explained
Background on hyperelliptic varieties and on the invariants computed on this site. The pages below go into detail; this index collects the terminology and the invariants in one place. See also the references.
Terminology
- Hyperelliptic (generalized hyperelliptic) variety. A quotient $X = T/G$ of a complex torus $T = V/\Lambda$ by a finite group $G$ acting freely and holomorphically, with no element acting as a translation. In dimensions $1$ and $2$ these are elliptic curves and bielliptic surfaces; the name has nothing to do with hyperelliptic curves.
- Holonomy group. The finite group $G$. Equivalently the image of $\pi_1(X)$ in $\mathrm{GL}(V)$: the deck group of the étale cover $T \to X$, acting linearly on the tangent space.
- Tangent representation $\rho\colon G \to \mathrm{GL}(V)$. The action of $G$ on the tangent space $V = \mathbb{C}^n$ of $T$ at $0$. Every invariant on this site is computed from $\rho$ alone (via character theory), independently of the translation parts of the action.
- Free action. $G$ acts without fixed points, so $X$ is smooth. A necessary condition is that every $g \neq 1$ has eigenvalue $1$ on $V$.
- Faithful / without translations. $\rho$ is injective: no nontrivial element acts purely by translation. This makes $G$ the genuine holonomy.
- Bielliptic surface. A hyperelliptic surface (dimension $2$); there are seven types, due to Bagnera–De Franchis.
- Bagnera–de Franchis variety. A generalized hyperelliptic variety whose holonomy $G$ is cyclic. In dimension $2$ every hyperelliptic surface is of this kind (holonomy $\mathrm{C}_2,\mathrm{C}_3,\mathrm{C}_4$ or $\mathrm{C}_6$), so "Bagnera–de Franchis" and "hyperelliptic" coincide there. In higher dimensions they are the cyclic-holonomy entries here; non-cyclic holonomy (already $\mathrm{D}_4$ in dimension $3$) gives the strictly generalized hyperelliptic varieties. See Demleitner, Classification of Bagnera–de Franchis varieties in small dimensions.
Invariants
All are functions of $(G,\rho)$; the formulas and the computation are on the invariants page. In brief:
- Hodge diamond $\mathrm{h}^{p,q}(X) = \dim(\bigwedge^p V^\ast \otimes \bigwedge^q \overline V^\ast)^G$.
- Order of the canonical bundle. $\omega_X$ is torsion; its order is $|\det\rho(G)|$.
- Irregularity $q = \dim V^G = \dim \operatorname{Alb}(X) = \dim \operatorname{Aut}^0(X)$.
- Number of moduli $\dim \mathrm{H}^1(X,\Theta_X)$, the dimension of the family.
- Polyvector fields $\mathrm{H}^q(X,\bigwedge^p T_X)$, whose anti-diagonals give the Hochschild cohomology $\mathrm{HH}^\bullet(X)$.
- Twisted Hodge numbers $\mathrm{h}^{p,q}(X,\omega_X^{\otimes j})$, a finite package since $\omega_X$ is torsion.
- Albanese fiber and the derived category $\mathbf{D}^{\mathrm{b}}(X)$: see the Albanese and indecomposability pages.
Pages
- completeness in dimension 4: why the dimension-4 list is complete but possibly redundant
- indecomposability: semiorthogonal indecomposability of the derived category
- invariants from representation theory: how the Hodge numbers and other invariants are computed
- not hyperelliptic curves: what the word "hyperelliptic" means here
- the Albanese morphism: the Albanese variety and its fibers
- twisted Hodge numbers: the finite package of twists by powers of the canonical bundle