( N , [[ ω ₂ ]]) < 0. An almost-Kӓhler metric on a smooth manifold M ²ⁿ of real dimension 2 n is a Riemannian metric g compatible with a symplectic structure ω , i.e. ω ( X , Y ) = g ( X , JY ) for an almost complex structure J , where X , Y are tangent vectors at a point of the manifold. Two symplectic forms ω ₀ and ω ₁ on M are called deformation equivalent , if there exists a diffeomorphism ψ of M such that ψ ^∗ ω ₁ and ω ₀ can be joined by a smooth homotopy of sympelctic forms, [5] . For a symplectic form ω , its deformation equivalence class shall be denoted by [[ ω ]]. We denote by Ω _[[ω]] the set of all almost Kӓhler metrics compatible with a symplectic form in [[ ω ]]. Examples of smooth manifolds with more than one symplectic deformation class have been an interesting subject to study; refer to [6] , [7] or [8] . For a smooth closed manifold M of dimension 2 n ≥ 4 which admits a symplectic structure ω , we have defined a symplectic invariant Z in [4] ; where dvol _g , s_g , Vol _g are the volume form, the scalar curvature and the volume of g respectively. In [4] , we presented a six dimensional non-simply connected closed manifold which admits two symplectic deformation classes [[ ω_i ]], i = 1, 2, such that their Z values have distinct signs. Then in [3] , we showed an eight dimensional simply connected closed manifold with the same property. The main result in this article is to present a simply connected manifold of dimension 4 k , for each k ∈ {2, 3, ⋯}, with the above property. Here we shall prove the following; Theorem 2.1. For each integer k ≥ 2, there exists a smooth closed simply connected 4 k - dimensional manifold N with symplectic deformation equivalence classes [[ ω_i ]], i = 1, 2 such that Z ( N , [[ ω ₁ ]]) = ∞ and Z ( N , [[ ω ₂ ]]) < 0. The manifold N is (diffeomorphic to) the product of k copies of a complex surface of general type with ample canonical line bundle which is homeomorphic to R ₈ , the blow up of the complex projective plane ℂℙ ₂ at 8 points in general position. This general type complex surface may be obtained as a small deformation of Barlow’s explicit complex surfaces [1] . When k = 2, the manifold N in the theorem can be the one studied by Catanese and LeBrun [2] . To prove this theorem, we need the following; Proposition 1. Let W be a complex surface of general type with ample canonical line bundle , homeomorphic to R ₈ . Consider a Kӓhler Einstein metric of negative scalar curvature on W with Kӓhler form ω_W on W . Set N := W × ⋯ × W , the k-fold product of W . Then , and it is attained by a Kӓhler Einstein metric . Proof . The argument here follows the scheme in [ 4 , Section 3] and is similar to that in [3] . We recall one known fact about W from [ 7 , Section 4]; there is a homeomorphism of W onto R ₈ which preserves the Chern class c ₁ . And there is a diffeomorphism of N onto , the k -fold product of R ₈ [ 2 , Section 4]. Note that R ₈ is well known to admit a Kӓhler Einstein metric of positive scalar curvature obtained by Calabi-Yau solution. Then, the first Chern class of W can be written as , where E_i , i = 0, · · · 8, is the Poincare dual of a homology class , i = 0, ⋯ 8 so that , i = 0, ⋯ 8, form a basis of H ₂ ( W ,ℤ) ≅ ℤ ⁹ and their intersections satisfy , where ε ₀ = 1 and ε_i = −1 for i ≥ 1. So, in this basis the intersection form becomes We have the orientation of W induced by the complex structure and the fundamental class [ W ] ∈ H ₄ ( W , ℤ ) ≅ ℤ . As ω_W is the Kӓhler form of a Kӓhler Einstein metric g_W of negative scalar curvature, we may get by scaling if necessary. By Künneth theorem , where π_j is the projection of N onto the j-th factor. Then, Consider any smooth path of symplectic forms ω_t , 0 ≤ t ≤ δ , on N such that ω ₀ = ω_W + ⋯ + ω_W . We may write for some continuous functions in t , i = 0, ⋯, 8. As { ω_t } is connected, their first Chern class c ₁ ( ω_t ) = c ₁ ( N ) does not depend on t . Using the intersection form we do a combinatorial computation; where . Set , so that . We put . As A_j (0) = [ ω_W ] ² [ W ] > 0 and from (2.1), we have A_j ( t ) > 0. Then and as , so . We also put . Since and , we get As , by combinatorial computation we obtain; Putting A = A ₁ ⋯ A_k and , from (2.1) and (2.3) we have; From the AM-GM (Arithmetic Mean - Geometric Mean) inequality; , setting , we get So, From (2.2), where . By calculus, for y ∈ [0, 1) with equality at . So, we get and . From this we have There is a basic inequality for any symplectic structure ω on a closed manifold M of dimension 2 n [4] ; As the expression is invariant under a change ω ↦ ϕ * ( ω ) by any diffeomorphism ϕ , so from (2.6) and the definition of Z , we get We consider the Kӓhler form ω_W +⋯+ ω_W of the product Kӓahler Einstein metric g_W + ⋯ + g_W of negative scalar curvature on N = W × ⋯ × W . One can readily check that this symplectic form satisfies the equality of both (2.6) and (2.7). So, we conclude .   ☐ Proof of Theorem 2.1. Consider the positive Kӓhler Einstein metric on R ₈ and let ω ₁ be the Kӓhler form of the product positive Kӓhler Einstein metric on R ₈ × ⋯ × R ₈ , which is diffeomorphic to N . We have Z ( N , [[ ω ₁ ]]) = ∞ (scaling by different constants on each factor gives ∞). And let ω ₂ be ω_W + ⋯ + ω_W . Then Z ( N , [[ ω ₂ ]]) < 0 from Proposition 1. From the fact that these values are different, we conclude that [[ ω ₁ ]] and [[ ω ₂ ]] are distinct symplectic deformation equivalence classes. This proves Theorem 2.1.   ☐ In this article I demonstrated examples in 4 k dimension. But by refining the argument of [4] , one may try to get, for each k ≥ 1, examples of closed symplectic (4 k + 2)-dimensional manifolds admitting two symplectic deformation equivalence classes with distinct signs of Z ( , [[ · ]]) invariants. So far we only used the Catanese-LeBrun manifold as building blocks. But one may use other 4-dimensional closed simply connected symplectic manifolds of smaller Euler characteristic.