2018 Lecture on Advanced Science in English (Mathematics 4)

Font size  SML

Register update notification mail Add to favorite lecture list
Academic unit or major
Honda Nobuhiro 
Class Format
Media-enhanced courses
Day/Period(Room No.)
Course number
Academic year
Offered quarter
Syllabus updated
Lecture notes updated
Language used
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.


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.


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.

Page Top