Semi-orthogonal decompositions seminar

In this seminar we study work on the derived categories of coherent sheaves on schemes, particularly focusing on methods related to semi-orthogonal decompositions. After building up the basic theory of derived categories, we will examine various cases of semi-orthogonal decompositions arising in geometry. Finally, we will see some examples of how these techniques are used in modern algebraic geometry research, with the specifics flexible depending on participant interest!

A more detailed summary of the seminar, including topics for the future talks and references, is available here (last updated February 23rd).

Setup

• Organizers: Amal Mattoo and Nicolás Vilches
• Location: Mathematics 528
• Time: Wednesdays, 1pm-2:30pm

Please contact either of the organizers to be added to the mailing list.

List of talks

• Week 1 (Jan 29): A fast introduction to derived categories (Nicolás Vilches)

  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

• Week 2 (Feb 5) Introduction to semi-orthogonal decompositions (Amal Mattoo)

  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

• Week 3 (Feb 12) SOD’s from geometry (Wenqi Li)

  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

• Week 4 (Feb 19) More examples on SOD’s (Kuan-Wen Chen)

  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)

• Week 5 (Feb 26) Mutations and Serre functors (Nicolás Vilches)

  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

• Week 6 (Mar 5) Kuznetsov components (Anna Abasheva)

  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)

• Week 7 (Mar 12) Surfaces with rational singularities (Vidhu Adhihetty)

  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.

• No meeting (Spring break)
• Week 8 (Mar 26) Computational results on SOD’s (Hechen Hu)
• Week 9 (Apr 2) Derived equivalences via SOD’s (Yoonjoo Kim)
• Week 10 (Apr 9) Categorical resolutions (Rafah Hajjar Muñoz)
• Week 11 (Apr 16) Stability conditions and SOD’s (Sofia Wood)
• Week 12 (Apr 23) TBD
• Week 13 (Apr 30) TBD