Mirror symmetry for Looijenga interiors and beyond

These talks will be followed by Mark Gross' talks.

Abstract: The moduli space of K3 surfaces of genus g is the quotient of a symmetric space of complex dimension 19 by the action of an arithmetic group. In this capacity, it has a natural infinite class of so-called "toroidal" compactifications. Is any one of these toroidal compactifications distinguished, in the sense that it parameterizes some generalized or "stable" K3 surfaces? I will describe joint work with V. Alexeev giving an affirmative answer to this question. The key combinatorial object encoding the stable K3 surfaces over the boundary is a family of "polarized integral-affine spheres" over a fan in the positive cone of R^{1,18}.

Abstract: We compactify the moduli space of Looijenga pairs via mirror symmetry. The first step is to produce an affine mirror family over an affine base, which can be achieved either by scattering diagrams as in Gross-Hacking-Keel, or by counting non-archimedean analytic curves. Next we compactify the fibers into Looijenga pairs, compactify the base into a complete toric variety, and extend the family over the compactified base. This extension step relies crucially on properties of non-archimedean curves, which is not clear from the approach via scattering diagrams or punctured log curves. Finally we show that all the fibers of the compactified family, together with the theta divisors, are KSBA stable pairs, and the base is a finite cover of the KSBA moduli space. We are currently working on generalizing various steps above to higher dimensions. Joint work with Hacking and Keel.

Abstract: We construct a non-finite type four-dimensional Weinstein domain M_{univ} and describe a HMS-type correspondence between certain birational transformations of P^2 preserving a standard holomorphic volume form and symplectomorphisms of M_{univ}. The space M_{univ} is universal in the sense it admits every Liouville four-manifold mirror to a log Calabi-Yau surface as a Weinstein subdomain; our construction recovers a mirror correspondence between the automorphism group of any open log Calabi-Yau surface and the group of symplectomorphisms of its mirror by restriction to these subdomains. This is joint work in progress with Ailsa Keating.

These talks will be followed by Mark Gross' talks.

Abstract: We will discuss an approach to homological mirror symmetry for Looijenga pairs that is closely tied to the tropical perspective. Namely, we will see how to construct a monomially admissible Fukaya category on the mirror to a Looijenga pair and tropical Lagrangian sections in this category in bijection with line bundles. Moreover, generators of the Lagrangian Floer cohomology of certain sections can be seen to correspond to integral points of polytopes that encode theta functions on the mirror. This is joint work with Abigail Ward.

Abstract: Gross, Hacking and Keel gave an algebro-geometric construction of cluster varieties: take a toric variety, blow up appropriate subvarieties in the boundary, and then remove the strict transform of the boundary. We work with a modification of this construction, which we call a truncated cluster variety--roughly, this comes from performing the same procedure on the toric variety with all the codimension 2 strata removed. The resulting variety differs from the cluster variety in codimension 2. I will describe a construction of a Weinstein manifold mirror to a truncated cluster variety and explain how to prove mirror symmetry in this case via Lagrangian skeleta. We hope that this can be extended to give an approach to mirror symmetry for the entire cluster variety. This is joint work with Benjamin Gammage.

This talk will be a sequel to Bernd Siebert's talks.

Abstract: Relative SH is a Hamiltonian Floer Theoretic invariant which associates a BV algebra to arbitrary compact subsets of a symplectic manifold M functorially with respect to inclusions. When the compact sets in question are pre-images of sufficiently small convex polygons in the base of an SYZ fibration satisfying some mild assumptions, their relative SH satisfies a number of remarkable properties such as freedom from torsion, a Mayer-Vietoris property, a Hartogs property, and more. Some of these properties are already in the literature, while some are still in progress. These properties allow to establish in a large variety of cases, including - but not limited to - SYZ fibrations with singularities of Gross-Siebert type, that relative SH for small polygons is the algebra of polyvector fields of some affinoid domain endowed with a Calabi-Yau structure which is canonical up to scaling. Moreover, in the case of a singularity of Gross-Siebert type, it is possible to fully compute this affinoid domain. This gives rise to a mirror construction which is not the end of a complicated computation, but its starting point. Moreover, for singularities of Gross-Siebert type, it is straightforward to compare with constructions via scattering diagrams. I will discuss work in progress joint with Umut Varolgunes which carries out this circle of ideas for the case of 2 dimensional symplectic cluster manifolds.

