Deformations of transversely holomorphic flows on spheres and deformations of hopf manifolds
Deformations of transversely holomorphic flows on spheres and deformations of hopf manifolds
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作者：
A. Haefliger
在这篇注记中，我们考虑s2n-L上的一维横切全纯叶F，它是通过与En上以零为压缩不动点的全纯流的轨道相交而得到的。证明了F(横切全纯叶面类)的任何形变仍然是由S~(2n-1)与全纯流的形变轨道相交得到的。我们使用类似于Kodaira-Spencer关于横切全纯叶化存在顶点变形的定理(见[6]或[7])和Arnold书[1]中所解释的全纯压缩向量场芽的分类(Poincaré-Dulac定理)。这本书是本文的主要灵感来源。在一个可以独立阅读的附录中，我们证明了并行考虑导致了Hopf流形的完整分类。这就是C.Borcea[2]的结果。1.主要定理1.1的陈述。±-共振矢量场。设03BB=(03BB1，…，03BBn)是具有严格负实部的复数序列。在Arnold([1]，第178页)之后，Cn中的加法*)03BB-共振单项向量场是形式为Azm~/~zs的向量场，其中m=(m1，…，Mn)是非负整数mi的多指标，使得这里(m，03BB)=03A3mi03BBi，zm=zm1…Zm和03B1~C。这个条件意味着m不全为零。定义：G03BB表示±-共振向量场的向量空间，即±-共振单项向量场之和的向量场。它是一个有限维向量空间。也可以刻画为C“上与Diago交换的全纯向量场的子空间。在附录中，我们将定义乘法p-共振向量场。
In this note we consider a transversely holomorphic foliation F of dimension one on s2n-l obtained by intersecting the orbits of a holomorphic flow on en having zero as a contracting fixed point. It is shown that any deformation of F (in the class of transversely holomorphic foliations) is still obtained by intersecting S2n-1 with the orbits of a deformation of the holomorphic flow. We use an analogue of the theorem of Kodaira-Spencer on the existence of a versal deformation for transversely holomorphic foliation (see [6] or [7]) and the classification of germs of holomorphic contracting vector fields (Poincaré-Dulac theorem) as explained in the book of Arnold [1]. This book was the main inspiration for this paper. In an appendix which can be read independently, we show that parallel considerations leads to a complete classification of Hopf manifolds. This completes results of C. Borcea [2]. 1. Statement of the main theorem 1.1. À-RESONANT VECTOR FIELDS. Let 03BB = (03BB1,...,03BBn) be a sequence of complex numbers with strictly negative real part. Following Arnold ([1], p. 178), an additive *) 03BB-resonant monomial vector field in C n is a vector field of the form azm~/~zs, where m = (m1,...,mn) is a multiindex of non negative integers mi such that Here (m, 03BB) = 03A3mi03BBi, z m = zml ... z m and 03B1~C. This condition implies that the m are not all zero. DEFINITION: g03BB denotes the vector space of À-resonant vector fields, i.e. vector fields which are sum of À-resonant monomial vector fields. It is a finite dimensional vector space. It can also be characterized as the subspace of holomorphic vector fields on C " commuting with the diago* In the appendix, we shall define multiplicative p-resonant vector fields.