2022 Special lectures on advanced topics in Mathematics D

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Academic unit or major
Graduate major in Mathematics
Nitta Yasufumi  Honda Nobuhiro 
Class Format
Lecture    (Face-to-face)
Media-enhanced courses
Day/Period(Room No.)
Intensive (本館2階201数学系セミナー室)  
Course number
Academic year
Offered quarter
Syllabus updated
Lecture notes updated
Language used
Access Index

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


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) Details will be provided during each class session.


None required

Reference books, course materials, etc.

Carlo Scarpa, The Hitchin-cscK system, PhD Thesis, Scuola Internazionale Superiore di Studi Avanzati

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


Not in particular

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