Group-Quark matrix and Leech-K3 analogueof del Pezzo surfaces
2/21/23(T) S. Mukai

Abstract: Groups are analyzed by taking normalizer of suitable
subgroups. For example, the (quaternionic) Wolf space and Weylgroup are attached to each simple Lie group in this way. As a discrete
analogy, Harada(2001) considered a 3 by 3 arrangement of certain
finite groups, terminated at the Monster, and asked their algebrogeometric meaning. The first row is one of Arnold's trinities and delPezzo surfaces play the main role there. I will discuss the groups in thesecond focusing on their (mostly conjectural) connection withsymplectic (or "quaternionic") varieties related to supersingular K3
surfaces.
§1 Background/Motivation
X : compact complex manifold, two long standing problems QK&HKQK

contact & Fano ⇨ homogeneous?

2 MHへ(X , ω) , ωハ (dω) nowhere vanishing, c > 0 ⇨ X = G/Pθ ?
11-form

1 は)
the unique closed orbit of GA P(P θ = (Heisenberg).G*

1 → C1 → H

2g・3

◦

2g 4・、→C→ 0, g: dual Coxeter number7.*」
Example 1. A -type G=SL(n+1) ⇨ ﬂag in P
x [P"

h2. E6 type, g=12"

凶P。 = C.C .SL(6)As 〈 EG:⊖2HK Classify irreducible holomorphic
simplectic manifolds (K3-like mfds).
2M(X , ω), ωω nowhere vanishing (hence c も =0)等

dim=2 ⇨ K3 surface, e.g., quartic surface in (P3( n)K3 n-th symmetric productmin. res. ofsingularities

へ

Ts ( v1↑

torus analogue

OG10
1024... ...
2n 67 82[n]def. class K3 K3
dim.
2 2nB222 23

Are these all (deformation types)?

( m ≥2)

3§2 Group-Quark matrix and its suggestion
Harada (2001)
W(E 6)

W(E7 )

41212

16.1.2L (4)Fi 「 がーM

ど八2

た。 (2)

TBFischerBaby monster

3lattice

W(E8)

※0)

2M

I 1,9

II 1,25no lattices
but VOA'sが1

Monstervertical: take normalizer of a suitable 2-extraspecial
1 +21てsubgroup 2 , discrete analogy of Heisenberg group.
horizontal: Arnold's trinities, say, E6 , E 7 , E8Result:Tetra
g=Coxeter # 12
g-8
4K3-like(g-8)-fold

[2]

K3

Octa
1810Icosa30
22

OG10 ? ヨ ?

TetraOcta

Icosa「G-Q matrix's suggestion:
Does there exist a K3-like
22- or 20-fold with B = 25?

4§3 Basic 1st line ⇄ del Pezzo surface of degree d=3, 2, 1
3(1, 1) S3 く)P smooth cubic surface
Well known: 9 27 lines on S 3Aut(conﬁg. of 27 lines) = W(E 6 )
33

3

3

G-Q understanding: consider special case x + y + z + t = 0
over iF , Fermat/Hermitian. Hence we have U 4(2) → W(E6 ),injective & image of index 2.
Standard way of understanding:

6 disjoint lines

h-_-,、

16"
S = Bl6 P門H (S,IZ)
7 2
2w. cup= (Zy , t - x -...-x6 )product
22

1c (S)1⇄ (3, 111111)
れc I (S)R

p、= EG -lattice

R(neg. deﬁnite)

P

4PO)な

1 = W(E ) is an isomorphism, and RHS isAut(config.) → O(E6 , Z)
62

generated by (-2)-reﬂections x → x + (x.a)a, with (a )=-2.
、

5§4 I 1 9 and II 1 25,,Both are unimodular hyperbolic lattices. I is odd and II is even, that is,2(x ) even for all x E II.
Both have beautiful fundamental domains & str. of orthog. grps

0 (Ia,,Z) ~{城Pby h)

refHi

り

で W ( ED・uneWIE )on s

Symmetry of afund. domain

Coxeter group(Conway ‘80’s)

0 位は R)~,positive cone○し)|'bySenatedに)refle.tns

丿^24 Cq
八八LeechlatticeO(Leech)区、'""

fund. domain、O6§5 From del Pezzo to K3-like

Pic X = Div X/ { (f), f: rat. fn}

dtPic Sd= H (Sd ,'Z) = (<-1> ) in I 1,9Div X = + Z [D]D CX

codim 1, irred.

Make analogy.

K3-like variety X is Leech-K3-analogue of del Pezzo ifd(Pic X, BBF-form) = (<-2> ) in II1 5 in narrow sense
」

,=R

2

2for root sublattice R〈II1,25in broad sense§6 Example (broad sense, Dolgachev-Kondo, supersingular, char. p=2)
332H

S = SDK : Σx yi= ΣxIy = 0 in [P x [P / F ⇄ R = D4.

i、:

ρ= B2 = 22i.

2def.

§7 Example (narrow sense)[2 ]」Tetra: X = S DK, Pic X = (<-2>+ <-2>+<-2>) in II1birational action of U6(2),~25, ρ=23

2

Octa: X = MS (2v) for suitable Mukai vector v with (v )=2 on S = S DK(conjecture, Co2 action not found yet, 1 Co 2-config. of (-2)-divisors)
Icosa : X might be 22- or 20dimensional?! (A reason given below.)7§8 Maximal subgroup and Lagrangian fibration
XLagrangian fibration = h. dim'l generalization of elliptic
fibrtaion of K3 surface
119Tetra case: 2 L 3 (4) is contained in U6 (2).
X has a Lagrangian fibration over P.

P。

2hn11(quasi-)abelianfinration」

229L 3(4) is Aut( [P ) over [F4 and 2 acts in fiber direction
Rem: L (4) = M (10Octa case: 2 .M 2 Lis contained in Co2.

2g5

X has a Lagrangian fibration over P .
-1on which M2 2acts.

(projectivization of irrep. of central extension 3.M )2211Icosa case: 2 .M 4is contained in Co 1 .
2Minimal projective space with action of M24is 10dimensional. So minimal possibility of dim X is it's twice,
20.
Space of Golay codes is a natural 12-dim'l rep. of M 4. In
this another possibility, X is 22 dim'l.
2

Postscript (3/3/23(F)) The possibility of dim X = 20 above wasoverlooked in my talk.

References§1
Beauville, A., Holomorphic symplectic geometry: a problem list, in"Complex and diﬀerential geometry", pp. 49-63, Springer-Verlag,
2011. (See Sect. 3 for the conjecture "contact+Fano implies
homogeneous".)§2
Harada, K., Group-quark matrix?, RIMS Kokyuroku, 1228(2001),
153-160.
Arnold, V.I., Polymathematics: is mathematics a single art or a setof arts?
§3Manin, Yu.I., Cubic forms: Algebra, Geometry, Arithmetic.
§4Conway, J.H., The automorphism group of the 26-dimensional even
unimodular Lorentzian lattice, J. Algebra, 1983. Chap. 27 of [SPLG].
§6Dolgachev, I. and Kondo, S., A supersingular K3surface incharacteristic 2 and the Leech lattice, Int'l. Math. Res. Notices,
2003, pp. 1-23.

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