Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Resolution of singularities via Tannaka duality

Jasper van de Kreeke

TL;DR

The paper develops a Tannakian framework for resolving finite quotient singularities by encoding the quotient stack $[*/Γ]$ via Clebsch–Gordan data, producing a Clebsch-Gordan variety $\mathrm{CG}_Γ$ and pairing it with $\mathbb{C}^2$ to realize both stacky and smooth resolutions as GIT quotients. It extends Abdelgadir–Segal's approach from the $A_n$ and $D_4$ cases to all Kleinian (ADE) types, offering a general construction and a six-stage strategy that connects $\mathrm{CG}_Γ\times\mathbb{C}^2$ to the classical quiver–preprojective model through a GL-equivariant map $R$ and invariant-theory comparisons. The framework yields explicit treatments for the $A_n$ and $D_n$ families, clarifies coherence and symmetry relations among Clebsch–Gordan data, and situates the stacky resolution in parallel with the smooth minimal resolution, widening the algebraic toolkit for quotient singularities and linking to physical interpretations via Clebsch–Gordan data. Overall, the approach provides a principled, scalable method to obtain both stacky and smooth resolutions universally for Kleinian singularities, with concrete constructions and proofs in the ADE setting.

Abstract

Resolving finite quotient singularities is a classical problem in algebraic geometry. Traditional methods of Geometric Invariant Theory (GIT) translate the singularity into a quiver representation space and take the GIT quotient with respect to a generic stability parameter. While this approach easily produces smooth resolutions, it fails to produce any stacky resolutions, as quiver representation spaces lack finite stabilizers. This paper provides an alternative framework which produces both smooth and stacky resolutions. Our framework is based on a trick of Abdelgadir and Segal, which deploys Tannaka duality to describe the points of the classifying stack of a finite group in terms of algebraic data. Abdelgadir and Segal successfully pursue this strategy and obtain smooth and stacky resolutions in the Kleinian $ D_4 $ case. We generalize this strategy to all Kleinian singularities and obtain a series of varieties which we refer to as Clebsch-Gordan varieties. We provide tools to work with these Clebsch-Gordan varieties, analyze their stable loci with respect to different stability parameters, and study the Kleinian $ A_n $ and $ D_n $ cases in detail.

Resolution of singularities via Tannaka duality

TL;DR

The paper develops a Tannakian framework for resolving finite quotient singularities by encoding the quotient stack via Clebsch–Gordan data, producing a Clebsch-Gordan variety and pairing it with to realize both stacky and smooth resolutions as GIT quotients. It extends Abdelgadir–Segal's approach from the and cases to all Kleinian (ADE) types, offering a general construction and a six-stage strategy that connects to the classical quiver–preprojective model through a GL-equivariant map and invariant-theory comparisons. The framework yields explicit treatments for the and families, clarifies coherence and symmetry relations among Clebsch–Gordan data, and situates the stacky resolution in parallel with the smooth minimal resolution, widening the algebraic toolkit for quotient singularities and linking to physical interpretations via Clebsch–Gordan data. Overall, the approach provides a principled, scalable method to obtain both stacky and smooth resolutions universally for Kleinian singularities, with concrete constructions and proofs in the ADE setting.

Abstract

Resolving finite quotient singularities is a classical problem in algebraic geometry. Traditional methods of Geometric Invariant Theory (GIT) translate the singularity into a quiver representation space and take the GIT quotient with respect to a generic stability parameter. While this approach easily produces smooth resolutions, it fails to produce any stacky resolutions, as quiver representation spaces lack finite stabilizers. This paper provides an alternative framework which produces both smooth and stacky resolutions. Our framework is based on a trick of Abdelgadir and Segal, which deploys Tannaka duality to describe the points of the classifying stack of a finite group in terms of algebraic data. Abdelgadir and Segal successfully pursue this strategy and obtain smooth and stacky resolutions in the Kleinian case. We generalize this strategy to all Kleinian singularities and obtain a series of varieties which we refer to as Clebsch-Gordan varieties. We provide tools to work with these Clebsch-Gordan varieties, analyze their stable loci with respect to different stability parameters, and study the Kleinian and cases in detail.
Paper Structure (29 sections, 28 theorems, 115 equations, 8 figures)

This paper contains 29 sections, 28 theorems, 115 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Tannakian approach
  4. Tannakian approach of Abdelgadir and Segal
  5. Construction of the variety $Z$
  6. Implementation of the ansatz $X = Z × ℂ^2$
  7. Resolution of finite quotient singularities
  8. Clebsch-Gordan data
  9. The Clebsch-Gordan variety
  10. Resolution of finite quotient singularities
  11. Resolution of Kleinian singularities
  12. Proof in the Kleinian case
  13. The $A_n$ case
  14. Stage 1: Construction of the Clebsch-Gordan variety
  15. Stage 2: Construction of the map $R$
  16. ...and 14 more sections

Key Result

Lemma 4.1

Let $U_i, U_j, U_k$ be a choice of simple $Γ$-representations. Then there exists a unique linear map $γ_{i, j, k}$ which renders the following diagram commutative for all $φ ∈ \mathrm{CG}_Γ$:

Figures (8)

  • Figure 2.1: This figure depicts the $A_n$ and $D_n$ singularities. We list the Kleinian group $Γ$, the equivalent description as hypersurface in $ℂ^3$ and the Kleinian quiver setting $(Q, α)$. Note that the Kleinian quiver with index $n$ has $(n+1)$-many vertices.
  • Figure 3.1: This figure depicts the quiver representation $R((β, A, B), x)$. The meaning of all symbols is elaborated in \ref{['sec:caseD-comparison']}.
  • Figure A.1: This figure visually depicts the rule for determining the exponents $α_i$ used in the proof of \ref{['th:caseA-theta2-semistable']}. Part (\ref{['fig:caseA-exponents-rule-1']}) concerns the case $x_n = 0$ and part (\ref{['fig:caseA-exponents-rule-2']}) concerns the case $x_1, x_n ≠ 0$. Each of the four graphics qualitatively depicts a representation of the $A_n$ quiver with $n = 15$. The solid arrows are meant to depict a nonzero arrow value, and all missing arrows are meant to depict the zero arrow value. The special vertex of the quiver is highlighted and positioned in the top of the images. The number at the vertex $i$ is the exponent $α_i$ which we associate with that vertex. The representations on the left have more zero arrows than the representations on the right, but both representations in each row have the same exponents $α_i$.
  • Figure A.2: This figure depicts the $(n+1)$-many one-parameter families of $θ_2$-representations that lie in the nullcone. We have expressed the representations $ρ_{a, b}$ in these families in terms of two parameters $a, b ∈ ℂ$ for notational convenience, even though one parameter would suffice due to gauging.
  • Figure A.3: This figure qualitatively depicts the exponents $α_j$ used in the proof of \ref{['th:caseA-SI-odd-pbrestr']}. The highlighted vertex in the top of the figure is the special vertex of the quiver. There are $(n-q)$-many vertices whose exponents are increasing negative multiples of $q$, and $q$-many vertices whose exponents are increasing negative multiples of $n-q$. The proof uses exponents which are slightly different from the ones depicted in this figure in order to ease the distinction of exponents that are responsible for divergence.
  • ...and 3 more figures

Theorems & Definitions (68)

  • Definition 4.1
  • Remark 4.1
  • Definition 4.2
  • Remark 4.2
  • Lemma 4.1
  • proof
  • Remark 4.3
  • Remark 4.4
  • Theorem 4.1
  • Definition 4.3
  • ...and 58 more