# Noncommutative Shapes: A conference in honor of Michel Van den Bergh's 60th birthday

University of Antwerp

September 12–16, 2022

## Abstracts

- Denis Auroux: Fukaya categories and functoriality in mirror symmetry
- Damien Calaque: Comparing Calabi-Yau and quasi-bisymplectic structures for multiplicative preprojective algebras
- Dmitry Kaledin: Non-commutative cristalline cohomology
- Amnon Neeman: Bounded t-structures on the category of perfect complexes
- Toby Stafford: Invariant holonomic systems for symmetric spaces
- Lenny Taelman: Deformations of Calabi–Yau varieties in mixed characteristic
- Gabriele Vezzosi: An analog of the Beilinson-Drinfeld Grassmannian for surfaces
- Michael Wemyss: Spherical objects in dimension 2 and 3

### Fukaya categories and functoriality in mirror symmetry

Denis Auroux (Harvard University)

The general theme of this talk will be functoriality in homological mirror symmetry, i.e. situations where naturally occurring functors between various types of Fukaya categories correspond under mirror symmetry to classical functors (induced by inclusions and projections) between derived categories of coherent sheaves (or matrix factorizations). The running examples will be very affine hypersurfaces on one side of mirror symmetry, and toric Calabi-Yau varieties on the other side. This talk is based on joint work with Abouzaid and Katzarkov, as well as work of Gammage and Jeffs.

### Comparing Calabi-Yau and quasi-bisymplectic structures for multiplicative preprojective algebras

Damien Calaque (University of Montpellier)

Multiplicative preprojective algebras first appeared in the work of Crawley-Boevey and Shaw, in the course of their study of the Deligne-Simpson problem. Their representation varities, known as multiplicative quiver varieties, naturally appear in various areas (character varieties, local systems on Riemann surfaces and perverse sheaves on nodal curves, integrable systems, etc...). Michel proved that these varieties can be obtained using a quasi-Hamiltonian reduction procedure (after Alekseev-Malkin-Meinreinken), and developed a noncommutative version of the quasi-Hamiltonian formalism so that all constructions actually hold directly at the level of the multiplicative preprojective algebras. In this talk I will explain explain that the moment map defining these multiplicative preprojective algebras carry a relative Calabi-Yau structure (a notion introduced by Brav and Dyckerhoff, following some earlier suggestion of Toën), and how the three notions (quasi-Hamiltonian structures, their non-commutative version, and relative Calabi-Yau structures) interact.

The talk is based on joint works with Tristan Bozec and Sarah Scherotzke.

### Non-commutative cristalline cohomology

Dmitry Kaledin (Steklov Mathematical Institute)

I am going to give an overview of several recent constructions of non-commutative versions of cristalling cohomology for DG algebras over a finite field (namely, a construction of Vologodsky and Petrov, and another construction of Tsygan-Nest). If time permits, I will also add some speculations as to what might be a possible counterpart of these constructions in characteristic zero.

### Bounded t-structures on the category of perfect complexes

Amnon Neeman (Australian National University)

Van den Bergh's work has greatly influenced mine. The talk will begin with a couple of papers by Van den Bergh that I am much indebted to, and then explain how the proof of a new result, about the non-existence of bounded t-structures on the category of perfect complexes on a scheme $X$, owes the key ideas to these papers by Van den Bergh.

### Invariant holonomic systems for symmetric spaces

Toby Stafford (University of Manchester)

Fix a complex reductive Lie group $G$ with Lie algebra $\mathfrak{g}$ and let $V$ be a symmetric space over $\mathfrak{g}$ with ring of differential operators $\mathcal{D}(V)$. A fundamental class of $\mathcal{D}(V)$-modules consists of the

admissible modules(these are natural analogues of highest weight $\mathfrak{g}$-modules). In this lecture I will describe the structure of some important admissible modules. In particular, when $V=\mathfrak{g}$ these results reduce to give Harish-Chandra's regularity theorem for $G$-equivariant eigendistributions and imply results of Hotta and Kashiwara on invariant holonomic systems. If I have time I will describe extensions of thes results to the more general polar $\mathfrak{g}$-representations.A key technique is relating (the admissible module over) invariant differential operators $\mathcal{D}(V)^G$ on $V$ to (highest weight modules over) Cherednik algebras.

This research is joint with Bellamy, Levasseur and Nevins.

### Deformations of Calabi–Yau varieties in mixed characteristic

Lenny Taelman (Universiteit van Amsterdam)

We study deformations of smooth projective varieties with trivial canonical bundle in positive and mixed characteristic. We show that (under suitable hypotheses) these are unobstructed. This is an analogue to the Bogomolov-Tian-Todorov theorem (in characteristic zero). We also show that “ordinary” varieties with trivial canonical bundle admit a preferred "canonical lift" to characteristic zero. This generalizes results of Serre-Tate (for abelian varieties) and Deligne (for K3 surfaces). Our proofs rely in an essential way on “derived” deformation theory as developed by Pridham and Lurie.

This is joint work with Lukas Brantner.

### An analog of the Beilinson-Drinfeld Grassmannian for surfaces

Gabriele Vezzosi (University of Firenze)

Fix an affine algebraic group $G$ over the complex numbers, and a proper smooth complex surface $X$. We can attach to a family of flags of closed subschemes in $X$ (parametrized by a scheme $S$) a generalization of the Beilinson-Drinfeld Grassmannian of $G$-bundles. The union of flags induces a simplicial structure that turns out to be 2-Segal, thus giving rise to a fusion monoidal structure on sheaves on this Grassmannian that also carries a flat connection and actions of suitable generalizations of the loop and positive loop groups. Time permitting, I will discuss the factorization property and some future directions. This is joint work with Benjamin Hennion and Valerio Melani.

### Spherical objects in dimension 2 and 3

Michael Wemyss (University of Glasgow)

I will explain how to classify spherical objects in various settings (Kleinian singularities, 3-fold flops) using simple modules for the corresponding noncommutative resolution. The main result is much more general, and also more surprising: in the null category $\mathcal{C}$, (1) all objects with no negative Ext groups belong to the heart of a bounded t-structure, and (2) the objects with no negative Ext groups, and whose self-Hom is one-dimensional, are precisely the simples. In fact, we can even classify all bounded t-structures on $\mathcal{C}$. There are various geometric corollaries. The main technique also works for finite dimensional algebras, where in the derived category of a silting discrete algebra, every semibrick complex can be completed to a simple minded collection. This is all joint work with Wahei Hara.