Calabi-Eckmann流形上度量的一些性质  

Some properties of the metric on Calabi-Eckmann manifolds

作　　者：甘宁[1] GAN Ning(School of Sciences,Jimei University,Xiamen 361021,China)

机构地区：[1]集美大学理学院,福建厦门361021

出　　处：《厦门大学学报（自然科学版）》2024年第6期1095-1098,共4页Journal of Xiamen University：Natural Science

基　　金：国家自然科学基金(12071386);福建省自然科学基金(2022J01341)。

摘　　要：[目的]Kaehler流形已经被广泛研究,但是非Kaehler流形还没有得到很大程度的研究.Calabi-Eckmann流形由Calabi和Eckmann引入,并首先研究其上的复结构及相关的性质.近年来有不少关于Calabi-Eckmann流形上的复子流形,上同调以及形变的研究.本文研究Calabi-Eckmann流形上度量的一些性质.将Hopf流形的相应结果推广到了Calabi-Eckmann流形上,在Hopf流形的研究中都利用了它是其万有覆盖空间C^(n){0}在其基本群作用下的商空间这个事实,但这个方法不能推广到Calabi-Eckmann流形,因为它是一个单连通的非Kaehler流形.[方法]利用Calabi-Eckmann流形具有S^(2m+1)×S^(2n+1)的形式,它可作为CP^(m)×CP^(n)上以椭圆曲线S^(1)×S^(1)为纤维的复解析纤维丛,构造了底空间CP^(m)×CP^(n)流形上整体定义的(1,1)Kaehler形式,由此得到整体定义的体积形式,并由CP^(m)×CP^(n)流形上Kaehler形式构造了Calabi-Eckmann流形上的Kaehler形式ω.[结果]证明了Calabi-Eckmann流形其底空间流形上的全纯淹没的拉回不是dd^(c)正合的,由此得到Calabi-Eckmann流形不是多重闭的;并证明了对于Calabi-Eckmann流形上的Kaehler形式ω成立dd^(c)ω≤0,从而得到Calabi-Eckmann流形是多重负定的.[结论]Calabi-Eckmann流形的度量还有一些值得进一步研究的性质,可以利用本文构造Calabi-Eckmann流形上整体的Kaehler形式ω研究由它诱导的度量是否是平衡和1-对称的.[Objective] Although Kaehler manifolds have been extensively studied,non-Kaehler manifolds have not been.Calabi-Eckmann manifolds were introduced by Calabi and Eckmann,and their complex structures and related properties were first studied.In recent years,numerous studies on complex submanifolds,cohomology,and deformation on Calabi-Eckmann manifolds have emerged.In this article,we study some properties of metrics on Calabi-Eckmann manifolds.Corresponding results of Hopf manifolds are extended to Calabi-Eckmann manifolds,and the fact that it is a quotient space of its universal covering space C^(n){0} under the action of its fundamental group is used in the study of Hopf manifolds.Unfortunately,this method cannot be extended to Calabi-Eckmann manifolds because it is a simply connected non-Kaehler manifold.[Methods] Herein,we use the Calabi-Eckmann manifold with the form of S^(2m+1)×S^(2n+1),which can be considered as a complex analytic fiber bundle with elliptic curves S^(1)×S^(1) as fibers over CP^(m)×CP^(n).Then,we construct a globally defined(1,1) Kaehler form on the base space manifold CP^(m)×CP^(n),obtain a globally defined volume form,and finally construct a Kaehler form on the Calabi-Eckmann manifold pullded back from a Kaehler form on CP^(m)×CP^(n).[Results] We prove that the holomorphic immersions of the Calabi-Eckmann manifold on its base space manifold are not dd^(c)-exact,and thus the Calabi-Eckmann manifold is non-pluriclosed.Additionally,we prove that for a Kaehler form ω on the Calabi-Eckmann manifold,dd^(c)ω≤0,and thus establish that the Calabi-Eckmann manifold is plurinegative.[Conclusion] Some properties of the metric of Calabi-Eckmann manifolds deserve further investigations.Hopefully,the proposed global Kaehler form can be used to study whether the induced metric is balanced and 1-symmetric.

关 键 词：Calabi-Eckmann流形 Kahler形式 多重闭流 多重负定流 

分 类 号：O174.56[理学—数学]