  We study the cohomology of the generic determinantal varieties $M_{m,n}^s =
\{ \varphi \in \mathbb C^{m\times n} : \mathrm{rank} \varphi <s \}$, their
polar multiplicities, their sections $D_k \cap M_{m,n}^s$ by generic
hyperplanes $D_k$ of various dimension $k$, and the real and complex links of
the spaces $(D_k\cap M_{m,n}^s,0)$. 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.
