Instantons, Moduli Spaces, and Hilbert Schemes of Points
This course develops the study of moduli spaces arising from gauge theory from an algebro-geometric viewpoint, with particular emphasis on their connections with algebraic geometry and representation theory.
Starting from anti-self-dual (ASD) instantons on four-manifolds, we study explicit descriptions of instanton moduli spaces. We begin with instantons on \mathbb{R}^{4} and introduce the ADHM construction, which identifies instanton moduli spaces with quotients of finite-dimensional algebraic data. The resulting quotient constructions can be understood both through geometric invariant theory and through hyperkähler reduction. In the rank one case, this leads to Hilbert schemes of points on algebraic surfaces.
We then study Hilbert schemes of points on algebraic surfaces based on Nakajima’s textbook Lectures on Hilbert Schemes of Points on Surfaces. We investigate the geometry and topology of Hilbert schemes, including their hyperkähler structures and Poincaré polynomials. Göttsche’s formula for the generating series of these invariants provides a fundamental connection between the geometry of Hilbert schemes and representations of Heisenberg algebras.
Through these developments, we explore the interplay among gauge theory, algebraic geometry, topology, and representation theory.
Starting from anti-self-dual (ASD) instantons on four-manifolds, we study explicit descriptions of instanton moduli spaces. We begin with instantons on \mathbb{R}^{4} and introduce the ADHM construction, which identifies instanton moduli spaces with quotients of finite-dimensional algebraic data. The resulting quotient constructions can be understood both through geometric invariant theory and through hyperkähler reduction. In the rank one case, this leads to Hilbert schemes of points on algebraic surfaces.
We then study Hilbert schemes of points on algebraic surfaces based on Nakajima’s textbook Lectures on Hilbert Schemes of Points on Surfaces. We investigate the geometry and topology of Hilbert schemes, including their hyperkähler structures and Poincaré polynomials. Göttsche’s formula for the generating series of these invariants provides a fundamental connection between the geometry of Hilbert schemes and representations of Heisenberg algebras.
Through these developments, we explore the interplay among gauge theory, algebraic geometry, topology, and representation theory.
Lecturer
Date
30th September, 2026 ~ 6th January, 2027
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Wednesday | 13:30 - 16:55 | Shuimo | ZOOM 02 | 518 868 7656 | BIMSA |
Prerequisite
Manifolds, vector bundles, principal bundles, connections, characteristic classes, Yang–Mills theory, anti-self-dual (ASD) instantons, moduli spaces, complex manifolds, algebraic varieties, coherent sheaves, moduli spaces of semistable sheaves, geometric invariant theory (GIT), and the Donaldson–Uhlenbeck–Yau theorem.
Syllabus
1. Course overview
2. Instantons on four-manifolds and moduli spaces
3. Instantons on \mathbb{R}^4
4. ADHM construction of instantons
5. Moduli spaces of framed sheaves
6. Hilbert schemes of points on surfaces
7. Hilbert–Chow morphism and the geometry of Hilbert schemes
8. GIT, moment maps, and hyperkähler quotients
9. Hyperkähler metrics on (\mathbb{C}^2)^{[n]}
10. Resolution of simple singularities
11. Poincaré polynomials of Hilbert schemes and Göttsche’s formula
12. Hilbert schemes of the cotangent bundle of a Riemann surface
13. Nakajima correspondences and representations of Heisenberg algebras
14. Further topics
2. Instantons on four-manifolds and moduli spaces
3. Instantons on \mathbb{R}^4
4. ADHM construction of instantons
5. Moduli spaces of framed sheaves
6. Hilbert schemes of points on surfaces
7. Hilbert–Chow morphism and the geometry of Hilbert schemes
8. GIT, moment maps, and hyperkähler quotients
9. Hyperkähler metrics on (\mathbb{C}^2)^{[n]}
10. Resolution of simple singularities
11. Poincaré polynomials of Hilbert schemes and Göttsche’s formula
12. Hilbert schemes of the cotangent bundle of a Riemann surface
13. Nakajima correspondences and representations of Heisenberg algebras
14. Further topics
Reference
[1] M. F. Atiyah, Geometry of Yang–Mills fields, Scuola Normale Superiore, Pisa, 1979.
