Volume 20, issue 1 (2016)

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Quantum periods for $3$–dimensional Fano manifolds

Tom Coates, Alessio Corti, Sergey Galkin and Alexander Kasprzyk

Geometry & Topology 20 (2016) 103–256
Abstract

The quantum period of a variety $X$ is a generating function for certain Gromov–Witten invariants of $X$ which plays an important role in mirror symmetry. We compute the quantum periods of all $3$–dimensional Fano manifolds. In particular we show that $3$–dimensional Fano manifolds with very ample anticanonical bundle have mirrors given by a collection of Laurent polynomials called Minkowski polynomials. This was conjectured in joint work with Golyshev. It suggests a new approach to the classification of Fano manifolds: by proving an appropriate mirror theorem and then classifying Fano mirrors.

Our methods are likely to be of independent interest. We rework the Mori–Mukai classification of $3$–dimensional Fano manifolds, showing that each of them can be expressed as the zero locus of a section of a homogeneous vector bundle over a GIT quotient $V∕∕G$, where $G$ is a product of groups of the form ${GL}_{n}\left(ℂ\right)$ and $V$ is a representation of $G$. When $G={GL}_{1}{\left(ℂ\right)}^{r}\phantom{\rule{0.3em}{0ex}}$, this expresses the Fano $3$–fold as a toric complete intersection; in the remaining cases, it expresses the Fano $3$–fold as a tautological subvariety of a Grassmannian, partial flag manifold, or projective bundle thereon. We then compute the quantum periods using the quantum Lefschetz hyperplane theorem of Coates and Givental and the abelian/non-abelian correspondence of Bertram, Ciocan-Fontanine, Kim and Sabbah.

Keywords
quantum cohomology, quantum period, Fano manifold, mirror symmetry
Mathematical Subject Classification 2010
Primary: 14J45, 14J33
Secondary: 14N35
Supplementary material

Table of Laurent polynomial mirrors for each of the $105$ deformation families of $3$--dimensional Fano manifolds.

Publication
Received: 12 February 2014
Revised: 2 April 2015
Accepted: 5 May 2015
Published: 29 February 2016
Proposed: Jim Bryan
Seconded: Richard Thomas, Yasha Eliashberg
Authors
 Tom Coates Department of Mathematics Imperial College London 180 Queen’s Gate London SW7 2AZ UK Alessio Corti Department of Mathematics Imperial College London 180 Queen’s Gate London SW7 2AZ UK Sergey Galkin Faculty of Mathematics and Laboratory of Algebraic Geometry National Research University Higher School of Economics 7 Vavilova str. Moscow 117312 Russia Alexander Kasprzyk Department of Mathematics Imperial College London 180 Queen’s Gate London SW7 2AZ UK