Stability Conditions and Homological Projective Duality

University of Michigan in Ann Arbor

April 7-8, 2018

The goal of this Mini-Workshop is to learn about the newest developments in the study of derived categories of algebraic varieties through two mini-courses taught by two leading experts in the subject. The target audience are graduate students and postdocs, but all others are welcome as well.

Main Speakers

  • Jack Huizenga (Penn State University): Birational geometry of moduli spaces of sheaves and Bridgeland stability
  • Alex Perry (Columbia University): Homological projective geometry

Registration

Please register via an informal email to ulirsch@umich.edu by March 10. Our aim is to provide accommodations for external participants, but we would encourage all participants to find their own funding for transportation.

Preparatory Lecture

Friday, April 6, East Hall 4096

  • 4:10-6 pm: Zili Zhang: Introduction to derived categories and t-structures

The goal is to acquaint local graduate students with the basic material that is needed in order to understand what is going on in the mini-courses. Other participants are welcome to join as well.

Schedule

Saturday, April 7, East Hall 1068

  • 9:30-10:30: Alex Perry I
  • 10:30-11:00: Coffee break
  • 11:00-12:00: Jack Huizenga I
  • 12:00-2:00 Break for Lunch
  • 2:00-3:00: Alex Perry II
  • 3:00-4:00: Coffee break
  • 4:00-5:00: Jack Huizenga II

Sunday, April 8, East Hall B844

  • 9:30-10:30: Alex Perry III
  • 10:30-11:00: Coffee break
  • 11:00-12:00: Jack Huizenga III

Abstracts

Jack Huizenga: Birational geometry of moduli spaces of sheaves and Bridgeland stability

Abstract: The subject of the birational geometry of moduli spaces seeks to understand the possible compactifications of a moduli space parameterizing "nice" objects. Compactifying a moduli space usually requires that we allow our space to parameterize certain "degenerate" objects, in addition to the objects we initially wanted to study. In recent years the concept of Bridgeland stability has lead to tremendous activity in the study of the birational geometry of moduli spaces of sheaves. We will begin by recalling the classical birational geometry of Hilbert schemes of points, which are some of the simplest moduli spaces of sheaves. Then we will see that Bridgeland stability conditions allow us to better understand this birational geometry and describe the "degenerate" objects that are added in new compactifications. Finally, we will see how the positivity lemma of Bayer and Macri allows us to prove new results about the birational geometry of moduli spaces of sheaves. In particular, we will focus on the computation of the ample cone of the moduli space of sheaves on a surface.

Alex Perry: Homological projective geometry

Abstract: This minicourse is concerned with the structure of derived categories of algebraic varieties. One of the most fundamental problems in this area is to determine when the derived categories of two varieties are equivalent, or more generally have a large subcategory in common. Besides being of intrinsic interest, such relations between derived categories often have strong geometric consequences.

Our goal is to explain some surprising "homological" counterparts of constructions and results in classical projective geometry, which give powerful tools for attacking the above problem. We will start by studying semiorthogonal decompositions -- a way to break up the derived category into smaller pieces -- through many examples. Then we will discuss Kuznetsov's theory of homological projective duality, which gives a mechanism for understanding the derived categories of linear sections of a fixed variety. Finally, we will discuss a categorical version of the join of two projective varieties, and explain how it can be used to greatly expand the applicability of homological projective duality. An underlying theme of the lectures will be the use of "noncommutative" varieties, especially as resolutions of singularities.

Zili Zhang: Introduction to derived categories and t-structures

Abstract: Derived categories were introduced to have a better foundation for the theory of derived functors. In this talk, we will start with the general construction of derived categories for abelian categories. Then we restrict to the bounded derived category derived categories of coherent sheaves on schemes and we will defined push-forward, pull-back and tensor product in the derived sense. We will focus on illustrating why working on chain complexes is better than single objects. As applications, the projection formula and Serre duality can be promoted to a derived version. We will also define t-structures of triangulated categories and their hearts. An interesting example will be presented showing that an injective morphism becomes surjective when t-structures change.

Organizer

Martin Ulirsch (University of Michigan)

Support

We acknowledge support from NSF Grant 1702114 (PI: Mircea Mustaţă).

Participants

Jack Huizenga, Penn State University

Alex Perry, Columbia University

Lizhen Ji, University of Michigan

Shizhuo Zhang, Indiana University

Bhargav Bhatt, University of Michigan

Jeffrey Lagarias, University of Michigan

Karen Smith, University of Michigan

Jun Wang, Ohio State University

Dylan Spence, Indiana University

Xuqiang QIN, Indiana University

Gilyoung Cheong, University of Michigan

Devlin Mallory, University of Michigan

Nancy Wang, University of Michigan

Emanuel Reinecke, University of Michigan

Yajnaseni Dutta, Northwestern

David Schwein, University of Michigan

Sayanta Mandal, UIC

Dmitry Zakharov, Central Michigan University

Aalong Li, Indiana University

Shaopeng Zhu, University of Maryland

Ghosh, Arkabrata, Central Michigan University

Sanal Prasad, University of Michigan

Wai Kit Yeung, Indiana University

Steffen Marcus, The College of New Jersey

Mircea Mustata, University of Michigan

Martin Ulirsch, University of Michigan

Eamon Quinlan, University of Michigan

Kannappan Sampath, University of Michigan

Shubhodip Mondal, University of Michigan

Haoyang Guo, University of Michigan

Yongbin Ruan, University of Michigan

Junyan Xu, Indiana University