Morse theory and Euler characteristic of sections of spherical varieties

Abstract

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 R^n and linear functionals. This will become related to stratification theory of Thom and Whitney as well as Palais-Smale generalized Morse theory.