Schedule for: 19w5179 - Mutations: Mirror Symmetry, Deformations, and Combinatorics

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

We provide LG mirrors for the family of del Pezzo surfaces of degree $8k+4$ in $P(2,2k+1,2k+1, 4k+1)$, first constructed by Johnson and Kollár. The main feature of these surfaces, which makes the mirror construction especially interesting, is that the anticanonical system is empty: because of this, our mirrors are not covered by any other construction known to us.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Let $X$ be a projective variety with projective coordinate ring $R$. By a result of Andersen, any full-rank homogeneous valuation on $R$ with finitely generated value semigroup leads to a degeneration of $X$ to a toric variety. Hence, it becomes important to understand which valuations have finitely generated value semigroup. In the first part of the talk, I will report on some results with Chris Manon and Milena Wrobel in which we study this problem for varieties with complexity-one torus actions. Given two such degenerations as above, how are the associated toric varieties related? In the second part of the talk, I will discuss how, via the use of complexity-one $T$-varieties, one can understand the relationships between such toric degenerations in terms of mutations of polytopes.

A dimer model is a bipartite graph described on the real two-torus. For a dimer model, we can assign the lattice polygon, and a dimer model enjoys rich information regarding toric geometry associated to such a polygon. On the other hand, there is the operation called the (combinatorial) mutation of a polygon, which makes a given lattice polygon another one. This mutation is important to understand mirror partners for Fano manifolds. Under these backgrounds, I expect that there is a certain operation for a dimer model that induces the mutation of the associated polygon. In my talk, I will introduce the operation which I call the deformation of a dimer model, and show that this operation realizes my expectation. This talk is based on a joint work with A. Higashitani (arXiv:1903.01636).

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In the first part of the talk I will focus on the Grassmannian of planes and the corresponding cluster algebra of type $A$. We show that universal coefficients (as introduced by Fomin-Zelevinsky) give rise to a Gröbner cone in the Gröbner fan of the Plücker ideal. Among the faces of the cone we identify one for every toric variety obtained from adding principal coefficients to the cluster algebra. All such "mutation-equivalent" toric degenerations arise in this way. In the second part of the talk I will apply similar ideas to the full flag variety. Using these methods one can show that the toric degeneration associated to the Feigin-Fourier-Littelmann-Vinberg polytope does not lie in the mutation equivalence class of toric degenerations associated with principal coefficients.

The theory of Newton-Okounkov bodies is a generalization of that of Newton polytopes for toric varieties, and it gives a systematic method of constructing toric degenerations of projective varieties. In this talk, we study Newton-Okounkov bodies of projective varieties from their cluster structures. More precisely, we construct Newton-Okounkov bodies from specific valuations which generalize g-vectors in cluster theory. In the case of flag varieties, we discuss how these bodies are related to string polytopes arising from representation theory. This is a joint work with Hironori Oya.

Let $\Delta$ be a canonical Fano $3$-polytope, i.e., a $3$-dimensional lattice polytope containing exactly one interior lattice point. Then the affine surface $Z_\Delta$ defined by a generic Laurent polynomial $f_\Delta$ with the Newton polytope $\Delta$ is birational to a smooth projective minimal surface $S_\Delta$ with $q=0$ and $p_g=1$. Using the classification of all $674,\!688$ canonical Fano $3$-polytopes obtained by Kasprzyk, we show that $S_\Delta$ is a $K3$-surface except for exactly $9,\!089$ canonical Fano $3$-polytopes $\Delta$. In the latter case, we obtain $9,\!040$ canonical Fano $3$-polytopes $\Delta$ defining minimal elliptic surfaces $S_\Delta$ of Kodaira dimension $1$ and $49$ canonical Fano $3$-polytopes $\Delta$ defining minimal surfaces $S_\Delta$ of general type with $|π_1(S_\Delta)|=K^2 \in \{1,2\}$ considered by Kynev and Todorov. This is a joint work with Alexander Kaspzryk and Victor Batyrev.

I will recall what is known for deformations of toric Fano varieties and I will explain connections with Mirror Symmetry for Fano varieties. I will also talk about a joint work with Alessio Corti and Paul Hacking about constructing smoothings of toric Fano threefolds with Gorenstein singularities.

A Calabi-Yau (CY) pair $(X,D)$ consists of a normal projective variety and a reduced anti-canonical integral divisor on it. Such pairs arise in a number of contexts; for example, cluster varieties are obtained by gluing CY pairs by volume preserving or crepant birational maps. A CY pair has maximal intersection if its dual complex has maximal dimension. Such pairs are degenerate objects and their geometry has a "Fano" flavor. Toric and cluster varieties have maximal intersection, and while in dimension $2$ these are essentially the only maximal intersection CY pairs, this does not hold in higher dimensions. In this talk, I will give some examples in dimension $3$, and focus on the birational geometry of maximal intersection CY pairs. In particular, I will explain how mutations fit in this picture.

We describe how to use to Gross-Siebert algorithm to construct three dimensional polarised tropical manifolds - and hence Calabi-Yau threefolds - from a class of four dimensional reflexive polytopes. These (simply connected) Calabi-Yau threefolds are expected to be anti-canonical sections of Fano fourfolds with isolated Gorenstein singularites. We explain how to compute topological invariants of these spaces using a constructible sheaf on the 1-skeleton of the polytope, and provide an example of a single polytope from which $91$ such tropical manifolds can be constructed, providing $22$ topological types of Calabi-Yau threefolds; $9$ of which are new topological types with $b_2=1$.

I will introduce the original McKay correspondence for rational double points in dimension two and show the similar correspondence in dimension three. The singularities are Calabi-Yau and there is an invariant in the superstring theory which was the first step to find Mirror pairs. Moreover, the invariant can be considered as a topological invariant of a crepant resolution of the singularity. I would like to explain the related topics with some examples in terms of toric geometry and give a comment on a relation with mutation.

Cluster varieties are blow up of toric varieties. They come in pairs (A,X), with A and X built from dual tori. Compactifications of A, studied by Gross, Hacking, Keel, and Kontsevich, generalize the polytope construction of toric varieties while the compactifications of X, studied by Fock and Goncharov, generalize the fan construction. The conjecture is that the A and the X cluster varieties are mirrors to each other. Together with Tim Magee, we have shown that there exists a positive polytope for the type A cluster varieties which give us a hint to the Batyrev-Borisov construction.

Using Bernstein Kushnirenko theorem one can describe the cohomology ring of a smooth projective toric variety via volume polynomial on the space of polytopes. One can extend this description to the case of equivariant compactifications of a torus principal bundle. In my talk I will formulate a version of Bernstein Kushnirenko theorem for topic bundles and will explain how it leads to the computation of the cohomology ring.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.