The hyperbolicity of phase curves of a generic polynomial vector field in \(\mathbb{C}^ n\)
The author's goal is to prove that phase curves of a generic polynomial vector field in complex euclidean space represent hyperbolic Riemann surfaces, that is, the universal covering space of any phase curve is conformally equivalent to the unit disc. In fact, an effective sufficient condition of hyperbolicity of phase curves is found and shown to be fulfilled on an open dense subset of the space of vector fields with vanishing linear part and fixed degrees of components. In order to formulate the main result recall that, according to Lojasiewicz theorem, for any isolated zero \(A\) of a holomorphic vector field \(v\) there exist \(\alpha\), \(C > 0\) such that sufficiently close to \(A\) one has \(|v(z) |\geq C |z - A |^\alpha\). The infimum of such \(\alpha - s\) is denoted by \(L(A)\) and called Lojasiewicz exponent of \(v\) at \(A\). Similarly, \(L (\infty)\) is defined as the supremum of all \(\alpha \geq 0\) that for sufficiently large \(|z |\) one has \(|v(z) |\geq C |z |^\alpha\). The main result is the following theorem. Suppose that a polynomial vector field in \(C^n\) has only isolated zeros \(Q_1, \dots, Q_k\) and \(\sum^k_{s = 1} (L (Q_s) - 1) < L (\infty) - 1\). Then all phase curves of this vector field represent hyperbolic Riemann surfaces. The author underlines that the idea of the proof belongs to Y. Il'ashenko. Namely, its crucial step is a construction of auxiliary positive function \(f\) such that the curvature of Hermitian metric \(fdt \overline {dt}\) on phase curves outside zeros is strictly negative. This construction results from a series of technical lemmas some of which have independent interest.
scientific article; zbMATH DE number 795611