2004/03/07 by Vladimir Lin, Lin, Vladimir · 1 citation
Mathematics · #14R99 #32H02 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #math.AG #math.CV #msc:14R99 #msc:32H02
paper · pdf · doi:10.48550/arxiv.math/0403120
89pp, 1 figure
openalex publication_date 2004/03/07 · arxiv created 2004/04/29 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study holomorphic self-maps of non-ordered n-point configuration spaces Cn(X), where X is either affine or projective complex line. The complex Lie group Aut(X) acts diagonally on Cn(X). We prove that for n>4 every endomorphism F of Cn(X) either is tame meaning that it is of the form F(Q)=T(Q)Q for a certain morphism T of Cn(X) to Aut(X) or carries the whole Cn(X) into one Aut(X) orbit Aut(X)Q0 in Cn(X). The first option happens if and only if the image of the induced endomorphism F_* of the fundamental group of Cn(X) (which is the braid group of X) is a non-cyclic group; otherwise F is of the second type. We also prove that for n>(dim(Aut(X))+1 a morphism F of Cn(X) to any Ck(X) always admits an n point subset Q of X whose intersection with its image F(Q) is non-empty. Finally, we give a complete description of unbranched k-coverings of Cn(x) for k<2n+1.