[2] S. K. Donaldson, Instantons and geometric invariant theory, Comm. Math. Phys. 93 (1984), 453–460.
[3] S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford Mathematical Monographs, Oxford University Press, New York, 1990.
[4] L. Göttsche, The Betti numbers of the Hilbert scheme of points on a smooth projective surface, Math. Ann. 286 (1990), 193–207.
[5] T. Gocho and H. Nakajima, Einstein–Hermitian connections on hyper-Kähler quotients, J. Math. Soc. Japan 44 (1992), 43–65.
[6] D. Huybrechts and M. Lehn, The geometry of moduli spaces of sheaves, Second edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2010.
[7] P. B. Kronheimer, The construction of ALE spaces as hyper-Kähler quotients, J. Differential Geom. 29 (1989), 665–683.
[8] P. B. Kronheimer and H. Nakajima, Yang–Mills instantons on ALE gravitational instantons, Math. Ann. 288 (1990), 263–307.
[9] D. Mumford, J. Fogarty and F. Kirwan, Geometric invariant theory, Third edition, Ergebnisse der Mathematik und ihrer Grenzgebiete (2), 34, Springer-Verlag, Berlin, 1994.
[10] H. Nakajima, Moduli spaces of anti-self-dual connections on ALE gravitational instantons, Invent. Math. 102 (1990), 267–303.
[11] H. Nakajima, Heisenberg algebra and Hilbert schemes of points on projective surfaces, Ann. of Math. (2) 145 (1997), 379–388.
[12] H. Nakajima, Lectures on Hilbert schemes of points on surfaces, University Lecture Series, 18, American Mathematical Society, Providence, RI, 1999.
[2] S. K. Donaldson, Instantons and geometric invariant theory, Comm. Math. Phys. 93 (1984), 453–460.
[3] S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford Mathematical Monographs, Oxford University Press, New York, 1990.
[4] L. Göttsche, The Betti numbers of the Hilbert scheme of points on a smooth projective surface, Math. Ann. 286 (1990), 193–207.
[5] T. Gocho and H. Nakajima, Einstein–Hermitian connections on hyper-Kähler quotients, J. Math. Soc. Japan 44 (1992), 43–65.
[6] D. Huybrechts and M. Lehn, The geometry of moduli spaces of sheaves, Second edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2010.
[7] P. B. Kronheimer, The construction of ALE spaces as hyper-Kähler quotients, J. Differential Geom. 29 (1989), 665–683.
[8] P. B. Kronheimer and H. Nakajima, Yang–Mills instantons on ALE gravitational instantons, Math. Ann. 288 (1990), 263–307.
[9] D. Mumford, J. Fogarty and F. Kirwan, Geometric invariant theory, Third edition, Ergebnisse der Mathematik und ihrer Grenzgebiete (2), 34, Springer-Verlag, Berlin, 1994.
[10] H. Nakajima, Moduli spaces of anti-self-dual connections on ALE gravitational instantons, Invent. Math. 102 (1990), 267–303.
[11] H. Nakajima, Heisenberg algebra and Hilbert schemes of points on projective surfaces, Ann. of Math. (2) 145 (1997), 379–388.
[12] H. Nakajima, Lectures on Hilbert schemes of points on surfaces, University Lecture Series, 18, American Mathematical Society, Providence, RI, 1999.
Audience
Advanced Undergraduate
, Graduate
, Postdoc
Video Public
No
Notes Public
No
Language
English
Lecturer Intro
My research interests are primarily centred on Gauge theory within mathematics. Recently, my focus has been on semistable Higgs sheaves on complex projective surfaces and associated gauge-theoretic invariants, employing algebro-geometric methods. However, I also have a strong interest in working within the analytic category.