2017 Special lectures on advanced topics in Mathematics H

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Academic unit or major
Graduate major in Mathematics
Instructor(s)
Honda Nobuhiro  Nishinou Takeo 
Course component(s)
Lecture     
Day/Period(Room No.)
Intensive (H201)  
Group
-
Course number
MTH.E532
Credits
2
Academic year
2017
Offered quarter
3Q
Syllabus updated
2017/3/17
Lecture notes updated
-
Language used
Japanese
Access Index

Course description and aims

This course deals with holomorphic curves on complex varieties. Although smooth complex varieties are easy to define, when we attempt to figure out explicit properties of them, the smoothness often prevents us from finding a foothold to start with. On the other hand, while singularities of varieties can itself be obstacles to the study, they can also be clues, since considerable amout of informations of varieties tend to concentrate in them. We will see how this idea is applied to the study of holomorphic curves through explicit examples.
We will begin with the basics of toric varieties, then give the construction of degeneration of varieties into singular toric varieties. The main example is a complex two dimensional torus. Using this example, we explain the relation between holomorphic curves on degenerate varieties and certain combinatorial object called tropical curves. Finally, we will see how such combinatorial data can be used to extract informations of the original varieties.

Student learning outcomes

・Be familiar with basic aspects of toric varieties
・Be familiar with the idea of degeneration
・Understand the construction of holomorphic curves on degenerate varieties
・Be familiar with deformations of holomorphic curves"

Keywords

complex varieties, holomorphic curves, toric varieties, toric degeneration, complex torus, deformation theory, obstruction

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 Basics of toric varieties 1 Details will be provided during each class session
Class 2 Basics of toric varieties 2 Details will be provided during each class session
Class 3 Toric degeneration of complex varieties 1 Details will be provided during each class session
Class 4 Toric degeneration of complex varieties 2 Details will be provided during each class session
Class 5 Construction of holomorphic curves on degenerate varieties 1 Details will be provided during each class session
Class 6 Construction of holomorphic curves on degenerate varieties 2 Details will be provided during each class session
Class 7 Basics of tropical curves 1 Details will be provided during each class session
Class 8 Basics of tropical curves 2 Details will be provided during each class session
Class 9 Basics of tropical curves 3 Details will be provided during each class session
Class 10 Deformation theory and obstruction 1 Details will be provided during each class session
Class 11 Deformation theory and obstruction 2 Details will be provided during each class session
Class 12 Deformation theory and obstruction 3 Details will be provided during each class session
Class 13 Calcualtion of obstruction in the case of deformations of holomorphic curves 1
Class 14 Calcualtion of obstruction in the case of deformations of holomorphic curves 2
Class 15 Calcualtion of obstruction in the case of deformations of holomorphic curves 3

Textbook(s)

None required

Reference books, course materials, etc.

None in particular

Assessment criteria and methods

Assignments (100%).

Related courses

  • MTH.B505 : Advanced topics in Geometry E1
  • MTH.B506 : Advanced topics in Geometry F1
  • MTH.A405 : Advanced topics in Algebra A1
  • MTH.A506 : Advanced topics in Algebra F1

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

Basic knowledge on geometry is expected

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