Quantum periods for 3–dimensional Fano manifolds

Coates, Tom¹ ; Corti, Alessio¹ ; Galkin, Sergey² ; Kasprzyk, Alexander¹

¹ Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK
² Faculty of Mathematics and Laboratory of Algebraic Geometry, National Research University Higher School of Economics, 7 Vavilova str., Moscow, 117312, Russia

Geometry & topology, Tome 20 (2016) no. 1, pp. 103-256.

Résumé

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. 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 ${GL}_n\left(ℂ\right)$ and $V$ is a representation of $G$. When $G={GL}_1{\left(ℂ\right)}^r$, 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 and Givental and the abelian/non-abelian correspondence of Bertram, Ciocan-Fontanine, Kim and Sabbah.

Détail

DOI : 10.2140/gt.2016.20.103

Classification : 14J45, 14J33, 14N35
Keywords: quantum cohomology, quantum period, Fano manifold, mirror symmetry

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