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

Abstract: For a germ $(X,0)$ of a normal complex analytic surface, let $$E:=H^{0({}^{p_+IC_X\mathbb}} Z)<em>0$$, where ${}^pIC_X\mathbb Z$ and ${}^p</em>+IC_X\mathbb Z$ 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 $H^2(L,\mathbb Z)<em>{\tors}$, 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 $E\cong \coker(T-\id)</em>{\tors}$, 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.

• The paper establishes that the finite abelian group E encapsulates the integral discrepancy between standard and dual middle intersection complexes in normal surface singularities.
• It employs a unified methodology linking perverse sheaf corrections with topological torsion, lattice discriminants from minimal resolutions, and monodromy determinants in hypersurface cases.
• The explicit computational formulas for |E| in various examples provide clear, actionable insights into local invariants and pave the way for further research in singularity theory.

Integral Perverse Obstructions for Normal Surface Singularities: Synthesis of Perverse, Topological, Lattice, and Monodromy Invariants

Introduction

The paper "Integral Perverse Obstructions for Normal Surface Singularities: Resolution Determinants and Monodromy" (2604.22132) addresses the problem of describing and computing the local discrepancy between the ordinary and dual middle-perversity intersection complexes with integral coefficients for a normal surface singularity. This discrepancy, encapsulated as a finite abelian group $E$, is shown to admit several equivalent but conceptually distinct manifestations: as a perverse-sheaf-theoretic correction, as torsion in the link cohomology, as the discriminant group of the exceptional lattice of a minimal resolution, and, in the hypersurface case, as the torsion in the cokernel of the Milnor monodromy variation.

Notably, the results establish a robust bridge between perverse sheaf theory, local topology, lattice theory, and singularity monodromy, yielding both explicit computational formulae and conceptual clarity on the origin and structure of integral obstructions in surface singularity theory.

Integral Middle Perversities and Local Obstruction Group

The authors study the difference between the ordinary and dual middle-perversity intersection complexes with $\mathbb{Z}$-coefficients, denoted $\mathrm{IC}_X^{\mathbb{Z}}$ and $"\mathrm{IC}_X^{\mathbb{Z}}$, respectively, for a normal complex analytic surface germ $(X,0)$. The critical object is the finite abelian group

$E := H^0("\mathrm{IC}_X^{\mathbb{Z}})_0,$

which captures the point-supported, purely integral obstruction to the coincidence of these two middle extensions.

The foundational result is that the perverse correction is concentrated at the singularity, is invisible rationally (i.e., over $\mathbb{Q}$), and constitutes a self-dual package in the derived category context. The structure of $E$ is shown to be invariant under analytic isomorphism of germs and to vanish if and only if the two middle extensions agree integrally.

Topological Realization: Link Cohomology

Passing to topology, the obstruction $E$ is concretely realized as the torsion subgroup of the second cohomology of the link $L$ of the singularity,

$\mathbb{Z}$0

where $\mathbb{Z}$1 is a compact, oriented, three-dimensional manifold determined by the link of the surface singularity. This identification follows from the analysis of the stalk cohomology of the extension complexes and Mayer-Vietoris arguments on the topology of a punctured neighborhood, with Lemma 3.2 making explicit the finite nature of this torsion.

Lattice-Theoretic Realization: Exceptional Lattice Discriminant

A powerful geometric interpretation emerges via resolution theory. For the minimal resolution $\mathbb{Z}$2, with exceptional divisor $\mathbb{Z}$3, the lattice $\mathbb{Z}$4 generated by the classes $\mathbb{Z}$5 inherits a negative definite intersection pairing. The discriminant group

$\mathbb{Z}$6

(where $\mathbb{Z}$7) is shown to be isomorphic to $\mathbb{Z}$8, with order

$\mathbb{Z}$9

where $\mathrm{IC}_X^{\mathbb{Z}}$0 is the intersection matrix. This explicit formula enables concrete calculations of the obstruction group in terms of the minimal resolution graph, establishing $\mathrm{IC}_X^{\mathbb{Z}}$1 as a local invariant that records the failure of the exceptional configuration to be unimodular.

