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SIMPLY CONNECTED MANIFOLDS OF DIMENSION 4k WITH TWO SYMPLECTIC DEFORMATION EQUIVALENCE CLASSES
SIMPLY CONNECTED MANIFOLDS OF DIMENSION 4k WITH TWO SYMPLECTIC DEFORMATION EQUIVALENCE CLASSES
Journal of the Korean Society of Mathematical Education Series B The Pure and Applied Mathematics. 2015. Nov, 22(4): 359-364
Copyright © 2015, Korean Society of Mathematical Education
  • Received : September 06, 2015
  • Accepted : November 17, 2015
  • Published : November 30, 2015
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JONGSU KIM

Abstract
We present smooth simply connected closed 4 k -dimensional manifolds N := Nk , for each k ∈ {2, 3, ⋯}, with distinct symplectic deformation equivalence classes [[ ωi ]], i = 1, 2. To distinguish [[ ωi ]]’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.
Keywords
1. INTRODUCTION
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] ;
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where dvol g , sg , 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 [[ ωi ]], 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.
2. EXAMPLES IN DIMENSION 4k
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 [[ ωi ]], 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 ωW on W . Set N := W × ⋯ × W , the k-fold product of W .
Then
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, 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
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, 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
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, where Ei , i = 0, · · · 8, is the Poincare dual of a homology class
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, i = 0, ⋯ 8 so that
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, i = 0, ⋯ 8, form a basis of H 2 ( W ,ℤ) ≅ ℤ 9 and their intersections satisfy
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, where ε 0 = 1 and εi = −1 for i ≥ 1. So, in this basis the intersection form becomes
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We have the orientation of W induced by the complex structure and the fundamental class [ W ] ∈ H 4 ( W , ℤ ) ≅ ℤ . As ωW is the Kӓhler form of a Kӓhler Einstein metric gW of negative scalar curvature, we may get
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by scaling if necessary.
By Künneth theorem
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, where πj is the projection of N onto the j-th factor. Then,
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Consider any smooth path of symplectic forms ωt , 0 ≤ t δ , on N such that ω 0 = ωW + ⋯ + ωW . We may write
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for some continuous functions
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in t , i = 0, ⋯, 8. As { ωt } is connected, their first Chern class c 1 ( ωt ) = c 1 ( N ) does not depend on t . Using the intersection form we do a combinatorial computation;
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where
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.
Set
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, so that
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. We put
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. As Aj (0) = [ ωW ] 2 [ W ] > 0 and
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from (2.1), we have Aj ( t ) > 0. Then
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and as
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, so
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.
We also put
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. Since
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and
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, we get
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As
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, by combinatorial computation we obtain;
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Putting A = A 1 Ak and
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, from (2.1) and (2.3) we have;
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From the AM-GM (Arithmetic Mean - Geometric Mean) inequality;
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, setting
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, we get
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So,
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From (2.2),
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where
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. By calculus,
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for y ∈ [0, 1) with equality at
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. So, we get
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and
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.
From this we have
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There is a basic inequality for any symplectic structure ω on a closed manifold M of dimension 2 n [4] ;
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As the expression
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is invariant under a change ω ϕ * ( ω ) by any diffeomorphism ϕ , so from (2.6) and the definition of Z , we get
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We consider the Kӓhler form ωW +⋯+ ωW of the product Kӓahler Einstein metric gW + ⋯ + gW 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
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.   ☐
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 ωW + ⋯ + ωW . 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.
Acknowledgements
This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MEST) (No.NRF-2010-0011704).
References
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