Quantum Periods for 3Dimensional Fano Manifolds
Abstract
The quantum period of a variety X is a generating function for certain GromovWitten invariants of X which plays an important role in mirror symmetry. In this paper we compute the quantum periods of all 3dimensional Fano manifolds. In particular we show that 3dimensional 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 MoriMukai classification of 3dimensional 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 3fold as a toric complete intersection; in the remaining cases, it expresses the Fano 3fold 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 CoatesGivental and the Abelian/nonAbelian correspondence of BertramCiocanFontanineKimSabbah.
 Publication:

arXiv eprints
 Pub Date:
 March 2013
 DOI:
 10.48550/arXiv.1303.3288
 arXiv:
 arXiv:1303.3288
 Bibcode:
 2013arXiv1303.3288C
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Symplectic Geometry;
 14J33;
 14J45 (Primary) 14N35 (Secondary)
 EPrint:
 104 pages. v2: references updated, minor changes to presentation. v3: some changes to exposition and minor mathematical corrections, plus much improved hyperlinking