Spring 2004 : MAT 645 Topics in Differential Geometry (Hyperkähler manifolds)

Monday and Wednesday 9:30-10:25am Mathematics 5-127 and Wednesday 1-1:55pm Mathematics P-131

Hyperkähler manifolds

The study of hyperkähler manifolds involves a wealth of ideas from various branches of mathematics, including Riemannian metrics, global analysis on manifolds, complex and Kähler structures, algebraic geometry, and the theory of lattices and quadratic forms. Many physically relevant moduli spaces (of instantons, monopoles, Higgs bundles, etc) inherit natural hyperkähler structures. In low dimensional topology, hyperkähler manifolds have been used to construct quantum invariants of links and 3-manifolds. They have also found applications in representation theory (highest weight modules of affine Lie algebras arise from Hilbert schemes and quiver varieties) and arithmetic geometry (where they are used to study rational density problems).

About the course

In this course I will focus on differential geometric and algebro geometric aspects of hyperkähler manifolds. After some quaternionic preliminaries my aim will be to cover:

Part I : Noncompact hyperkähler manifolds

• The Eguchi-Hanson and Taub-NUT metrics, the Gibbons-Hawking ansatz, Calabi's metrics
• Twistor spaces and applications
• Hyperkähler quotients: ALE metrics and instantons on ALE spaces, Nakajima's quiver varieties, instanton moduli spaces and dimensional reductions (monopole moduli spaces, Higgs moduli spaces), coadjoint orbits
• Resolution of symplectic singularities

Part II : Compact hyperkähler manifolds

• Statement of Yau's theorem
• K3 surfaces and their moduli space
• Beauville's examples and Mukai's moduli spaces of stable sheaves, the examples of O'Grady
• Deformations, moduli spaces of hyperkähler metrics, the period map
• Structure of the cohomology of compact hyperkähler manifolds

Prerequisites:

• Basic differential geometry (MAT568 should be more than enough)
• Complex geometry (MAT545 or equivalent)
• In some sense this course will complement Claude LeBrun's MAT682 Riemannian manifolds of special holonomy, by expanding on hyperkähler manifolds, but I will not assume anything from that course.

References:

• M. Gross, D. Huybrechts, and D. Joyce, "Calabi-Yau manifolds and related geometries", Springer-Verlag (2003) ISBN 3540440593 (available from the University Book Store)
• N. Hitchin, "Hyperkähler manifolds", Seminaire Bourbaki, 748 (1991-92)
• A.S. Dancer, "Hyperkähler manifolds", in Lectures on Einstein Manifolds, LeBrun and Wang eds., International Press (2000)

Course notes (will be updated periodically): postscript, pdf.

My office is MAT 4-104. You can email me on sawon@math.sunysb.edu to make an appointment.

Americans with Disabilities Act: If you have a physical, psychological, medical or learning disability that may impact your course work, please contact Disability Support Services, ECC (Educational Communications Center) Building, room 128, (631) 632-6748. They will determine with you what accommodations are necessary and appropriate. All information and documentation is confidential. Students requiring emergency evacuation are encouraged to discuss their needs with their professors and Disability Support Services. For procedures and information, please visit the Environmental Health & Safety webpage.

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This page last modified by Justin Sawon
Sunday, 08-Feb-2004 10:05:08 EST
Email corrections and comments to sawon@math.sunysb.edu