Contents

# Contents

## Idea

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

## Properties

### Resolution by spheres touching along a Dynkin diagram

The blow-up of an ADE-singularity is given by a union of Riemann spheres that touch each other such as to form the shape of the Dynkin diagram whose A-D-E label corresponds to that of the given finite subgroup of SU(2).

This statement is originally due to (duVal 1934 I, p. 1-3 (453-455)). A description in terms of hyper-Kähler geometry is due to Kronheimer 89a.

Quick survey of this fact is in Reid 87, a textbook account is Slodowy 80.

In string theory this fact is interpreted in terms of gauge enhancement of the M-theory-lift of coincident black D6-branes to an MK6 at an ADE-singularity (Sen 97):

graphics grabbed from HSS18

See at M-theory lift of gauge enhancement on D6-branes for more.

$\,$

Dynkin diagram via
McKay correspondence
Platonic solidfinite subgroups of SO(3)finite subgroups of SU(2)simple Lie group
$A_n$cyclic group
$\mathbb{Z}_{n+1}$
cyclic group
$\mathbb{Z}_{n+1}$
special unitary group
$D_{n+4}$dihedron,
hosohedron
dihedral group
$D_{n+2}$
binary dihedral group
$2 D_{n+2}$
special orthogonal group
$E_6$tetrahedrontetrahedral group
$T$
binary tetrahedral group
$2T$
E6
$E_7$cube,
octahedron
octahedral group
$O$
binary octahedral group
$2O$
E7
$E_8$dodecahedron,
icosahedron
icosahedral group
$I$
binary icosahedral group
$2I$
E8

## References

### General

Original articles include

• Patrick du Val, (1934a), “On isolated singularities of surfaces which do not affect the conditions of adjunction. I”, Proceedings of the Cambridge Philosophical Society, 30 (4): 453–459, doi:10.1017/S030500410001269X

• Patrick du Val, (1934b), “On isolated singularities of surfaces which do not affect the conditions of adjunction. II”, Proceedings of the Cambridge Philosophical Society, 30 (4): 460–465, doi:10.1017/S0305004100012706

• Patrick du Val, (1934c), “On isolated singularities of surfaces which do not affect the conditions of adjunction. III”, Proceedings of the Cambridge Philosophical Society, 30 (4): 483–491, doi:10.1017/S030500410001272X

Textbook accounts include

• Alan H. Durfee, Fifteen characterizations of rational double points and simple critical points, L’Enseignement Mathématique Volume: 25 (1979) (doi:10.5169/seals-50375, pdf)

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

• Klaus Lamotke, chapter IV of Regular Solids and Isolated Singularities, Vieweg, Braunschweig, Wiesbaden 1986.

• Miles Reid, Young persons guide to canonical singularities, in Spencer Bloch (ed.),Algebraic geometry – Bowdoin 1985, Part 1, Proc. Sympos. Pure Math. 46 Part 1, Amer. Math. Soc., Providence, RI, 1987, pp. 345-414 (pdf)

(The last formula on page 409 has a typo: there should be no $r$ in the denominator.)

Discussion in terms of hyper-Kähler geometry:

Reviews and lecture notes include

• Classification of singularities

• Igor Burban, Du Val Singularities (pdf)

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

• Anda Degeratu, Crepant Resolutions of Calabi-Yau Orbifolds, 2004 (pdf)

• Fabio Perroni, Orbifold Cohomology of ADE-singularities (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

Riemannian geometry of manifolds with ADE singularities is discussed in