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

For a germ $(X,0)$ of a normal complex analytic surface, let $E:=H^0({}_{+}^{p}I{C}_X\mathbb{Z}{)}_0$, where ${}^{p}I{C}_X\mathbb{Z}$ and ${}_{+}^{p}I{C}_X\mathbb{Z}$ 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 $H^2(L,\mathbb{Z}{)}_{\text{tors}}$, 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 $\mathrm{\mid }E\mathrm{\mid }=\mathrm{\mid }\det (M)\mathrm{\mid }$. If $(X,0)$ is an isolated hypersurface surface singularity, we further prove that $E\cong \text{coker}(T-\text{id}{)}_{\text{tors}}$, where $T$ is the Milnor monodromy on integral vanishing cohomology. Under the additional hypothesis that $(T-\text{id}){\otimes }_{\mathbb{Z}}\mathbb{Q}$ is an isomorphism, this yields $\mathrm{\mid }E\mathrm{\mid }=\mathrm{\mid }\det (T-\text{id})\mathrm{\mid }$. Thus the same local integral obstruction admits compatible perverse, topological, resolution-theoretic, and monodromy-theoretic realizations.