Invariants from representation theory
Because $G$ acts freely on $T$, the quotient map is étale and
$$ \mathrm{H}^k(X,\mathbb{C}) = \mathrm{H}^k(T,\mathbb{C})^G . $$The translation parts $\tau(g)$ act trivially on cohomology, so every invariant below depends only on the tangent representation $\rho\colon G \to \mathrm{GL}(V)$, not on the group action.
- Hodge numbers. $$ \mathrm{h}^{p,q}(X) = \dim\big(\textstyle\bigwedge^p V^\ast \otimes \bigwedge^q \overline{V}^\ast\big)^{G}, $$ computed as the multiplicity of the trivial character in the character of $\bigwedge^p V^\ast \otimes \bigwedge^q \overline{V}^\ast$.
- Order of the canonical bundle. $\omega_X$ is torsion of order $|\det \rho(G)| = \operatorname{lcm}_{g} \operatorname{ord}(\det \rho(g))$.
- Number of moduli. $\dim \mathrm{H}^1(X, \Theta_X) = \dim(V \otimes \overline{V}^\ast)^G$; the deformations are unobstructed.
- Irregularity. $q = \dim V^G = \dim \operatorname{Alb}(X) = \dim \operatorname{Aut}^0(X)$.
- Polyvector fields. $\mathrm{H}^q(X, \bigwedge^p T_X) = (\bigwedge^p V \otimes \bigwedge^q \overline{V}^\ast)^G$, displayed as a parallelogram on each object page; its anti-diagonals sum to the Hochschild cohomology $\mathrm{HH}^m(X) = \bigoplus_{p+q=m} \mathrm{H}^q(X, \bigwedge^p T_X)$.
- Holomorphic Poisson structures. These are the bivector fields $\mathrm{H}^0(X, \bigwedge^2 T_X) = (\bigwedge^2 V)^{G}$ (the left edge of the parallelogram). A holomorphic bivector on $X$ lifts to a $G$-invariant one on $T$, which is translation-invariant, hence constant, hence automatically Poisson ($[\pi,\pi]=0$). So $X$ carries a non-trivial holomorphic Poisson structure exactly when $(\bigwedge^2 V)^{G} \neq 0$. This never happens for the bielliptic surfaces (there $\bigwedge^2 V$ is the non-trivial determinant character), but it does for some threefolds and fourfolds.
All of these are computed with OSCAR from the character table of $G$; see the source.
The formulas and the general framework are due to Demleitner.
References
- A. Demleitner, The classification of hyperelliptic groups in dimension 4. arXiv:2211.07998
- H. Lange, Hyperelliptic varieties, Tôhoku Math. J. (2) 53 (2001) 491–510. MR1862215 doi
- C. Birkenhake, H. Lange, Complex abelian varieties, 2nd ed., Grundlehren 302, Springer (2004). MR2062673 doi