Hyperkahler 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 hyperkahler metrics given by Calabi, we will then discuss the hyperkahler quotient construction of Kronheimer’s ALE spaces, the Taub–NUT metric and the construction of complete hyperkahler 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.
References
[1] Hitchin, N., Hyper-Kahler 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 Lindstr?m, U. and Ro?ek, M., Hyper-Kahler 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.