Bagnera–De Franchis type 5
The generator translates the first factor by a $4$-torsion point and multiplies the harmonic curve $E_i$ by $i$.
- holonomy group
- $\mathrm{C}_{4}$, order $4$ · character table
- full acting group (with translations)
- $\mathbb{Z}/4$
- torus
- $E_\tau \times E_i$
- order of $\omega_X$
- 4
- number of moduli
- 1
- irregularity $q = \dim \operatorname{Aut}^0(X)$
- 1
- Albanese
- an elliptic curve; the general fiber is an elliptic curve (details)
- $\mathbf{D}^{\mathrm{b}}(X)$
- indecomposable
- Hochschild cohomology $\dim \mathrm{HH}^\bullet$
- $(1, 2, 1, 0, 0)$
Hodge diamond
1
1 1
0 2 0
1 1
1
1 1
0 2 0
1 1
1
polyvector fields $\mathrm{H}^q(X,\bigwedge^p T_X)$
1
11
010
00
0
11
010
00
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
020
11
1
11
020
11
1
$\omega_X^{\otimes 1}$
0
01
011
01
0
01
011
01
0
$\omega_X^{\otimes 2}$
0
00
000
00
0
00
000
00
0
$\omega_X^{\otimes 3}$
0
10
110
10
0
10
110
10
0
References
- G. Bagnera, M. de Franchis, Le superficie algebriche le quali ammettono una rappresentazione parametrica mediante funzioni iperellittiche di due argomenti, Mem. Soc. Ital. Sci. (3) 15 (1908)
- A. Demleitner, Classification of Bagnera–de Franchis varieties in small dimensions, Ann. Fac. Sci. Toulouse Math. (6) 29 (2020) 111–133. MR4809689 doi
- H. Lange, Hyperelliptic varieties, Tôhoku Math. J. (2) 53 (2001) 491–510. MR1862215 doi