Classification of indecomposable reflexive modules on quotient singularities through Atiyah--Patodi--Singer theory

TL;DR

The paper addresses the long-standing question of whether indecomposable reflexive modules on quotient surface singularities can be classified by topological invariants, namely their rank and first Chern class, together with the $\tilde{\xi}$-invariant from Atiyah–Patodi–Singer theory. The authors unify singularity theory with flat bundle theory over spherical 3-manifolds, establishing that every spherical 3-manifold arises as a link of a quotient singularity and that indecomposable flat bundles over these manifolds are classified by rank, $c_1$, and $\tilde{\xi}$ (with a notable exception in rank-2 dihedral cases). They provide explicit irreducible representations for all small finite subgroups of $GL(2,\mathbb{C})$, compute CCS-numbers and $\tilde{\xi}$-invariants, and use full-sheaf techniques to transfer these invariants to reflexive modules on the singularities. The main result yields an almost complete classification: isomorphism classes of indecomposable reflexive $\mathcal{O}_{X}$-modules are determined by their rank, $c_1$, and $\tilde{\xi}$-invariant, bringing a 37-year open problem to a near-resolution and linking singularity theory with refined topological invariants of 3-manifolds. The work has significant implications for understanding moduli of full sheaves and for the interplay between singularity theory and 3-manifold CCS invariants, with an explicit conjectural gap for certain dihedral-type rank-2 cases.

Abstract

In [20] Esnault asked whether on a general quotient surface singularity the rank and the first Chern class distinguish isomorphism classes of indecomposable reflexive modules. Wunram gave a contraexample in [46] showing two different full shaves on a quotient singularity, with the same rank and the same first Chern class. In this article, we prove that irreducible reflexive modules over quotient surface singularities are determined by the rank, first Chern class and the Atiyah-Patodi-Singer $\tildeξ$-invariant [5], except for the case of rank $2$ indecomposable reflexive modules over dihedral quotient surface singularities $\mathbb{D}_{n,q}$ with $\gcd(m,2)=2$, which we conjecture to follow the same pattern. To prove the classification theorem, first we prove that every spherical $3$-manifold with non-trivial fundamental group appears as the link of a quotient surface singularity, and that indecomposable flat vector bundles over spherical $3$-manifolds are classified by their rank, first and second Cheeger-Chern-Simons classes, with the exception of the aforementioned case.

Theorems & Definitions (77)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 2.1
• proof
• Theorem 2.2
• Remark 2.3
• Remark 2.4
• Theorem 2.5
• Remark 2.6