The vanishing homology of isolated determinantal singularities

An isolated determinantal singularity $(X_0,0)\subset (\mathbf C^N,0)$
of type $(m,n,t)$ is described by the $t$-minors of a matrix $A$ of
size $m \times n$ with holomorphic entries. One can show that there is a
unique determinantal Milnor fiber $X_\delta$ arising from deformations
of the singularity $(X_0,0)$ coming from perturbations of the defining
matrix $A$. Contrary to the case of isolated complete intersection
singularities, there are isolated determinantal singularities, whose
Milnor fiber has nontrivial homology groups below the middle degree.
This was first observed by J. Damon and B. Pike, who computed the Euler
characteristic. We will present results that allow the computation of
the distinguished Betti numbers for these singularities and give a
characterization for the appearing ``unusual vanishing cycles''.