Index theorems for meromorphic self-maps of the projective space

2011/06/13 by Marco Abate, Abate, Marco
Mathematics · #14M20 #32H50 #37F75 #37F99 #58J20 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Primary 32H04 #math.AG #math.CV #math.DS #msc:14M20 #msc:32H04 #msc:32H50 #msc:37F75 #msc:37F99 #msc:58J20

paper · pdf · doi:10.48550/arxiv.1106.2394

arxiv created 2011/06/13 · arxiv updated 2011/06/14

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.

Related

• Index theorems for holomorphic self-maps
• Index theorems for couples of holomorphic self-maps
• Index theorems for holomorphic maps and foliations
• Indices of holomorphic foliations and the bifurcation conjecture
• Meromorphic maps from \Bbb C^p into semi-abelian varieties and general projective varieties