The Geometry of the Space of Holomorphic Maps from a Riemann Surface into Complex Projective Space

1997/10/20 by Sadok Kallel, S. Kallel, Kallel, S. +3
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #alg-geom #math.AG

paper · pdf · doi:10.48550/arxiv.alg-geom/9710024

47 pages, plain TeX. To appear in J. Diff. Geo

arxiv created 1997/10/20 · openalex publication_date 1997/10/20 · arxiv updated 2009/11/30 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28

Abstract

In this paper we study the topology of the spaces Hol(M,Pn,k) of (basepoint preserving) holomorphic maps of a given degree k from a Riemann surface M of genus g>0 into the n-th complex projective space Pn, n>0. Using symmetric products of the surface as well as the geometry of the associated Abel-Jacobi maps, we construct an explicit topological compactification of Hol(M,Pn,k) which allow us to construct spectral sequences with known E₁-terms that can be used to determine the homology groups of Hol(M,Pn,k). Complete calculations are given for the elliptic case (g=1). Complete rational calculations are also given for hyperelliptic curves.

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