Time: January 13-21, 2026
Venue: TBD
This workshop is to facilitate the communication of recent developments of complex geometry. It consists of several lecture series on analytic methods in complex geometry delivered by young experts.
Organizers:
Xin Fu, Westlake University
Jiyuan Han, Westlake University
Lectures:
January 13-16
Wangjian Jian, Chinese Academy of Sciences
Title: The Li-Yau type estimates along the Ricci flow
Abstract:
In this serie of lecture, we will first recall the original Li-Yau estimate of heat equation. Then we will talk about Hamilton's Li-Yau type estimates along the Ricci flow, which are fundamental in the Ricci flow theory. Then we will review Perelman's Li-Yau type estimates along the Ricci flow, and it's applications. Finally, we will talk about the Li-Yau type estimates in the Kahler-Ricci flow theory.
Kewei Zhang, Beijing Normal University
Title：Canonical metrics in Kahler geometry
Abstract:
Searching for canonical metrics on compact Kahler manifolds is one of the central problems in geometric analysis. The (still open) Yau-Tian-Donaldson conjecture predicts that existence of canonical metrics is equivalent to certain algebro-geometric conditions. In these lectures we mainly focus on the existence of two kinds of canonical metrics: Kahler-Einstein metrics and the more general constant scalar curvature Kahler (cscK) metrics. We will use variational methods to show that these metrics exist iff the manifold is algebraically stable in a suitable sense. The audience should be prepared with some basic knowledge of Kahler geometry and pluripotential theory. Very minimal amount of knowledge for algebraic geometry is also required.
References：
[1] Geometric pluripotential theory on Kähler manifolds, Tamas Darvas, Advances in complex geometry, 1-104, Contemp. Math. 735, Amer. Math. Soc., Providence, RI, 2019.
[2] Canonical metrics in Kähler geometry, Tian, Gang, Lectures Math. ETH Zürich.
[3] An introduction to extremal Kähler metrics, Székelyhidi, Gábor, Grad. Stud. Math., 152.
Siqi He, Chinese Academy of Sciences
Title: Z/2 Harmonic 1-Forms and Related Problems in Geometry
Abstract:
This mini-course will focus on Z/2 harmonic 1-forms and their connections to gauge theory, topology, and compactification problems. The lectures will be divided into four parts:
Gauge Theory with SL(2,C) structure group: Gauge-theoretic equations with SL(2,C) structure group, including flat connection equations and the Kapustin–Witten equations. We will discuss their basic properties and related geometric and topological problems.
Compactness of Flat SL(2,C) Connections: Taubes' compactness theorem and the role of Z/2 harmonic 1-forms in describing the ideal boundary. Basic properties of Z/2 harmonic 1-forms will also be introduced.
Deformation of Z/2 Harmonic 1-Forms: Donaldson's work on the deformation of Z/2 harmonic 1-forms, along with possible geometric applications of these deformations.
Relations to Low-Dimensional Topology: Connections between Z/2 harmonic 1-forms and classical objects in low-dimensional topology, including Thurston's compactification of Teichmüller space, measured foliations, and the Morgan–Shalen compactification.
The lectures are intended for PhD students and early-career researchers with a background in differential geometry or gauge theory.
References:
S. K. Donaldson, Deformations of multivalued harmonic functions, arXiv:1912.08274.
C. H. Taubes, PSL(2,C) connections on 3-manifolds with L² bounds on curvature, arXiv:1205.0514.
C. H. Taubes, Compactness theorems for SL(2,C) generalizations of the 4-dimensional anti-self-dual equations, arXiv:1307.6447.
H. Siqi, R. Wentworth, B. Zhang, Z/2 harmonic 1-forms, R-trees, and the Morgan–Shalen compactification, arXiv:2409.04956.
January 18-21
Junchao Shentu, University of Science and Technology of China
Title: Hodge Theory and Selected Applications
Abstract:
This lecture series offers a comprehensive introduction to Hodge structures on complex projective varieties, including both pure and mixed types, and examines their variations. Some applications in birational geometry and the moduli theory of algebraic varieties will be presented. The lectures are structured as follows:
Lecture 1 (2 hours): Foundations of Hodge Theory—pure Hodge structures on compact Kähler manifolds and mixed Hodge structures on algebraic varieties.
Lecture 2 (2 hours): Extended realizations of mixed Hodge structures—in intersection cohomology and Borel–Moore homology—with applications to vanishing theorems.
Lecture 3 (2 hours): Variations of Hodge structures in both pure and mixed contexts.
Lecture 4 (2 hours): Applications in birational geometry and moduli theory, including the moduli spaces of KSBA (Kollár–Shepherd-Barron–Alexeev) pairs and stable minimal models as developed by Birkar.
Juanyong Wang, Chinese Academy of Sciences
Title: Positivity of vector bundles and applications to geometry
Abstract:
Positivity of vector bundles and applications to geometry
Abstract: In this series of talks, we first review the positivity notions for vector bundles (and more generally, for torsion free sheaves). In the second part, we concentrate on the positivity results for the (twisted) relative (pluri)canonical bundles and their direct images. In the last part, we show how to apply these positivity results to geometric problems.
Shengxuan Zhou, University of Toulouse
Title: Some Aspects of Bergman Kernels
Abstract:
The Bergman kernel plays a central role in complex and algebraic geometry, linking analysis with the geometry of Kähler manifolds. In recent decades, it has become a key tool in understanding canonical metrics, quantization, and arithmetic geometry. This mini-course provides an introduction to the theory and applications of Bergman kernels, covering both classical foundations and recent developments. Topics include:
1.Basics of Kähler Geometry
2.Tian's Peak Section Method and the Tian–Yau–Zelditch Expansion
3.Uniform Estimates for Bergman Kernels
4.Applications of Bergman Kernels in Arakelov Theory
Application:
We encourage graduate students and young reseachers to apply and of course all applications are welcome.
https://www.wjx.cn/vm/QZAC5CH.aspx#
Application deadline: December 15 2025
