Geometry & Topology

Quantum periods for $3$–dimensional Fano manifolds

Tom Coates, Alessio Corti, Sergey Galkin, and Alexander Kasprzyk

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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 VG, where G is a product of groups of the form GLn() and V is a representation of G. When G = GL1()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

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

Received: 12 February 2014
Revised: 2 April 2015
Accepted: 5 May 2015
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14J45: Fano varieties 14J33: Mirror symmetry [See also 11G42, 53D37]
Secondary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]

quantum cohomology quantum period Fano manifold mirror symmetry


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.

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Supplemental materials

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