Vector bundles appear in a natural way in many different areas of mathematics, such as algebraic geometry, differential geometry, and mathematical physics. In all these contexts the notion of the stability of a vector bundle plays a key role. In this course we will study properties of stable vector bundles in the simplest, most instructive setting: for vector bundles on algebraic curves (equivalently, on Riemann surfaces). We will start from the very beginning and proceed to more complicated results near the end.

The main goal of this class is the Narasimhan–Seshadri theorem. This theorem shows that on algebraic curves stable vector bundles arise from the topological data, rather than algebraic or holomorphic, as one could expect based purely on the definition. Along the way we will discuss some properties of moduli spaces of vector bundles on curves. Due to a rather wide range of topics encountered in the study of vector bundles on curves, this course serves as a gentle introduction to many foundational concepts: stable vector bundles, boundedness properties for moduli spaces, the Riemann-Hilbert correspondence, deformation theory, compactifications of moduli spaces, etc.

Prerequisites: basics of algebraic or complex geometry (varieties, vector bundles, cohomology of coherent sheaves, Serre duality) and algebraic topology (the fundamental group of a Riemann surface, local systems, de Rham cohomology).

Lecturer
Pirozhkov Dmitrii Vladimirovich

Organizations
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center