On a conjecture of Varchenko

Abstract

In this note we generalize and prove a recent conjecture of Varchenko concerning the number of critical points of a (multivalued) meromorphic function $ϕ$ on an algebraic manifold. Under certain conditions, this number turns out to coincide with the Euler characteristic (up to a sign) of the complement of the divisor of $ϕ$. A few variants of this basic situation are also discussed. Two independent proofs are given, respectively using Chern classes and Morse theory In its original form, this counting problem arises (in special cases) in evaluating the asymptotics of correlation functions of Wess--Zumino--Witten conformal field theory.