We give an introduction to the Fukaya category of singularities and mirror symmetry relations to the derived category of coherent sheaves.

Given a crepant contraction to a singularity, we may expect a symmetry of the derived category of coherent sheaves on the source. Focussing on surface and 3-fold examples, I will explain general constructions of such symmetries for hypersurface singularities (arXiv:2409.19555). I will discuss relations with existing constructions, including the noncommutative deformation theory appearing in my previous work with Wemyss on flopping contractions.

When we consider deformations of a coherent sheaf on a projective variety, it is natural not to assume that the parameter ring is commutative. This is because the DG algebra controlling the deformations is non-commutative. The existence of a versal formal NC deformation is easily proved, but it is difficult to describe it for a specific problem. I will review some cases where versal NC deformations are known. The convergence is also known when the variety is smooth. I will also talk about a global moduli space.

Minimal surfaces with pg=q=0 admit non-trivial semiorthogonal decompositions (SODs) by exceptional collections, meaning that SODs do not always come from the minimal model program. On the other hand it has been a folklore conjecture that these are the only exceptions: Namely, the derived category of a minimal surface is indecomposable unless pg=q=0. This conjecture was finally solved in the affirmative in a very recent preprint by Pirozhkov. In my talks I will give a sketch of the proof of the conjecture and mention some related topics, while emphasizing how people who got interested in the subject inspired each other.

I will explain the construction of the moduli space of augmented stability conditions on a triangulated category. This space gives a partial compactification of Bridgeland's space of stability conditions, modulo the action of the additive group of complex numbers. I will explain how this space plays a prominent role in the construction of canonical semiorthogonal decompositions of D^b(X). Particular attention will be paid to the example of P^1. This is based on joint works with Daniel Halpern-Leistner. 

On the projective plane, full exceptional collections and Q-Gorenstein degenerations to weighted projective planes are both controlled by Markov equations, exhibiting an intriguing parallelism between these objects. This fits into a more general principle, due to Hacking, that under suitable conditions Wahl singularities give rise to exceptional vector bundles on a Q-Gorenstein smoothing. It is natural to ask what happens when a surface contains other types of singularities. In this talk, I will discuss several extensions of Hacking's result in this direction.

A noncommutative crepant resolution (NCCR) of a commutative ring R is defined to be the endomorphism algebra End(M) of a certain R-module M, which is called a maximal modifying (MM) module. We give a classification of MM modules over a certain 3-dimensional quotient singularity. If time permits, we explain that MM modules induce subspaces of Bridgeland stability conditions, and the union of these spaces is a regular covering space of the complexified Tits cone. This talk is based on ongoing joint work with Wahei Hara.

In 2003, Eisenbud and Schreyer introduced Ulrich bundles as geometric analogues of maximal Cohen-Macaulay modules having the maximal number of generators. In the same paper they proposed two questions: (1) whether a given projective variety X embedded in a projective space supports an Ulrich sheaf; (2) if yes, what is the smallest possible rank of it. The first question is nowadays often called the Eisenbud-Schreyer conjecture, in a connection with the study of Boij-Soederberg cones. The second question is called the Ulrich complexity problem, which is widely open even for projective surfaces and cubic hypersurfaces. In this talk, we first discuss how to construct an Ulrich bundle of rank 6 on a smooth cubic fourfold X by using deformation theory on the Kuznetsov component of the derived category of X. And then we also discuss a computational approach to achieve Ulrich bundles of various ranks on some special cubic fourfolds. If time permits, I would like to also discuss a relationship between instanton bundles and an instantonic approach to Ulrich problems, appeared as in Kuznetsov, Cho-Kim-Lee, and Casnati-Faenzi-Galluzi for instance. A part of the talk is based on a joint work with Daniele Faenzi.

An Ulrich bundle is a vector bundle with very strong cohomology vanishing conditions. Eisenbud and Schreyer conjectured that every smooth projective variety possesses an Ulrich bundle. Despite many results on low dimensional varieties and special varieties, the general existence is unknown. In this talk, I will describe recent work in progress with Kyoung-Seog Lee and Jiwan Jung on the construction of Ulrich bundles on an intersection of quadrics.

Ulrich vector bundles are one of interesting objects on projective varieties. In this talk, I will discuss Ulrich bundles on prime Fano threefolds V22 of degree 22 in terms of the exceptional collections in the bounded derived category of coherent sheaves on V22. It is proven that every Ulrich bundle on V22 arises from a monad consisting of explicit sequence of vector bundles appearing naturally from geometric descriptions of V22. As a result, we can relate a stable Ulrich bundle on V22 with a stable quiver representation of the coefficient quiver associated with the corresponding monad. This talk is based on joint work with Woohyuck Choi and Kyoung-Seog Lee.

We discuss several applications of Markman’s recent construction of hyperholomorphic bundles on products of hyper-Kähler varieties of K3^[n] type. These include the algebraicity of certain natural Hodge classes (Markman’s own work), the D-equivalence conjecture (joint work with Davesh Maulik, Junliang Shen, and Ruxuan Zhang), and, potentially, a better understanding of the Chow ring of K3^[n] type varieties.

A classical result states that a smooth index one prime Fano threefold of degree 14 is uniquely determined by the (geometrically) corresponding cubic threefold and an instanton bundle on it. As a result, the fiber of period map for such a Fano threefold is the locus of rank two instanton bundles on the cubic threefold. In my talk, I will give a categorical re-interpretation of this result and show that such an index one prime Fano threefold is uniquely determined by its Kuznetsov component--a subcategory of bounded derived category and a distinguished object. Then I will talk about its generalization in two directions: smooth Fano threefolds of other genus and nodal Fano threefolds. For the first case, acyclic extension of instanton bundles will appear, for the second case,non-locally free instanton sheaves appear. Then I will describe a correspondence between degeneration of instanton bundles and degeration of smooth Fano threefolds. This is based on a joint work with Zhiyu Liu,Jacovskis Augustinas and a very recent work with Xun Lin and Daniele Faenzi.