an overview of the literature listing Fano fourfolds
The invariants that appear across the Fano-fourfold literature, grouped by type. Where an
invariant is also stored in the fano-fourfolds corpus, the
data.json field name is shown. The coverage matrix at the bottom shows which invariant each
source reports.
Which invariant each source reports (from its invariants_provided). Column headers link to the explanations below.
| code | construction | hodge | anticanonical | index | fan | numerical | quantum_period | volume | hilbert_series | contractions | lefschetz_defect | picard_rank | chi_T | base_locus |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| SW90a | ✓ | · | · | · | · | · | · | · | · | · | · | · | · | · |
| SSW91 | ✓ | · | · | · | · | · | · | · | · | · | · | · | · | · |
| SW90b | ✓ | · | · | · | · | · | · | · | · | · | · | · | · | · |
| Kuc95 | ✓ | ✓ | ✓ | ✓ | · | · | · | · | · | · | · | · | · | · |
| Bat99 | ✓ | · | · | · | ✓ | ✓ | · | · | · | · | · | · | · | · |
| CKP15 | ✓ | · | · | · | · | · | ✓ | · | · | · | · | · | · | · |
| Ben18 | ✓ | · | · | · | · | · | · | · | · | · | · | · | · | · |
| Kal19 | ✓ | · | · | · | · | · | ✓ | · | · | · | · | · | · | · |
| CGKS20 | · | · | · | · | · | · | ✓ | · | · | · | · | · | · | · |
| HLM22 | · | ✓ | · | · | · | · | · | ✓ | ✓ | ✓ | · | · | · | · |
| CRS22 | ✓ | ✓ | ✓ | · | · | · | · | ✓ | · | · | ✓ | ✓ | ✓ | · |
| Sec23 | ✓ | ✓ | · | · | · | ✓ | · | ✓ | · | · | · | · | · | · |
| HS23 | ✓ | · | · | · | · | ✓ | · | · | · | · | · | · | · | ✓ |
| Pas25 | ✓ | ✓ | · | · | · | ✓ | · | · | · | · | ✓ | · | · | · |
| Kaw89 | · | · | · | · | · | · | · | · | · | ✓ | · | · | · | · |
| AW96 | · | · | · | · | · | · | · | · | · | ✓ | · | · | · | · |
| Tak99 | · | · | · | · | · | · | · | · | · | ✓ | · | · | · | · |
| BCW02 | · | · | · | · | · | · | · | · | · | ✓ | · | · | · | · |
| Tsu10a | · | · | · | · | · | · | · | · | · | ✓ | · | · | · | · |
| Tsu10b | · | · | · | · | · | · | · | · | · | ✓ | · | ✓ | · | · |
| Fuj14 | · | · | · | · | · | · | · | · | · | ✓ | · | · | · | · |
| Ou17 | · | · | · | · | · | · | · | · | · | ✓ | · | · | · | · |
| $\rho_X = \mathrm{h}^{1,1}$ | Picard rankhodge[1][1] | Rank of the Picard group; equals $\mathrm{h}^{1,1}$ for these fourfolds. The primary sorting invariant of most classifications (Picard number 1, 2, 3, …). |
| $\delta_X$ | Lefschetz defect | Casagrande's Lefschetz defect. Classifications by $\delta_X = 3$ (CRS22) and $\delta_X = 2$ (Pas25). |
| $r_X$ | Fano index | Largest integer dividing $-\mathrm{K}_X$ in $\operatorname{Pic}(X)$. Küchle's fourfolds are the index-1, Picard-number-1 prime case. |
| $\mathrm{h}^{p,q}$ | Hodge diamondhodge | The full Hodge diamond. For Fano fourfolds $\mathrm{h}^{2,2}$, $\mathrm{h}^{1,1}$, $\mathrm{h}^{1,2}$, $\mathrm{h}^{1,3}$ carry the content; Betti numbers and the Euler characteristic follow. Reported by Küchle, HLM22, Sec23, Pas25. |
| $(-\mathrm{K}_X)^4$ | Anticanonical degree (volume)volume | Top self-intersection of $-\mathrm{K}_X$. The coarse first key for matching. |
| $\mathrm{h}^0(-\mathrm{K}_X)$ | Anticanonical sectionsanticanonical | Dimension of the anticanonical system. |
| $H_X(t)$ | Hilbert series | Anticanonical Hilbert series / polynomial (provided by HLM22). |
| $\operatorname{Bs}|-\mathrm{K}_X|$ | Anticanonical base locus | Dimension/geometry of the base locus of $|-\mathrm{K}_X|$, the classifying datum of HS23 (base locus of dimension two). |
| $\chi(T_X)$ | Euler characteristic of the tangent sheaftangent | Not generally tabulated in the source papers, but computed for the FanoBase corpus; a useful discriminant. |
| $\mathrm{HH}^k$, $\mathrm{h}^q(\wedge^p T_X)$ | Hochschild cohomology / polyvector parallelogramhh, polyvector | Hochschild cohomology and the HKR polyvector-field parallelogram. Computed in FanoBase; a fine discriminant beyond Hodge data. |
| $G_X(t)$ | Regularized quantum period | Regularized quantum-period coefficients / J-function. The organizing invariant of the mirror-symmetry lists (CKP15, Kal19, CGKS20) and an independent check on identifications. |
Ambient / defining dataambient, bundle | The explicit model: ambient variety + defining bundle, Cox ring, quiver, or projective-bundle presentation. | |
| $\Sigma$ | Toric fan | Rays and maximal cones of the fan (Batyrev's toric list). |
| Elementary contractions | Structure of extremal/elementary contractions (exceptional loci, centres). The organizing datum of the extremal-contraction papers and of HLM22 §6. | |
| Assorted numerical invariants | Catch-all for the numerical-invariant tables (degrees, Chern numbers, contraction data) provided by Bat99, Sec23, HS23, Pas25. |