On the classification of nilpotent singularities
This paper deals with the analytic classification of germs of singular holomorphic foliations in a neighborhood of the origin of \(\mathbb{C}^2\) having nonzero linear part in the ``nilpotent'' case. Several works have already dealt with the ``semisimple'' case: \textit{A. D. Bryuno} [Trans. Moscow Math. Soc. 25, 131-288 (1973); translation from Tr. Mosk. Mat. O.-va 25, 119-262 (1971; Zbl 0263.34003)]; \textit{J. Ecalle} [Publ. Math. Orsay 85-05, 1-585 (1985; Zbl 0602.30029]; \textit{J. Martinet} and \textit{J.-P. Ramis} [Inst. Hautes Etud. Sci., Publ. Math. 55, 63-164 (1982; Zbl 0546.58038); Ann. Sci. Éc. Norm. Supér., IV. Ser. 16, 571-621 (1983; Zbl 0534.34011)]; \textit{R. Perez-Marco} and \textit{J.-C. Yoccoz} [Astérisque 222, 345-371 (1994; Zbl 0809.32008)]. In the ``semisimple'' case, the ratio of the eigenvalues of the linear part is closely related to the classification, and the moduli are essentially the ones of a holonomy diffeomorphism. Now the authors suppose that both eigenvalues vanish. They first consider a ``saddle-node singularity'' with two separatrices, where the strong separatrix is the \(x\)-axis and the central one is the \(y\)-axis. Then they give a new proof of the result that the holonomy diffeomorphism associated to the strong separatrix gives the classification \textit{J. Martinet} and \textit{J.P. Ramis} [loc. cit.]. The germ of a foliation is called nilpotent by the authors if its 1-jet is linearly equivalent to \(y dy\), which possesses normal formal form of the type \({\Omega}^{n,p}=d(y^2+x^n) +x^pU(x)dy\), where \(n-1 \geq 2\) is the Milnor number, \(p \geq 2\) is an integer and \(U\) is a an element of \({\mathbb C}x\) with \(U(0)\neq 0\). For this kind of germs, if \(2p<n\) and a desingularization process produces a saddle-node singularity, then they can characterize the holomorphic equivalence and the rigidity (rigidity holds precisely when holonomy group is non-abelian).
scientific article; zbMATH DE number 1331661