Abstract: Strominger-Yau-Zaslow conjecture predicts Calabi-Yau manifolds admits special Lagrangian fibrations. In this talk, I will explain the existence of the special Lagrangian fibrations in some log Calabi-Yau surfaces and their dual fibrations in their expected mirrors. The journey leads us to the study of the moduli space of Ricci-flat metrics with certain asymptotics on these geometries and the discovery of new semi-flat metrics. I will also explain various applications of the SYZ fibrations. The talk is based on joint works with T. Collins, A. Jacob, S.-C. Lau and T.-J. Lee.

Abstract: Given a log Calabi--Yau pair (X,D), consisting of a smooth projective variety X together with a normal crossings anti-canonical divisor D, we first provide a combinatorial algorithm for solving the enumerative problem of computing rational stable maps to (X,D) touching D at a single point. We then explain how to use the solution to write explicit equations for mirrors to such pairs at arbitrary dimensions. Part of this is joint work with Hulya Arguz.

Abstract: A Lagrangian torus in a toric manifold is called exotic if it is not symplectomorphic to a fibre of the toric system. One way of constructing exotic tori is coming up with other interesting Lagrangian torus fibrations on the same space and investigate their fibres. For example, Vianna tori in the complex projective plane arise as fibres of so-called almost toric fibrations. In this talk, we illustrate this interplay by some explicit constructions inspired by almost toric fibrations. In particular, we focus on a construction (different from that given by Auroux) of infinitely many monotone tori in~$\mathbb{C}^3$ and discuss the corresponding Lagrangian torus fibrations.

Abstract: Let f be a linear combination of theta functions on a cluster variety X. Let D be an irreducible component of the boundary of a log CY compactification of X. I will outline a proof that the valuation of f along D is equal to the minimum of the valuations along D of the theta functions contributing to f. The proof is based on new characterizations of tropical theta functions. As an application, we find theta function bases for the global sections of line bundles on compactifications of X. We also prove that certain theta functions are well-behaved under the operation of unfreezing indices, a finding with applications to Fock and Goncharov's moduli of local systems. This is based on upcoming joing work with M.-W. Cheung, T. Magee, and G. Muller.

Abstract: Cluster varieties come in pairs: for any X-cluster variety there is an associated Fock--Goncharov dual A-cluster variety. On the other hand, in the context of mirror symmetry, associated with any log Calabi--Yau variety is its mirror dual, which can be constructed using the enumerative geometry of rational curves in the framework of the Gross--Siebert program. I will explain how to bridge the theory of cluster varieties with the algebro-geometric framework of Gross--Siebert mirror symmetry, and show that the mirror to the X-cluster variety is a degeneration of the Fock-Goncharov dual A-cluster variety. To do this, we investigate how the cluster scattering diagram of Gross-Hacking-Keel-Kontsevich compares with the canonical scattering diagram defined by Gross-Siebert to construct mirror duals in arbitrary dimensions. This is joint work with Hülya Argüz.

Abstract: A well-known conjecture states that the (one-parameter) quantum connection on a Fano manifold M has a singularity of unramified exponential type at q=\infty. This conjecture is a pre-requisite to being able to state the Gamma conjectures or Dubrovin's conjecture and is also part of the conjecture that QH^*(M) carries a non-commutative Hodge structure.
Suppose M carries a smooth ample anti-canonical divisor D and let U=M\D be its complement. I will explain how to reduce the above conjecture on the quantum connection to an S^1-equivariant version of the conjecture of Borman-Sheridan-Varolgunes that "QH^*(M) is a deformation of SH^*(U)." Part of the argument is purely categorical and uses the recent result of Petrov, Vaintrob, and Vologodsky that the Gauss-Manin connection on the periodic cyclic homology of a smooth family of categories over C[t] has regular singularities. This is joint work in progress with Paul Seidel.

Abstract: We aim to give an explicit example for a Floer-theoretic T-duality. More importantly, the dual analytic fibration, including the singular fibers, will be explicitly presented in my talk. Its construction modifies a non-archimedean singular model of Kontsevich-Soibelman and also follows the work of Gross-Hacking-Keel. Surprisingly, we will see the dual singular fiber can be larger than the Maurer-Cartan set. We will start with a review of the family Floer mirror construction and briefly explain why the ideas and technologies in my thesis are necessary. Also, we will explain the motivations for our explicit dual analytic fibration and propose a concrete SYZ mirror statement. If time allowed, we will describe a version of SYZ converse and make some computations for a folklore conjecture for the critical values of the mirror Landau-Ginzburg superpotential.