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2016　Special courses on advanced topics in Mathematics N

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Academic unit or major
Mathematics
Instructor(s)
Akutagawa Kazuo  Konno Hiroshi 
Class Format
Lecture     
Media-enhanced courses
Day/Period(Room No.)
Intensive (H201)  
Group
-
Course number
ZUA.E338
Credits
2
Academic year
2016
Offered quarter
4Q
Syllabus updated
2016/4/27
Lecture notes updated
-
Language used
Japanese
Access Index

Course description and aims

The main subject of this course is geometry of symplectic quotients, which are quotient spaces in symplectic geometry. After recalling basic knowledge on symplectic manifolds and group actions, we introduce moment maps and symplectic quotients. Next, we explain symplectic toric manifolds, which are important example of symplectic quotients. Moreover, we explain the relation between symplectic quotients and GIT quotients, which are quotient spaces in algebraic geometry. Finally, we introduce hyperkähler quotients, which are analogues of symplectic quotients in hyperkähler geometry.
Moduli spaces of various objects in differential geometry are often constructed as symplectic quotients or hyperkähler quotients. On the other hand, moduli space of various objects in algebraic geometry are often constructed as GIT quotients. In many cases, symplectic quotients can be identified with GIT quotients. This fact has been given many guiding principles in the existence problem of canonical metrics of holomorphic vector bundles and complex manifolds. Moreover, these quotient spaces can be investigated not only by geometric methods but also by algebraic methods. The main purpose of this course is to describe the geometry of these quotient spaces.

Student learning outcomes

・Understand basic properties of symplectic manifolds and group actions
・Understand with basic properties of moment maps and symplectic quotients
・Be familiar with many examples of symplectic toric manifolds
・Understand the relation between symplectic quotients and GIT quotients
・Be familiar with basic examples of hyperkähler quotients

Keywords

symplectic manifold, moment map, symplectic quotient, toric manifold, prequantum line bundle, GIT quotient, hyperkähler quotient, toric hyperkähler manifold

Competencies that will be developed

✔ Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

Class flow

This is a standard lecture course. There will be some assignments.

Course schedule/Required learning

  Course schedule Required learning
Class 1 symplectic manifolds Details will be provided during each class
Class 2 actions of Lie groups
Class 3 backgrounds from physics
Class 4 moment maps
Class 5 symplectic quotients
Class 6 examples of symplectic quotients
Class 7 symplectic toric manifolds
Class 8 toric manifolds as complex manifolds
Class 9 properties of toric manifolds
Class 10 prequantum line bundles and moment maps
Class 11 quotient spaces in algebraic geometry(GIT quotients)
Class 12 relations between symplectic and GIT quotients
Class 13 hyperkähler quotients
Class 14 toric hyperkähler manifold
Class 15 properties of hyperkähler quotients

Textbook(s)

none required

Reference books, course materials, etc.

none required

Assessment criteria and methods

reports 100%

Related courses

  • MTH.B301 ： Geometry I
  • MTH.B302 ： Geometry II
  • MTH.B331 ： Geometry III
  • MTH.B341 ： Topology

Prerequisites (i.e., required knowledge, skills, courses, etc.)

manifold theory, cohomology theory, theory of Lie group

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