On first order deformations of homogeneous foliations

TL;DR

This paper investigates first order deformations of homogeneous holomorphic foliations in complex affine space, classifying how these deformations behave within specific foliation types and degrees.

Contribution

It characterizes first order homogeneous deformations of various classes of homogeneous foliations in complex affine space, including polynomial, rational, and logarithmic types.

Findings

Deformations of the same degree stay within the same foliation class.

The structure of deformations depends on the foliation type.

Existence of deformations with different degrees is also studied.

Abstract

We study analytic deformations of holomorphic foliations given by homogeneous integrable one-forms in the complex affine space $\mathbb C^n$. The deformation is supposed to be of first order (order one in the parameter). We also assume that the deformation is given by homogeneous polynomial one-forms. The deformation takes place in the affine space since we are not assuming that the foliations descent to the projective space. We describe the space of such deformations in three main situations: (1) the given foliation is given by the level hypersurfaces of a homogeneous polynomial. (2) the foliation is rational, ie., has a first integral of type $P^r/Q^s$ for some homogeneous polynomials $P,Q$. (3) the foliation is logarithmic of a generic type. We prove that, for each class above, the first order homogeneous deformations of same degree are in the very same class. We also investigate the existence of such deformations with different degree.