A six-dimensional compact symplectic solvmanifold without Kähler structures
A six-dimensional compact symplectic solvmanifold without Kähler structures
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无凯勒结构的六维紧辛求解流形
DOI：
10.18910/9398
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发表时间：
1996
影响因子：
0.4
通讯作者：
Martín Saralegui
中科院分区：
文献类型：
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作者：
M. Fernández;M. León;Martín Saralegui
Gordon 和 Benson 已经证明，如果紧致尼勒流形承认卡勒结构，则它是环面 [5] 更准确地说，他们证明了条件 (iv) 对于非托勒尼勒流形 M 上的任何辛结构都失败。这个结果由 Hasekawa [12] 独立证明，表明 (v) 对于 M 失败。对于维度 4 的紧致求解流形 M，已知 M 具有卡勒结构当且仅当它是复数时环面或超椭圆面。事实上，Auslander 和 Szczarba 在[4]中证明，如果 M 的第一个 Betti 数 bι(M) 为 2，则 M 是 T 上具有纤维 T 的纤维束。那么根据 Ue [19]，只有当 M 是超椭圆曲面或紧致尼尔流形的初等 Kodaira 曲面时，M 才具有复杂结构。因此，如果 M 是卡勒流形，则它必定是超椭圆曲面。由于 !<&ι(M)<4，仅当 M 是复环面或超椭圆面时，它才可以是卡勒流形。超椭圆曲面是求解流形这一事实源自 Auslander [3]。上述结果可以概括为以下猜想：紧致求解流形具有卡勒结构当且仅当它是复环面的有限商。与紧尼尔流形的情况相反，有紧辛
Gordon and Benson have proved that if a compact nilmanifold admits a Kahler structure then it is a torus [5] more precisely they proved that the condition (iv) fails for any symplectic structure on a non-toral nilmanifold M. This result was independently proved by Hasegawa [12] by showing that (v) fails for M. For a compact solvmanifold M of dimension 4 it is known that M has a Kahler structure if and only if it is a complex torus or a hyperelliptic surface. In fact, Auslander and Szczarba in [4] proved that if the first Betti number bι(M) of M is 2, M is a fiber bundle over T with fiber T. Then by Ue [19] M has a complex structure only if it is a hyperelliptic surface or a primary Kodaira surface which is a compact nilmanifold. Thus, if M is a Kahler manifold, it must be a hyperelliptic surface. Since !<&ι(M)<4, M can be a Kahler manifold only if it is a complex torus or a hyperelliptic surface. The fact that a hyperelliptic surface is a solvmanifold follows from Auslander [3]. The above result may be generalized as the following conjecture : A compact solvmanifold has a Kahler structure if and only if it is a finite quotient of a complex torus. In contrast to the case of compact nilmanifolds there are compact symplectic
Masaki Kami：“瑟斯顿和塞弗特 4 流形 I 意义上的几何 4 流形”
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