On the topology of determinantal links

Abstract

We study the cohomology of the generic determinantal varieties M s m,n = {ϕ ∈ C m×n : rank ϕ < s}, their polar multiplicities, their sections D k ∩ M s m,n by generic hyperplanes D k of various codimension k, and the real and complex links of the spaces (D k ∩ M s m,n , 0). Such complex links were shown to provide the basic building blocks in a bouquet decomposition for the (determinantal) smoothings of smoothable isolated determinantal singularities. The detailed vanishing topology of such singularities was still not fully understood beyond isolated complete intersections and a few further special cases. Our results now allow to compute all distinct Betti numbers of any determinantal smoothing.