Floer Theory of Higher Rank Quiver 3-folds

We study threefolds Y fibred by $$A_m$$ A m -surfaces over a curve S of positive genus. An ideal triangulation of S defines, for each rank m, a quiver $$Q(\Delta _m)$$ Q ( Δ m ) , hence a $$CY_3$$ C Y 3 -category $$\mathcal {C}(W)$$ C ( W ) for any potential W on $$Q(\Delta _m)$$ Q ( Δ m ) . We show that for $$\omega $$ ω in an open subset of the Kähler cone, a subcategory of a sign-twisted Fukaya category of $$(Y,\omega )$$ ( Y , ω ) is quasi-isomorphic to $$(\mathcal {C},W_{[\omega ]})$$ ( C , W [ ω ] ) for a certain generic potential $$W_{[\omega ]}$$ W [ ω ] . This partially establishes a conjecture of Goncharov (in: Algebra, geometry, and physics in the 21st century, Birkhäuser/Springer, Cham, 2017) concerning ‘categorifications’ of cluster varieties of framed $${\mathbb {P}}GL_{m+1}$$ P G L m + 1 -local systems on S, and gives a symplectic geometric viewpoint on results of Gaiotto et al. (Ann Henri Poincaré 15(1):61–141, 2014) on ‘theories of class $${\mathcal {S}}$$ S ’.

Compactifications of Cluster Varieties and Convexity

Gross–Hacking–Keel–Kontsevich [13] discuss compactifications of cluster varieties from positive subsets in the real tropicalization of the mirror. To be more precise, let ${\mathfrak{D}}$ be the scattering diagram of a cluster variety $V$ (of either type– ${\mathcal{A}}$ or ${\mathcal{X}}$), and let $S$ be a closed subset of $\left (V^\vee \right )^{\textrm{trop}} \left ({\mathbb{R}}\right )$—the ambient space of ${\mathfrak{D}}$. The set $S$ is positive if the theta functions corresponding to the integral points of $S$ and its ${\mathbb{N}}$-dilations define an ${\mathbb{N}}$-graded subalgebra of $\Gamma (V, \mathcal{O}_V){ [x]}$. In particular, a positive set $S$ defines a compactification of $V$ through a Proj construction applied to the corresponding ${\mathbb{N}}$-graded algebra. In this paper, we give a natural convexity notion for subsets of $\left (V^\vee \right )^{\textrm{trop}} \left ({\mathbb{R}}\right )$, called broken line convexity, and show that a set is positive if and only if it is broken line convex. The combinatorial criterion of broken line convexity provides a tractable way to construct positive subsets of $\left (V^\vee \right )^{\textrm{trop}} \left ({\mathbb{R}}\right )$ or to check positivity of a given subset.

Toric degenerations of cluster varieties and cluster duality

We introduce the notion of a $Y$-pattern with coefficients and its geometric counterpart: an $\mathcal {X}$-cluster variety with coefficients. We use these constructions to build a flat degeneration of every skew-symmetrizable specially completed $\mathcal {X}$-cluster variety $\widehat {\mathcal {X} }$ to the toric variety associated to its g-fan. Moreover, we show that the fibers of this family are stratified in a natural way, with strata the specially completed $\mathcal {X}$-varieties encoded by $\operatorname {Star}(\tau )$ for each cone $\tau$ of the $\mathbf {g}$-fan. These strata degenerate to the associated toric strata of the central fiber. We further show that the family is cluster dual to $\mathcal {A}_{\mathrm {prin}}$ of Gross, Hacking, Keel and Kontsevich [Canonical bases for cluster algebras, J. Amer. Math. Soc. 31 (2018), 497–608], and the fibers cluster dual to $\mathcal {A} _t$. Finally, we give two applications. First, we use our construction to identify the toric degeneration of Grassmannians from Rietsch and Williams [Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians, Duke Math. J. 168 (2019), 3437–3527] with the Gross–Hacking–Keel–Kontsevich degeneration in the case of $\operatorname {Gr}_2(\mathbb {C} ^{5})$. Next, we use it to link cluster duality to Batyrev–Borisov duality of Gorenstein toric Fanos in the context of mirror symmetry.

Constructing Fano 3-folds from cluster varieties of rank 2

Cluster algebras give rise to a class of Gorenstein rings which enjoy a large amount of symmetry. Concentrating on the rank 2 cases, we show how cluster varieties can be used to construct many interesting projective algebraic varieties. Our main application is then to construct hundreds of families of Fano 3-folds in codimensions 4 and 5. In particular, for Fano 3-folds in codimension 4 we construct at least one family for 187 of the 206 possible Hilbert polynomials contained in the Graded Ring Database.