Cours spécialisé

Noncompact complex symplectic and hyperkähler manifolds

P. BOALCH

Email : philip point boalch à ens point fr

Noncompact hyperkähler manifolds feature prominently in various parts of mathematics, for example in Nakajima's work on the representation theory of quantum algebras and in the approach of Witten and collaborators to the geometric Langlands program. The aim of this course is to introduce some aspects of the geometry of hyperkähler manifolds (and more general complex symplectic manifolds) focusing on basic ideas and examples.

1. Introductory definitions and examples
2. Quotients (symplectic, Kähler, complex symplectic, hyperkähler)
3. ALE spaces and Nakajima quiver varieties
4. Examples of gauge theory equations as moment maps (e.g. flat connections, instantons, monopoles, Higgs bundles, Nahm's equations)
5. Quasi-Hamiltonian construction of some complex symplectic manifolds

Prerequisites

Differential geometry (also some knowledge of Lie groups would be useful)

References

• V. Guillemin and S. Sternberg, Symplectic techniques in physics, C.U.P. 1984
• N. J. Hitchin, Hyperkähler manifolds, Séminaire Bourbaki 748, 1991 (available on Numdam)
• N. J. Hitchin, A. Karlhede, U. Lindström and M. Rocek, Hyperkähler metrics and supersymmetry, Comm. Math. Phys. 108 (1987), 535–589
• P. B. Kronheimer, The construction of ALE spaces as hyperKähler quotients J. Differential Geom., 29, 1989, pp. 665–683