Minkowski Polynomials and Mutations

Akhtar, Mohammad

doi:10.3842/sigma.2012.094

Cited by 85 publications

(221 citation statements)

References 22 publications

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“…See [KN12] for an overview of Fano polytopes. We briefly recall the notation of [ACGK12,§3]. Any choice of primitive vector w P M determines a lattice height function w : N Ñ Z which naturally extends to N Q Ñ Q.…”

Section: Mutations Of Fano Polytopesmentioning

confidence: 99%

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Mutations of Fake Weighted Projective Planes

Coates

Gonshaw

Kasprzyk

et al. 2015

Proceedings of the Edinburgh Mathematical Society

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Abstract. In previous work by Coates, Galkin, and the authors, the notion of mutation between lattice polytopes was introduced. Such a mutation gives rise to a deformation between the corresponding toric varieties. In this paper we study one-step mutations that correspond to deformations between weighted projective planes, giving a complete characterisation of such mutations in terms of T -singularities. We show also that the weights involved satisfy Diophantine equations, generalising results of Hacking-Prokhorov.

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Section: Mutations Of Fano Polytopesmentioning

confidence: 99%

“…In [ACGK12] it was also shown that mutations have a natural description as a piecewise linear transformation of the lattice M . We require the following definition.…”

Section: Mutations Of Fano Polytopesmentioning

confidence: 99%

Mutations of Fake Weighted Projective Planes

Coates

Gonshaw

Kasprzyk

et al. 2015

Proceedings of the Edinburgh Mathematical Society

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“…where the P k are polynomials. There are 3747 Minkowski polynomials (up to monomial change of variables) but Akhtar-Coates-Galkin-Kasprzyk showed that these Laurent polynomials together generate only 165 periods [1]. That is, Minkowski polynomials fall into 165 equivalence classes where f and g are equivalent if and only if they have the same period.…”

Section: A Introductionmentioning

confidence: 99%

“…where G acts as in (1), and C × acts trivially on the first factor and by rescaling on the second factor. (3) Let F = P(E) be as in (2).…”

mentioning

confidence: 99%

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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“…Using this it is straightforward to check that the period condition for Proof. One can check that polygons ∆ 1 and ∆ 2 differ by (a sequence of) mutations (see, say, [ACGK12]). These mutations agree with fiberwise birational isomorphisms of toric Landau-Ginzburg models modulo change of basis in H 2 (S, Z) by the construction.…”

Section: Proposition 21 the Laurent Polynomial F Sd Is A Toric Landmentioning

confidence: 99%