2022 Students Enrolled in or before 2015 School of Science Mathematics
Special courses on advanced topics in Mathematics D
- Academic unit or major
- Mathematics
- Instructor(s)
- Yasufumi Nitta / Nobuhiro Honda
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - Intensive (本館2階数学系201セミナー室)
- Class
- -
- Course Code
- ZUA.E334
- Number of credits
- 200
- Course offered
- 2022
- Offered quarter
- 4Q
- Syllabus updated
- Jul 10, 2025
- Language
- Japanese
Syllabus
Course overview and goals
In Kähler geometry, the existence problem of canonical Kähler metrics such as Kähler-Einstein metrics or Kähler metrics with constant scalar curvature (cscK metric) is one of the central problems and is still being studied. In this course, we will discuss the existence problem and related topics of the new canonical Kähler metric called the HcscK metric recently introduced by Scarpa and Stoppa. The aim of this course is to provide an understanding of the basics.
Course description and aims
・Understand the definition and basic properties of Kähler manifolds
・Be familiar with curvature
・Understand that the scalar curvature can be regarded as a moment map
・Understand the definition and basic properties of the HcscK system and HcscK metrics
Keywords
Kähler manifolds, cscK metrics, Hyperkähler manifolds, Hyperkähler moment map, HcscK system, HcscK metrics, K-stability
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
This is a standard lecture course. There will be some assignments.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | The following topics will be covered in this order : -- Basics on Kähler manifolds -- Curvature of Kähler manifolds, cscK metrics -- Fujiki-Donaldson's moment map picture -- A hyperkähler structure on the cotangent bundle and hyperkähler moment maps -- HcscK system and HcscK metrics -- Relationship to stability (If time allows) | to be specified in each lecture |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
None
Reference books, course materials, etc.
Carlo Scarpa, The Hitchin-cscK system, PhD Thesis, Scuola Internazionale Superiore di Studi Avanzati
(arXiv:2010.07728)
Evaluation methods and criteria
Assignments (100%).
Related courses
- MTH.B301 ： Geometry I
- MTH.B302 ： Geometry II
- MTH.B331 ： Geometry III
- MTH.C301 ： Complex Analysis I
- MTH.C302 ： Complex Analysis II
Prerequisites
Good understanding on the materials in the "related courses" is expected
Other
Not in particular