×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI