Introduction to the Donaldson-Uhlenbeck-Yau theorem


Lecturer

Yuuji Tanaka (田中祐二, Assistant Professor, BIMSA)
Date

2025-09-24 ~ 2026-01-07
Schedule
Weekday    Time    Venue    Online    ID    Password
Wed    10:40-12:15    A14-203    Zoom    928 682 9093    BIMSA
Wed    13:30-15:05    A14-203    Zoom    928 682 9093    BIMSA

Prerequisite

complex analysis, manifolds, differential forms, de Rham cohomology, introductory knowledge of vector bundles and connections
Introduction

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 K?hler 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.
Syllabus

1. Course overview
2. Review of manifolds, vector bundles
3. Complex manifolds
4. Connections on vector bundles
5. Holomorphic vector bundles and Chern connections
6. Chern classes of complex vector bundles
7. Vanishing theorem
8. Hermitian Yang-Mills metrics
9. Stable vector bundles
10. The Donaldson-Uhenbeck-Yau theorem
11. Advanced topics
Reference

D. D. Joyce, Riemannian holonomy groups and calibrated geometry, Oxford Graduate Texts in Mathematics, 12, Oxford University Press, 2007.
S. Kobayashi, Differential geometry of complex vector bundles, Princeton University Press, 2014.
M. Lübke and A. Teleman, The Kobayashi-Hitchin correspondence, World Scientific, 1995.
R. O. Wells, Jr., Differential analysis on complex manifolds, Springer, 2008.
Video Public

No
Notes Public

No
Audience

Advanced Undergraduate, Graduate, Postdoc, Researcher
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.
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FROM 202.120.11.*