Cours spécialisé Paris VI M2 2008-2009
Noncompact complex symplectic and hyperkähler manifolds
P. BOALCH
Email : philip point boalch à ens point fr
Mardi 9h-11h et jeudi 9h-11h, 0D7 (Chevalaret).
Notes for lectures 1-5
Preliminary incomplete notes for lectures 1-10
(Comments/corrections welcome.)
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.
Provisional plan:
- Introduction and basic examples: Calogero-Moser spaces and Hilbert schemes
- Real and complex symplectic geometry, moment maps and symplectic quotients
- Very quick review of Kahler geometry, Kahler quotients.
- Quaternions, hyperkahler geometry, hyperkahler quotients, twistor space
- Nakajima quiver varieties, ALE gravitational instantons, McKay correspondence
- Algebraic approach via geometric invariant theory, Kempf-Ness theorem
- Examples of gauge theory equations as moment maps (e.g. flat
connections, instantons, monopoles, Higgs bundles, Nahm's equations)
- More on Higgs bundles, nonabelian Hodge theory on curves as a hyperkahler rotation
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
Paris VI webpage for this course