[Paper Review] Crepant Property of Fujiki-Oka Resolutions for Gorenstein Abelian Quotient Singularities

Kohei Sato, Yusuke Sato|arXiv (Cornell University)|Apr 7, 2020

Algebraic Geometry and Number Theory17 references4 citations

TL;DR

This paper establishes a necessary and sufficient condition for Fujiki-Oka resolutions of Gorenstein abelian quotient singularities to be crepant in any dimension using Ashikaga’s multidimensional continued fractions. It proves that all three-dimensional Gorenstein abelian quotient singularities admit a crepant Fujiki-Oka resolution, offering a simple computational alternative to prior case-by-case proofs.

ABSTRACT

We show a sufficient condition for Fujiki-Oka resolutions of Gorenstein abelian quotient singularities to be crepant in all dimensions by using Ashikaga's continuous fractions. Moreover, we prove that all three dimensional Gorenstein abelian quotient singularities possess a crepant resolution as a corollary. This alternative proof of existence needs only simple computations comparing with the results ever known.

Motivation & Objective

• To determine a necessary and sufficient condition for Fujiki-Oka resolutions of Gorenstein abelian quotient singularities to be crepant in arbitrary dimensions.
• To extend the crepant resolution condition from cyclic to abelian quotient singularities using iterated Fujiki-Oka resolutions.
• To provide a simple, computational proof of the existence of crepant resolutions in dimension three, contrasting with prior case-by-case approaches.
• To explore the relationship between Fujiki-Oka resolutions and $G$-Hilbert schemes in three dimensions.
• To investigate connections between economic resolutions and Fujiki-Oka resolutions in terminal threefold quotient singularities.

Proposed method

• Utilizes Ashikaga’s multidimensional continued fractions, decomposing them into remainder and round-down polynomials to analyze resolution steps.
• Applies the Fujiki-Oka resolution process, which is always smooth, and checks crepant property by verifying canonical divisor preservation at each step.
• Introduces iterated Fujiki-Oka resolutions for abelian groups, recursively resolving subquotients to handle higher-dimensional abelian actions.
• Defines the age of coefficients in remainder polynomials as a key invariant to determine crepantness: resolution is crepant iff all such ages equal 1.
• Employs toric geometry and lattice constructions (e.g., $N'$, vectors ${m{v}}_i$) to model resolutions and compare with $G$-Hilbert schemes.
• Compares Fujiki-Oka resolutions with $A$-Hilbert schemes and economic resolutions via moduli space constructions and isomorphism criteria.

Experimental results

Research questions

• RQ1 What is the necessary and sufficient condition for a Fujiki-Oka resolution of a Gorenstein abelian quotient singularity to be crepant in arbitrary dimension?
• RQ2 Can the existence of crepant resolutions in dimension three be proven via a uniform, simple method rather than case-by-case analysis?
• RQ3 Under what conditions does a Fujiki-Oka resolution coincide with the $A$-Hilbert scheme in three dimensions?
• RQ4 How do Fujiki-Oka resolutions relate to economic resolutions in terminal threefold quotient singularities?
• RQ5 When do the continued fraction fans $\mathrm{CFF}_{{\bm{e}}_1}(\sigma)$, $\mathrm{CFF}_{{\bm{e}}_2}(\sigma)$, and $\mathrm{CFF}_{{\bm{e}}_3}(\sigma)$ coincide, implying isomorphism to $A$-Hilb$(\mathbb{C}^3)$?

Key findings

• A Fujiki-Oka resolution of a cyclic quotient singularity $\frac{1}{r}(1,a_2,\dots,a_n)$ is crepant if and only if the ages of all coefficients in the corresponding remainder polynomial $\mathcal{R}_*\left(\frac{(1,a_2,\dots,a_n)}{r}\right)$ are exactly 1.
• For abelian quotient singularities, an iterated Fujiki-Oka resolution is crepant if and only if the ages of all coefficients in the remainder polynomials at each resolution step are 1.
• All three-dimensional Gorenstein abelian quotient singularities admit a crepant Fujiki-Oka resolution, providing a uniform, computationally simple proof of existence.
• In the case $G = \langle \frac{1}{r}(1,1,r-2) \rangle \subset SL(3,\mathbb{C})$, the Fujiki-Oka resolution is isomorphic to $A$-Hilb$(\mathbb{C}^3)$, confirming agreement with the $G$-Hilbert scheme.
• For terminal threefold quotient singularities $\mathbb{C}^3/G$ with $G = \langle \frac{1}{r}(1,a,r-a) \rangle$, the Fujiki-Oka resolution coincides with the economic resolution, and thus is isomorphic to a moduli space of $\theta$-stable $G$-constellations.
• The conjecture is supported that if the Fujiki-Oka resolution $X(N', \mathrm{CFF}_{{\bm{e}}_1}(\sigma))$ is isomorphic to $A$-Hilb$(\mathbb{C}^3)$, then the projective toric crepant resolution is unique up to isomorphism.