# Contents

## Idea

An ADE singularity is a orbifold fixed point locally of the form $\mathbb{C}^2/\Gamma$ with $\Gamma \hookrightarrow SU(2)$ a finite subgroup of the special unitary group given by the ADE classification (and $SU(2)$ is understood with its defining linear action on the complex vector space $\mathbb{C}^2$).

These singularities have crepant resolutions, obtained by repeatedly blowing up at singular points. The resulting exceptional fiber (the blow-up of the singular point, an ALE space) is a union of Riemann spheres that touch each other such as to form the shape of the corresponding Dynkin diagram.

(graphics grabbed from Wijnholt 14, part III)

## References

### General

• Peter Slodowy, Simple singularities and simple algebraic groups, in Lecture Notes in Mathematics 815, Springer, Berlin, 1980.

• Classification of singularities

• Miles Reid, The Du Val Singularities $A_n$, $D_n$, $E_6$, $E_7$, $E_8$ (pdf)

• Kyler Siegel, section 6 of The Ubiquity of the ADE classification in Nature , 2014 (pdf)

• MathOverflow, Resolving ADE singularities by blowing up

Families of examples of G2 orbifolds with ADE singularities are constructed in