Related papers: The Frobenius structure theorem for affine log Cal…

For a log Calabi Yau pair (X,D) with X\D smooth affine, satisfying either assumption 1.1 of "The canonical wall structure and intrinsic mirror symmetry" or contains a Zariski dense torus, we prove under the condition that D is the support…

We construct the mirror algebra to a smooth affine log Calabi-Yau variety with maximal boundary, as the spectrum of a commutative associative algebra with a canonical basis, whose structure constants are given as naive counts of…

Cluster varieties come in pairs: for any $\mathcal{X}$ cluster variety there is an associated Fock-Goncharov dual $\mathcal{A}$ cluster variety. On the other hand, in the context of mirror symmetry, associated with any log Calabi-Yau…

Using recent results of Battistella, Nabijou, Ranganathan and the author, we compare candidate mirror algebras associated with certain log Calabi-Yau pairs constructed by Gross-Siebert using log Gromov-Witten theory and Tseng-You using…

In previous work, the first three authors conjectured that the ring of regular functions on a natural class of affine log Calabi-Yau varieties (those with maximal boundary) has a canonical vector space basis parameterized by the integral…

Using heuristics from mirror symmetry, combinations of Gross, Hacking, Keel, Kontsevich, and Siebert have given combinatorial constructions of canonical bases of "theta functions" on the coordinate rings of various log Calabi-Yau spaces,…

We work in the setting of Calabi-Yau mirror symmetry. We establish conditions under which Kontsevich's homological mirror symmetry (which relates the derived Fukaya category to the derived category of coherent sheaves on the mirror) implies…

Given a log Calabi--Yau surface $(Y,D)$, Bousseau has constructed a quantization of the mirror algebra of this pair. We give a formula for structure constants of this quantization in terms of higher genus descendant logarithmic…

Gross, Hacking, and Keel have constructed mirrors of log Calabi-Yau surfaces in terms of counts of rational curves. Using $q$-deformed scattering diagrams defined in terms of higher genus log Gromov-Witten invariants, we construct…

We show that the ring of regular functions of every smooth affine log Calabi-Yau surface with maximal boundary has a vector space basis parametrized by its set of integer tropical points and a $\mathbb{C}$-algebra structure with structure…

We discuss how the Gross-Siebert reconstruction theorem applies to the local mirror symmetry of Chiang, Klemm, Yau and Zaslow. The reconstruction theorem associates to certain combinatorial data a degeneration of (log) Calabi-Yau varieties.…

We establish a comparison result relating non-archimedean cylinder counts and logarithmic cylinder counts in a smooth affine log Calabi-Yau variety. Using the decomposition theorem and the gluing formula from log Gromov-Witten theory, we…

We study integral dlt models of a proper C((t))-variety X along a toric stratum of the special fiber. We prove that the associated Berkovich retraction - from the non-archimedean analytification of X onto the dual complex of the model - is…

We describe in purely combinatorial terms dual pairs of integral affine structures on spheres which come from the conjectural metric collapse of mirror families of Calabi-Yau toric hypersurfaces. The same structures arise on the base of a…

This paper is the first arising from our project announced in math.AG/0211094, "Affine manifolds, log structures, and mirror symmetry." We aim to study mirror symmetry by studying the log structures of Illusie-Fontaine and Kato on…

We construct a generalization of the variations of Hodge structures on Calabi-Yau manifolds. It gives a Mirror partner for the theory of genus=0 Gromov-Witten invariants

The paper studies three classes of Frobenius manifolds: Quantum Cohomology (topological sigma-models), unfolding spaces of singularities (K. Saito's theory, Landau-Ginzburg models), and the recent Barannikov-Kontsevich construction starting…

Let $U$ be a smooth affine curve over a number field $K$ with a compactification $X$ and let $\mathbb L$ be a rank $2$, geometrically irreducible $\bar{\mathbb Q}_\ell$-local system on $U$ with cyclotomic determinant that extends to an…

In this paper we study initial topological properties of the (non-)finitely-generated locus of Frobenius Algebra coming from Stanley-Reisner rings defined through face ideals. More specifically, we will give a partial answer to a conjecture…

We define Calabi-Yau and periodic Frobenius algebras over arbitrary base commutative rings. We define a Hochschild analogue of Tate cohomology, and show that the "stable Hochschild cohomology" of periodic CY Frobenius algebras has a…