Realization in Hypersurface Monodromy

When the singularity $\mathrm{IC}_X^{\mathbb{Z}}$2 arises as an isolated hypersurface, the paper leverages Milnor fibration theory and the Wang sequence to relate $\mathrm{IC}_X^{\mathbb{Z}}$3 to the Milnor monodromy $\mathrm{IC}_X^{\mathbb{Z}}$4 acting on the integral vanishing cohomology of the Milnor fiber $\mathrm{IC}_X^{\mathbb{Z}}$5:

$\mathrm{IC}_X^{\mathbb{Z}}$6

Under the algebraic condition that $\mathrm{IC}_X^{\mathbb{Z}}$7 is an isomorphism rationally, the order of $\mathrm{IC}_X^{\mathbb{Z}}$8 attains the refinement

$\mathrm{IC}_X^{\mathbb{Z}}$9

Thus, in these cases, $"\mathrm{IC}_X^{\mathbb{Z}}$0 can be calculated entirely through monodromy data.

Explicit Examples and Numerical Values

The theory is substantiated with a range of explicit examples:

• For $"\mathrm{IC}_X^{\mathbb{Z}}$1 surface singularities, $"\mathrm{IC}_X^{\mathbb{Z}}$2, with $"\mathrm{IC}_X^{\mathbb{Z}}$3 matching link cohomology, lattice discriminant, and monodromy determinant.
• For rational double points of types $"\mathrm{IC}_X^{\mathbb{Z}}$4, $"\mathrm{IC}_X^{\mathbb{Z}}$5, $"\mathrm{IC}_X^{\mathbb{Z}}$6, and $"\mathrm{IC}_X^{\mathbb{Z}}$7, the respective groups are $"\mathrm{IC}_X^{\mathbb{Z}}$8 or $"\mathrm{IC}_X^{\mathbb{Z}}$9, $(X,0)$0, $(X,0)$1, and $(X,0)$2 reflecting the discriminant of the associated root lattices.
• For cyclic quotient and Brieskorn-Pham singularities, the obstruction group is computed explicitly, highlighting the persistence of nontrivial torsion invariants beyond the ADE context.

Broader Theoretical Implications

The identification and computation of $(X,0)$3 have substantive implications:

• From a perverse sheaf perspective, the results clarify the precise nature and source of integral corrections, with relevance to the study of torsion phenomena in intersection homology, factorization, and the behavior of class groups versus divisor-theoretic structures.
• In topology, $(X,0)$4 provides a bridge between the geometry of resolutions and the link’s cohomology, unifying approaches using the topology of the boundary, algebraic geometry, and sheaf-theoretic tools.
• For hypersurface singularities, the tie to the integral variation map and monodromy opens avenues to study torsion effects in vanishing cycles and Picard–Lefschetz theory.

Future Directions

Potential extensions include:

• Generalization to higher codimension or to different classes of singularities, tracking how perverse obstructions manifest in more general resolution-theoretic contexts.
• Applications to global questions of (Q-)factoriality and the arithmetic of divisor class groups, leveraging the explicit local description of obstructions.
• Deeper study of the interplay between integral and torsion-sensitive perverse sheaves, especially in birational geometry and singularity theory.

Conclusion

This work achieves a comprehensive, unifying description of the integral perverse correction group $(X,0)$5 for normal surface singularities, yielding a suite of equivalent, computable invariants across sheaf theory, topology, and singularity theory. The explicit formulae for $(X,0)$6 in terms of natural geometric and topological data upgrade the understanding of local singularity invariants, provide rigorous answers to conjectures in the literature, and lay the foundation for further inquiries into the arithmetic and geometric structures of surface singularities.