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.

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Quantum periods for 3-dimensional Fano manifolds

We use cluster structures and mirror symmetry to explicitly describe a natural class of Newton-Okounkov bodies for Grassmannians. We consider the Grassmannian $X=Gr_{n-k}(\mathbb C^n)$, as well as the mirror dual Landau-Ginzburg model $(\check{X}^\circ, W_q:\check{X}^\circ \to \mathbb C)$, where $\check{X}^\circ$ is the complement of a particular anti-canonical divisor in a Langlands dual Grassmannian $\check{X} = Gr_k((\mathbb C^n)^*)$, and the superpotential W_q has a simple expression in terms of Pl\"ucker coordinates. Grassmannians simultaneously have the structure of an $\mathcal{A}$-cluster variety and an $\mathcal{X}$-cluster variety. Given a cluster seed G, we consider two associated coordinate systems: a $\mathcal X$-cluster chart $\Phi_G:(\mathbb C^*)^{k(n-k)}\to X^{\circ}$ and a $\mathcal A$-cluster chart $\Phi_G^{\vee}:(\mathbb C^*)^{k(n-k)}\to \check{X}^\circ$. To each $\mathcal X$-cluster chart $\Phi_G$ and ample `boundary divisor' $D$ in $X\setminus X^{\circ}$, we associate a Newton-Okounkov body $\Delta_G(D)$ in $\mathbb R^{k(n-k)}$, which is defined as the convex hull of rational points. On the other hand using the $\mathcal A$-cluster chart $\Phi_G^{\vee}$ on the mirror side, we obtain a set of rational polytopes, described by inequalities, by writing the superpotential $W_q$ in the $\mathcal A$-cluster coordinates, and then "tropicalising". Our main result is that the Newton-Okounkov bodies $\Delta_G(D)$ and the polytopes obtained by tropicalisation coincide. As an application, we construct degenerations of the Grassmannian to toric varieties corresponding to these Newton-Okounkov bodies. Additionally, when $G$ corresponds to a plabic graph, we give a formula for the lattice points of the Newton-Okounkov bodies, which has an interpretation in terms of quantum Schubert calculus.

Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians

We state a number of conjectures that together allow one to classify a broad class of del Pezzo surfaces with cyclic quotient singularities using mirror symmetry. We prove our conjectures in the simplest cases. The conjectures relate mutation-equivalence classes of Fano polygons with Q-Gorenstein deformation classes of del Pezzo surfaces.

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Mirror symmetry and the classification of orbifold del Pezzo surfaces

Mirror Symmetry and the Classification of Orbifold del Pezzo Surfaces

In this article we use cluster structures and mirror symmetry to explicitly describe a natural class of Newton–Okounkov bodies for Grassmannians. We consider the Grassmannian X=Grn−k(Cn), as well as the mirror dual Landau–Ginzburg model (Xˇ∘,W:Xˇ∘→C), where Xˇ∘ is the complement of a particular anticanonical divisor in a Langlands dual Grassmannian Xˇ=Grk((Cn)∗) and the superpotential W has a simple expression in terms of Plucker coordinates. Grassmannians simultaneously have the structure of an A-cluster variety and an X-cluster variety; roughly speaking, a cluster variety is obtained by gluing together a collection of tori along birational maps. Given a plabic graph or, more generally, a cluster seed G, we consider two associated coordinate systems: a network or X-cluster chart ΦG:(C∗)k(n−k)→X∘ and a Plucker cluster or A-cluster chart ΦG∨:(C∗)k(n−k)→Xˇ∘. Here X∘ and Xˇ∘ are the open positroid varieties in X and Xˇ, respectively. To each X-cluster chart ΦG and ample boundary divisor D in X∖X∘, we associate a Newton–Okounkov body ΔG(D) in Rk(n−k), which is defined as the convex hull of rational points; these points are obtained from the multidegrees of leading terms of the Laurent polynomials ΦG∗(f) for f on X with poles bounded by some multiple of D. On the other hand, using the A-cluster chart ΦG∨ on the mirror side, we obtain a set of rational polytopes—described in terms of inequalities—by writing the superpotential W as a Laurent polynomial in the A-cluster coordinates and then tropicalizing. Our first main result is that the Newton–Okounkov bodies ΔG(D) and the polytopes obtained by tropicalization on the mirror side coincide. As an application, we construct degenerations of the Grassmannian to normal toric varieties corresponding to (dilates of ) these Newton–Okounkov bodies. Our second main result is an explicit combinatorial formula in terms of Young diagrams, for the lattice points of the Newton–Okounkov bodies, in the case in which the cluster seed G corresponds to a plabic graph. This formula has an interpretation in terms of the quantum Schubert calculus of Grassmannians.

Newton–Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians

Given a Laurent polynomial f, one can form the period of f: this is a function of one complex variable that plays an important role in mirror symmetry for Fano manifolds. Mutations are a particular class of birational transformations acting on Laurent polynomials in two variables; they preserve the period and are closely connected with cluster algebras. We propose a higher-dimensional analog of mutation acting on Laurent polynomials f in n variables. In particular we give a combinatorial description of mutation acting on the Newton polytope P of f, and use this to establish many basic facts about mutations. Mutations can be understood combinatorially in terms of Minkowski rearrangements of slices of P, or in terms of piecewise-linear transformations acting on the dual polytope P* (much like cluster transformations). Mutations map Fano polytopes to Fano polytopes, preserve the Ehrhart series of the dual polytope, and preserve the period of f. Finally we use our results to show that Minkowski polynomials, which are a family of Laurent polynomials that give mirror partners to many three-dimensional Fano manifolds, are connected by a sequence of mutations if and only if they have the same period.

Minkowski Polynomials and Mutations

We show that a cyclic quotient surface singularity S can be decomposed, in a precise sense, into a number of elementary T-singularities together with a cyclic quotient surface singularity called the residue of S A normal surface X with isolated cyclic quotient singularities {S_i} admits a Q-Gorenstein partial smoothing to a surface with singularities given by the residues of the S_i We define the singularity content of a Fano lattice polygon P: this records the total number of elementary T-singularities and the residues of the corresponding toric Fano surface X_P We express the degree of X_P in terms of the singularity content of P; give a formula for the Hilbert series of X_P in terms of singularity content; and show that singularity content is an invariant of P under mutation

Singularity content

We show that Fano lattice polygons define a class of balanced quivers with interesting properties. The combinatorics of these quivers is related to singularities of the underlying toric Fano surface. This allows us to show that every Fano polygon defines a point on a certain family of algebraic hypersurfaces. Our quivers admit a generalized mutation which preserves balancing and coincides with combinatorial mutation of Fano polygons whenever both operations are defined. We characterize balanced quivers arising from Fano polygons and discuss generalizations to higher dimensions.

Mohammad Akhtar

Papers

Minkowski Polynomials and Mutations

Mirror symmetry and the classification of orbifold del Pezzo surfaces

Mirror Symmetry and the Classification of Orbifold del Pezzo Surfaces

Singularity content

Polygonal Quivers.