Embedded Files
We will be reading The Geometry of Moduli Spaces of Sheaves by Huybrechts & Lehn. The first edition of this book is available online.
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 Liu, Terry Song
Format: We meet weekly online. Teams link
Timetable & Notes
Timetable & Notes
8 Aug 2023 — Terry: Homological algebra of coherent sheaves.
(Section 1.1) Notes.
16 Aug 2023 — Peize: Stability & Harder-Narasimhan filtration.
(Section 1.2-1.3) Notes.
27 Aug 2023 — Terry: Stability of cotangent bundle of projective space;
boundedness and regularity. (Section 1.4, 1.7) Notes.
7 Sept 2023 — Terry: Flat family 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.
New Year Day — Terry: GIT construction of the moduli space; representability;
deformation. (Chapter 4) Notes.
7 Jan 2024 — Peize: Low-dimensional Moduli spaces on K3 surfaces.
(Section 5.3, 6.1) Notes.
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
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 survey by Sawon and original paper (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;
...
Page updated
Google Sites
Report abuse