DOI: 10.1515/advgeom-2026-0019 ISSN: 1615-7168

On the exceptional set of crepant resolutions of abelian singularities

Ugo Bruzzo, Fábio Arceu Ferreira
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Abstract

Let G be a finite abelian subgroup of SL( n , ℂ), and suppose there exists a toric crepant resolution ϕ : X ⟶ ℂ n / G of the quotient variety ℂ n / G . Let Exc( ϕ ) = E 1 ∪⋅ ⋅ ⋅ ∪ E s be the decomposition of the exceptional set of ϕ into irreducible components. We prove that for every i there exists an open subset U i of X such that E i ⊂ U i , and U i is isomorphic to the total space of the canonical bundle

ω E i $\omega_{E_{i}}$
of E i . Furthermore, X = U 1 ∪⋅ ⋅ ⋅ ∪ U s . This contributes to the collection of results aimed at solving a classical problem, i.e., to determine which submanifolds of a complex manifold have a neighborhood isomorphic to a neighborhood of the zero section of their normal bundle.

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