  Suppose that the critical locus $\Sigma$ of a complex analytic function $f$
on affine space is, itself, a space with an isolated singular point at the
origin $\0$, and that the Milnor number of $f$ restricted to normal slices of
$\Sigma-\{\0\}$ is constant. Then, the general theory of perverse sheaves puts
severe restrictions on the cohomology of the Milnor fiber of $f$ at $\0$, and
even more surprising restrictions on the cohomology of the Milnor fiber of
generic hyperplane slices.
