The linearization of webs of \(\mathbb{C}^ 2\)
A \(d\)-web in the space germ \((\mathbb{C}^ 2,0)\) is a configuration given by \(d\) foliations \((d \geq 2)\) of smooth complex analytic curves in \((\mathbb{C}^ 2,0)\) in general position. That means, the occuring curves are integral curves of \(d\) holomorphic vector fields \[ X_ i = a_ i \cdot \partial_ x + b_ i \cdot \partial_ y,\quad 1 \leq i \leq d, \] with coefficients in \(\mathbb{C} \{x,y\}\), such that, for \(1 \leq i<j \leq d\), \(a_ i \cdot b_ j-a_ j \cdot b_ i\) is a unit in \(\mathbb{C} \{x,y\}\). Such a \(d\)-web \({\mathcal W}\) is called linearizable, iff there exists a local isomorphism \(f: (\mathbb{C}^ 2,0) \to (\mathbb{C}^ 2, f(0))\) which transforms \({\mathcal W}\) into a linear web, i.e., into a \(d\)-web defined by foliations of (pieces of) lines in \(\mathbb{C}^ 2\). The study of the geometry of webs is a classical topic of both topology and (complex) differential geometry, which has been systematically developed and particularly cultivated mainly by W. Blaschke and G. Bol [cf. \textit{W. Blaschke} und \textit{G. Bol}, Geometrie der Gewebe, Springer-Verlag Berlin (1938; Zbl 0020.06701)] and S.-S. Chern [cf. \textit{S.-S. Chern}, Bull. Am. Math. Soc., New Ser. 6, 1-8 (1982; Zbl 0483.53013)]. The present paper is devoted to the equally classical problem of finding criteria of linearizability for \(d\)-webs in \((\mathbb{C}^ 2,0)\). The main result provides a necessary and sufficient condition for linearizability, which is described by the existence of analytic solutions of a certain associated system of nonlinear partial differential equations. Moreover, this criterion allows, in the affirmative case, to determine all possible linearizing local isomorphisms \(f:(\mathbb{C}^ 2,0) \to (\mathbb{C}^ 2,f(0))\) for a linearizable \(d\)-web \({\mathcal W}\). As an application, the author shows how some classical results (in special cases) can be derived and reproved by means of his new criterion (e.g., the well-known theorem of Mayrhofer and Reidemeister on the linearizability of certain 4-webs in \((\mathbb{C}^ 2,0))\). The link between differential systems and webs in \((\mathbb{C}^ 2,0)\) has been already discussed in the author's foregoing paper [Math. Scand. 66, No. 2, 161-172 (1990; Zbl 0724.32007)].
scientific article; zbMATH DE number 488110