Topology of exceptional orbit hypersurfaces of prehomogeneous spaces (Q2826646)

This paper studies the topology of highly nonisolated singularities which arise as the hypersurface of exceptional orbits for a representation of a complex linear algebraic group with open orbit. In particular, the determinantal hypersurface in the space of \(m\times m\) (symmetric, skew symmetric) matrices is studied, and the case of equidimensional representations \(\rho: G \to\mathrm{GL}(V)\), meaning that \(\dim G=\dim V\). This includes certain quiver representations. The exceptional orbit variety is a linear free (or sometimes slightly weaker a linear free\(^*\)) divisor. The defining equation \(f\) is homogeneous, so as Milnor fiber one can take the global affine hypersurface \(f^{-1}(1)\). Starting from the topology of the complement, the cohomology of the Milnor fiber and the link are determined. For the cases of general \(m\times m\) matrices the Milnor fiber is \(\mathrm{SL}_m(\mathbb C)\), for symmetric matrices \(\mathrm{SL}_m(\mathbb C)/\mathrm{SO}_m(\mathbb C)\). From this follows that the cohomology of the Milnor fibers and complements are isomorphic as algebras to exterior algebras or for one family, modules over exterior algebras.NEWLINENEWLINEIn the final section, Thom-Sebastiani sums of these types of hypersurfaces are studied, and also sums with weighted homogeneous nonisolated singularities with Milnor fiber a bouquet of spheres. The resulting Milnor fibers are bouquets of spaces, each of which are suspensions of joins of compact manifolds.