Quantum periods for 3-dimensional Fano manifolds

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

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), https://doi.org/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: https://nottingham-repository.worktribe.com/output/774714
• Publisher URL: http://msp.org/gt/2016/20-1/p03.xhtml