On fixed points of rational self-maps of complex projective plane

TL;DR

This paper constructs examples of degree d rational self-maps of the complex projective plane with no fixed points, contrasting with the one-dimensional case, and studies the topological closure of such fixed point free maps.

Contribution

It provides explicit examples of fixed point free rational maps on b6^2 of any degree d and analyzes their topological properties within the space of all such maps.

Findings

Existence of fixed point free rational self-maps for all degrees d

The set of fixed point free maps is closed in a natural topology

Contrast with fixed points behavior in one-dimensional case

Abstract

For any given natural $d\ge 1$ we provide examples of rational self-maps of complex projective plane $\pp^2$ of degree $d$ without (holomorphic) fixed points. This makes a contrast with the situation in one dimension. We also prove that the set of fixed point free rational self-maps of $\pp^2$ is closed (modulo "degenerate" maps) in some natural topology on the space of rational self-maps of $\pp^2$.