# Non-commutative, derived and homotopical methods in geometry

University of Antwerp

September 19–24, 2016

## Abstracts

### Categorical measures for equivariant varieties

Daniel Bergh (University of Bonn)

The categorical measure was introduced by Bondal, Larsen and Lunts in 2004. This is a ring homomorphism from the Grothendieck group of varieties to the Grothendieck group of saturated dg categories. We study this measure in an equivariant setting. In particular, we prove a formula suggested by Galkin and Shinder relating the motivic zeta functions on the respective Grothendieck groups. Related to this is a conjecture by Polishchuk and Van den Bergh regarding the existence of a certain semi-orthogonal decomposition of the equivariant derived category for a variety with an effective action by a finite group. We give an example showing that such a decomposition cannot exist if the effectiveness assumption is removed.

This is a joint work (in progress) with S. Gorshinsky, M. Larsen and V. Lunts.

### Hermitian Yang–Mills in derived categories

Jonathan Block (University of Pennsylvania)

We formulate a Hermitian Yang–Mills equation in the derived category of coherent sheaves. A derived Calabi–Yau object naturally arises.

### Canonical local generators for birational morphisms

Alexey Bondal (Steklov Mathematical Institute)

In his paper on flops of relative dimension 1, Van den Bergh studied t-structures, introduced by Bridgeland, in derived categories related to flopping contractions of relative dimension 1 and constructed local projective generators for the hearts of two of the t-structures, by generalizing an earlier construction of Artin and Verdier. These projective generators turned out to be an important tool for studying flops, because, in particular, their local endomorphism algebras can be regarded as non-commutative resolutions.

The construction of the generators was inherently non-canonical. We discuss joint results with Agnieszka Bodzenta towards constructing new t-structures related to birational morphisms of smooth varieties, surprisingly simple canonical local projective generators related to them and their endomorphism algebras.

### Tilting in the graded singularity category of a quasihomogeneous Gorenstein curve

Ragnar-Olaf Buchweitz (University of Toronto)

In this joint work with Iyama and Yamaura we show that the stable category of graded maximal Cohen-Macaulay modules over a quasi-homogeneous reduced Gorenstein curve singularity admits an explicit tilting object.

This means, in particular, that that triangulated category is exact equivalent to the derived category of modules of finite length over a (usually non-commutative) Artin algebra. That Artin algebra is of finite global dimension and one can use this to construct any graded torsionfree module over the curve singularity through finitely many extensions of finitely many known modules and their (co-)syzygies.

### Derived symplectic and Poisson geometry

Damien Calaque (Université de Montpellier)

We will present an overview of the recent progress on symplectic and Poisson structures in the setting of derived algebraic geometry. We will point out several interesting connections with mathematical physics, for instance with topological field theories.

### A secondary Dold-Kan correspondence

Tobias Dyckerhoff (Hausdorff Center for Mathematics)

Various recent developments, in particular in the context of topological Fukaya categories, seem to be glimpses of an emerging theory of secondary, or categorified, homotopical and homological algebra. The increasing number of meaningful examples and constructions make it desirable to develop such a theory systematically. In this talk, we discuss a step towards this goal: a secondary analog of the Dold–Kan correspondence.

### On some constructions of $\mathrm{L}_\infty$-algebras

Yaël Frégier (University of Artois)

Gerstenhaber and Schack have introduced a cohomology governing infinitesimal deformations of diagrams of algebras. It recovers Hochschild and Kodaira–Spencer theories in specific cases. However, more structure is needed to describe the formal deformations as a Maurer–Cartan equation. We will explain how one can build a $\mathrm{L}_\infty$ algebra governing such deformations in two ways: by resolution of operads or by derived brackets. Examples encompass some simultaneous deformation problems such as morphisms and algebras. The derived bracket approach can find applications in geometry, where the operadic approach does not apply.

### An elementary LG model without projective mirrors

Elizabeth Gasparim (Universidad Católica del Norte, Chile)

I will discuss deformations of local surfaces and their moduli of vector bundles. This is joint work with Severin Barmeier. I will then present a Landau-Ginzburg model that has no projective mirrors.

