## Geometry & Topology

### Quantum periods for $3$–dimensional Fano manifolds

#### 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 $GLn(ℂ)$ and $V$ is a representation of $G$. When $G = GL1(ℂ)r$, 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.

#### Article information

Source
Geom. Topol., Volume 20, Number 1 (2016), 103-256.

Dates
Revised: 2 April 2015
Accepted: 5 May 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.gt/1510858924

Digital Object Identifier
doi:10.2140/gt.2016.20.103

Mathematical Reviews number (MathSciNet)
MR3470714

Zentralblatt MATH identifier
1348.14105

#### Citation

Coates, Tom; Corti, Alessio; Galkin, Sergey; Kasprzyk, Alexander. Quantum periods for $3$–dimensional Fano manifolds. Geom. Topol. 20 (2016), no. 1, 103--256. doi:10.2140/gt.2016.20.103. https://projecteuclid.org/euclid.gt/1510858924

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#### Supplemental materials

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