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Construction of singular holomorphic vector fields and foliations in dimension two. (English) Zbl 0625.57012
This paper concerns the construction of singular holomorphic vector fields and foliations in $${\mathbb{C}}^ 2$$. There are two main results. The first is as follows: Let $$G=\{g_ 1,...,g_ k\}$$ be a set of germs at $$0\in {\mathbb{C}}$$ of holomorphic diffeomorphisms which leave 0 fixed and such that $$g_ 1,...,g_ k$$, $$g_ 1\circ...\circ g_ k$$ are linearizable. Then there is a germ of a holomorphic vector field X, with singularity at $$0\in {\mathbb{C}}^ 2$$, such that its projective holonomy is conjugate (holomorphically) to the group generated by G.
The second result concerns Riccati equations of the form $dx/dt=p(x),\quad dy/dt=a(x)+b(x)y+c(x)y^ 2,$ where x,y,t$$\in {\mathbb{C}}$$, and a,b,c,p are polynomials. The author shows that if $$f_ 1,...,f_ k$$ are Moebius transformations, then there is a Riccati equation with foliation $${\mathcal F}$$, whose holonomy is conjugate to the group generated by $$\{f_ 1,...,f_ k\}$$.
Reviewer: E.Bedford

##### MSC:
 57R30 Foliations in differential topology; geometric theory 32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects) 34M99 Ordinary differential equations in the complex domain 32M99 Complex spaces with a group of automorphisms
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