### How to glue derived categories

Dmitry Kaledin (Steklov Mathematical Institute)

It is well-known that one cannot glue triangulated categories unless they are equipped with some enhancement. In practice, people use either DG enhancement, or the machinery of stable infinity-categories. Unfortunately, for quite a few applications, DG enhancements are not flexible enough, while infinity-categories are much too heavy. We are going to sketch an intermediate approach based on the notion of a “stable model pair”, and show how it can be applied to practical problems such as deformations of abelian categories.

### Perverse schobers on surfaces via Ran categories

Mikhail Kapranov (Yale University)

We give a definition of perverse schobers on an oriented topological surface using a relative version of the category of exit paths of the Ran space. It incorporates the periodicity properties of the relative Waldhausen S-construction of a spherical functor. This allows us to define the topological Fukaya category with coefficients in a schober. Joint work with T. Dyckerhoff, V. Schechtman and Y. Soibelman.

### Local duality for representations of finite group schemes

Henning Krause (University of Bielefeld)

The modular representations of a finite group scheme enjoy a Gorenstein property which implies a “local duality” for the representations supported at a point of the projective variety of the cohomology ring. To explain this new type of duality is the aim of my talk. All this is based on recent work with D. Benson, S. Iyengar, and J. Pevtsova.

### Categorical joins and homological projective duality

Alexander Kuznetsov (Steklov Mathematical Institute)

In the talk I will discuss the notion of a categorical join and its relation to homological projective duality. In particular, I will describe some interesting semiorthogonal decompositions and equivalences of categories that can be obtained by using the method of joins. This is a joint work with Alex Perry.

### Spaces of locally finite triangulated categories

Fernando Muro (University of Sevilla)

Locally finite triangulated categories have been extensively studied from a classical perspective by Xiao and Zhu, Amiot, and Krause. In this talk we will consider their spaces of differential graded enhancements. We will see that the obstruction theory for the existence and uniqueness of enhancements works particularly well with these triangulated categories, allowing for explicit computations.

### Strong generation in derived categories of schemes

Amnon Neeman (Australian National University)

Bondal and Van den Bergh proved in 2003 that the bounded derived category of coherent sheaves on a scheme smooth over a field $k$ is strongly generated. Rouquier proved in 2008 that this is also true without the smoothness hypothesis. And a tremendous amount of work has gone into studying the phenomenon, and estimating the Rouquier dimension of these categories. We will review some of the work.

Based on an idea from a beautiful paper by Orlov the speaker was able to improve an approximation theorem by Lipman and himself. The relevance to the results in the first paragraph is that one can generalise greatly: it turns out that the 2003 theorem by Bondal and Van den Bergh extends to say that the category of perfect complexes on any quasi compact, separated scheme is strongly generated if and only if the scheme is locally of finite global dimension, while the bounded derived category of coherent sheaves is strongly generated for any scheme of finite type over an excellent 3-fold.

But it turns out that the approximation result has other applications—time permitting I hope to say something about them.

### Compact moduli of marked non-commutative del Pezzo surfaces

Shinnosuke Okawa (Osaka University)

I will introduce certain construction of compactified moduli spaces of marked non-commutative del Pezzo surfaces. The construction is based on full strong exceptional collections of derived categories. I will also discuss relationship with non-commutative projective planes and their blowups. The talk will be based on my joint work with Tarig Abdelgadir and Kazushi Ueda.

### Descent and equivalences in non-commutative geometry

Tony Pantev (University of Pennsylvania)

I will describe a descent formalism in categorical non-commutative geometry which is geared towards constructions of Fourier–Mukai functors. The formalism allows one to carry out descent constructions in general algebraic and analytic frameworks without resorting to generators. I will discuss various applications, such as the connection to the classical Zariski and flat descents, constructions of Fukaya categories, and homological mirror symmetry. This is a joint work with Katzarkov and Kontsevich.

### Moduli spaces of $\mathrm{A}_\infty$-structures

Alexander Polishchuk (University of Oregon)

In this talk I will present a criterion on the existence of a nice moduli space parametrizing $\mathrm{A}_\infty$-structures on a given finite-dimensional graded associative algebra. I will consider two examples involving such moduli spaces: one is related to moduli spaces of curves and another to solutions of the associative Yang–Baxter equation.

