Crepant Terminalisations and Orbifold Euler Numbers for SL(4) Singularities

TL;DR

This paper investigates the existence of Euler terminalisations for certain complex orbifolds and singularities, extending known results to higher dimensions and specific group actions, with implications for orbifold Euler number conjectures.

Contribution

It proves the existence of Euler terminalisations for toric and certain $ ext{SL}(4)$ singularities, advancing the understanding of orbifold resolutions in four dimensions.

Findings

Euler terminalisations exist for toric varieties in any dimension.

Existence of Euler terminalisations for 4-dimensional toroidal varieties.

Existence of Euler terminalisations for $ ext{C}^4/G$ singularities with specific $ ext{SL}(4)$ subgroups.

Abstract

Let $X$ and $Y$ be two analytic canonical Gorenstein orbifolds. A resolution of singularities $Y\to X$ 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.