RTG Lectures, Winter semester 2024-2025

Florent Schaffhauser, Heidelberg University

By Tazerenix - Own work, CC BY-SA 4.0

By Tazerenix - Own work, CC BY-SA 4.0

This series of 6 lectures is part of the RTG Days between Heidelberg and Kalrsruhe. The official announcement of the lectures is available here.

Vector bundles on Riemann surfaces

Vector bundles on compact Riemann surfaces are of interest for at least two reasons:

  1. They arise naturally in the study of analytic differential equations.
  2. They provide non-Abelian analogues of the classical Jacobian variety in complex geometry.

In these lectures, we provide an introduction to these objects, focusing on their classification and the topology of their moduli spaces.

The first three lectures will focus on the general theory of holomorphic vector bundles on a compact Riemann surface, while the last three will present the gauge-theoretic approach developed by Atiyah, Bott and Donaldson in the 1980s.

Outline of the lectures1

  • Lecture 1 (29.10.2024, Heidelberg). The Riemann-Roch formula.
    1. Vector bundles, sections, degree.
    2. The Riemann-Roch formula, Serre duality.
  • Lecture 2 (12.11.2024, Karlsruhe). Vector bundles on the Riemann sphere.
    1. Sub-bundles, filtrations, direct sums.
    2. Grothendieck-Birkhoff decomposition.
  • Lecture 3 (26.11.2024, Heidelberg). The Harder-Narasimhan filtration.
    1. Slope semistability.
    2. Existence of a canonical filtration.
  • Lecture 4 (14.01.2025, Heidelberg). Yang-Mills connections.
    1. The Atiyah-Bott symplectic form.
    2. Donaldson’s proof of the Narasimhan-Seshadri theorem.
  • Lecture 5 (14.01.2025, Heidelberg). Moduli spaces of vector bundles.
    1. The Yang-Mills flow.
    2. The Shatz stratification.
  • Lecture 6 (28.01.2025, Karlsruhe). Betti numbers of the moduli space.
    1. The classifying space of the gauge group.
    2. Recursive and closed formulas for the Betti numbers.

References

Atiyah, M. F., and R. Bott. 1983. “The Yang-Mills Equations over Riemann Surfaces.” Philos. Trans. Roy. Soc. London Ser. A 308 (1505): 523–615. https://doi.org/10.1098/rsta.1983.0017.
Donaldson, S. K. 1983. “A New Proof of a Theorem of Narasimhan and Seshadri.” J. Differential Geom. 18 (2): 269–77. http://projecteuclid.org/getRecord?id=euclid.jdg/1214437664.
Grothendieck, A. 1957. “Sur La Classification Des Fibrés Holomorphes Sur La Sphère de Riemann.” Amer. J. Math. 79: 121–38. https://doi.org/10.2307/2372388.
Harder, G., and M. S. Narasimhan. 1974/75. “On the Cohomology Groups of Moduli Spaces of Vector Bundles on Curves.” Math. Ann. 212 (1974/75): 215–48.
Hitchin, N. J., G. B. Segal, and R. S. Ward. 1999. Integrable systems. Twistors, loop groups, and Riemann surfaces. Based on lectures given at the instructional conference on integrable systems, Oxford, UK, September 1997. Oxford: Clarendon Press.
Narasimhan, M. S., and C. S. Seshadri. 1965. “Stable and Unitary Vector Bundles on a Compact Riemann Surface.” Ann. Of Math. (2) 82: 540–67.
Schaffhauser, F. 2013. “Differential Geometry of Holomorphic Vector Bundles on a Curve.” In Geometric and Topological Methods for Quantum Field Theory, Proceedings of the 2009 Villa de Leyva Summer School, 39–80. Cambridge University Press.

Footnotes

  1. Note that there is no lecture on 10.12.2024, but two lectures on 14.01.2025.↩︎