Morse theory and Euler characteristic of sections of spherical varieties

Details Morse theory and Euler characteristic of sections of spherical varieties Morse theory and Euler characteristic of sections of spherical varieties

2001-12-06

arxiv.org/abs/math/0112064v1

Mathematics

Algebraic Geometry

69 pages

Scientific paper

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.

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