Semi-orthogonal decompositions seminar

We will introduce the basics of derived categories and derived functors. Our aim is to give concrete examples on how to operate with these objects, as a build-up to the goals of the seminar. Notes 

I will present the basic definitions and properties of semi-orthogonal decompositions and work out some key examples. I may also touch on some other ideas which will be important going forward, such as Fourier-Mukai transforms, Serre functors, and reconstruction theorems. Notes 

We will study derived categories in the situation of a closed immersion and of a blow-up. The goal is to construct fully faithful functors whose images are admissible subcategories of a blow-up, and to show that they form a semi-orthogonal decomposition. Finally we will see these results in action in some examples. Notes 

In this talk, we will discuss Kapranov’s constructions of semi-orthogonal decompositions for Grassmannians and quadrics. If time permits, we will also cover the case of intersections of two quadrics as studied by Bondal-Orlov. Unofficial notes (courtesy of Amal Mattoo).

We will start by recalling the definition of a Serre functor, together with some examples. We will use them to construct adjoints of functors in various contexts. In particular, this will give us a way to produce new semi-orthogonal decompositions from old ones, a process known as a mutation. We will see these in action in various concrete scenarios. Notes 

Kuznetsov components are admissible subcategories inside derived categories of hypersurfaces of small degree. I'll define them and prove that they are Calabi–Yau categories. In particular, we'll understand why Kuznetsov components of cubic fourfolds are "derived K3 surfaces" and time permitting we'll look at some cases when a Kuznetsov component of a cubic fourfold is actually isomorphic to the derived category of a (twisted) K3 surface. Unofficial notes (courtesy of Amal Mattoo)

We will explore how to construct semi-orthogonal decompositions for normal surfaces with rational singularities under favorable conditions. The key idea will be to pass to resolution of the surface, construct a "compatible" semi-orthogonal decomposition of the bounded-above derived category there, descend it down to a semi-orthogonal decomposition for the bounded-above derived category for the original surface, and use it to construct a "nice" semi-orthogonal decomposition for the bounded derived category (we will explain the terms in quotes in the talk). We will conclude with some examples of this procedure, and (time-permitting) discuss some Brauer group obstructions to running this process. Unofficial notes (courtesy of Amal Mattoo)

I'll discuss the notion of classical generators and weak generators. After setting up some lemmas, I'll prove that the derived category of complexes of O_X-modules with quasi-coherent cohomology sheaves on a quasi-compact, quasi-separated scheme is generated by a single perfect object. Finally, I'll illustrate this result in the case of projective spaces and its quasi-compact locally closed subschemes. Notes 

Every two birational Calabi-Yau manifolds are conjecturally derived equivalent. By Kawamata, a birational map between Calabi-Yau manifolds decomposes into a finite sequence of flops, so it is enough to show that a single flop induces a derived equivalence. Two special instances of flops are a standard flop and Mukai flop. In this talk, we will see that these two flops indeed induce derived equivalences, following Bondal–Orlov, Kawamata, and Namikawa's results. Unofficial notes (courtesy of Amal Mattoo)

Having seen how a resolution of rational singularities leads to an elegant semi-orthogonal decomposition, we now examine the case of arbitrary singularities. We outline how the derived category of a singular variety can be embedded into a smooth triangulated category which, while not necessarily itself the derived category of a smooth variety, admits a semi-orthogonal decomposition into components that are. To do so we will need to sketch some machinery of dg categories. Time permitting we will introduce other related notions to categorical resolutions, such as refined blowups and categorical absorption. Notes. 

We define Bridgeland stability and make contact with representation theory via highest weight categories. We will do a couple of very fun examples, including something Amal discussed last week. Unofficial notes (courtesy of Amal Mattoo)

I will give an introduction to Kuznetsov's homological projective duality (HPD), which is an extension (to the context of derived categories) of the classical projective duality between subvarieties of a projective space and subvarieties of the dual space. I will discuss a few of the many examples of derived equivalences which have come out of it. Unofficial notes (courtesy of Amal Mattoo)

Many interesting varieties in algebraic geometry can be constructed using Geometric Invariant Theory, and appear as an open substack X^ss of a quotient stack X=Y/G. The complement of X^ss has a special form of stratification, called a Theta-stratification. We will explain, following Halpern-Leistner, how the Theta-stratification induces a semiorthogonal decomposition of D^b(X), with one of the pieces, the window, being D^b(X^ss). We will see this in action with simple examples. Under some special circumstances, the window remains unchanged when we change the stability parameter, producing interesting derived equivalences. Halpern-Leistner used this idea to establish the D-equivalence conjecture in some cases. Unofficial notes (courtesy of Amal Mattoo)