Integral Perverse Obstructions for Normal Surface Singularities: Resolution Determinants and Monodromy

TL;DR

This paper investigates the integral discrepancy in normal surface singularities, linking topological, geometric, and monodromy invariants through intersection complexes and resolution data.

Contribution

It establishes a topological and geometric realization of the integral invariant E, relating it to the link's torsion homology and the resolution's discriminant group, and connects it to monodromy for hypersurface singularities.

Findings

E is isomorphic to the torsion part of the second homology of the link.

|E| equals the absolute value of the determinant of the resolution's intersection matrix.

For hypersurface singularities, E is isomorphic to the cokernel of (T - id), linking monodromy to the invariant.

Abstract

For a germ $(X,0)$ of a normal complex analytic surface, let $E:=H^0({}^p_+IC_X\mathbb Z)_0$, where ${}^pIC_X\mathbb Z$ and ${}^p_+IC_X\mathbb Z$ denote the ordinary and dual middle-perversity intersection complexes with integral coefficients. This finite abelian group measures the integral discrepancy between the two middle extensions. Motivated by work of Jung--Saito, we study $E$ as a local invariant of the singularity. We prove that $E$ admits a topological realization as $H^2(L,\mathbb Z)_{\tors}$, where $L$ is the link of the singularity, and a geometric realization as the discriminant group of the exceptional lattice of the minimal resolution. In particular, if $M$ is the intersection matrix of the irreducible exceptional curves, then $|E|=|\det(M)|$. If $(X,0)$ is an isolated hypersurface surface singularity, we further prove that $E\cong \coker(T-\id)_{\tors}$, where $T$ is the Milnor monodromy on integral vanishing cohomology. Under the additional hypothesis that $(T-\id)\otimes_{\mathbb Z}\mathbb Q$ is an isomorphism, this yields $|E|=|\det(T-\id)|$. Thus the same local integral obstruction admits compatible perverse, topological, resolution-theoretic, and monodromy-theoretic realizations.