A PROOF OF THE STRATIFIED MORSE INEQUALITIES FOR SINGULAR COMPLEX ALGEBRAIC CURVES USING THE WITTEN DEFORMATION

摘要

The Witten deformation is an analytic method proposed by Wit-ten which, given a Morse function f : M→R on a smooth compact manifold M, allows to prove the Morse inequalities. The aim of this article is to generalise the Witten deformation to stratified Morse functions (in the sense of stratified Morse theory as developed by Goresky and MacPherson) on a singular complex algebraic curve. In a previous article the author developed the Witten deformation for the model of an algebraic curve with cone-like singularities and a certain class of func-tions called admissible Morse functions. The perturbation arguments needed to understand the Witten deformation on the curve with its metric induced from the Fubini-Study metric of the ambient projective space and for any stratified Morse function are presented here.