#### 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.

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##### 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.