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.
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.
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
standasrd lecture course
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.
P. Griffiths, J. Harris, "Principles of Algebraic Geometry", Wiley-Interscience.
Assignments (100%)
Basic knowledge on geometry (manifolds, differential forms, homology group) is required.