Hypersurface normalizations and numerical invariants

Abstract

We define a new perverse sheaf, the comparison complex, naturally associated to any locally reduced complex analytic space $X$ on which the (shifted) constant sheaf $\mathbb{Q}_X^\bullet[\dim X]$ is perverse. In the hypersurface case, this complex is isomorphic to the perverse eigenspace of the eigenvalue one for the Milnor monodromy action on the vanishing cycles; we also examine how the characteristic polar multiplicities of this complex behave in certain one-parameter families of deformations of hypersurfaces with codimension-one singularities, and generalize a classical formula for the Milnor number of a plane curve singularities in terms of double-points. In general, the vanishing of the cohomology sheaves of the comparison complex provide a criterion for determining if the normalization of the space X is a rational homology manifold. When the normalization is a rational homology manifold, we can also compute several terms in the weight filtration of the constant sheaf $\mathbb{Q}_X^\bullet[\dim X]$ in those cases for which this perverse sheaf underlies a mixed Hodge module. In the surface case $V(f) \subseteq \C^3$, this produces a new numerical invariant, the weight zero part of the constant sheaf, which is a perverse sheaf concentrated on a single point. We then prove two special cases of a conjecture of Javier Fern'{a}ndez de Bobadilla for hypersurfaces with $1$-dimensional critical loci (Corollary 4.2.0.2 and Theorem 4.3.0.2). We do this via a new numerical invariant for such hypersurfaces, called the beta invariant, first defined and explored by the Massey in 2014. The beta invariant is an algebraically calculable invariant of the local ambient topological-type of the hypersurface, and the vanishing of the beta invariant is equivalent to the hypotheses of Bobadilla's conjecture. Bobadilla's Conjecture is related to a more well-known conjecture by L\^{e} D\~{u}ng Tr'{a}ng (\conjref{conj:leconj}) regarding the equsingularity of parameterized surfaces in $\C^3$.