超复诣零流形的代数维数与复子簇
Algebraic dimension and complex subvarieties of hypercomplex nilmanifolds
摘要：
幂零流形是幂零李群与余紧格的（左）商。流形上的超复结构是满足四元数关系的复结构算子的三元组。超复零流形是幂零李群的紧商，它具有左不变的超复结构。这样的流形允许由四元数诱导的复结构的整个2维球$S^2$。证明了对任意超复零流形$M$和一般复结构$L\in S^2$，复流形$(M,L)$的代数维数为0.当超复零流形是阿贝尔流形时，证明了一个更强的结果。考虑关于复结构$I\in S^2$的幂零李群上的Hodge型（1，0）左不变向量场的李代数.当一个超复零流形是交换的时，这个李代数称为交换的。我们证明了超复阿贝尔流形上的广义$L\in S^2$的$(M,L)$的所有复子簇也是超复流形。
英文摘要：
A nilmanifold is a (left) quotient of a nilpotent Lie group by a cocompact lattice. A hypercomplex structure on a manifold is a triple of complex structure operators satisfying the quaternionic relations. A hypercomplex nilmanifold is a compact quotient of a nilpotent Lie group equipped with a left-invariant hypercomplex structure. Such a manifold admits a whole 2-dimensional sphere $S^2$ of complex structures induced by quaternions. We prove that for any hypercomplex nilmanifold $M$ and a generic complex structure $L\in S^2$, the complex manifold $(M,L)$ has algebraic dimension 0. A stronger result is proven when the hypercomplex nilmanifold is abelian. Consider the Lie algebra of left-invariant vector fields of Hodge type (1,0) on the corresponding nilpotent Lie group with respect to some complex structure $I\in S^2$. A hypercomplex nilmanifold is called abelian when this Lie algebra is abelian. We prove that all complex subvarieties of $(M,L)$ for generic $L\in S^2$ on a hypercomplex abelian nilmanifold are also hypercomplex nilmanifolds.
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