Pure and Applied Mathematics Quarterly

Volume 15 (2019)

Number 4

Special Issue in Honor of Simon Donaldson: Part 2 of 2

Guest Editor: Richard Thomas (Imperial College London)

Laurent inversion

Pages: 1135 – 1179

DOI: https://dx.doi.org/10.4310/PAMQ.2019.v15.n4.a5


Tom Coates (Department of Mathematics, Imperial College London, United Kingdom)

Alexander Kasprzyk (School of Mathematical Sciences, University of Nottingham, United Kingdom)

Thomas Prince (Mathematical Institute, University of Oxford, United Kingdom)


We describe a practical and effective method for reconstructing the deformation class of a Fano manifold $X$ from a Laurent polynomial $f$ that corresponds to $X$ under Mirror Symmetry. We explore connections to nef partitions, the smoothing of singular toric varieties, and the construction of embeddings of one (possibly-singular) toric variety in another. In particular, we construct degenerations from Fano manifolds to singular toric varieties; in the toric complete intersection case, these degenerations were constructed previously by Doran–Harder. We use our method to find models of orbifold del Pezzo surfaces as complete intersections and degeneracy loci, and to construct a new four-dimensional Fano manifold.


mirror symmetry, Fano manifolds, toric degenerations

2010 Mathematics Subject Classification

Primary 14J33. Secondary 14J45, 52B20.

T. Coates was supported by ERC Starting Investigator Grant 240123, ERC Consolidator Grant 682603, and EPSRC Program Grant EP/N03189X/1.

A. Kasprzyk was supported by ERC Starting Investigator Grant 240123, and by EPSRC Fellowship EP/N022513/1.

T. Prince was supported by an EPSRC Prize Studentship, an EPSRC Doctoral Prize Fellowship, and a Fellowship by Examination at Magdalen College, Oxford.

Received 26 March 2018

Published 20 March 2020