Holonomy for affine holomorphic actions
DOI:
https://doi.org/10.17268/sel.mat.2026.01.04Keywords:
Quasi-homogeneous vector fields, Lie bracket, holonomy of a foliationAbstract
The holonomy of a leaf of a foliation was introduced by C. Ehresmann and is the version of the Poincaré first return map of a real flow in a neighborhood of a periodic orbit. In general, holonomy allows us to see the behavior of leaves near a fixed leaf. In particular, we will focus on the following problem: Let X and Y be two holomorphic vector fields defined in the three-dimensional complex space with singularity at the origin such that X is a diagonal linear vector field and Y is a quasi-homogeneous vector field, and furthermore these vector fields are related by the Lie bracket through [X, Y] = -Y. We will study the holonomy of X, Y and of the integrable 1-form determined by these vector fields.
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