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.
・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"
complex varieties, holomorphic curves, toric varieties, toric degeneration, complex torus, deformation theory, obstruction
|✔ Specialist skills||Intercultural skills||Communication skills||Critical thinking skills||Practical and/or problem-solving skills|
This is a standard lecture course. There will be some assignments.
|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|
|Class 3||Toric degeneration of complex varieties 1|
|Class 4||Toric degeneration of complex varieties 2|
|Class 5||Construction of holomorphic curves on degenerate varieties 1|
|Class 6||Construction of holomorphic curves on degenerate varieties 2|
|Class 7||Basics of tropical curves 1|
|Class 8||Basics of tropical curves 2|
|Class 9||Basics of tropical curves 3|
|Class 10||Deformation theory and obstruction 1|
|Class 11||Deformation theory and obstruction 2|
|Class 12||Deformation theory and obstruction 3|
|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|
None in particular
Basic knowledge on geometry is expected