A construction of compact pseudo-Kahler solvmanifolds with no Kahler structures
A construction of compact pseudo-Kahler solvmanifolds with no Kahler structures
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无卡勒结构的紧凑伪卡勒求解流形的构造
DOI：
10.21099/tkbjm/1496164895
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发表时间：
2005
影响因子：
0.7
通讯作者：
Takumi Yamada
中科院分区：
文献类型：
--
作者：
Takumi Yamada
In this paper we investigate the Hard Lefschetz property on certain compact symplectic solvmanifolds and construct compact pseudo-Kahler solvmanifolds which do not have the Hard Lefschetz property. We also construct holomorphic symplectic structures, hypercomplex structures and pseudo占yperkahlerstructures on certain compact solvmanifolds. Introduction Let (MJ勺ρ)be a compact symplectic manifold. We say that (M勺ο) has the Hard Lefschetz property, if the Lefschetz mapping Lk : HsRk(M)→ HZ7/(M) de註nedby Lk([α])口 [ α 八ωk]is an isomorphism for any k壬m.It is well known that the Hard Lefschetz property is a necessary condition for the existence of a Kahler structure. Benson and Gordon [2] proved that non-toral compact nilmanifolds do not have the Hard Lefschetz property. They also conjecture the following: BENSON-GORDON CONJECTURE [3]. Let G be a si, 加'mpl) かyルμ幽幽-叩 S却01んυαωblたeLie group and r a latice of G. Then Gjr has a Ka・hlerstructureザand only if it is a torus Moreover, since a hyperelliptic surface has a Kahler structure and a structure of solvmanifold (not completely solvable solvmanifold), there exists the following generalized co吋ecture(see [6] or [12]): A compact solvmanifold admits a Kahler structure if and only if it is a finite quotient of a complex torus, which has also a structure of complex torus bundle over a complex torus. A solvable Lie algebra 9 2000 Mathematics Subject Classification. 53D05 (53C55).