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A Calabi-Yau database: threefolds constructed from the Kreuzer-Skarke list. (English) Zbl 1388.53071
Summary: M. Kreuzer and H. Skarke [Adv. Theor. Math. Phys. 4, No. 6, 1209–1230 (2000; Zbl 1017.52007)] famously produced the largest known database of Calabi-Yau threefolds by providing a complete construction of all 473,800,776 reflexive polyhedra that exist in four dimensions. These polyhedra describe the singular limits of ambient toric varieties in which Calabi-Yau threefolds can exist as hypersurfaces. In this paper, we review how to extract topological and geometric information about Calabi-Yau threefolds using the toric construction, and we provide, in a companion online database (see http://nuweb1.neu.edu/cydatabase), a detailed inventory of these quantities which are of interest to physicists. Many of the singular ambient spaces described by the Kreuzer-Skarke list can be smoothed out into multiple distinct toric ambient spaces describing different Calabi-Yau threefolds. We provide a list of the different Calabi-Yau threefolds which can be obtained from each polytope, up to current computational limits. We then give the details of a variety of quantities associated to each of these Calabi-Yau such as Chern classes, intersection numbers, and the Kähler and Mori cones, in addition to the Hodge data. This data forms a useful starting point for a number of physical applications of the Kreuzer-Skarke list.

##### MSC:
 53C55 Global differential geometry of Hermitian and Kählerian manifolds 53-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to differential geometry 14-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to algebraic geometry 14J32 Calabi-Yau manifolds (algebro-geometric aspects)
##### Keywords:
differential and algebraic geometry; superstring vacua
##### Software:
Calabi-Yau database; PALP; SageMath; TOPCOM
Full Text:
##### References:
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