Quantum Periods for 3-Dimensional Fano Manifolds
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- March 2013
- DOI:
- 10.48550/arXiv.1303.3288
- arXiv:
- arXiv:1303.3288
- Bibcode:
- 2013arXiv1303.3288C
- Keywords:
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- Mathematics - Algebraic Geometry;
- Mathematics - Symplectic Geometry;
- 14J33;
- 14J45 (Primary) 14N35 (Secondary)
- E-Print:
- 104 pages. v2: references updated, minor changes to presentation. v3: some changes to exposition and minor mathematical corrections, plus much improved hyperlinking