Abstract
Suppose that X is a Fano manifold that corresponds under Mirror Symmetry to a Laurent polynomial f, and that P is the Newton polytope of f. In this setting it is expected that there is a family of algebraic varieties over the unit disc with general fiber X and special fiber the toric variety defined by the spanning fan of P. Building on recent work and conjectures by Corti–Hacking–Petracci, who construct such families of varieties, we determine the topology of the general fiber from combinatorial data on P. This provides evidence for the Corti–Hacking–Petracci conjectures, and verifies that their construction is compatible with expectations from Mirror Symmetry.
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Notes
- 1.
We learned the statement of Theorem 1 from Andrea Petracci.
- 2.
The fiber \(F_v\) is homologous in Y to the sum of toric curves corresponding to 2-dimensional cones in \(\Sigma \) that contain the ray spanned by v and that lie entirely on one side of the hyperplane defined by the edge e. The choice of side does not matter here, as the resulting sums are homologous.
- 3.
- 4.
The number of Minkowski polynomials here differs slightly from the count in [2], because there the authors required Minkowski decompositions of facets to satisfy an additional lattice condition (ibid., Definition 7) and here we do not regard \({{\,\mathrm{GL}\,}}(3,\mathbb {Z})\)-equivalent Minkowski polynomials as the same.
- 5.
The seven three-dimensional Fano manifolds without very ample canonical bundle are not expected to admit Laurent polynomial mirrors with reflexive Newton polytopes, and so fall outside the range of the Corti–Hacking–Petracci construction.
- 6.
Betti numbers for three-dimensional Fano manifolds can be found in [15].
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Acknowledgements
This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 682603), and from the EPSRC Programme Grant EP/N03189X/1, Classification, Computation, and Construction: New Methods in Geometry. We thank Paul Hacking and Andrea Petracci for many extremely useful conversations, and thank Andy Thomas, Matt Harvey, and the Imperial College Research Computing Service team for invaluable technical assistance. TC thanks Alexander Kasprzyk for a number of very useful conversations about computations.
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Coates, T., Corti, A., da Silva, G. (2022). On the Topology of Fano Smoothings. In: Kasprzyk, A.M., Nill, B. (eds) Interactions with Lattice Polytopes. ILP 2017. Springer Proceedings in Mathematics & Statistics, vol 386. Springer, Cham. https://doi.org/10.1007/978-3-030-98327-7_6
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