论文标题
向量场和Meromormorthic溶液的集成性
Integrability of vector fields and meromorphic solutions
论文作者
论文摘要
令$ \ Mathcal {f} $为复杂的射歧管$ m $ dimension $ n $定义的叶面，并在某些非空的Zariski-open集合下接纳了holomorphic vector field $ x $切线。 In this paper we prove that if $X$ has sufficiently many integral curves that are given by meromorphic functions defined on $\mathbb{C}$ then the restriction of $\mathcal{F}$ to any invariant complex $2$-dimensional analytic set admits a first integral of Liouvillean type.特别是，在$ \ mathbb {c}^3 $上，每个有理矢量字段的解决方案是$ \ mathbb {c} $在$ \ mathbb {c}上定义的函数，承认了一个非空的不变分析集$ 2 $，其中矢量场的限制会产生一个可产生的集成元素。
Let $\mathcal{F}$ be a foliation defined on a complex projective manifold $M$ of dimension $n$ and admitting a holomorphic vector field $X$ tangent to it along some non-empty Zariski-open set. In this paper we prove that if $X$ has sufficiently many integral curves that are given by meromorphic functions defined on $\mathbb{C}$ then the restriction of $\mathcal{F}$ to any invariant complex $2$-dimensional analytic set admits a first integral of Liouvillean type. In particular, on $\mathbb{C}^3$, every rational vector fields whose solutions are meromorphic functions defined on $\mathbb{C}$ admits a non-empty invariant analytic set of dimension $2$ where the restriction of the vector field yields a Liouvillean integrable foliation.
