### 2018　Special Lecture on Science in English (Mathematics 4)

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Instructor(s)
Honda Nobuhiro
Class Format
Lecture
Media-enhanced courses
Day/Period(Room No.)
-
Group
-
Course number
MTH.E442
Credits
1
2018
Offered quarter
2Q
Syllabus updated
2018/6/3
Lecture notes updated
-
Language used
English
Access Index

### Course description and aims

In this lecture series, we will begin by reviewing some basic material from Kahler geometry. We will then discuss basic examples of Calabi-Yau manifolds, and Yau's Theorem. In dimension 4, these metrics are hyperkahler, and the Gibbons-Hawking ansatz is an important tool for producing noncompact examples, such as ALE, ALF, ALG, ALH, etc. Noncompact examples also arise from a construction of Tian-Yau. We will then outline various gluing results, which give a picture of the degenerations of Yau's metrics on K3 surfaces.

### Student learning outcomes

To know some basic concepts of Kahler geometry
To understand basic examples of Calabi-Yau manifolds and hyperkahler metrics on K3 surfaces.

### Keywords

Kahler manifold, Calabi-Yau metrics, hyperkahler geometry.

### Competencies that will be developed

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

### Class flow

standasrd lecture course

### Course schedule/Required learning

Course schedule Required learning
Class 1 Kahler geometry, basics Details will be provided during each class session
Class 2 Kahler metrics, Yau's Theorem.
Class 3 Calabi-Yau manifolds, examples.
Class 4 Hyperkahler metrics in dimension 4.
Class 5 Gibbons-Hawking ansatz.
Class 6 This will be a general colloquium talk about Calabi-Yau metrics on K3 surfaces.
Class 7 del Pezzo surfaces, rational elliptic surfaces and Tian-Yau Theorem.
Class 8 Noncompact hyperkahler metrics, ALE, ALF, ALG, ALH.
Class 9 Nilmanifolds and ALH_b geometry
Class 10 Examples of collapsing hyperkahler metrics on K3 surfaces.

### Textbook(s)

P. Griffiths, J. Harris, "Principles of Algebraic Geometry", Wiley-Interscience.

### Reference books, course materials, etc.

P. Griffiths, J. Harris, "Principles of Algebraic Geometry", Wiley-Interscience.

### Assessment criteria and methods

Assignments (100%)

### Related courses

• MTH.B301 ： Geometry I
• MTH.B302 ： Geometry II

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

Basic knowledge on geometry (manifolds, differential forms, homology group) is required.