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Quantum periods for 3-dimensional Fano manifolds

Coates, Tom; Corti, Alessio; Galkin, Sergey; Kasprzyk, Alexander M.

Authors

Tom Coates

Alessio Corti

Sergey Galkin

Alexander M. Kasprzyk



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. In this paper 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(C) and V is a representation of G. When G=GL_1(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-Givental and the Abelian/non-Abelian correspondence of Bertram-Ciocan-Fontanine-Kim-Sabbah.

Citation

Coates, T., Corti, A., Galkin, S., & Kasprzyk, A. M. (2016). Quantum periods for 3-dimensional Fano manifolds. Geometry and Topology, 20(1), doi:10.2140/gt.2016.20.103

Journal Article Type Article
Publication Date Feb 29, 2016
Deposit Date Mar 24, 2016
Publicly Available Date Mar 24, 2016
Journal Geometry & Topology
Print ISSN 1465-3060
Electronic ISSN 1364-0380
Publisher Mathematical Sciences Publishers
Peer Reviewed Peer Reviewed
Volume 20
Issue 1
DOI https://doi.org/10.2140/gt.2016.20.103
Public URL http://eprints.nottingham.ac.uk/id/eprint/32511
Publisher URL http://msp.org/gt/2016/20-1/p03.xhtml
Copyright Statement Copyright information regarding this work can be found at the following address: http://eprints.nottingham.ac.uk/end_user_agreement.pdf

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Copyright Statement
Copyright information regarding this work can be found at the following address: http://eprints.nottingham.ac.uk/end_user_agreement.pdf





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