# 2014年几何暑期学校系列课程【Lorezo Foscolo】

教室1208

The aim of the course is to explore a number of different constructions of non-compact hyper kähler manifolds, in particular in dimension 4. Hyperkähler geometry, in Joyce’s words, “form(s) a beautiful and rich branch of mathematics” and the course is an attempt to describe some aspects of this theory.

We will start with the first examples of hyperkähler metrics given by Calabi, we will then discuss the hyperkähler quotient construction of Kronheimer’s ALE spaces, the Taub?NUT metric and the construction of complete hyperkähler manifolds as moduli spaces of solutions to the Yang?Mills anti-self-duality equations.

One motivation for studying complete non-compact Ricci-flat manifolds is the understanding of the moduli space of compact Einstein metrics and their degenerations. As an illustration of this point, we will describe the construction of the Calabi-Yau metric on the Kummer surface by gluing methods.

Introduction to hyperkähler geometry. The group Sp(n) as a Riemannian holonomy group, the curvature of Riemannian 4?manifolds, compact examples, Calabi’s metric on T*CPn

Hyperkähler quotient construction and ALE spaces. Symplectic and Kähler quotients, hy perkähler quotients, Kronheimer’s ALE spaces.

Gibbons?Hawking ansatz, twistor spaces. Gibbons?Hawking ansatz, the Taub?NUT metric, gravitational instantons, the twistor space of a hyperkähler manifold.

Infinite dimensional hyperkähler quotients. The Yang?Mills anti-self-duality equations as a hyperkähler moment map, Kronheimer’s hyperkähler metrics on coadjoint orbits.

The Kummer construction. Donaldson’s gluing construction of the Calabi-Yau metric on the Kummer surface.

References
[1] Hitchin, N., Hyper-Kähler manifolds, Séminaire Bourbaki, Vol. 1991/92, Astérisque 206, 1992, Exp. No. 748,
3,137?166.
[2] Joyce, D., Riemannian holonomy groups and calibrated geometry, Chapter 10, Oxford Graduate Texts in Math ematics, 12, Oxford University Press, 2007.
[3] Hitchin, N., and Karlhede, A. and Lindström, U. and Roček, M., Hyper-Kähler metrics and supersymmetry, Comm. Math. Phys., Communications in Mathematical Physics, 108, 1987, 4, 535?589.
[4] Donaldson, S., Calabi-Yau metrics on Kummer surfaces as a model gluing problem, Advances in geometric analysis, Adv. Lect. Math. (ALM), 21, 109?118, Int. Press, Somerville, MA, 2012.