### Matlis–Greenlees–May duality, covariant Serre–Grothendieck duality, and comodule–contramodule correspondence

Leonid Positselski (Higher School of Economics)

Covariant Serre–Grothendieck duality is an equivalence between the coderived and the contraderived category of modules. To construct such an equivalence, one needs an additional piece of data, namely, a dualizing complex. Matlis–Greenlees–May duality is an equivalence between the conventional derived categories of comodules and contramodules. To construct such an equivalence, one needs a piece of data called a dedualizing complex. Comodule–contramodule correspondence is an equivalence between the coderived category of comodules and the contraderived category of contramodules. This equivalence is canonical and does not need any additional data.

### Homological projective duality for Pfaffians

Ed Segal (Imperial College London)

A Pfaffian variety is a space of anti-symmetric matrices with some fixed upper bound on the rank, and the projective dual of a Pfaffian variety is a another Pfaffian variety. Kuznetsov conjectured that these varieties should have non-commutative resolutions, with the derived categories of projectively-dual pairs satisfying a nice relationship called “homological projective duality”. I will discuss the construction of these non-commutative resolutions (which is due to Špenko and Van den Bergh) and a proof that they do indeed satisfy the duality. Our proof is motivated by a physical duality of non-abelian GLSMs, proposed by Hori. This is joint work with Jørgen Rennemo.

### Non-commutative minimal surfaces

Susan Sierra (University of Edinburgh)

In the classification of (commutative) projective surfaces, one first classifies minimal models for a given birational class, and then shows that any surface can be blown down at a finite number of curves to obtain a minimal model.

Artin has proposed a similar programme for non-commutative surfaces (that is, domains of Gelfand–Kirillov dimension 3). In the generic “rational” case of rings birational to a Sklyanin algebra, the likely candidates for minimal models are the Sklyanin algebra itself and Van den Bergh's quadric surfaces. We show, using our previously developed non-commutative version of blowing down, that these algebras are minimal in a very strong sense: given a Sklyanin algebra or quadric $R$, if $S$ is a connected graded, noetherian overring of $R$ with the same graded ring of fractions, then $S=R$.

This is joint work with Rogalski and Stafford.

### Semi-orthogonal decomposition for GIT quotient stacks

Špela Špenko (University of Edinburgh)

We will present a semi-orthogonal decomposition of the derived category of a GIT quotient stack $\mathbf{D}(X/G)$ consisting of derived categories of coherent sheaves of rings on $X/\!/G$ which are locally of finite global dimension. This is joint work with Michel Van den Bergh.

### Non-commutative rational surfaces and subalgebras of Sklyanin algebras

Toby Stafford (University of Manchester)

One of the major open problems in non-commutative algebraic geometry is to classify non-commutative surfaces (or domains of Gelfand–Kirillov dimension 3). The non-commutative projective planes are known by the classic Artin–Tate–Van den Bergh theorem, with the “generic” planes been given by the Sklyanin algebras Skl. But can one classify all generic rational surfaces, (or algebras birational to Skl)? Subalgebras of Skl that are maximal orders have been classified in recent work with Rogalski and Sierra, using a version of blowing-up, which we will discuss in the first part of this lecture. The next step is to understand blowing down, which will be covered in the second half of the lecture. This will also be used in Sierra’s lecture on minimal models.

This is joint work with Rogalski and Sierra.

### Bridgeland stability for semiorthogonal decompositions

Paolo Stellari (University of Milan)

Various interesting results concerning the geometry of smooth cubic threefolds point to the relevance of constructing Bridgeland stability conditions on semiorthogonal decompositions. We illustrate some of them with the aim of introducing a new method to induce stability conditions on semiorthogonal decompositions. We prove that it provides Bridgeland stability conditions on the Kuznetsov component of the derived category of some Fano threefolds and cubic fourfolds. This is joint work in progress with Arend Bayer, Martí Lahoz and Emanuele Macrì.

