Shorter Notes: Some Simple Examples of Symplectic Manifolds
Shorter Notes: Some Simple Examples of Symplectic Manifolds
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简短笔记：辛流形的一些简单例子
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发表时间：
1976
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通讯作者：
W. Thurston
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作者：
W. Thurston
这是一个没有Kaehler结构的闭辛流形的构造。辛流形是具有闭2-形式a的2k维流形，使得Ak是非奇异的。如果M2k是一个闭辛流形，则a的上同调类是非平凡的，并且它通过k的所有幂都是非平凡的。M也有一个几乎复杂的结构，与a，to-to同伦有关。是否每个闭辛流形都有Kaehler结构(反之亦然)。Kaehler流形具有奇数维Betti数为偶数的性质。古根海默在文[1]、[2]中指出，辛流形也有偶奇Betti数。在对文献[1]的回顾[3]中，利伯曼指出，证据是不完整的。通过构造古根海默断言的反例，我们产生了不是Kaehler的辛流形的初等例子。在由(1，0)-P4id定义的T2的微分同胚群中存在Z E Z的表示p，(0，1I0O11l其中[81]表示T2的变换被R2的线性变换所覆盖.该表示确定T上具有光纤T2的丛M4：M4=T2 XZ9Z T2，其中Z E Z通过覆盖变换作用于T2，并通过p作用于T2(M4也可被视为以一组仿射变换为模的R4)。设Q1为T2的标准体积形式。由于p保留21，这在M4上定义了闭合的2-形式i2，其在每根光纤上是非奇异的。设p是到基的投影，则可以证明S21+P*‘1是辛形式。(一般来说，对于任何闭合的Q‘1，它是每根光纤的体积形式，并且K足够大，’j+Kp*21是辛形式。)但H1(M4)=Z@Z@Z，所以M4不是Kaehler流形。可以构建更多的例子。同样地，如果M2K是一个闭辛流形，并且如果M2K上的N2K+2个纤维的基本类在N中不同调为零，那么N也是一个辛流形。例如，如果光纤的欧拉特性不是零，编辑们于1974年7月31日收到了这一消息。AMS(MOS)主题分类(1970)。初级57D15，58H05。C 1976年美国数学学会
This is a construction of closed symplectic manifolds with no Kaehler structure. A symplectic manifold is a manifold of dimension 2k with a closed 2-form a such that ak is nonsingular. If M2k is a closed symplectic manifold, then the cohomology class of a is nontrivial, and all its powers through k are nontrivial. M also has an almost complex structure associated with a, up-to homotopy. It has been asked whether every closed symplectic manifold has also a Kaehler structure (the converse is immediate). A Kaehler manifold has the property that its odd dimensional Betti numbers are even. H. Guggenheimer claimed [1], [2] that a symplectic manifold also has even odd Betti numbers. In the review [3] of [1], Liberman noted that the proof was incomplete. We produce elementary examples of symplectic manifolds which are not Kaehler by constructing counterexamples to Guggenheimer's assertion. There is a representation p of Z E Z in the group of diffeomorphisms of T2 defined by (1, 0) -P4 id, (0,1 I 0o 1l where [81 ]" denotes the transformation of T2 covered by the linear transformation of R2. This representation determines a bundle M4 over T with fiber T2: M4 = T2 XZ9Z T2, where Z E Z acts on T2 by covering transformations, and on T2 by p (M4 can also be seen as R4 modulo a group of affine transformations). Let Q1 be the standard volume form for T2. Since p preserves 21, this defines a closed 2-form i2 on M4 which is nonsingular on each fiber. Let p be projection to the base: then it can be checked that S21 + P*' 1 is a symplectic form. (It is, in general, true that "'j + Kp* 21 is a symplectic form, for any closed Q'1 which is a volume form for each fiber, and K sufficiently large.) But H1 (M4) = Z @ Z @ Z, so M4 is not a Kaehler manifold. Many more examples can be constructed. In the same vein, if M2k is a closed symplectic manifold, and if N2k+2 fibers over M2k with the fundamental class of the fiber not homologous to zero in N, then N is also a symplectic manifold. If, for instance, the Euler characteristic of the fiber is not zero, this Received by the editors July 31, 1974. AMS (MOS) subject classifications (1970). Primary 57D15, 58H05. C American Mathematical Society 1976