hyperelliptic(.ncag).info

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

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

holonomy group
$\mathrm{C}_{4}$, order $4$, SmallGroup $[4,1]$ · character table
tangent representation $\rho$
$\rho(g_{1}) = \operatorname{diag}(1, i, -1)$
order of $\omega_X$
4
number of moduli
2
irregularity $q = \dim \operatorname{Aut}^0(X)$
1
Albanese
the Albanese variety has dimension $1$; the general fiber has dimension $2$ and is an abelian variety or a hyperelliptic variety (details)
$\mathbf{D}^{\mathrm{b}}(X)$
indecomposable
Hochschild cohomology $\dim \mathrm{HH}^\bullet$
$(1, 2, 2, 4, 5, 2, 0)$
Hodge diamond
1
1 1
0 3 0
0 2 2 0
0 3 0
1 1
1
polyvector fields $\mathrm{H}^q(X,\bigwedge^p T_X)$
1
11
020
0220
131
11
0

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

Since $\operatorname{ord} \omega_X = 4$, the twists by powers of the canonical bundle form a finite package of $4$ diamonds (explained).

$\omega_X^{\otimes 0}$
1
11
030
0220
030
11
1
$\omega_X^{\otimes 1}$
0
10
121
0321
121
10
0
$\omega_X^{\otimes 2}$
0
11
121
0220
121
11
0
$\omega_X^{\otimes 3}$
0
01
121
1230
121
01
0

References

  1. K. Uchida, H. Yoshihara, Discontinuous groups of affine transformations of $\mathbb{C}^3$, Tôhoku Math. J. (2) 28 (1976) 89–94. MR400271 doi
  2. H. Lange, Hyperelliptic varieties, Tôhoku Math. J. (2) 53 (2001) 491–510. MR1862215 doi
  3. F. Catanese, A. Demleitner, The classification of hyperelliptic threefolds, Groups Geom. Dyn. 14 (2020) 1447–1454. MR4186481 doi
  4. F. Catanese, A. Demleitner, Hyperelliptic threefolds with group $\mathrm{D}_4$. arXiv:1805.01835
  5. P. Belmans, A. Demleitner, P. Núñez, The Albanese morphism for hyperelliptic varieties, Indag. Math. (N.S.) 37 (2026) 1450–1475. MR5103244 doi