Communications in Number Theory and Physics

Volume 12 (2018)

Number 3

Mirror symmetry for lattice polarized del Pezzo surfaces

Pages: 543 – 580

DOI: https://dx.doi.org/10.4310/CNTP.2018.v12.n3.a3

Authors

Charles F. Doran (Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, Canada)

Alan Thompson (Mathematics Institute, University of Warwick, Coventry, United Kingdom; and Department for Pure Mathematics and Mathematical Statistics, University of Cambridge, United Kingdom)

Abstract

We describe a notion of lattice polarization for rational elliptic surfaces and weak del Pezzo surfaces, and describe the complex moduli of the former and the Kähler cone of the latter. We then propose a version of mirror symmetry relating these two objects, which should be thought of as a form of Fano-LG correspondence. Finally, we relate this notion to other forms of mirror symmetry, including Dolgachev–Nikulin–Pinkham mirror symmetry for lattice polarized K3 surfaces and the Gross–Siebert program.

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The first author was supported by the Natural Sciences and Engineering Research Council of Canada, the Pacific Institute for the Mathematical Sciences, and the Visiting Campobassi Professorship at the University of Maryland.

The second author was supported by the Engineering and Physical Sciences Research Council programme grant “Classification, Computation, and Construction: New Methods in Geometry”.

Received 8 November 2017

Accepted 27 March 2018

Published 25 September 2018