The Kobayashi-Hitchin Correspondence (Fall 2022)
This is the reading group webpage on the Kobayashi-Hitchin correspondence (also known as Donaldson-Uhlenbeck-Yau theorem), i.e. the natural isomorphy of the moduli spaces of stable holomophic structures respectively irreducible Hermitian-Einstein connections in a differentiable complex vector bundle over a compact complex manifold (Lübke & Teleman). The first meeting will be on September 7th (W) and the last one will be around December 5th (M). Please email me (qiuyu_ren@berkeley.edu) or Yueqing (fyq@berkeley.edu) if you want to present in a future talk or if you have any additional questions.
References:
[D1] Donaldson, A new proof of a theorem of Narasimhan and Seshadri;
[D2] Donaldson, Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles;
[D3] Donaldson, Infinite determinants, stable bundles and curvature;
[DK] Donaldson & Kronheimer, The Geometry of Four-Manifolds, Chapter 6, 10;
[H] N. Hitchin, The self-duality equations on a Riemann surface;
[LT] Lübke & Teleman, The Kobayashi-Hitchin Correspondence;
[Si] Y.T. Siu, Lectures on Hermitian-Einstein Metrics for Stable Bundles and Kähler-Einstein Metrics;
[Sz] Székelyhidi, An introduction to extremal Kähler metrics, Chapter 5;
[UY] Uhlenbeck & Yau, On the existence of Hermitean Yang-Mills-connections on stable bundles over Kähler manifolds.
Time: MW 10-11am, except days with scheduled MSRI programs.
Location: Evans Hall 959, UC Berkeley.
Schedule:
Aug 31, Sep 1, Sep 2: (At MSRI) Overview: Gauge theory and complex geometry I~III. Speaker: Song Sun.
Sep 7: Organizational meeting/Preliminary I. [LT] Chapter 1~2. Speaker: Yueqing.
Sep 19: Preliminary II. Speaker: Yueqing.
Sep 21: Preliminary III. Speaker: Yueqing.
Sep 26: Proof of the easy direction: stability. [LT] Chapter 2. Speaker: Garrett.
Sep 28: Uhlenbeck-Yau's proof of existence I. [LT] Chapter 3, [UY]. Speaker: Garrett.
Oct 3: Uhlenbeck-Yau's proof of existence II. Speaker: Garrett.
Oct 5: Uhlenbeck-Yau's proof of existence III. Speaker: Zhongkai.
Oct 10: Uhlenbeck-Yau's proof of existence III. Speaker: Zhongkai.
Oct 12: Proof of moduli space level correspondence I. [LT] Chapter 4.1-4.4, [DK] Chapter 6.4. Speaker: Qiuyu.
Oct 17: Proof of moduli space level correspondence II. Speaker: Qiuyu.
Oct 19: Motivation: Kempf-Ness theorem. [DK] Chapter 6.5, [Sz] Chapter 5. Speaker: Carlos.
Oct 31: Motivation: Kempf-Ness theorem. Speaker: Carlos.
Nov 2: Cancelled.
Nov 7: Donaldson's proof of existence I: Riemann surface case. [D1]. Speaker: Garrett.
Nov 9: Application: Class VII surfaces I. [LT] Chapter 5.4. Speaker: Yueqing.
Nov 18: (1-3pm, on zoom, complex geometry seminar) Donaldson's proof of existence II: complex algebraic surface case. [D2]. Speaker: Yueqing.
Nov 21: (on zoom) Application: Class VII surfaces II. Speaker: Yueqing.
Nov 23: Thanksgiving!
Nov 28: Application: Donaldson theory, failure of h-cobordism theorem in dimension 4. [DK] Chapter 9.1. Speaker: Qiuyu.
Nov 30: Examples from Kähler surfaces I. [DK] Chapter 10. Speaker: Jiakai.
Dec 5: Examples from Kähler surfaces II. Speaker: Jiakai.