### Bounded derived categories, normalisations, and duality

Greg Stevenson (University of Bielefeld)

Given a ring, one is frequently led to consider the bounded derived category of suitably finite modules; this category, from some points of view, has much better properties than the perfect complexes and admits interesting Verdier quotients such as the singularity category. I'll discuss joint work with John Greenlees proposing an analogue of the bounded derived category for certain augmented ring spectra, in terms of a kind of Noether normalisation. In particular, I'll highlight the way in which, given a sufficiently nice normalisation, one obtains a pleasingly symmetric situation upon passing to Koszul duals, together with examples demonstrating that this applies in many cases of interest.

### Voevodsky's nilpotence conjecture via non-commutative geometry

Goncalo Tabuada (Massachusetts Institute of Technology)

Voevodsky conjectured in the nineties that the smash nilpotent equivalence relation on algebraic cycles agrees with the classical numerical equivalence relation. Making use of non-commutative geometry, I will prove some new cases of this important conjecture.

### Bloch's conductor formula and matrix factorizations

Bertrand Toën (University of Toulouse)

The Bloch's conductor formula states that the change of Euler characteristics of a degenerating family of proper schemes is the sum of a certain intersection number and the Swan conductor. It has been proved in some special case by Kato-Saito but remains open in general. In this talk I will explain how to prove the conjecture using matrix factorizations, non-commutative motives and their $\ell$-adic realizations.

### Affine actions from 3-fold flops, and tilings of the plane

Michael Wemyss (University of Edinburgh)

Iterating flops (via cluster theory) gives a “birational geometry” action on the derived category of 3-folds, and I will explain how to understand this group from the viewpoint of the topology of hyperplane arrangements. They are very similar to pure braid groups. However, the expectation, now proved in many cases, is that this action be extended to an “affine pure braid group” acting on the derived category. I will explain some Coxeter combinatorics background behind this, which is completely independent of the geometric motivation. The remarkable thing is that the geometry predicts unseen algebraic phenomenon, like an affine version of the symmetries of the pentagon (which does not exist!), and most of the talk will be motivated by the simplest case of two intersecting flopping curves. Towards the end I will explain how this case is related to certain tilings of the plane, and indeed every choice of two nodes of a Dynkin diagram induces a tiling of the plane. These tilings seem to be new, and are somewhat mysterious.

### Weak proregularity, weak stability and the non-commutative MGM equivalence

Amnon Yekutieli (Ben Gurion University)

Let $A$ be a commutative ring, and let $\mathfrak{a}$ be a finitely generated ideal in it. It is known that a necessary and sufficient condition for the derived $\mathfrak{a}$-torsion and the derived $\mathfrak{a}$-adic completion functors to be nicely behaved is the weak proregularity of the ideal $\mathfrak{a}$. In particular, the MGM equivalence holds under this condition.

Because weak proregularity is defined in terms of elements of the ring (specifically, it involves limits of Koszul complexes), it is not suitable for non-commutative ring theory.

Consider a torsion class $\mathcal{T}$ in the category $\mathop{\mathrm{Mod}}A$ of left modules over a ring $A$. We introduce a new condition on $\mathcal{T}$: weak stability. Our first main theorem is that when $A$ is commutative, an ideal $\mathfrak{a}$ in $A$ is weakly proregular if and only if the corresponding torsion class $\mathcal{T}$ in $\mathop{\mathrm{Mod}}A$ is weakly stable.

It turns out that when the ring $A$ is non-commutative, one must impose two more conditions on the torsion class $\mathcal{T}$: quasi-compactness and finite dimensionality (these are new names for old conditions). We prove that for a torsion class $\mathcal{T}$ that is weakly stable, quasi-compact and finite dimensional, the right derived $\mathcal{T}$-torsion functor is isomorphic to a left derived tensor functor. This result involves derived categories of bimodules. Some examples will be given.

The third main theorem is the non-commutative MGM equivalence, under the same assumptions on $\mathcal{T}$. Finally, there is a theorem about derived left-sided and right-sided torsion for complexes of bimodules. This last theorem is a generalization of a result of Van den Bergh from 1997, and it corrects an error in a paper of Yekutieli-Zhang from 2003.

We expect that the approach outlined in this talk will open up the way to a useful theory of rigid dualizing complexes in the arithmetic non-commutative setting (namely without a base field).

The work above is joint with Rishi Vyas.