2022　Special lectures on current topics in Mathematics D

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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
2022
Offered quarter
4Q
Syllabus updated
2022/4/20
Lecture notes updated
-
Language used
Japanese
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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.