Mirror Symmetry and Fano Manifolds

Coates, Tom; Corti, Alessio; Galkin, Sergey; Golyshev, Vasily; Kasprzyk, Alexander

doi:10.4171/120-1/16

Cited by 85 publications

(226 citation statements)

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“…We conjectured, jointly with Golyshev, that the 98 Minkowski periods of manifold type 2 correspond under mirror symmetry to the 98 deformation families of 3-dimensional Fano manifolds with very ample anticanonical bundle [10]. That is, there is a one-to-one correspondence between deformation families of 3-dimensional Fano manifolds X with very ample anticanonical bundle and equivalence classes of Minkowski polynomials f , such that 3 the Fourier-Laplace transform G X of the quantum period of X coincides with the period π f of f .…”

Section: A Introductionmentioning

confidence: 85%

“…CoatesGalkin-Kasprzyk have computed the Picard-Fuchs operators for the Minkowski polynomials numerically [14]. Their results, which are computer-assisted rigorous and which pass a number of stringent checks, show that exactly 98 of the 165 Minkowski periods are of manifold type.We conjectured, jointly with Golyshev, that the 98 Minkowski periods of manifold type 2 correspond under mirror symmetry to the 98 deformation families of 3-dimensional Fano manifolds with very ample anticanonical bundle [10]. That is, there is a one-to-one correspondence between deformation families of 3-dimensional Fano manifolds X with very ample anticanonical bundle and equivalence classes of Minkowski polynomials f , such that 3 the Fourier-Laplace transform G X of the quantum period of X coincides with the period π f of f .…”

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confidence: 85%

Section: Quantum Periods For Fano Manifoldsmentioning

confidence: 99%

“…In joint work with Golyshev [10] we introduced Minkowski polynomials: these are a collection of Laurent polynomials f in three variables such that the Newton polytope ∆ of f is a reflexive polytope, defined 1 in terms of Minkowski decompositions of the facets of ∆. Given a Laurent polynomial f , one can define the 1 Some of these Laurent polynomials correspond under mirror symmetry to 3-dimensional Fano manifolds which admit a small toric degeneration [4].…”

Section: A Introductionmentioning

confidence: 99%

“…Our results suggest, too, that one should search for higher-dimensional Fano manifolds systematically by searching for their Laurent polynomial mirrors. This is discussed in our joint work with Golyshev, where we outline a program to classify 4-dimensional Fano manifolds using these ideas [10].We know of no a priori reason why every 3-dimensional Fano manifold admits a construction as in Theorem A.1. At present this can be proven only post-classification, by a case-by-case analysis.…”

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Quantum periods for 3–dimensional Fano manifolds

et al. 2016

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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 GLn(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. A. IntroductionThe quantum period of a Fano manifold X is a generating function for Gromov-Witten invariants. It is a deformation invariant of X that carries detailed information about quantum cohomology. In this paper we give closed formulas for the quantum periods for all 3-dimensional Fano manifolds. As a consequence we prove a conjecture, made jointly with Golyshev, that identifies Laurent polynomials which correspond under mirror symmetry to each of the 98 deformation families of 3-dimensional Fano manifolds with very ample anticanonical bundle. We also exhibit Laurent polynomial mirrors for the remaining 7 deformation families. Our arguments rely on the classification of 3-dimensional Fano manifolds, due to Iskovskikh and Mori-Mukai: this is a difficult theorem whose proof, even today, requires delicate arguments in explicit birational geometry. On the other hand our mirror Laurent polynomials have a simple combinatorial definition and classification. Given a suitable mirror theorem this classification would give a straightforward, combinatorial, and uniform alternative proof of the classification of 3-dimensional Fano manifolds. The general outlines of such a mirror theorem are beginning to emerge [2,3,38,40,70], as are some promising approaches to proving it [24][25][26][27]42,43].Let X be a Fano manifold, that is, a smooth projective variety such that the anticanonical bundle −K X is ample. The quantum period G X (t) of X, defined in §B below, is a generating function for certain genus-zero Gromov-Witten invariants of X. It satisfies a differential equation:where D = t d dt and the p k are polynomials, called the quantum differential equation for ...

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Section: A Introductionmentioning

confidence: 85%

mentioning

confidence: 85%

Section: Quantum Periods For Fano Manifoldsmentioning

confidence: 99%

Section: A Introductionmentioning

confidence: 99%

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confidence: 99%

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