SIMPLY CONNECTED MANIFOLDS OF DIMENSION 4k WITH TWO SYMPLECTIC DEFORMATION EQUIVALENCE CLASSES

Abstract

We present smooth simply connected closed 4k-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, [[ω₁]]) = ∞ and Z(N, [[ω₂]]) < 0.

1. INTRODUCTION

An almost-Kӓhler metric on a smooth manifold M2n of real dimension 2n 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 ω0 and ω1 on M are called deformation equivalent, if there exists a diffeomorphism ψ of M such that ψ∗ω1 and ω0 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 2n ≥ 4 which admits a symplectic structure ω, we have defined a symplectic invariant Z in [4];

where dvolg, sg, Volg 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 4k, 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 4k-dimensional manifold N with symplectic deformation equivalence classes [[ωi]], i = 1, 2 such that Z(N, [[ω1]]) = ∞ and Z(N, [[ω2]]) < 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 R8, the blow up of the complex projective plane ℂℙ2 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 R8. 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 , 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 R8 which preserves the Chern class c1. And there is a diffeomorphism of N onto , the k-fold product of R8 [2, Section 4].

Note that R8 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 Ei , i = 0, · · · 8, is the Poincare dual of a homology class , i = 0, ⋯ 8 so that , i = 0, ⋯ 8, form a basis of H2(W,ℤ) ≅ ℤ 9 and their intersections satisfy , where ε0 = 1 and εi = −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] ∈ H4(W, ℤ ) ≅ ℤ . As ωW is the Kӓhler form of a Kӓhler Einstein metric gW of negative scalar curvature, we may get by scaling if necessary.

By Künneth theorem , where πj is the projection of N onto the j-th factor. Then,

Consider any smooth path of symplectic forms ωt , 0 ≤ t ≤ δ, on N such that ω0 = ωW + ⋯ + ωW . We may write

for some continuous functions in t, i = 0, ⋯, 8. As {ωt} is connected, their first Chern class c1(ωt) = c1(N) does not depend on t. Using the intersection form we do a combinatorial computation;

where .

Set , so that . We put . As Aj (0) = [ωW ]2[W] > 0 and from (2.1), we have Aj(t) > 0. Then and as , so .

We also put . Since and , we get

As , by combinatorial computation we obtain;

Putting A = A1 ⋯ Ak 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 2n [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 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 .   ☐

Proof of Theorem 2.1. Consider the positive Kӓhler Einstein metric on R8 and let ω1 be the Kӓhler form of the product positive Kӓhler Einstein metric on R8 × ⋯ × R8, which is diffeomorphic to N. We have Z(N, [[ω1]]) = ∞ (scaling by different constants on each factor gives ∞). And let ω2 be ωW + ⋯ + ωW . Then Z(N, [[ω2]]) < 0 from Proposition 1. From the fact that these values are different, we conclude that [[ω1]] and [[ω2]] are distinct symplectic deformation equivalence classes. This proves Theorem 2.1.   ☐

In this article I demonstrated examples in 4k dimension. But by refining the argument of [4], one may try to get, for each k ≥ 1, examples of closed symplectic (4k + 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.