Total rigidity of generic quadratic vector fields

TL;DR

This paper proves that generic quadratic vector field foliations on the complex projective plane exhibit total rigidity, meaning they are topologically equivalent to only finitely many other such foliations, with an upper bound of 240.

Contribution

It establishes the total rigidity property for generic quadratic foliations, showing they are topologically equivalent to finitely many others, extending understanding of foliation classification.

Findings

Generic quadratic foliations are finitely topologically equivalent to others.

Total rigidity property is proven for quadratic foliations.

The maximum number of topologically equivalent foliations is at most 240.

Abstract

We consider a class of foliations on the complex projective plane that are determined by a quadratic vector field in a fixed affine neighborhood. Such foliations, as a rule, have an invariant line at infinity. Two foliations with singularities on $\mathbb C P^2$ are topologically equivalent provided that there exists a homeomorphism of the projective plane onto itself that preserves orientation both on the leaves and in $\mathbb C P^2$ and brings the leaves of the first foliation to that of the second one. We prove that a generic foliation of this class may be topologically equivalent to but a finite number of foliations of the same class, modulo affine equivalence. This property is called \emph{total rigidity}. Recent result of Lins Neto implies that the finite number above does not exceed 240. This is the first of the two closely related papers. It deals with the rigidity properties of quadratic foliations, whilst the second one studies the foliations of higher degree.