2020/02/10 by Zach, Matthias
#Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2002.04009
We describe a generalization of Milnor's formula for the Milnor number of an isolated hypersurface singularity to the case of a function f whose restriction f|(X,0) to an arbitrarily singular reduced complex analytic space (X,0) ⊂ (\mathbb Cn,0) has an isolated singularity in the stratified sense. The corresponding analogue of the Milnor number, μf(α;X,0), is the number of Morse critical points in a stratum \mathscr Sα of (X,0) in a morsification of f|(X,0). Our formula expresses μf(α;X,0) as a homological index based on the derived geometry of the Nash modification of the closure of the stratum \mathscr Sα. While most of the topological aspects in this setup were already understood, our considerations provide the corresponding analytic counterpart. We also describe how to compute the numbers μf(α;X,0) by means of our formula in the case where the closure \mathscr Sα ⊂ X of the stratum in question is a hypersurface.