Rational surface maps with invariant meromorphic two forms

TL;DR

This paper studies rational maps on complex surfaces with invariant meromorphic two forms, showing how to simplify the form via birational transformations and characterizing algebraic stability through pole behavior, with applications to the projective plane.

Contribution

It introduces a method to eliminate zeros of invariant forms via birational changes and characterizes algebraic stability through pole dynamics, linking it to rotation numbers.

Findings

Zeros of invariant forms can be removed by birational transformations.

Algebraic stability is equivalent to pole behavior of the map.

Stability criterion relates to the rationality of a rotation number.

Abstract

We consider a rational map f:S->S of a complex projective surface together with an invariant meromorphic two form. Under a mild topological assumption on the map, we show that the zeroes of the invariant form can be eliminated by birational change of coordinate. In this context, when the form has no zeroes, we investigate the notion of algebraic stability for f. We show in particular that algebraic stability is equivalent to a more tractable condition involving the behavior of f on the poles of the form. Finally, we illustrate our results in the particular case where S is the projective plane and the invariant form is dx dy / xy, showing that our criterion for stability translates to whether or not the rotation number for a certain circle homeomorphism is rational.