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Published November 17, 2021 | Version v1.0
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Regularized quantum periods for four-dimensional Fano manifolds

  • 1. Imperial College London
  • 2. University of Nottingham


The database smooth_fano_4

This is a database of regularized quantum periods for four-dimensional Fano manifolds. The database will be updated as new four-dimensional Fano manifolds are discovered and new regularized quantum periods computed.

Each entry in the database is a key-value record with keys and values as described in the paper [CK2021]. If you make use of this data, please cite that paper and the DOI for this data:



The database describes Fano varieties via names, as follows:

Names of Fano manifolds
Name Description
P1 one-dimensional projective space
P2 two-dimensional projective space
dP(k) the del Pezzo surface of degree k given by the blow-up of P2 in 9-k points
P3 three-dimensional projective space
Q3 a quadric hypersurface in four-dimensional projective space
B(3,k) the three-dimensional Fano manifold of Picard rank 1, Fano index 2, and degree 8k
V(3,k) the three-dimensional Fano manifold of Picard rank 1, Fano index 1, and degree k
MM(r,k) the k-th entry in the Mori-Mukai list of three-dimensional Fano manifolds of Picard rank r, ordered as in [CCGK2016]
P4 four-dimensional projective space
Q4 a quadric hypersurface in five-dimensional projective space
FI(4,k) the four-dimensional Fano manifold of Fano index 3 and degree 81k
V(4,k) the four-dimensional Fano manifold of Picard rank 1, Fano index 2, and degree 16k
MW(4,k) the k-th entry in Table 12.7 of [IP1999] of four-dimensional Fano manifolds of Fano index 2 and Picard rank greater than 1
Obro(4,k) the k-th four-dimensional Fano toric manifold in Obro's classification [O2007]
Str(k) the k-th Strangeway manifold in [CGKS2020]
CKP(k) the k-th four-dimensional Fano toric complete intersection in [CKP2015]
CKK(k) the k-th four-dimensional Fano quiver flag zero locus in Appendix B of [K2019]

A name of the form "S1 x S2", where S1 and S2 are names of Fano manifolds X1 and X2, refers to the product manifold X1 x X2.


[CCGK2016] Quantum periods for 3-dimensional Fano manifolds; Tom Coates, Alessio Corti, Sergey Galkin, Alexander M. Kasprzyk; Geometry and Topology 20 (2016), no. 1, 103-256.

[CGKS2020] Quantum periods for certain four-dimensional Fano manifolds; Tom Coates, Sergey Galkin, Alexander M. Kasprzyk, Andrew Strangeway; Experimental Math. 29 (2020), no. 2, 183-221.

[CK2021] Databases of quantum periods for Fano manifolds; Tom Coates, Alexander M. Kasprzyk; 2021.

[CKP2015] Four-dimensional Fano toric complete intersections; Tom Coates, Alexander M. Kasprzyk, Thomas Prince; Proc. Royal Society A 471 (2015), no. 2175, 20140704, 14.

[IP1999] Fano varieties; V.A. Iskovskikh, Yu. G. Prokhorov; Encyclopaedia Math. Sci. vol. 47, Springer, Berlin, 1999, 1-247.

[K2019] Four-dimensional Fano quiver flag zero loci; Elana Kalashnikov; Proc. Royal Society A 275 (2019), no. 2225, 20180791, 23. 

[O2007] An algorithm for the classification of smooth Fano polytopes; Mikkel Obro; arXiv:0704.0049 [math.CO]; 2007.



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Additional details


GWT – Gromov-Witten Theory: Mirror Symmetry, Birational Geometry, and the Classification of Fano Manifolds 682603
European Commission
Classification, Computation, and Construction: New Methods in Geometry EP/N03189X/1
UK Research and Innovation
The Combinatorics of Mirror Symmetry EP/N022513/1
UK Research and Innovation