$\mathrm{C}_{4}$ in dimension 3
- holonomy group
- $\mathrm{C}_{4}$, order $4$, SmallGroup $[4,1]$ · character table
- tangent representation $\rho$
- $\rho(g_{1}) = \operatorname{diag}(1, i, -i)$
- order of $\omega_X$
- 1
- number of moduli
- 3
- 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 $\dim \mathrm{HH}^\bullet$
- $(1, 2, 5, 8, 5, 2, 1)$
Hodge diamond
1
1 1
1 3 1
1 3 3 1
1 3 1
1 1
1
1 1
1 3 1
1 3 3 1
1 3 1
1 1
1
polyvector fields $\mathrm{H}^q(X,\bigwedge^p T_X)$
1
11
131
1331
131
11
1
11
131
1331
131
11
1
References
- K. Uchida, H. Yoshihara, Discontinuous groups of affine transformations of $\mathbb{C}^3$, Tôhoku Math. J. (2) 28 (1976) 89–94. MR400271 doi
- H. Lange, Hyperelliptic varieties, Tôhoku Math. J. (2) 53 (2001) 491–510. MR1862215 doi
- F. Catanese, A. Demleitner, The classification of hyperelliptic threefolds, Groups Geom. Dyn. 14 (2020) 1447–1454. MR4186481 doi
- F. Catanese, A. Demleitner, Hyperelliptic threefolds with group $\mathrm{D}_4$. arXiv:1805.01835
- P. Belmans, A. Demleitner, P. Núñez, The Albanese morphism for hyperelliptic varieties, Indag. Math. (N.S.) 37 (2026) 1450–1475. MR5103244 doi