Institute for Mathematical Sciences

Preprint ims95-11

Roberto Silvotti
On a conjecture of Varchenko

Abstract: Varchenko conjectured that, under certain genericity conditions, the number of critical points of a product $\phi$ of powers of linear functions on $\Bbb C^n$ should be given by the Euler characteristic of the complement of the divisor of $\phi$ (i.e., a union of hyperplanes). In this note two independent proofs are given of a direct generalization of Varchenko's conjecture to the case of a generalized meromorphic function on an algebraic manifold whose divisor can be any (generally singular) hypersurface. The first proof uses characteristic classes and a formula of Gauss--Bonnet type for affine algebraic varieties. The second proof uses Morse theory.
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