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

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 $\tilde{\xi}$-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.