Quantum periods for 3-dimensional Fano manifolds
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 a...
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Mathematical Sciences Publishers
2016
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| Online Access: | https://eprints.nottingham.ac.uk/32511/ |
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| author | Coates, Tom Corti, Alessio Galkin, Sergey Kasprzyk, Alexander M. |
| author_facet | Coates, Tom Corti, Alessio Galkin, Sergey Kasprzyk, Alexander M. |
| author_sort | Coates, Tom |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | 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. |
| first_indexed | 2025-11-14T19:16:00Z |
| format | Article |
| id | nottingham-32511 |
| institution | University of Nottingham Malaysia Campus |
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| last_indexed | 2025-11-14T19:16:00Z |
| publishDate | 2016 |
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| spelling | nottingham-325112020-05-04T17:34:51Z https://eprints.nottingham.ac.uk/32511/ Quantum periods for 3-dimensional Fano manifolds Coates, Tom Corti, Alessio Galkin, Sergey Kasprzyk, Alexander M. 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. Mathematical Sciences Publishers 2016-02-29 Article PeerReviewed Coates, Tom, Corti, Alessio, Galkin, Sergey and Kasprzyk, Alexander M. (2016) Quantum periods for 3-dimensional Fano manifolds. Geometry & Topology, 20 (1). pp. 103-256. ISSN 1364-0380 http://msp.org/gt/2016/20-1/p03.xhtml doi:10.2140/gt.2016.20.103 doi:10.2140/gt.2016.20.103 |
| spellingShingle | Coates, Tom Corti, Alessio Galkin, Sergey Kasprzyk, Alexander M. Quantum periods for 3-dimensional Fano manifolds |
| title | Quantum periods for 3-dimensional Fano manifolds |
| title_full | Quantum periods for 3-dimensional Fano manifolds |
| title_fullStr | Quantum periods for 3-dimensional Fano manifolds |
| title_full_unstemmed | Quantum periods for 3-dimensional Fano manifolds |
| title_short | Quantum periods for 3-dimensional Fano manifolds |
| title_sort | quantum periods for 3-dimensional fano manifolds |
| url | https://eprints.nottingham.ac.uk/32511/ https://eprints.nottingham.ac.uk/32511/ https://eprints.nottingham.ac.uk/32511/ |