hyperelliptic(.ncag).info

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

$\mathrm{D}_4$ in dimension 3

holonomy group
$\mathrm{D}_4$, order $8$, SmallGroup $[8,3]$ · character table
tangent representation $\rho$
$\rho(g_{1}) = \operatorname{diag}(1, -1, -1)$
$\rho(g_{2}) = \operatorname{diag}(1, -1, -1)$
$\rho(g_{3}) = \operatorname{diag}(1, -1, -1)$
$\rho \cong$ $\chi_{ 4 }$ $\oplus$ $\chi_{ 5 }$
order of $\omega_X$
1
number of moduli
2
irregularity $q = \dim \operatorname{Aut}^0(X)$
0
Albanese
the Albanese variety is a point; the fiber is $X$ itself (details)
$\mathbf{D}^{\mathrm{b}}(X)$
indecomposable
Hochschild cohomology $\dim \mathrm{HH}^\bullet$
$(1, 0, 2, 6, 2, 0, 1)$
Hodge diamond
1
0 0
0 2 0
1 2 2 1
0 2 0
0 0
1
polyvector fields $\mathrm{H}^q(X,\bigwedge^p T_X)$
1
00
020
1221
020
00
1

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