hyperelliptic(.ncag).info

the classification of hyperelliptic varieties in dimensions 2, 3 and 4

$\mathrm{C}_{3} \times \mathrm{C}_{3}$ in dimension 4

holonomy group
$\mathrm{C}_{3} \times \mathrm{C}_{3}$, order $9$, SmallGroup $[9,2]$ · character table
tangent representation $\rho$
$\rho(g_{1}) = \operatorname{diag}(\zeta_{3}, 1, \zeta_{3}, \zeta_{3}^{2})$
$\rho(g_{2}) = \operatorname{diag}(1, \zeta_{3}, \zeta_{3}, \zeta_{3})$
order of $\omega_X$
3
number of moduli
0
irregularity $q = \dim \operatorname{Aut}^0(X)$
0
Albanese
the Albanese variety is a point; the fiber is $X$ itself (details)
$\mathbf{D}^{\mathrm{b}}(X)$
indecomposability open (why)
Hochschild cohomology $\dim \mathrm{HH}^\bullet$
$(1, 0, 0, 8, 12, 6, 1, 0, 0)$
Hodge diamond
1
0 0
0 4 0
1 3 3 1
0 0 6 0 0
1 3 3 1
0 4 0
0 0
1
polyvector fields $\mathrm{H}^q(X,\bigwedge^p T_X)$
1
00
000
1331
03630
0330
010
00
0

twisted Hodge numbers $\mathrm{h}^{p,q}(X,\omega_X^{\otimes j})$

Since $\operatorname{ord} \omega_X = 3$, the twists by powers of the canonical bundle form a finite package of $3$ diamonds (explained).

$\omega_X^{\otimes 0}$
1
00
040
1331
00600
1331
040
00
1
$\omega_X^{\otimes 1}$
0
01
030
0330
01601
0330
030
01
0
$\omega_X^{\otimes 2}$
0
10
030
0330
10610
0330
030
10
0

References

  1. A. Demleitner, The classification of hyperelliptic groups in dimension 4. arXiv:2211.07998
  2. A. Demleitner, C. Gleissner, The classification of rigid hyperelliptic fourfolds, Ann. Mat. Pura Appl. (4) 202 (2023) 1425–1450. MR4576947 doi
  3. P. Belmans, A. Demleitner, P. Núñez, The Albanese morphism for hyperelliptic varieties, Indag. Math. (N.S.) 37 (2026) 1450–1475. MR5103244 doi