# On the multipliers at fixed points of self-maps of the projective plane

## TL;DR

This paper investigates algebraic relations among eigenvalues at fixed points of quadratic holomorphic self-maps of the complex projective plane, revealing new constraints and classification results, especially for maps with invariant lines.

## Contribution

It characterizes all algebraic relations among eigenvalues for quadratic self-maps with invariant lines and shows generic maps are determined by these eigenvalues.

## Key findings

- All algebraic relations for quadratic maps with invariant lines are obtained.
- Generic quadratic maps are uniquely determined by their fixed point eigenvalues.
- Relations among eigenvalues translate to relations among Kowalevski exponents in vector fields.

## Abstract

This paper deals with holomorphic self-maps of the complex projective plane and the algebraic relations among the eigenvalues of the derivatives at the fixed points. These eigenvalues are constrained by certain index theorems such as the holomorphic Lefschetz fixed-point theorem. A simple dimensional argument suggests there must exist even more algebraic relations that the ones currently known. In this work we analyze the case of quadratic self-maps having an invariant line and obtain all such relations. We also prove that a generic quadratic self-map with an invariant line is completely determined, up to linear equivalence, by the collection of these eigenvalues. Under the natural correspondence between quadratic rational maps of $\mathbb{P}^2$ and quadratic homogeneous vector fields on $\mathbb{C}^3$, the algebraic relations among multipliers translate to algebraic relations among the Kowalevski exponents of a vector field. As an application of our results, we describe the sets of integers that appear as the Kowalevski exponents of a class of quadratic homogeneous vector fields on $\mathbb{C}^3$ having exclusively single-valued solutions.