Compact locally conformal Kähler nilmanifolds
Compact locally conformal Kähler nilmanifolds
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紧凑的局部共形 Kähler nilmanifolds
DOI：
10.1007/bf00182906
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发表时间：
1986
影响因子：
0.5
通讯作者：
Manuel Deleón
中科院分区：
文献类型：
--
作者：
L. A. Cordero;M. Fernández;Manuel Deleón
广义Hopf流形[16]，[17]是局部共形的K/ihler流形，其Lee形式是平行的，但不是精确的（参见[8]，[15]）。这类流形的主要非K/ihler例子是S1 × S2 k + i（k>/1）。本文给出了一类2 r + 2维的紧致广义Hopf流形N（r，1）× S1.这里N（r，1）是幂零李群的紧商，即[5]，[10]中考虑的广义海森堡群H（r，1）。H（r，1）是[11]中考虑的H型幂零李群的一个例子。Thurston紧辛流形[14]是N（1，1）× S1.此外，考虑N（r，1）的复数模拟Nc（r，1）。则Nc（1，1）是岩泽流形[2，p.4]，实际上Hc（1，1）是H型6维李群。证明了：（1）当r>/2时，N（r，1）× S1不可能有辛结构;（2）Nc（r，1）不可能有广义Hopf流形的结构;（3）当r>/2时，Nc（r，1）不可能有辛结构;（4）N（r，1）× S1的de Rham上同调环的极小模型不是形式的.这后一事实扩大了紧广义Hopf流形和紧K/ihler流形之间的拓扑差异（比较[6]，[15]）。
A generalized Hopf manifold [16],[17] is a locally conformal K/ihler manifold whose Lee form is parallel but not exact (see also [8],[15]). The main non-K/ihler example of such a manifold is S 1× S 2k+ i (k>/1). In the present paper we exhibit a large family of compact generalized Hopf manifolds N (r, 1)× S 1 of dimension 2r+ 2. Here N (r, 1) is a compact quotient of a nilpotent Lie group, namely the generalized Heisenberg group H (r, 1) considered in [5],[10]. H (r, 1) is an example of a nilpotent Lie group of type H considered in [11]. Furthermore, Thurston's compact symplectic manifold [14] is N (1, 1)× S 1. Also, the complex analog Nc (r, 1) of N (r, 1) is considered. Then Nc (1, 1) is the Iwasawa manifold [2, p. 4], and in fact Hc (1, 1) is the 6-dimensional Lie group of type H. Moreover, we prove:(1) N (r, 1)× S 1 can have no symplectic structure for r>/2;(2) Nc (r, 1) can have no structure of generalized Hopf manifold;(3) Nc (r, 1) can have no sympletic structure for r>/2;(4) the minimal model of the de Rham cohomology ring of N (r, 1)× S~ is not formal. This latter fact enlarges the topological differences between compact generalized Hopf manifolds and compact K/ihler manifolds (compare [6],[15]).