hyperelliptic(.ncag).info

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

$\mathrm{C}_{5}$ in dimension 3

holonomy group
$\mathrm{C}_{5}$, order $5$, SmallGroup $[5,1]$ · character table
tangent representation $\rho$
$\rho(g_{1}) = \operatorname{diag}(1, \zeta_{5}, \zeta_{5}^{2})$
order of $\omega_X$
5
number of moduli
1
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 $\dim \mathrm{HH}^\bullet$
$(1, 2, 1, 2, 4, 2, 0)$
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
010
0110
121
11
0

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

Since $\operatorname{ord} \omega_X = 5$, the twists by powers of the canonical bundle form a finite package of $5$ diamonds (explained).

$\omega_X^{\otimes 0}$
1
11
030
0220
030
11
1
$\omega_X^{\otimes 1}$
0
10
111
0211
111
10
0
$\omega_X^{\otimes 2}$
0
01
021
0130
021
01
0
$\omega_X^{\otimes 3}$
0
10
120
0310
120
10
0
$\omega_X^{\otimes 4}$
0
01
111
1120
111
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