Reading Group: Moduli of Sheaves
We will be reading The Geometry of Moduli Spaces of Sheaves by Huybrechts & Lehn. The first edition of this book is available hereLink opens in a new window.
As a preliminary plan, we may go through Chapter 1, 2 (Quot-scheme), 3 (Bogomolov inequality), 4 (local properties), 5, 6, 7, 10.
-
Organisers: Peize LiuLink opens in a new window, Terry SongLink opens in a new window
- Format: We meet weekly online. Feel free to contact me if you are interested to join us. Teams linkLink opens in a new window
Timetable & Notes
8 Aug 2023 — Terry: Homological algebra of coherent sheaves (Section 1.1) NotesLink opens in a new window.
16 Aug 2023 — Peize: Stability & Harder–Narasimhan filtration (Section 1.2-1.3) NotesLink opens in a new window.
27 Aug 2023 — Terry: Stability of cotangent bundle of ; boundedness and regularity (Section 1.4, 1.7) NotesLink opens in a new window.
7 Sept 2023 — Terry: Flat families of sheaves; Quot-scheme and its deformations (Section 2.1-2.2).
27 Dec 2023 — Peize: Serre's construction; elementary transformations (Section 5.1-5.2) Notes.Link opens in a new window
New Year Day — Terry: GIT construction of the moduli space; representability; deformation (Chapter 4) NotesLink opens in a new window.
7 Jan 2024 — Peize: Low-dimensional moduli spaces on K3 surfaces (Section 5.3, 6.1) NotesLink opens in a new window.
21 Jan 2024 — Terry: Deformation equivalence of moduli spaces on K3 surfaces (elliptic case).
TBD — Peize: Deformation equivalence of moduli spaces on K3 surfaces (general case).
Tentative Future Plan
Phase I — continuing through Huybrechts–Lehn:
-
§6 — statement and sketches of various results on moduli of sheaves on a K3 surface; the surveyLink opens in a new window by Sawon and original paperLink opens in a new window (large scanned PDF) by Mukai are also worth looking at? (Could split into two sessions if necessary.)
-
§7 — how stability is preserved upon restricting to curves in surfaces; focus on Bogolomov's results in §7.3, and skim through results of Mehta–Ramanathan and Flenner in previous sections if time permits.
-
§10–11 — skimming through results on symplectic structures and birationality of moduli of sheaves.
Phase II — skimming papers on related topics, maybe we can choose from:
-
wall-crossing of moduli of sheaves (on surfaces) via Bridgeland stability;
-
one-dimensional sheaves, Hilbert–Chow morphism, and Gopakumar–Vafa invariants;
-
Euler characteristics and cut-and-paste invariants of moduli of sheaves;
-
...
