Index theorems for meromorphic self-maps of the projective space

TL;DR

This paper develops index theorems linking local residues at fixed and indeterminacy points of meromorphic self-maps of complex projective space with Chern classes, using techniques from local dynamics of holomorphic germs.

Contribution

It introduces a novel approach to study global meromorphic maps via local dynamics and establishes new index theorems connecting local residues to global geometric invariants.

Findings

Holomorphic foliation induced by the meromorphic map in Riemann surfaces

Index theorems relating residues to Chern classes

Characterization of fixed and indeterminacy points

Abstract

In this short note we would like to show how it is possible to use techniques introduced in the theory of local dynamics of holomorphic germs tangent to the identity to study global meromorphic self-maps of the complex projective space. In particular we shall show how a meromorphic self-map of a complex projective space induces a holomorphic foliation of the projective space in Riemann surfaces, whose singular points are exactly the fixed points and the indeterminacy points of the map; and we shall prove three index theorems, relating suitably defined local residues at the fixed and indeterminacy points with Chern classes of the projective space.