A moment map is a generalisation of the angular momentum in classical mechanics, and plays an important role in symplectic geometry. The first half of this course concerns the definition of moment maps and the symplectic quotient, which is the quotient space of the zero of the moment map by the group action. The latter half of the course is devoted to the Kempf-Ness theorem, which states that a point in a smooth projective variety is polystable in the sense of Geometric Invariant Theory (GIT) if and only if it is a zero of the moment map. This is an important theorem that exhibits a nontrivial relationship between the stability defined in terms of algebraic geometry and the moment maps defined in terms of symplectic geometry, and its infinite-dimensional version has been an active topic of research in recent years; time permitting, the infinite-dimensional version may be briefly commented on. The lecture plans are only approximate.
The first half of this course is an introduction to symplectic manifolds, Hamiltonian vector fields, and moment maps. We shall also define the symplectic quotient by means of the moment maps, and see that it has a natural symplectic structure. In the latter half of the course, we shall prove the Kempf-Ness theorem, after necessary preparations, by studying an energy functional called the Kempf-Ness functional. Moreover, we shall prove that the GIT quotient is homeomorphic to the symplectic quotient in certain good circumstances.
Students will learn the basics of symplectic manifolds and Hamiltonian vector fields, as well as the definition (and significance) of moment maps and symplectic quotients, both in terms of theory and explicit calculations in elementary examples. They will also learn that the convexity of the Kempf-Ness functional plays a key role in the proof of the Kempf-Ness theorem.
Hamiltonian vector fields, moment maps, symplectic quotients, Kempf-Ness theorem.
|✔ Specialist skills||Intercultural skills||Communication skills||Critical thinking skills||Practical and/or problem-solving skills|
Standard lectures, problems for the homework assignments will be given during the course.
|Course schedule||Required learning|
|Class 1||Symplectic manifolds and Hamiltonian vector fields||Details will be provided during each class session.|
|Class 2||Moment maps||Details will be provided during each class session.|
|Class 3||Symplectic quotients||Details will be provided during each class session.|
|Class 4||Introduction to Kähler manifolds||Details will be provided during each class session.|
|Class 5||Reductive Lie groups||Details will be provided during each class session.|
|Class 6||Kempf-Ness functional||Details will be provided during each class session.|
|Class 7||Proof of the Kempf-Ness theorem||Details will be provided during each class session.|
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
None, but typeset notes will be distributed as appropriate.
A. Cannas da Silva. Lectures on symplectic geometry. Lecture Notes in Mathematics, 1764. Springer-Verlag, Berlin, 2001. xii+217 pp. ISBN: 3-540-42195-5
S. Donaldson, P. Kronheimer. The geometry of four-manifolds. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1990. x+440 pp. ISBN: 0-19-853553-8
R. Thomas. Notes on GIT and symplectic reduction for bundles and varieties. Surveys in differential geometry. Vol. X, 221--273, Int. Press, Somerville, MA, 2006. (available at https://arxiv.org/abs/math/0512411)
Graded by homework assignments.
Students are assumed to be familiar with the theory of smooth manifolds and Lie groups (e.g. correspondence between Lie groups and Lie algebras, and the exponential maps). Familiarity with Riemannian metrics and linear connections on vector bundles is also desirable, but not essential as they will be recalled briefly in the lectures. It should be possible to understand this course without attending Advanced topics in Geometry E, as long as students can accept the definition of GIT stability that will be given in the lectures; there will be, however, no detailed explanation of GIT itself in this course.
Not specifically, but extra discussions can be arranged by email or after each class.