Kyle Broder -- Recent Developments Concerning the Schwarz Lemma
A lecture I gave at the Beijing International Center for Mathematical Research geometric analysis seminar. The title being Recent Developments Concerning the Schwarz Lemma with applications to the Wu--Yau Theorem. This contains some recent results concerning the Bochner technique for the Gauduchon connections in the Hermitian category. More details can be found in the present form of my Ph.D. thesis, which is available at https://www.kylebroder.com
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0:00 Introduction
2:08 Classical Bochner Technique
3:45 Complex-Analytic Bochner Formula
5:51 Connections on Holomorphic Vector Bundles
13:33 Royden's Polarization Argument
14:54 Royden's Schwarz Lemma
16:02 The Wu-Yau Theorem
18:08 The Kobayashi Conjecture
19:56 Curvature Heirarchy
20:43 Examples
22:37 Sketch of Wu-Yau Theorem Proof
29:43 Proof of Ampleness
31:01 The Schwarz Lemma Revisited
32:43 Target curvature term
33:58 The Real Bisectional Curvature
35:31 Hermitian Schwarz Lemma
37:10 Gauduchon connections
38:32 Constructing a Schwarz Lemma
39:16 Novel Bochner Formula
40:51 Incorporating Torsion
42:47 Bochner formula with torsion
43:53 The Schwarz Lemma with torsion
48:00 Gauduchon-Schwarz Lemma with Kähler source
49:12 Monotonicity Theorem for the Holomorphic Sectional Curvature
51:13 Refined Schwarz Lemma
52:21 Refined Wu-Yau Theorem
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