### 2021　Advanced topics in Geometry G1

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Academic unit or major
Graduate major in Mathematics
Instructor(s)
Honda Nobuhiro
Class Format
Lecture
Media-enhanced courses
Day/Period(Room No.)
Fri5-6()
Group
-
Course number
MTH.B507
Credits
1
Academic year
2021
Offered quarter
3Q
Syllabus updated
2021/3/19
Lecture notes updated
-
Language used
English
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### Course description and aims

Overviewing basic notions in differential geometry and complex geometry, some advanced topics such as relevance of Ricci curvature with the topology of manifolds will be treated. Basic properties of K3 surfaces might be also included.

### Student learning outcomes

・To understand special properties of compact Kaehler manifolds.
・To understand basic properties of "positive" line bundle over complex manifolds
・To understand basic properties of compact Kaehler surfaces
・To understand "special" Riemannian manifolds such as Kaehler manifolds and Ricci flat Kaehler manifolds, in terms of holonomy group
・To understand Calabi conjecture and its consequences

### Keywords

Chern class, curvature, kaehler manifold, positive line bundle, harmonic form, Ricci form, holonomy group, Hodge decomposition, Calabi conjecture, Bochner principle

### Competencies that will be developed

 ✔ Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

### Class flow

regular lecture course

### Course schedule/Required learning

Course schedule Required learning
Class 1 harmonic theory on compact Riemannian manifolds Definitions and properties
Class 2 connections, curvature and Chern class of complex vector bundle Definitions and properties
Class 3 Kaehler manifolds and Hodge decomposition 1 Definitions and properties
Class 4 Kaehler manifolds and Hodge decomposition 2 Definitions and properties
Class 5 compact Kaehler surfaces Definitions and properties
Class 6 holonomy group Definitions and properties
Class 7 Ricci curvature and holonomy group, Calabi conjecture Definitions and properties

### Out-of-Class Study Time (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.

No textbook

### Reference books, course materials, etc.

D. Joyce, "Compact manifolds with special holonomy", Oxford University Press
A. Besse, "Einstein manifolds" Springer
P. Griffiths, J. Harris, "Principles of Algebraic Geometry" Wiley-Interscience
R. O. Wells, "Differential analysis on complex manifolds" Springer Graduate Texts in Mathematics

### Assessment criteria and methods

Homework assignments (100％)

### Related courses

• MTH.B202 ： Introduction to Topology II
• MTH.B301 ： Geometry I
• MTH.B302 ： Geometry II

### Prerequisites (i.e., required knowledge, skills, courses, etc.)

This course is not entirely introductory. We expect audience to be familiar with basic notions in differential geometry and complex geometry to some extent.