Monodromy of Primitive Vanishing Cycles for Hypersurfaces in $\mathbb P^4$

Abstract

Let X be a complex submanifold of projective space. Schnell showed that the middle-dimensional primitive cohomology of X is generated by tube classes, which arise from the monodromy of the vanishing homology on hyperplane sections. Clemens asks if the theorem is still true when we restrict the generating set to the tube classes over the class of a single vanishing sphere of nodal degeneration. We prove this is true for hypersurfaces in CP^4. The proof is based on the degeneration of a hypersurface to the union of hypersurfaces of lower degrees.