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- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Nitta Yasufumi Honda Nobuhiro
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- Day/Period(Room No.)
- Intensive (本館2階数学系201セミナー室)
- Group
- -
- Course number
- MTH.E634
- Credits
- 2
- Academic year
- 2022
- Offered quarter
- 4Q
- Syllabus updated
- 2022/4/20
- Lecture notes updated
- -
- Language used
- Japanese
- Access Index
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Course description and aims
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.
Student learning outcomes
・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 that will be developed
| ✔ 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/Required learning
| Course schedule | Required learning | |
|---|---|---|
| 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 |
Textbook(s)
none in particular
Reference books, course materials, etc.
Carlo Scarpa, The Hitchin-cscK system, PhD Thesis, Scuola Internazionale Superiore di Studi Avanzati
(arXiv:2010.07728)
Assessment criteria and methods
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 (i.e., required knowledge, skills, courses, etc.)
Good understanding on the materials in the "related courses" is expected.
