Quantum Periods for 3-Dimensional Fano Manifolds
File(s)
Accepted version
Author(s)
Coates, T
Corti, A
Galkin, S
Kasprzyk, A
Type
Journal Article
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.
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.
Date Issued
2016-02-29
Date Acceptance
2015-05-05
Citation
Geometry & Topology, 2016, 20 (1), pp.103-256
ISSN
1465-3060
Publisher
Mathematical Sciences Publishers (MSP)
Start Page
103
End Page
256
Journal / Book Title
Geometry & Topology
Volume
20
Issue
1
Copyright Statement
© 2016 Mathematical Sciences Publishers. All rights reserved
Sponsor
Commission of the European Communities
The Royal Society
The Leverhulme Trust
The Royal Society
Engineering & Physical Science Research Council (EPSRC)
Engineering & Physical Science Research Council (EPSRC)
Identifier
http://arxiv.org/abs/1303.3288v3
Grant Number
240123
UF090056
MATH_P36759
516002.K5822/kk
EP/I008128/1
EP/G06170X/1
Subjects
math.AG
math.AG
14J33, 14J45 (Primary) 14N35 (Secondary)
Notes
104 pages. v2: references updated, minor changes to presentation. v3: some changes to exposition and minor mathematical corrections, plus much improved hyperlinking
Date Publish Online
2016-02-29