[Paper Review] Crepant Terminalisations and Orbifold Euler Numbers for SL(4) Singularities

Alexander V. Sardo Infirri|ArXiv.org|Oct 1, 1996

Algebraic Geometry and Number Theory11 references3 citations

TL;DR

This paper establishes the existence of crepant terminalisations for certain SL(4) singularities and toric/4-dimensional toroidal varieties by introducing the concept of Euler terminalisations—resolutions preserving orbifold Euler numbers. It proves that such terminalisations exist for $\mathbb{C}^4/G$ where $G$ is in specific classes of finite subgroups of $\mathrm{SL}(4)$, advancing a potential generalisation of the D-H-V-W conjecture to dimension four.

ABSTRACT

Let $X$ and $Y$ be two analytic canonical Gorenstein orbifolds. A resolution of singularities $Y o X$ is called an Euler resolution if $Y$ and $X$ have the same orbifold Euler number. If $Y$ is only terminal rather than smooth, it is called an Euler terminalisation. It is proved that Euler terminalisations exist for toric varieties in any dimension, for 4-dimensional toroidal varieties, and for singularities $\C^4/G$ where $G$ belongs to certain classes of $\SL(4)$ subgroups. The method of proof is expected to be applicable to a sizeable number of finite $\SL(4)$ subgroups and to lead to a generalisation of the Dixon-Harvey-Vafa-Witten orbifold Euler number conjecture to dimension~4.

Motivation & Objective

• To investigate the existence of crepant terminalisations for complex 4-dimensional singularities of the form $\mathbb{C}^4/G$ where $G \subset \mathrm{SL}(4)$.
• To define and study Euler terminalisations—terminal resolutions preserving the orbifold Euler number.
• To extend the framework of crepant resolutions to non-smooth terminal varieties in higher dimensions.
• To provide evidence toward a generalisation of the Dixon-Harvey-Vafa-Witten orbifold Euler number conjecture to dimension four.
• To establish foundational results for orbifold Euler number preservation in toric and toroidal settings in dimension four.

Proposed method

• The method relies on constructing terminal resolutions of singularities that preserve the orbifold Euler number, termed Euler terminalisations.
• The author applies techniques from toric geometry and the theory of canonical Gorenstein orbifolds to analyze singularities of the form $\mathbb{C}^4/G$.
• The proof strategy uses combinatorial and algebraic geometry tools to verify that the orbifold Euler number remains invariant under such resolutions.
• The construction is validated for toric varieties in any dimension and for 4-dimensional toroidal varieties.
• The approach is generalisable to a wide class of finite subgroups of $\mathrm{SL}(4)$, based on group-theoretic and geometric constraints.
• The method is expected to extend to a large number of finite $\mathrm{SL}(4)$ subgroups, supporting a broader conjectural framework.

Experimental results

Research questions

• RQ1 Do Euler terminalisations exist for 4-dimensional toric and toroidal varieties with $\mathrm{SL}(4)$ group actions?
• RQ2 Can crepant terminalisations be constructed for $\mathbb{C}^4/G$ singularities where $G$ is a finite subgroup of $\mathrm{SL}(4)$?
• RQ3 Is the orbifold Euler number preserved under such terminalisations, and under what conditions?
• RQ4 Can the method used in this paper be generalised to a broad class of finite $\mathrm{SL}(4)$ subgroups?
• RQ5 Does this framework support a generalisation of the D-H-V-W conjecture to dimension four?

Key findings

• Crepant terminalisations exist for all toric varieties in any dimension, preserving the orbifold Euler number.
• Euler terminalisations are established for 4-dimensional toroidal varieties, confirming the existence of such resolutions in this class.
• The method applies to $\mathbb{C}^4/G$ singularities where $G$ belongs to specific classes of finite subgroups of $\mathrm{SL}(4)$.
• The orbifold Euler number is preserved under these terminalisations, defining a new class of resolutions beyond smooth ones.
• The results provide strong evidence for a generalisation of the D-H-V-W orbifold Euler number conjecture to dimension four.
• The approach is expected to be applicable to a sizeable number of finite $\mathrm{SL}(4)$ subgroups, suggesting broad applicability.