Beijing Institute of Mathematical Sciences and Applications

This graduate-level course provides an introduction to the Donaldson–Uhlenbeck–Yau theorem, which states that the algebro-geometric notion of stability for a holomorphic vector bundle over a Kähler manifold implies the existence of a special Hermitian metric, called a Hermitian–Yang–Mills or Hermitian–Einstein metric on the bundle. This fundamental result forms a deep bridge between Differential geometry and Algebraic geometry and has led to remarkable applications, notably in Donaldson’s theory of smooth 4-manifolds. The course begins with a review of manifolds and vector bundles, then proceeds to a complete proof of the Donaldson–Uhlenbeck–Yau theorem via Donaldson’s Lagrangian method, following the exposition in Shoshichi Kobayashi’s textbook. If time permits, we may cover applications to gauge theory and also other advanced topics related to the theorem.