Moduli spaces of vector bundles: curve case
This course provide a comprehensive introduction to the theory of moduli spaces of vector bundles on algebraic curves. Topics covered include moduli problems and functors, Quot schemes, Harder--Narasimhan filtration, vector bundles on elliptic curves, geometric invariant theory, and the construction and geometry of moduli spaces of semistable vector bundles.
Lecturer
Date
13th October ~ 30th December, 2026
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Tuesday,Wednesday | 10:40 - 12:15 | Shuimo | - | - | - |
Prerequisite
AGI and II
Syllabus
Week 1: Moduli problems, fine and coarse moduli spaces, unboundedness, jumping phenomena, and semistable vector bundles.
Week 2: Flat families, Hilbert polynomials, Castelnuovo–Mumford regularity, Kleiman's boundedness criterion, and boundedness of semistable bundles.
Week 3: Grassmannian functor, Plücker embedding, Quot scheme construction, properness, tangent space, and deformation theory of quotients.
Week 4: Grothendieck's classification on ℙ¹, Harder–Narasimhan filtrations (existence, uniqueness, polygons), and elementary deformation theory.
Week 5: Openness of semistability, existence of stable bundles, flag schemes, deformation of flags, schematic HN stratification, and relative HN filtrations.
Week 6: Indecomposable bundles, Krull–Schmidt theorem, Atiyah's classification on elliptic curves , and moduli spaces as symmetric products / projective spaces.
Week 7: Affine algebraic groups, actions, orbits, stabilisers, reductive groups, Hilbert's 14th problem, Reynolds operator, finite generation, and affine GIT quotients.
Week 8: Linearised line bundles, projective GIT semistable/stable points, construction of projective quotients, and Hilbert–Mumford criterion (statement).
Week 9: Hilbert–Mumford weights, numerical criteria, Picard group of G-varieties, GIT stability on Grassmannians, and application to Quot schemes.
Week 10: Semistability criterion via global sections, embedding into Quot scheme with GIT linearisation, equality of stabilities, and GIT construction of moduli spaces.
Week 11: Irreducibility of moduli spaces,Luna's étale slice theorem, smoothness of stable locus, tangent space , and singular locus.
Week 12: Existence of universal family , descent of vector bundles, Picard group, and determinant line bundles / Theta divisors.
Week 2: Flat families, Hilbert polynomials, Castelnuovo–Mumford regularity, Kleiman's boundedness criterion, and boundedness of semistable bundles.
Week 3: Grassmannian functor, Plücker embedding, Quot scheme construction, properness, tangent space, and deformation theory of quotients.
Week 4: Grothendieck's classification on ℙ¹, Harder–Narasimhan filtrations (existence, uniqueness, polygons), and elementary deformation theory.
Week 5: Openness of semistability, existence of stable bundles, flag schemes, deformation of flags, schematic HN stratification, and relative HN filtrations.
Week 6: Indecomposable bundles, Krull–Schmidt theorem, Atiyah's classification on elliptic curves , and moduli spaces as symmetric products / projective spaces.
Week 7: Affine algebraic groups, actions, orbits, stabilisers, reductive groups, Hilbert's 14th problem, Reynolds operator, finite generation, and affine GIT quotients.
Week 8: Linearised line bundles, projective GIT semistable/stable points, construction of projective quotients, and Hilbert–Mumford criterion (statement).
Week 9: Hilbert–Mumford weights, numerical criteria, Picard group of G-varieties, GIT stability on Grassmannians, and application to Quot schemes.
Week 10: Semistability criterion via global sections, embedding into Quot scheme with GIT linearisation, equality of stabilities, and GIT construction of moduli spaces.
Week 11: Irreducibility of moduli spaces,Luna's étale slice theorem, smoothness of stable locus, tangent space , and singular locus.
Week 12: Existence of universal family , descent of vector bundles, Picard group, and determinant line bundles / Theta divisors.
Reference
1. Geometric invariant theory, D. Mumford, J. Fogarty, and F. Kirwan;
2. Lectures on vector bundles, J. Le Potier;
3. The geometry of moduli spaces of sheaves, second edition, D. Huybrechts and M. Lehn.
4.https://www.bimsa.net/activity/Modspaofvecbunoncur/
2. Lectures on vector bundles, J. Le Potier;
3. The geometry of moduli spaces of sheaves, second edition, D. Huybrechts and M. Lehn.
4.https://www.bimsa.net/activity/Modspaofvecbunoncur/
Audience
Advanced Undergraduate
, Graduate
Video Public
Yes
Notes Public
Yes
Language
Chinese