Kummer quartics and associatedsymplectic 6-folds

5/25/23(Th) Univ. Milano
Shigeru MUKAICorrected on 8/18/24(S).

Abstract: Kondo(1998) described the automorphism group of thegeneric Jacobean Kummer surface Km(C) using a Conway
chamber in the nef cone. As a sample of higher dimensional
analogue, we describe the binational automorphism group of acertain holomorphic symplectic 6-fold associated with Km(C)using the Borcherds(2000) reflection group. Bir-Aut is generated
by 864 involutions and an extended extraspecial group 2^{1+8} .2
whose center is Rapagnetta(2007)'s involution.

§1 Motivation/Introduction
General Study Bir-Aut(IHS) similarly to Aut(K3), replacing
the nef cone Nef(K3) with the movable cone Mov(IHS)
Peculiar (today) 1. Higher dimensional analogue ofKondo(1998)'s description of Aut(generic Jacobian Kummer)
2. 5 reflection groups in R^{1,17}
L := II_{1,17}(2^{+2a}), a = 0, 1, 2, 3, 4
even integral lattice of signature (1, 17), 2-elementary, and2aq : L /L = Z/2Z

Q/2Z is evena = 0, 1, 2 ⇄ L is realized as the Picard lattice of a K312aL0
unimodular
U+E8+E8
realization S
P1with 2E8〜# of (-2)'sand (-4)'s2
4
U+E8+D8 U+D8+D8double
P1×P1

19020
22today

Tablev

68U+Kum U+BW

Km(E1×E2) today
24
2464896

not (yet)0∞branchBarth-Peters
surface

Fact: Reflections generate a subgroup of finite index in O已(L)in each case.
Remark II_{1, 17}(2^{+2}), 2a=2, belongs to the Conway-Vinberg
chain, on which I gave a talk in the last JES(2022).
Notation U = (Z^2,0110),〜An, Dn, E6, E7, E8: negative definite root lattices, An etc.: aﬃnenegative definite Kum コ 16A1 Kummer lattice{ BW Barnes-Wall lattice, (Leech lattice) A rinv
rank 162

Borcherds(2000, 2a=6)(*)

O已(II_{1,17}(2^{+6}) =group generated by960 reflections-.

G.L (2)

G. L∠(2) = (symmetry of fundamental domain), #G = 2^10, L (2) = VT8~Today Give a geometric interpretation of (*), that is, describe BirAut(X) for holomorphic symplectic 6-fold X=X(S, h) associatedwith (Picard general) Kummer quartic surface (S, h).

「3S = Km(C)ρ(S)=17, h: pull-back of OB(1)Jac C/±14 P^3|2Θ|Kum is an overlattice of 16A1 = + Zei, (e 主^2) = -2
iGT22

T = C^2/Γ, Γ = Z^4, T( ) : group of 2-torsions〜2Kum = + Zei+ Z

Ui ε T( z{ 計Σ e H T subgroup of order 8, 16}
iccn

Qe i: GTiz,)*)BW sublattice of index 2 and of Leech type, i.e., 日 (-2)-element§2 Another background
X = Mδ(0, h, a), a = 0, 1, holomorphic symplectic 6-folda[3すbirationally equivalent to S

{ not equivalent

when a = 1
when a = 0〜

Birational double cover of KJac (0, 2Θ, 2a), which is OG6(symplectic resolution of Albanese fibration
M(2v)

Jac>

Jac × Jac, v = (0, Θ, a)of moduli of sheaves on Jac = Jac C. (Studied by
Rapagnetta(2004, thesis), Mongardi-Rapagnetta-Sacca(2018),etc.)女) BW is also constructed from the Reed-Muller code of length 16 (cf. [Conway-Sloane]).

/ξ E Ms (0, h, a)4torsion sheaf on SSupp ξ E lhl = P^3v

母D,pa(D) = 3

This morphism is a Lagrangian fibration. Generic fiber is Pica D(Abelian 3-fold).

(Pic X , Beauville form) = (0, h, a) in Z + Pic S + Z
when a = 1
泛 UPic+ S(h +in<-4>
Pic S) = U + Kum when a = 0〜

X:= XO = Mδ (0, h, 0) from now on
Advantage of X: Fourier-M. transform with Kernel

D Poincaréline bdl

6descends to S × S = S × M (2, - 辶 Σ e , -2)
定δinterchangedby FMtransf.

√Jac × JacΦI = Φ:XP^3,*S,ξSupp ξΦ2= Φ:XP^3S = S, ξSupp ξIhilIklal

Pic X っ U(2) + BW
<H , H2>.

Rapagnetta's
symplectic
birational
involution[3]

K3
covering involution of ↓OG66+1-1× O ( -- Σ ei)
S2,5§3 Main Theorem
960 reflections = 64 (-2) refl's and 896 (-4) refl's♢ (-2)'s : ョ eﬀective irreducible divisors D c X with Beauville norm(D^2)=-2
Kummer(166 - 166)
configuration〜16 (-2)P1's c S nodes, Φ1 : XP^3,* has 16A1

16 (-2)P1's c S tropes, ΦZ: XP^3 has 16A1〜〜

64 (-2) eﬀective divisors = 32 A1 ○○♢ (-4)'s: 兄 (-4)-divisor classes with divisibility 2, #=896=32 28
864 corresponding to birational involutions of X
{ 32 represented by irreducible eﬀective divisors on X
Main Theorem(S, h): Picard general Kummer, X = Mδ(0, h, 0)(1) Bir-Aut(X) = group generated
by 864 involutionsgroup of order 2^10.

G, where G is Borcherds'(Corrected on Aug. 18, 2024.)(2) (Geom. interpretation of G) G is the semi-direct product H.C2of an extraspecial group, i.e., finite analogue of Heisenberg group,1

Rapagnetta's inv.

H(Jac × Jac)[2 )0by C generated by FM transf. (cf. Mongardi-Wandel(2017)).
(3) 生 28 curves of genus 2 C=C , C , ... , C8 such that X =MSi(0, hi , 0) for i=1, ..., 28 (by Torelli type theorem). Each Cこ,i=1, gives 32 birational involutions of X. Hence we have 864
involutions. 16 (co-)tropes of Km(C) yields 32 irred. divisors.
〜References§1 Motivation/Introduction
Kondo, S.: The automorphism group of a generic Jacobian
Kummer surface, J. Alg. Geom. 7(1998), 589-609.
Borcherds, R.: Reflection groups of Lorentzian lattices, Duke Math.
J., 104(2000), 319-366.
Nikulin, V.V.: On Kummer surfaces, Math. USSR Izv., 9(1975),
261-275.
Nikulin, V.V.: Factor groups of groups of automorphisms ofhyperbolic forms with respect to subgroups generated by 2reflections. Algebrogeometric applications, J. Soviet Math.,
22(1983), 14-1-1475.
Conway, J.H. and Sloane, N.J.A.: Sphere packings, lattices andgroups, Springer-Verlag, 1988.
§2 Another background
Rapagnetta, A.: Topological invariants of O'Grady's six
dimensional irreducible synplectic variety, Math. Z., 256(2007),
1-34.
Mongardi, G., Rapagnetta, A. and Sacca, G.: The Hodge diamondof O’Grady’s six-dimensional example, Comp. Math., 154(2018),
984-1013.
§3 Main Theorem
Mongardi, G. and Wandel, M.: Automorphisms of O'Grady's
manifolds acting trivially on cohomology, Alg. Geom., 4(2017),
104-119.

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