hyperelliptic(.ncag).info

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

$\mathrm{C}_{2}$ in dimension 4

This tangent representation satisfies the necessary conditions for a hyperelliptic fourfold, but it is not known whether it is realized by an actual variety. See completeness in dimension 4.
holonomy group
$\mathrm{C}_{2}$, order $2$, SmallGroup $[2,1]$ · character table
tangent representation $\rho$
$\rho(g_{1}) = \operatorname{diag}(1, 1, -1, -1)$
order of $\omega_X$
1
number of moduli
8
irregularity $q = \dim \operatorname{Aut}^0(X)$
2
Albanese
the Albanese variety has dimension $2$; the general fiber has dimension $2$ and is an abelian variety or a hyperelliptic variety (details)
$\mathbf{D}^{\mathrm{b}}(X)$
indecomposable
Hochschild cohomology $\dim \mathrm{HH}^\bullet$
$(1, 4, 12, 28, 38, 28, 12, 4, 1)$
Hodge diamond
1
2 2
2 8 2
2 12 12 2
1 8 20 8 1
2 12 12 2
2 8 2
2 2
1
polyvector fields $\mathrm{H}^q(X,\bigwedge^p T_X)$
1
22
282
212122
182081
212122
282
22
1

References

  1. A. Demleitner, The classification of hyperelliptic groups in dimension 4. arXiv:2211.07998
  2. A. Demleitner, C. Gleissner, The classification of rigid hyperelliptic fourfolds, Ann. Mat. Pura Appl. (4) 202 (2023) 1425–1450. MR4576947 doi
  3. P. Belmans, A. Demleitner, P. Núñez, The Albanese morphism for hyperelliptic varieties, Indag. Math. (N.S.) 37 (2026) 1450–1475. MR5103244 doi