Bonn

MODULI OF VECTOR BUNDLES. (MODULE V4A5)

Lecturer: Dr. Yajnaseni Dutta
Time and location Mondays, 16:15 - 17:45 and Rm. 1.008. Hybrid (zoom details on eCampus).
ANNOUNCEMENT: Special office hours: Feb 4, 2022. 16-17:30h Rm. 3.003.
Office hours: Appointment (in person or zoom) by Email.

Prerequisites.Voraussetzung
Algebraic Geometry sequence 1 & 2 or equivalently Hartshorne Ch. I, II & III and Riemann–Roch on curves.

Overview.
Given a smooth projective variety, the set of vector bundles (after fixing some invariants) on it has a lot of structures. Moduli problems are the formal machinery to access these structures. One of the first examples is the isomorphism classes of line bundles, denoted by Pic(X). It is not only a group but also admits the structure of a separated and locally of finite type scheme. The Picard functor was formally introduced by Grothendieck in his 1962 Bourbaki talks. However in higher ranks, even for smooth projective algebraic curves, the moduli of vector bundles with fixed rank and degree behaves badly. There are two major recourses in the literature. We will focus on one of them which is to restrict the class of vector bundle by imposing the so-called stability criterion. This gives rise to a fairly well-behaved coarse moduli space. After reviewing some basic properties of vector bundles, coherent sheaves and homological algebra, we will discuss various examples of stable bundles, which are interesting in their own right. The final goal would be to discuss the construction of the moduli space, what can be said about universal families on them, and understand the case of K3 surfaces explicitly. The construction builds upon Grothendieck’s Quot schemes, which are interesting and indispensable in the study of moduli problems in general. We will follow The geometry of moduli spaces of sheaves. Cambridge University Press. 2nd edition., 2010 by D. Huybrechts and M. Lehn closely.

Outline.
Throughout this course we will also learn a few techniques for proving representability of functors. Some of you may have already seen some interesting ways. Think about them and bring them to the class! I plan to cover the following topics which will evolve as we go forward:

Flat family of sheaves, flattening stratification.
Grassmannian functor (using this representability criterion).
Quot functor (using the Grassmannian to guess a subscheme which represents)
Semistability (we will mostly focus on Gieseker stability) Harder-Narasimhan filtration, Jordan-Hölder filtration and their relative versions
Moduli functor of semistable sheaves on Curves and universal family
Moduli functor of semistable sheaves on Surfaces
Constructing semistable sheaves on surfaces using Serre correspondence and elementary transformations.

After this point the plan is to discuss moduli of sheaves on K3 surfaces. We will explore some examples and perhaps see the symplectic structure on it or may be discuss its cohomology ring using chern classes of the universal family. We will see.

Logistics and Covid update

First day of lecture: 11.10.2021.
Covid Update (Nov 22, 2021): University has appealed to everyone to keep masks (medical masks or FFP2 masks) on at seats.
Lectures will be held in hybrid format regularly. This means there is a possibility to join it remotely. For everyone's safety if you are feeling any flu-like symptom whatsoever, you are requested to join the class on zoom. This applies to me too, which is when the classes will go completely online.
Emails will be sent from eCampus. Please sign-up if you want to get future emails.
Discussion forums have been set up on eCampus where you can ask and answer questions throughout the semester.

References.

Daniel Huybrechts and Manfred Lehn. The geometry of moduli spaces of sheaves. Cambridge Mathematical Library of Cambridge University Press. 2nd edition., 2010.
Steven L. Kleiman. The Picard scheme Fundamental algebraic geometry. Grothendieck’s FGA explained, Mathematical Surveys and Monographs 123, Amer. Math. Soc., 2005. pp. 235–321
Nitin Nitsure. Construction of Hilbert and Quot schemes. Fundamental algebraic geometry. Grothendieck’s FGA explained, Mathematical Surveys and Monographs 123, Amer. Math. Soc., 2005. pp. 105–137
Mihnea Popa. Moduli of vector bundles on curves and generalized theta divisors. Lectures from summer school on Moduli Spaces and Arcs in Algebraic Geometry, Cologne, 2006.

Lecture notes.

Notes will be updated here

Exam.
Oral. First exam date Monday February 6, 2022.