2001/12/06 by Kiumars Kaveh, Kaveh, Kiumars
Mathematics · #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AG
paper · pdf · doi:10.48550/arxiv.math/0112064
69 pages
arxiv created 2001/12/06 · openalex publication_date 2001/12/06 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A theorem due to D. Bernstein states that Euler characteristic of a hypersurface defined by a polynomial f in (C\0)n is equal (upto a sign) to n! times volume of the Newton polyhedron of f. This result is related to algebaric torus actions and toric varieties. In this thesis, I prove that one can generalize the above result to actions of reductive groups with spherical orbits. That is, if a reductive group acts linearly on a vector space such that generic orbits are spherical, one can compute the Euler characteristic of generic hyperplane sections of a generic orbit in terms of combinatorial data. Our main tool is Morse theory. We begin with developing a variant of classical Morse theory for algebraic submanifolds of Rn and linear functionals. This will become related to stratification theory of Thom and Whitney as well as Palais-Smale generalized Morse theory.