Locally conformal Kähler manifolds with potential
Locally conformal Kähler manifolds with potential
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具有潜力的局部共形凯勒流形
DOI：
10.1007/s00208-009-0463-0
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发表时间：
2010
影响因子：
1.4
通讯作者：
M. Verbitsky
中科院分区：
文献类型：
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作者：
L. Ornea;M. Verbitsky
局部共形Kähler（LCK）流形M是被Kähler流形复盖的流形，其甲板变换群共形地作用于M上，如果M上有全纯流，共形地作用于M上，则称之为Vaisman流形。LCK流形和Vaisman流形在小变形下都不是稳定的。我们定义了一类新的LCK-流形，称之为位势LCK流形，它在小变形下是闭的。所有的Vaisman流形都是具有位势的LCK流形。我们证明了一个具有势的LCK流形存在一个覆盖，通过增加一个点，该覆盖可以紧化为Stein簇。这被用来证明任何具有位势dimM≥ 3的LCK流形M都可以嵌入到一个Hopf流形中，从而改进了关于Vaisman流形Ornea和Verbitsky的类似结果（Math Ann 332：121-143，2005）。
A locally conformally Kähler (LCK) manifoldMis one which is covered by a Kähler manifoldwith the deck transformation group acting conformally on. IfMadmits a holomorphic flow, acting onconformally, it is called a Vaisman manifold. Neither the class of LCK manifolds nor that of Vaisman manifolds is stable under small deformations. We define a new class of LCK-manifolds, called LCK manifolds with potential, which is closed under small deformations. All Vaisman manifolds are LCK with potential. We show that an LCK-manifold with potential admits a covering which can be compactified to a Stein variety by adding one point. This is used to show that any LCK manifoldMwith potential, dimM≥ 3, can be embedded into a Hopf manifold, thus improving similar results for Vaisman manifolds Ornea and Verbitsky (Math Ann 332:121–143, 2005).