Construction of singular holomorphic vector fields and foliations in dimension two.

*(English)*Zbl 0625.57012This 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 \(\{f_ 1,...,f_ k\}\).

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 |